46 GS09 Abstracts IP1 The Use of Geophysical Methods to Characterize Hydrogeologic Systems There is growing use of geophysical methods to obtain in- formation about subsurface properties and processes in hy- drogeologic systems. Critical to advancing this use is the need to better understand the link between measurements made remotely of geophysical parameters and the prop- erties and processes governing fluid flow and contaminant transport. Captured in the geophysical data is informa- tion about hydrogeologic systems across a range of scales from the pore scale to the facies scale. Laboratory ex- periments best reveal the complex relationships between geophysical parameters and various physical and chemical properties/processes at the pore scale. Field studies best reveal the relationship between geophysical images and the larger facies-scale hydrogeologic structure. At all scales of measurement we need to develop a quantitative framework that allows us to accurately characterize hydrogeologic sys- tems using data acquired with geophysical methods. Rosemary Knight Stanford University [email protected]IP2 Decision and Control with Uncertainty: What Problems Should We Aim to Solve? There are four basic types of activity involving combina- tions of uncertainty and optimisation: * Uncertainty prop- agation where the problem is to predict the behaviour of an uncertain system, where the uncertainty is in the initial state of the system or factors, such as properties, influenc- ing the system evolution. * Data assimilation, also known as history matching, system identification or inverse prob- lems. * Decision making. Here a choice must be made between competing courses of action. For each choice of action the outcome is uncertain. * Optimal control of an uncertain system. A system is only known in a probabilis- tic way. One has to design a control policy that optimises the system. The problem is particularly difficult when op- timisation of the measurement system is included in the problem. These problems are closely related to one an- other. However, the subject of decision theory is perhaps the most fundamental and well developed theory as it has a firm axiomatic basis and an extensive literature. In this talk we will examine the formulation of the problems and then discuss the matter of how close we are to solving any of these problems in a satisfactory way. The main points that will be defended are (i) we should always optimise our expected utility and (ii) sequential formulations are the most fruitful for practical progress. Chris L. Farmer Oxford Centre for Collaborative Applied Mathematics University of Oxford [email protected]IP3 Career Awardees Talk: Challenges in Modeling Subsurface Complex Phenomena Today we are witnessing the beginnings of an ongoing in- tellectual paradigm shift in that computational simulations are being (1) included routinely in scientific analyses and (2) used to make engineering design decisions. These devel- opments involve the merging of (1) high performance, ad- vanced numerical computation and (2) field and laboratory experimental results with a properly validated and verified simulation tool for prediction. This result is a necessary prerequisite for wide-spread acceptance of computational simulation and enhancement of computer driven discov- ery and innovation in the geosciences, and, ultimately, in geo-engineering. A fundamental difficulty in understanding and predicting large-scale fluid movements in porous media is that these movements depend upon phenomena occur- ing on small scales in space and/or time. The differences in scale can be staggering. Aquifers and reservoirs extend for thousands of meters, while their transport properties can vary across centimeters, reflecting the depositional and dia- genetic processes that formed the rocks. In turn, transport properties depend on the distribution, correlation and con- nectivity of micron sized geometric features such as pore throats, and on molecular chemical reactions. Seepage and even pumped velocities can be extremely small compared to the rates of phase changes and chemical reactions. The coupling of flow simulation with mechanical deformations is also important in addressing the response of reservoirs located in structurally weak geologic formations. An ex- ample of a subsurface grand challenge is the sequestration of carbon in saline aquifers. Here one needs to accurately predict the fate of injected carbon dioxide in conditions governed by multiphase flow, rock mechanics, multicom- ponent transport, thermodynamic phase behavior, chemi- cal reactions within both the fluid and the rock, and the coupling of all these phenomena over multiple time and spatial scales. In this presentation we will discuss several multiscale and multiphysics approaches for addressing the modeling and simulation of complex subsurface phenomena such as carbon sequestration. Mary Wheeler The University of Texas at Austin [email protected]IP4 Junior Scientist Award Lecture: Geologic Stor- age of Carbon Dioxide: A Challenge for the Geo- sciences Geologic storage of CO2 presents the geosciences commu- nity with a renewed challenge in terms of spatial and tem- poral scales, non-linearly interacting physical processes, data uncertainty, and urgency of answers to key regulatory questions. In this talk, we will outline the development and implementation of a modeling approach to CO2 stor- age. We discuss dominant processes and temporal scales, properties of numerical methods and analytical solutions, and design of new solution approaches. In the context of applications to real datasets we conclude by opening a dis- cussion on whether simulating CO2 storage is a well posed challenge. Jan M. Nordbotten Department of Mathematics University of Bergen [email protected]IP5 Multiple Scales Analyses for Atmospheric Flows Asymptotic techniques generalize the classical approach of scale analysis in theoretical meteorology. Through, e.g., matched asymptotic and multiple scales expansions they allow us to systematically study interactions across sepa- rated length and times scales. This will be demonstrated
Transcript
46 GS09 Abstracts
IP1
The Use of Geophysical Methods to CharacterizeHydrogeologic Systems
There is growing use of geophysical methods to obtain in-formation about subsurface properties and processes in hy-drogeologic systems. Critical to advancing this use is theneed to better understand the link between measurementsmade remotely of geophysical parameters and the prop-erties and processes governing fluid flow and contaminanttransport. Captured in the geophysical data is informa-tion about hydrogeologic systems across a range of scalesfrom the pore scale to the facies scale. Laboratory ex-periments best reveal the complex relationships betweengeophysical parameters and various physical and chemicalproperties/processes at the pore scale. Field studies bestreveal the relationship between geophysical images and thelarger facies-scale hydrogeologic structure. At all scales ofmeasurement we need to develop a quantitative frameworkthat allows us to accurately characterize hydrogeologic sys-tems using data acquired with geophysical methods.
Decision and Control with Uncertainty: WhatProblems Should We Aim to Solve?
There are four basic types of activity involving combina-tions of uncertainty and optimisation: * Uncertainty prop-agation where the problem is to predict the behaviour ofan uncertain system, where the uncertainty is in the initialstate of the system or factors, such as properties, influenc-ing the system evolution. * Data assimilation, also knownas history matching, system identification or inverse prob-lems. * Decision making. Here a choice must be madebetween competing courses of action. For each choice ofaction the outcome is uncertain. * Optimal control of anuncertain system. A system is only known in a probabilis-tic way. One has to design a control policy that optimisesthe system. The problem is particularly difficult when op-timisation of the measurement system is included in theproblem. These problems are closely related to one an-other. However, the subject of decision theory is perhapsthe most fundamental and well developed theory as it hasa firm axiomatic basis and an extensive literature. In thistalk we will examine the formulation of the problems andthen discuss the matter of how close we are to solving anyof these problems in a satisfactory way. The main pointsthat will be defended are (i) we should always optimiseour expected utility and (ii) sequential formulations arethe most fruitful for practical progress.
Chris L. FarmerOxford Centre for Collaborative Applied MathematicsUniversity of [email protected]
IP3
Career Awardees Talk: Challenges in ModelingSubsurface Complex Phenomena
Today we are witnessing the beginnings of an ongoing in-tellectual paradigm shift in that computational simulationsare being (1) included routinely in scientific analyses and(2) used to make engineering design decisions. These devel-opments involve the merging of (1) high performance, ad-
vanced numerical computation and (2) field and laboratoryexperimental results with a properly validated and verifiedsimulation tool for prediction. This result is a necessaryprerequisite for wide-spread acceptance of computationalsimulation and enhancement of computer driven discov-ery and innovation in the geosciences, and, ultimately, ingeo-engineering. A fundamental difficulty in understandingand predicting large-scale fluid movements in porous mediais that these movements depend upon phenomena occur-ing on small scales in space and/or time. The differences inscale can be staggering. Aquifers and reservoirs extend forthousands of meters, while their transport properties canvary across centimeters, reflecting the depositional and dia-genetic processes that formed the rocks. In turn, transportproperties depend on the distribution, correlation and con-nectivity of micron sized geometric features such as porethroats, and on molecular chemical reactions. Seepage andeven pumped velocities can be extremely small comparedto the rates of phase changes and chemical reactions. Thecoupling of flow simulation with mechanical deformationsis also important in addressing the response of reservoirslocated in structurally weak geologic formations. An ex-ample of a subsurface grand challenge is the sequestrationof carbon in saline aquifers. Here one needs to accuratelypredict the fate of injected carbon dioxide in conditionsgoverned by multiphase flow, rock mechanics, multicom-ponent transport, thermodynamic phase behavior, chemi-cal reactions within both the fluid and the rock, and thecoupling of all these phenomena over multiple time andspatial scales. In this presentation we will discuss severalmultiscale and multiphysics approaches for addressing themodeling and simulation of complex subsurface phenomenasuch as carbon sequestration.
Junior Scientist Award Lecture: Geologic Stor-age of Carbon Dioxide: A Challenge for the Geo-sciences
Geologic storage of CO2 presents the geosciences commu-nity with a renewed challenge in terms of spatial and tem-poral scales, non-linearly interacting physical processes,data uncertainty, and urgency of answers to key regulatoryquestions. In this talk, we will outline the developmentand implementation of a modeling approach to CO2 stor-age. We discuss dominant processes and temporal scales,properties of numerical methods and analytical solutions,and design of new solution approaches. In the context ofapplications to real datasets we conclude by opening a dis-cussion on whether simulating CO2 storage is a well posedchallenge.
Jan M. NordbottenDepartment of MathematicsUniversity of [email protected]
IP5
Multiple Scales Analyses for Atmospheric Flows
Asymptotic techniques generalize the classical approach ofscale analysis in theoretical meteorology. Through, e.g.,matched asymptotic and multiple scales expansions theyallow us to systematically study interactions across sepa-rated length and times scales. This will be demonstrated
GS09 Abstracts 47
drawing from recent work on hurricane-like concentratedvortices and on cloud–internal wave interactions. Whereasthe classical theory of anelastic mostions by Ogura andPhillips (1962) is naturally captured in an asymptotics-based framework, its subsequent extensions, e.g., by Dut-ton and Fichtl (1969), Lipps and Hemler (1982), Bannon(1996), as well as Durran’s pseudo-incompressible model(1989,2008) pose a particular challenge. I will show thattheir systematic theoretical justification will require tech-niques that go beyond scale analysis and single or multiplescales expansions.
Rupert KleinPotsdam Institute for Climate Impact ResearchFree University of [email protected]
IP6
Sea Ice Modelling
On the geophysical scale sea ice is a thin, broken layer onthe polar oceans which is modified in thickness and com-pactness by dynamic and thermodynamic processes. Seaice represents the boundary between the two much largergeophysical fluids, the atmosphere and the ocean, and,therefore, influences their interaction considerably. Sea iceis described as a two-dimensional deformable solid mate-rial with a specific geophysical rheology. The presentationwill focus on the optimization of sea ice dynamics, and onmodel verification using satellite data and buoy trajecto-ries. Applications to past and current sea ice anomalieswill be shown, including the interaction with atmosphereand ocean.
In the wake of recent Gulf of Mexico hurricanes (Katrina,Rita, Gustav and Ike), efforts to accurately model stormsurge in the Gulf of Mexico have intensified. A modelingsystem has been developed that simulates hurricane winds,wind-waves, storm surge, tides and river flow in this com-plex region. This is accomplished by defining a domain andcomputational resolution appropriate for the relevant pro-cesses, specifying realistic boundary conditions, and imple-menting accurate, robust, and highly parallel unstructuredgrid algorithms for both the wind waves and the long wavecurrent/storm surge/tide model.
Joannes WesterinkDepartment of Civil Engineering and Geological SciencesUniversity of Notre [email protected]
IP8
Design of Carbon Dioxide Storage
We propose design strategies for CO2 injection to maxi-mize storage in aquifers and to maximize both CO2 stor-age and enhanced oil recovery (EOR) in oil reservoirs. Wepropose a carbon storage strategy where CO2 and brineare injected into an aquifer together followed by brine in-jection alone. This renders 80-95% of the CO2 immobilein pore-scale droplets within the porous rock. The favor-
able mobility ratio between injected and displaced fluidsleads to a more uniform sweep of the aquifer leading to ahigher storage efficiency than injecting CO2 alone. Thisdesign was demonstrated through one-dimensional simula-tions that were verified through comparison with analyticalsolutions. We then performed simulations of CO2 storagein a North Sea aquifer. We then extended our study tooil fields. We propose to inject more water than the tra-ditional optimum that maximizes only oil recovery. Thiscauses the CO2 to remain in the reservoir, increases thefield life and leads to improved storage of CO2 as a trappedphase. Again, a short period of chase brine injection at theend of the process traps most of the remaining CO2.
Martin BluntDept. Earth Science and EngineeringImperial College [email protected]
CP1
A Lagrangian-Averaged Air-Water Coupled Sys-tem for Wind-Driven Sea Surface Waves
Based on the Lagrangian-averaging modeling concept weformulate equations for the generation of wind-driven seasurface waves. The equations are an air-water coupledsystem which includes viscosity and the surface tension.We apply the system to simplified two-dimensional lam-inar flows, and compare the results with those from themodels based on the exact Navier-Stokes equations. Es-pecially, we discuss its stability with laminar base profile.Growth rate, the zones of amplification and zones of decaywith respect to different wave lengths and wind speeds arefound.
Bong-Sik KimMathematics Division, George Mason University - RAK,[email protected]
CP1
Parameterization of Turbulent Transport in Ran-dom Vortex Models
We employ homogenization theory to develop a systematicparameterization strategy for quantifying the transport ef-fects of mesoscale coherent structures in the ocean whichcannot be well resolved by large-scale weather and climatesimulations. We work from the ground up with simplekinematic models and study in particular how the effectivediffusivity depends on the governing parameters, such asStrouhal number and Peclet number, in a class of dynam-ical random vortex flows.
Peter R. KramerRensselaer Polytechnic InstituteDepartment of Mathematical [email protected]
Shafer SmithCourant InstituteCenter for Atmosphere Ocean [email protected]
48 GS09 Abstracts
CP1
Modeling Wave Breaking and Its Dissipative Ef-fects on Planetary-Scale Wave-Current Interac-tions
The equations for basin scale currents and waves are mod-ified to include wave breaking via velocity perturbationsof random strength at random locations. The accelerationfield of individual perturbations is parametrized as to fitobservations. The statistics of the perturbations are de-rived from realizations of a Gaussian ocean surface. Sim-ulated wave groups are tracked and their growth rate ofwave group energy is computed as a predictor of the onsetof wave breaking.
A Weak Solution and a Numerical Solution of theCoupled Navier-Stokes and Darcy Equations.
In this work we propose a multi-numerics scheme formodeling the coupling of the Navier-Stokes and Darcyequations using the continuous finite element method inthe incompressible flow region and discontinuous Galerkinmethod in the porous medium region. We first show exis-tence and uniqueness of a weak solution. We then proveconvergence of the numerical scheme and show some nu-merical simulations.
On the Simulation of Flow in Large-Scale FracturedMedia
We propose to simulate flow within fractures lying in animpervious rock matrix. Each fracture is modeled by anellipse of random distribution of eccentricity, length, posi-tion and orientation. For efficiency purpose and easy meshrefinement, fractures are meshed independently. The chal-lenge comes then to guarantee the continuity of the fluxesand heads at the fracture intersections by the means of aMortar method for any crossing configurations of the frac-tures.
Jean-Raynald De DreuzyGeosciences Rennes, UMR6118 CNRSUniversite de Rennes [email protected]
CP2
Some Results Concerning a Model for Fractureswith Forchheier Flow
Flow of a single phase, incompressible fluid in a porousmedium with a fracture is usually modeled using a conser-vation law together with Darcy’s law both in the fractureand in the surrounding rock matrix. However for somefractures in which the flow is more rapid, the flow in thefracture is more accurately described using Forchheimer’sequation. Here we are interested in a model in which thefracture is treated as an interface between two subdomainsof rock matrix and in which the flow in the fracture, isgoverned by Forchheimer’s equation, while that in the sur-rounding matrix is governed by Darcy’s law. We showexistence and uniqueness of the solution of the resultingsystem. Some numerical results are also presented.
Coupling Multi-Component Porous Media Flowwith Free Flow
Up to now, the coupling of free flow with porous media flowhas been often considered only for a single-phase system.We extend this classical concept to two-component non-isothermal flow with two phases inside the porous mediumand a single phase inside the free flow region. Our modelalso takes into account evaporation and condensation pro-cesses. We discuss the coupled model and its iterative so-lution by means of several numerical examples.
Downscaling of Pressure and Saturation Maps forAn Improved Modelling of 4D Seismic Data
History-matching workflows usually use pressure and sat-uration maps provided by fluid-flow simulations to com-pute 4D-seismic attributes. Unfortunately the resolutionof these maps is often too coarse to represent precisely thefluid displacements at the geological scale. We introducehere downscaling algorithms which compute fine pressureand saturation maps from geological data and reservoir-simulation results. These algorithms are fast, conservativeand improve significantly the 4D-seismic response of thereservoir model.
Finite Difference Modeling the Acoustic Logging inPorous Media
A finite-difference time-domain (FDTD) algorithm is pro-posed to simulate the axisymmetric acoustic logs in fluid-saturated porous formation. In this algorithm, differentforms of acoustic wave equations in fluid, solid and porousmedia are unified as the form of Biots equations governingwave propagation in poroelastic media. The algorithm isvalidated by the comparisons with the analytical method.Simulations of the acoustic logging in radial layering andhorizontally layering porous formations are given by thealgorithm.
A Geostatistical Study of Geophysical Data In-side Rock Blocks Bordered by Brittle Structuresin Crystalline Bedrock
Interpolated gravity and magnetic maps are used togetherwith elevation models in the interpretation of large geo-logical structures such as folds and faults in crystallinebedrock. In the present study, geophysical data from studysites in southern and eastern Finland were analysed and in-terpolated to maps inside bedrock blocks bordered by frac-ture zones which were defined by using geological observa-tions and elevation models. The geophysical data consistedof gravity, magnetic and seismic surveys. As a result, geo-statistical measures such as different kinds of variograms
show characteristic anisotropies for different rock blocks inthe study areas. The results were applied in analysing therelative movements of rock blocks and, especially, in 3Dmodelling of discontinuous ore bodies.
On Semivariogram Fitting by TLS Adjustmentwith a New Weighting Scheme for AeromagneticData
In order to fit a nonlinear mathematical model to the em-pirical semivariogram values that belong to a set of spa-tially distributed data, the Total Least-Squares (TLS) ad-justment method is employed with a new extended weight-ing scheme. In contrast to the standard Least-Squares (LS)method, TLS approximation is particularly suited in thiscase since it treats the lag distance and the empirical semi-variogram values symmetrically, assuming random errorsin both variables. Beside the development of a (simplified)weighting scheme, proper measures for goodness of fit arealso considered in this more general case, the applicabil-ity and performance of the relatively new method will bedemonstrated by using a set of aeromagnetic data.
Burkhard SchaffrinSchool of Earth SciencesThe Ohio State [email protected]
Evolution of Rotation of a Satellite with CavityFilled with a Viscous Fluid under the Action ofLight Pressure Torque
We investigate fast rotational motion of dynamically asym-metric satellite with cavity filled with viscous fluid underthe action of gravitational and light pressure torque. Thisproblem is similar to the motion of the planet composedof a liquid core and rigid mantle. The system obtained af-ter averaging with respect to Euler–Poinsot motion is ana-lyzed. Numerical analysis shows that the kinetic energy ismonotonically decreasing. Analytical analysis is conductedin neighboring of axial motion.
Dmytro D. LeshchenkoOdessa State Academy of Civil Engineering andArchitectureleshchenko [email protected]
Leonid D. AkulenkoInstitute for Problems in Mechanics [email protected]
On Resolving Exponentially Large Viscosity Vari-ations in Convective Mantle Flow with High OrderDiscontinuous Galerkin Methods
The equations governing planetary mantle convection in-volve a strongly temperature dependent viscosity co-efficient in the elliptic problem for the fluid velocity.The viscosity is typically Arrhenius or more genericallyexp(f(T(x))) and needs to be resolved accurately in orderto calculate the correct velocity and temperature fields. Ananalytic discussion showing how viscosity influences numer-ical calculations is followed by some numerical examplesof convective mantle flow solved with the discontinuousGalerkin formulation on unstructured meshes.
Daniel T. PaulsenBrown UniversityDivision of Applied Mathematicsdaniel [email protected]
Jan S. HesthavenBrown UniversityDivision of Applied [email protected]
Marc ParmentierBrown UniversityGeological Sciencesem [email protected]
CP4
Generation, Retention and Expulsion of Hydrocar-bons
Hydrocarbon generation and expulsion are strictly interre-lated processes and at a first approximation it is reasonableto assume a Darcy flow description of the expulsion pro-cess. The proposed modelling combines a two-phase Darcyflow, with time dependent porosity, with a set of Ahrreniusreactions to describe hydrocarbon generation and crackingand with a differential retention process in kerogen that af-fects petroleum fractionation. The modelling approach hasbeen compared and validated with laboratory experimentsresults.
Paolo RuffoEni, E&P Div. - GEBA DeptVia Emilia 1, S. Donato Mil.se (MI), [email protected]
CP4
Applications of Full Three-Dimensional DynamicStructural Modelling
An innovative full three-dimensional dynamic structuralmodelling software has been developed in order to modelbasin geological evolution. The proposed approach is ableto mimic the principal geological processes: sediment com-
paction, non-Newtonian behaviour of salt and sedimentaryrocks, basement evolution (due to isostasy and/or tectonicmovements) and fault induced displacements. Some syn-thetic and real application examples are used to validateand demonstrate the robustness of the approach.
Matteo Longoni, Cristiano MalossiMOX, Department of Mathematics; Politecnico di MilanoVia Bonardi 9, Milan, [email protected],[email protected]
Andrea VillaUniversita’ degli Studi di MilanoDepartment of [email protected]
CP5
A Local Discontinuous Galerkin Discretization forIncompressible Hydrostatic Flows with Free Sur-face Using σ-Transformed Coordinates
The Local Discontinuous Galerkin (LDG) method is ahigher order discretization method for advection-diffusionequations. In this talk we present a conservative LDGmethod for the simulation of incompressible hydrostaticfree surface flows using σ-transformed coordinates. Thetime-dependent computational domain representing thefluid region is transformed onto a domain that is fixed intime. The advantage of solving the resulting transformedsystem of partial differential equations is that unlike inmoving mesh strategies there is no need to smooth the dis-crete surface elevation for grid adaptation.
Christoph GersbacherDivision of Applied MathematicsUniversity of [email protected]
Andreas DednerDivision of Applied MathematicsUniversity of Freiburg, [email protected]
CP5
Representation of Linear Terrain Features in a 2DFlood Model with Regular Cartesian Mesh
The failure of water control infrastructures leads to poten-tially destructive floods. Accurate numerical simulationof such flows is fundamental in performing risk analysesand planning for emergency management. This paper de-scribes a cut-cell boundary approach to represent subgridlinear features (dams, rivers,...) inside a 2D structuredFV code; a special version of this method is developed torepresent river flows allowing the coupling of 1D and 2Dcomputations to simulate levee overtopping floods.
Nonlinear Shallow Water Equation in Polar Coor-dinates
An interaction of two water waves in a circular basin isstudied within quadratic approximation. When the polarcoordinates are used, the usual perturbation techniques inseparation of variables method inevitably lead to a seriesof overdetermined systems of linear algebraic equations forunknown coefficients (in contrast with the Cartesian coor-dinates). However, if we formally introduce a new functionsatisfying the first system of this series, all these overdeter-mined systems become compatible (remaining overdeter-mined) for the special case of the nonlinear shallow waterequation. Using the new function and quadratic polyno-mials of the Bessel functions of radius, we explicitly ex-press the coefficients of the resulting harmonics. It givessolutions describing the two-waves interaction which arefound with the same accuracy as the nonlinear shallowwater equation is derived. As a consequence, a generalboundary problem can be explicitly solved in these terms.
Numerical Simulation of Multiphase Flows inPorous Media with a Degenerate Parabolic System
In this work is introduced a numerical scheme for solv-ing degenerate parabolic systems for porous media flow bymeans of mixed ?nite elements formulation when a dis-tinct number of ?uid phases ?ows in distinct ?ow regionsin the physical domain. By means of an operator splittingand domain decomposition techniques conformity condi-tions were obtained to bypass such degeneracy in order toobtain a non-singular system for their numerical solution.
A Comparison of Two Numerical Models for Sec-ondary Oil Migration and Entrapment
In this work, we compare the solutions obtained by twodifferent methods for a synthetic trapping scenario withrealistic input parameters. The first method is an invasionpercolation model based on the following assumptions : oilmigration is a rapid process limited by oil generation rateand capillary pressure barriers. The second method is atwo phase flow finite volume method with a flux definitionbased on local extended pressure continuity conditions.
Outflow Boundary Conditions in Porous MediaFlow Equations
To model the flow in porous media one typically uses ei-ther the Richards equation if the flow can be describedeffectively by one phase, or the two-phase flow equationsif two phases must be modeled. Due to the wide rangeof applications, the interest in these two systems is enor-mous. Rigorous analytical results became available in the80ies, when strong nonlinearities in time-dependent prob-lems were treated systematically. In most contributionson the subject, the analysis is simplified by restricting toDirichlet boundary conditions, even though a physicallymore appropriate boundary condition is the outflow con-dition. After motivating and explaining this condition wepresent the corresponding qualitative analytical problems,a regularization technique and existence results.
Formulation of Compressible Immiscible Two-Phase Flow Model by Means of Global Pressure
We consider a compressible immiscible two-phase flowmodel in porous media formulated in terms of a global pres-sure. Two possible definitions of global pressure are con-sidered: a simplified one which is a generalization of incom-pressible flow global pressure, and based on an approxima-tive calculation of mass densities, and a new one introducedby B. Amaziane and M. Jurak which does not assume anyapproximation. We present analytical and computationalcomparison of the two models.
Numerical Modelling of Dispersion in ComplexSystems
We consider dispersion processes taking place in complexsystems where particles can be transported over large dis-tances by the fluctuations of the velocity field (e.g. turbu-
52 GS09 Abstracts
lent flows) or the heterogeneity of the of system (e.g. sub-surface hydrology). For such situations, dispersion modelsbased on Ficks law might not be accurate. These modelsimplicitly assume that the particles are following a Brow-nian motion, where the dispersion of a cloud of particlesis only driven by the local interactions between particles.Such a model prevents random displacements of large am-plitude that could result from external forcings or hetero-geneities. A more accurate dispersion pattern can be ob-tained by assuming that random fluctuations follow a Levydistribution, which results in so-called Levy flights. In thatcase, the distribution of particles is no longer the solutionof a second-order diffusion equation. Instead, it can beshown that it is the solution of a fractional-order equationwhose exponent is generally comprised between 1 and 2.Fractional-order differential operators are global differen-tial operators in the sense that they take the whole behav-ior of the function into account and not just the local valueof the slope or convexity. In this work, we discuss numericalmethods to solve the fractional-order dispersion equation.Numerical methods like the finite difference and finite el-ement methods are not well suited to solve this equationas they are generally of low order and thus require a lotof grid points to obtain an accurate solution. For localdifferential operators, this results in large sparse matricesthat can be handled easily. However, for global differen-tial operators, like the ones considered here, the resultingmatrix is full since the global behavior of the function hasto be taken into account. High-order, global numericalmethods like the spectral method therefore appear to be abetter choice to solve this problem as they naturally takethe global behavior of the solution into account and usea limited number of degrees of freedom. Both approacheswill be compared in terms of accuracy and computationalefficiency.
Emmanuel HanertUCLouvainDepartment of Environmental Sciences and Land [email protected]
Attractor for Random Boussinesq-Glover Equationwith Colored Noise
We consider the random Boussinesq-Glover equation withmonotone operators in Banach or Hilbert spaces, drivenby a colored noise and with random initial condition. Thenoise is defined as stationary solution of a stochastic differ-ential equation in finite dimensional (or Hilbert) spaces.Under suitable assumptions, we prove the existence ofglobal attractor. This attractor is independent on proba-bilitary variable. Similar results arise in random reaction-diffusion equations.
The Simulation of Contaminant Transport inGroundwater and Environmental Risk Assessment
The problem of assessing groundwater pollution has be-
come a matter of considerable concern. For proper ground-water management, it is necessary to model the contam-ination mathematically in order to assess the effects ofcontamination and predict the transport of contaminants.Several deterministic models have been proposed and nu-merical procedures developed. Because of aquifer hetero-genity, the the spatial variation of flow properties is er-ratic. Therefore a stochastic model of flow regime andtransport processes is more realistic. In this talk we usea new method (A. Beskos and G.O. Roberts), Exact sim-ulation of diffusions, The Annals of Applied Probability2005, vol. 15(4), 2422-2444) for modelling contaminanttransport. Furthermore we adress sensivity analysis of ex-ceedance probabili- ties with respect to variations of thetransmissivity field, porosity and dispersivity
Franz KonecnyBOKU-University of Natural Resources and AppliedLife Sciences, [email protected]
CP7
Models for Transport in Porous Media
Tracer transport is most commonly described with the ad-vection/dispersion equation (ADE). However even in sim-ple experiments on homogenous systems, the ADE maydemonstrate incorrect behavior. In context of a specificexperiment, data is compared with numerical solutionsof advection/dispersion and dual-permeability models. Apore scale network model introduces a numerical labo-ratory for transport in homogeneous porous media withtime dependent boundary conditions. Finally we discuss amodel based on the continuous transport that covers gen-eral transport phenomena in porous media.
Ensemble Kalman Filter (enkf) for An OregonCoastal Transition Zone (octz)
A primitive-equation numerical ocean model is used tostudy the upwelling circulation of OCTZ where shelf flowsinteract with the northern California Current. The presentOCTZ simulation results show realistic features such ascoastal jet separation and eddy formation offshore of CapeBlanco. An Ensemble Kalman Filter is formulated to im-prove forecasts and representations of the flow, and tostudy sensitivities to uncertainty in forcing and boundaryconditions.
Uncertainty Quantification for An Oregon CoastalTransition Zone Ocean Model
We explore the practical effect of perturbing the bound-aries and wind forcing in a realistic-grid model of the Ore-gon Coastal Transition Zone, a region in which mesoscaleeddies play an important role. The shelf flow is stronglywind-driven, and dynamically balanced perturbations ofthe boundary data produce different solutions in variousperturbed model simulations. Integrated effects on thesimulation results are assessed, and the predictability anduncertainty are quantified both in a Bayesian frameworkand by other standard statistical quantities.
Numerical studies using Regional Ocean Modeling Systemare presented. We investigate the formation of oceanicwakes, generated by obstacle of the shape of Madeira Is-land without consideration of bathymetry (i.e., with verti-cal sides). The results are compared with ones, obtainedby [Dong, McWilliams, Shchepetkin, 2007: Island Wakes inDeep Water. J. Phys. Oceanogr., 37, 962-981] where theyinvestigate the formation and evolution of wakes aroundan idealized cylindrical island.
Euclides A. LuisCEMAT (Center for Mathematics and its Applications) –IST,Technical University of Lisbon, [email protected]
Dmitri BoutovInstitute of Oceanography, Faculty of Sciences,University of [email protected]
CP8
Numerical Method for Solving the UnsteadyGravity-Capillary Wave Problem
The Boundary Integral Equation Method (BIEM) basedon the Mixed-Eulerian-Lagrangian (MEL) appraoch is pre-sented in this talk. We apply the method to numericallysolve the unsteady free surface flows in water of finitedepth. Here, the fluid is assumed to be inviscid and in-compressible, while the flow is irrotational. The gravityand surface tension effects are included in the dynamicboundary condition. The main propose is to study thestability of steady gravity-capillary waves. Some numeri-cal results for both steady and unsteady flows are shown.
Connections between them are also made.
Montri MaleewongDepartment of Mathematics, Faculty of ScienceKasetsart [email protected]
CP8
On Some Models of the Equipotential Gravity Sur-faces in the Coastal Areas.
This presentation deals with mathematical models of thegeoid equipotential surfaces in the coastal areas. At thismoment the greatest errors in the local (small scale) Geoiddetermination are either in the complex mountain terrainsor in the coastal areas, and it is quite important to eval-uate precise Geoid in the short wavelength (small scale)for the local communities in the coastal areas as to predictimpacts of tropical storms and floods. Recent studies andmodels of the Geoid demonstrated that the greatest errorsin the Geoid surface have been made in the coastal areawhere there are border lines between two huge volumes ofmasses with rather different densities. In this presentationof the ongoing research we are considering some statisticalmodels based on the observed anomaly data as well as themodel based on the Stokes-Helmert integral for the poten-tial of the gravitational field in the neighborhood of theshoreline, namely, we consider some ideal boundary of thewater and the land masses. We have used MatLab softwareto simulate some models and to visualize equipotential sur-faces.
Alexey L. SadovskiTexas A&M University-Corpus ChristiDept of Comp & Math [email protected]
Using Energy Minimizing Basis Functions As aMultiscale Method, with Applications to Multigridand Domain Decomposition Solvers
We demonstrate the applicability of energy minimizing ba-sis functions for two-phase flow simulations, and we high-light their numerous benefits. We show how they can beimplemented to obtain efficient serial (algebraic multigrid)and parallel (Additive Schwarz with coarse space correc-tion) linear solvers for large-scale heterogeneous problems.Moreover, they possess the same advantages of other mul-tiscale techniques, but without the need of constructing acoarse mesh, and they are well-suited for adaptive algebraiccoarsening.
An Unfitted Discontinuous Galerkin Finite Ele-ment Method for Numerical Upscaling in PorousMedia
Porescale simultations usually requirer a trade-off betweencomputational costs, simplification of the modell or the ge-ometry. A new scheme is presented, avoiding most of thesedraw-backs. It combines the idea of Unfitted Finite Ele-ments with Discontinuous Galerkin methods. The minimalnumber of unknowns is not determined by the shape of thedomain, still a maximum of the geometric information isaccounted. The method is verified by computing the per-meability for a domain with known properties.
Iterative Multiscale Finite Volume Method forCompressible Multiphase Flow in Porous Media
We introduce an iterative MSFV (i-MSFV) method forparabolic problems arising from compressible multiphaseflow in porous media as an extension of recently pub-lished i-MSFV method for incompressible (elliptic) prob-lems. Convergence studies are presented including appli-cations for multiphase flow. We show that only a few itera-tions per time step are sufficient in order to obtain good re-sults; even for highly anisotropic heterogeneous reservoirs.In any case, however, the resulting solution is conservativeat the fine scale.
The Heterogeneous Multiscale Finite ElementMethod for Advection-Diffusion Problems withRapidly Oscillating Coefficients and Large Ex-pected Drift
In this contribution we analyze a new version of the het-erogeneous multiscale finite element method (HM-FEM;originally introduced by E and Engquist [1]) for solvingmultiscale advection-diffusion problems. These types ofproblems have a variety of applications in geoscience, es-pecially in modelling transport of solutes in groundwaterand surface water. We give a-priori and a-posteriori errorestimates for the method and provide corresponding nu-merical experiments that underline the analytical results.For details see [2,3,4]. [1] W. E and B. Enquist. The het-erogeneous multiscale methods. Commun. Math. Sci.1 (2003), no. 1, 87–132. [2] P. Henning, M. Ohlberger.The heterogeneous multiscale finite elements method foradvection-diffusion problems with rapidly oscillating coef-ficients and large expected drift. Munster University, inpreparation. [3] P. Henning, M. Ohlberger. The heteroge-neous multiscale finite element method for elliptic homog-enization problems in perforated domains. Munster Uni-versity, Preprint 01/08 - N. [4] M. Ohlberger. A-Posteriorierror estimates for the heterogeneous multiscale finite ele-ment method for elliptic homogenization problems. Multi-
scale Modeling and Simulation 4 (2005) 1, 88 - 114.
Patrick HenningUniversity of Munster,Institut fur Numerische und Angewandte [email protected]
CP9
A Multiscale Mass Conservative Domain Decompo-sition Preconditioner for Solving Elliptic Problemson Perturbed Grids
We present a multiscale mass conservative domain decom-position preconditioner for solving elliptic problems on per-turbed grids. An important feature for the multiscalemethods, is the localisation approximations, for reducingthe size of the global problem. We study different proper-ties of the fine scale solution. We have looked at lack ofmonotonicity on the coarse scale, and studied circulationsin fine scale velocity field.
Ivar AavatsmarkCenter for Integrated Petroleum ResearchUniversity of [email protected]
Jan M. NordbottenDepartment of MathematicsUniversity of [email protected]
CP10
Hybrid Method for Low-Frequency Electromag-netic Scattering from a Resistive Underground Tar-get
We present a fast and accurate hybrid method for comput-ing low-frequency electromagnetic scattering from a resis-tive underground target. The method consists of solving afinite volume problem in a localized region containing thetarget, and using the integral equation (IE) method to ob-tain the field outside that region. The hybrid method thusreplaces the dense-matrix part of the rigorous IE methodby sparse-matrix calculations based on an approximationof Maxwell’s equations.
Shaaban A. BakrPH.D. researcher, centre for Integrated [email protected]
Trond MannsethCentre for Integrated Petroleum ResearchUniversity of Bergen, [email protected]
CP10
Impact of Time-Lapse Seismic Data on Permeabil-ity Estimation
Time-lapse pseudo-seismic (saturation) data are integratedwith production data to estimate the permeability field.We investigate the impact of pseudo-seismic data withfairly large uncertainty on the final estimates by apply-
GS09 Abstracts 55
ing an error analysis to a particular estimation technique.Numerical experiments show that the over-parametrizationproblem is more due to measurement errors than represen-tation bias. A quantity is derived to study the impact ofperturbations in different types of measurements on theestimate.
Trond MannsethCentre for Integrated Petroleum ResearchUniversity of Bergen, [email protected]
CP10
Locating Point Diffractors in Layered Media bySpatial Dynamics
We present a new approach to the problem of detectingpoint diffractors from active source surface seismic data.We formulate an optimization problem in the configura-tion space of possible collections of scatterers and con-struct a birth-and-death spatial dynamic, which convergesto the optimal solution. By design, this dynamic doesnot have resolution limits typical of migration based tech-niques, which allows for subwavelength sensing.
Non-Invasive Methods and Modelling Approachesto Study the Impact of Subsurface Structures onFlow and Transport
Geophysical methods may play an important role inmanaging our terrestrial environment and in maintainingecosystem functioning and services. Especially, the appli-cation and further development of hydrogeophysical meth-ods combined with mathematical models seem very promis-ing to maintain and protect soil and groundwater quality.Hydrogeophsical methods may help to improve our controlon storage, filter and buffer functions of soils and ground-water systems. Moreover, methods are needed that willhelp us to bridge the gap between the scale of measure-ments and observations and the scale at which manage-ment of terrestrial systems takes place. In this presenta-tion several examples will be presented showing how hydro-geophysical research can contribute in meeting these chal-lenges and may be used to characterize subsurface waterflow and transport.
Harry VereeckenInsitute of Chemistry and Dynamics of the GeosphereForschungszentrum [email protected]
J.A. HuismanInstitute of Chemistry and Dynamics of the GeosphereForschungszentrum [email protected]
J. van Der KrukInstitute of Chemistry and Dynamics of the GEosphereForschungszentrum [email protected]
Recovery of Active Faults from Surface Displace-ment Fields.
The goal of this research project is to process measure-ments of surface displacements in such a way to use themas data for the inverse problem consisting of locating faultsand portraying their geometry. Our research is also aim-ing at determining whether a measured displacement fieldon the surface is indicative of the onset of a destabiliza-tion phase. We have already entirely solved a two dimen-sional problem associated to the strike slip model, whichessentially reduces displacement fields to two dimensionalscalar fields. Deriving the inversion method involved a rig-orous mathematical eigenvalue asymptotic analysis, lead-ing to closed form inversion formulas. Those formulas werethen tested for robustness in numerical simulations. Asthe strike slip model is limited in scope (it captures onlyone of the textbook examples of faults), we have workedon extending our results to fully three dimensional faultproblems. In this much more difficult case, we have al-ready obtained very promising closed form formulas (validfor the dominant part of the asymptotic behavior), and wehave tested their use on numerical data. Nonetheless, acomplete mathematical analysis of the eigenvalue problemobtained by studying destabilization is still being inves-tigated. This is joint work with I. R. Ionescu, with thesupport of NSF grant DMS 0707421.
An Experimental Setup of Ground-Based and Air-borne Systems to Study Spatio-Temporal Struc-tures in Atmosphere-Land Surface Energy, Waterand Co2 Exchange
Exchange processes between land surface, vegetation andatmosphere over structured, inhomogeneous regions are in-vestigated in an experimental approach. For that purposeground-based long-term monitoring and dedicated cam-paigns are combined within the Rur catchment, Germany.Four campaigns covering different vegetation periods havebeen performed with instrumentation ranging from leaflevel gas exchange, eddy correlation, scanning remote sen-sors to aircraftobservations. Synergistic data analysis shallreveal spatio-temporal structures in the exchange processesand their relation to external parameters.
Christoph SelbachInstitute for Geophysics and MeteorologyCologne [email protected]
CP11
One-Shot Parameter Optimization in ClimateModeling
We present an iterative method to solve data assimilationproblems to identify parameters of parts of the climate sys-tem as for example ocean or atmospheric models. The goalis to obtain feasibility and optimality by simultaneously up-dating the state, adjoint and parameter values. Numericalresults are shown on the basis of a least-squares fit of theRahmstorf North Atlantic THC box model to given data.
Andreas GriewankHU Berlin, MATHEON Research Center, [email protected]
CP11
Three-Dimensional Dynamics in Non-ParallelShear Stratified Flows
The instabilities of non-parallel flows such as those in-duced by polarized inertia-gravity waves embedded in astably stratified environment are analyzed in the contextof the 3D Euler-Boussinesq equations. We derive a suf-ficient condition for shear stability and a necessary con-dition for instability in the case of non-parallel velocityfields. Three dimensional numerical simulations of the fullnonlinear equations are conducted to characterize the re-spective modes of instability, their topology and dynamics,and subsequent breakdown into turbulence. We investi-gate three-dimensional characteristics and present compu-tational results on Lagrangian particle dynamics.
Alex MahalovDepartment of MathematicsArizona State [email protected]
Mohamed Moustaouidept. of mathematics and statisticsArizona State University, Tempe, AZ [email protected]
CP11
Evaluation of Numerical Schemes Intended forGlobal Atmospheric Climate Models
The products of climate models are temporal statisticsbased on a calculated evolution of an atmosphere modelin a forced-dissipative equilibrium. Evaluation techniquesbeyond deterministic test cases are needed. Because of un-certainties in the nonlinear, interactive forcing, absolute er-rors of the statistics are difficult to attribute to the dynam-ics. However comparative statements can be made whichaddress the question: from a set of schemes which pro-duce the same quality statistical solution, which requires
the least computer time.
David L. WilliamsonNational Center for Atmospheric [email protected]
CP12
A Priori Convergence Analysis of a Coupled Ge-omechanics and Reservoir Flow Model with Stress-Dependent Permeability
We consider the numerical solution of a coupled geome-chanics and ?a stress-sensitive reservoir flow model.? Wecombine a ? mixed finite element for Darcy flow andGalerkin finite element for elasticity. ? ?This work fo-cuses on deriving ?convergence results for the numericalsolution of this nonlinear partial differential system. TheCG/mixed method produces optimal convergence rateswith respect to regularity. The theoretical error estimateswe derived include the possibility for the displacement andthe flow variables being calculated on different grids. Weperform numerical experiments for verifying our theory andmodeling some engineering applications.
Silvia BarbeiroDepartment of Mathematics, University of CoimbraCSM, ICES, University of Texas at [email protected]
Large Deformation in Viscoelastic Solid Bodies –Numerical Simulation of Salt Migration.
We consider instability of a two layered solid body of adenser material on top of a lighter one. This problem iswidely known to geophysicists in sediment-salt migrationas salt diapirism. In the literature, this problem has oftenbeen treated as Raleigh-Taylor instability in viscous fluidsinstead of solid bodies. In this presentation, we proposea successive incremental method for large deformation inviscoelastic solids as a model for salt migration.
I-Shih LiuInstituto de MatematicaUniversidade Federal do Rio de [email protected]
Rolci CipolattiInstituto de MatematicaUniversidade Federal do Rio de [email protected]
Mauro A. RinconInstituto de Matematica (DCC/IM)Universidade Federal do Rio de [email protected]
CP12
An Asymptotic Model of Carbon Dioxide Dissolu-tion and Mineral Carbonation Kinetics
We formulate a nonlinear o.d.e. model that describes thekinetics of the dissolution of carbon dioxide in water, andsubsequent chemical reactions through to the formation of
GS09 Abstracts 57
calcium carbonate, a process central to CO2 sequestration.An asymptotic analysis reveals seven different timescaleswithin this system, and approximate expressions for theevolution of each species on each timescale. These approx-imations are used to derive uniformly valid composite ex-pressions for the evolution of the system to equilibrium.
The Riemann Solution for the Injection of Super-critical Carbon Dioxide and Steam in a PorousMedium
We solve a model for flow of CO2, vapor and water inporous media, neglecting compressibility, heat losses andcapillary effects. We study the dynamics of CO2 seques-tration in brine aquifers assuming constant pressure andtemperature, emulating long time scale scenarios. Due tohigh pressures and temperatures, CO2 appears in super-critical state. To solve the Riemann Problem, we analyzemathematical structures such as rarefactions, shock wavesand their bifurcations.
Capillary Effects on the Dynamic Sequestration ofCo2
We present a series of models of the dynamics of a plumeof CO2 being sequestered in a confined inclined permeablerock. We describe the effects of capillary retention in thepore spaces both on the advancing CO2 front and, in thelate stages, on the receding CO2 front. We also discussthe role of capillary forces in controlling the entry pressureof CO2 into a neighbouring, lower permeability layer, andhow this can enhance the lateral dispersion of the flow asit thins. Simplified analytical models are presented alongwith numerical solutions of the governing equations to il-lustrate the key effects.
Andy WoodsBP Institute for Multiphase FlowUniversity of [email protected]
CP13
On An Optimal Number of Time Steps For a Se-quential Solution of An Elliptic-Hyperbolic System
We are interested in efficient sequential solution of thepressure-saturation formulation of two-phase flow equa-tions in porous media. We estimate an optimal numberof pressure updates in sense of saving computational ef-forts at given accuracy. An analytical solution to a spe-cial initial boundary value problem for the 1D formulation
of the coupled elliptic-hyperbolic system is used for suchoptimization. We show that global and local optimizationprocedures yield better results than with equidistant steps.
An Efficient Algorithm for Modelling DynamicDrainage
The high computational cost involved in modelling viscouseffects in multiphase flow simulations using large pore net-work models comes from the requirement to solve a newlarge linear system every time one phase displaces anotherfrom a pore. To address this problem, we propose the useof an algorithm that performs a low-rank sparse Choleskyupdate. This direct sparse solver is faster than the pre-conditioned conjugate gradient (PCG) iterative solver, andeliminates the slowing down associated with the PCG inthe case of adverse viscosity ratios. We apply the methodto a simple description of two-phase dynamic drainage inunstructured pore network models.
We discuss the development of a new, genuinely multi-dimensional second order semi-discrete central scheme forsolving hyperbolic conservation laws arising in the simula-tion of multiphase flow problems in heterogeneous porousmedia. The scheme produces accurate solutions, partic-ularly in the presence of high permeability flow channels,which lead to strong restrictions on the time step selection.Numerical simulations are presented and discussed for ap-plications in CO2 sequestration and oil reservoir simula-tion.
Felipe PereiraDepartment of MathematicsUniversity of [email protected]
Marcos MendesDepartment of MathematicsUniversity of [email protected]
Simone RibeiroDepartment of MathematicsUniversidade Federal Fluminense, [email protected]
58 GS09 Abstracts
CP13
Testing Nonlinear Solver Techniques on DifficultFinite Element Problems in Unsaturated Flow inPorous Media
Discritizations of unsaturated flow in porous media createextremely difficult nonlinear systems of equations to solve.This presentation will show results of testing combinationsof Newton only and Newton with Picard nonlinear itera-tions with bisection, quadratic, and forcing function onlyline searches. A three-dimensional research groundwaterfinite element computer program written for the parallelenvironment was used for the testing. Results for both ananalytical solution and a laboratory test problem will bepresented.
Fred T. TracyEngineer Research and Development CenterWaterways Experiment [email protected]
CP14
Finite Volume Approximation ofa Multi-Component Multi-Phase Reservoir Modelwith Heat Transfer
In order to interpret recorded temperatures, we are in-terested in the thermal simulation of a multi-component,multi-phase (oil, gas and water) flow in petroleum reser-voirs. To do, we have extended an existing isothermalsimulator (GPRS : General Puropse Reservoir Simulator)by adding an exhaustive energy equation and correspond-ing thermodynamics. Finite volumes are employed for thespace discretization and the nonlinear system obtained issolved by Newton-Raphson’s method. Numercal tests in-cluding real test-cases will be presented.
Stability of combustion in a porous medium is studied ina simplified model that takes into account the balance be-tween heat generation with temperature dependence givenby Arrhenius law and heat losses due to conduction to therock formation. The system evolution is described by in-finitely many nonlinear modes. Its long time behavior isdictated by the two dominant modes, whose phase diagramcontains two attractors and a saddle, as in classical chem-ical engineering.
Pablo Castaneda
Instituto Nacional de Matematica Pura e Aplicada(IMPA)[email protected]
Dan MarchesinInstituto Nacional de Matematica Pura e [email protected]
CP14
Rarefaction Waves for Flow in Porous Media withMass Transfer Between Phases
Using singular perturbation methods, we study the longtime behavior of rarefaction waves appearing in stiff non-hyperbolic system of balance laws modelling thermal flowin porous media for several chemical species and phases.By enforcing thermodynamical equilibrium laws, they re-duce to systems of conservation laws, where non-isothermalevaporation or condensation rarefactions appear. Howeverunder thermodynamical equilibrium there should be nomass transfer between phases: we discuss here the contra-dictory existence of these mass transfer rarefaction waves.
Dan MarchesinInstituto Nacional de Matematica Pura e [email protected]
Mathematical Modelling of Crown Forest Fire Ini-tiation
Mathematical model of forest fire was based on an anal-ysis of known experimental data and using concept andmethods from reactive media mechanics. The forest andcombustion products are considered as a homogeneous twotemperatures, reacting, non - deformed medium. The re-search is done by means of mathematical modeling of physi-cal processes. It is based on numerical solution of Reynoldsequations for chemical components and equations of en-ergy conservation for gaseous and condensed phases. Theboundary-value problem is solved numerically. As a re-sult of mathematical modeling the fields of temperatures,mass concentrations of components of gaseous phase, vol-ume fractions of components of solid phase, as well as vec-torial fields of velocity at different instants of time withtaking into account mutual influence of the layer of atmo-sphere and a crown fire on each other was obtained.
Valeriy A. PerminovBelovo Branch of Kemerovo State Universityp [email protected]
CP14
Nonlinear Operator Splitting for Thermal Multi-
GS09 Abstracts 59
Phase Multi-Component Displacements
The characteristics associated with temperature and com-positions are closely related in thermal-compositional flowproblems. However, use of a tie-line based CompositionalSpace Parameterization (CSP) approach, these character-istics can be separated at the nonlinear level with appropri-ate assumptions. For more than two phases, tie-simplexes(tie-triangles for three-phase) can be used for the parame-terization. We demonstrate that for thermal-compositionalsimulations, we can employ an efficient sequential couplingscheme, thus avoiding a fully coupled strategy.
A Numerical Method for System of Wave, Current,and Seabed Morphology in Coastal Processes
In order to accurately simulate multi-physics coastal oceanprocesses, the wave action equation, the shallow-waterequations, and the Exner equation are coupled in a simulta-neous manner. A flux-limited version of the Roe scheme isderived to discretize the coupled system for high-resolutionsolutions. Numerical examples and prediction of actualproblems will also be presented to demonstrate the per-formance in comparison with convention approaches andinteraction among wave, current, and seabed morphology.
Hansong TangDept. of Civil Eng., City College of NewYork, [email protected]
Morphodynamics Discontinuous Galerkin Mod-elling with Applications to Stratigraphy
Morphodynamics is the study of the time-dependent inter-action and adjustment of the seafloor (or riverbed) topog-raphy and hydrodynamic processes. To deal with problemsin which complex domains and a range of scales betweenthe physical processes are involved, the application of ro-bust numerical methods is of paramount importance. Inthis work, we introduce a discontinuous Galerkin methodto solve the hydrodynamics and the seafloor evolution andapply it to stratigraphy modelling.
Maximum Likelihood Parameer Estimation byEnsemble-Based Filters
The maximum likelihood (MLE) parameter estimation isa mathematically rigorous and practically robust method,and ensemble-based filters can be an efficient tool for MLEparameter estimation. These filters calculate the like-lihood function of the observations. I will discuss theMLE method, its implementation for ensemble-based fil-ters, and advantages and disadvantages. I will also presentthe results of estimating the noise strength of a stochasticPDE model that conceptually represents an Atlantic OceanThermohaline Circulation.
Estimation of Non-Gaussian Random Fields withthe Ensemble Kalman Filter and Kernel PrincipalComponent Analysis
Because history matching is an ill-posed problem with non-unique solutions, additional prior information, usually inthe form of geological constraints on the history match-ing problem, is generally required to obtain geologicallyrealistic history matched models that have good predic-tive capability. However, history matching is a computa-tionally intensive process, especially for large scale simu-lation models, as it usually requires numerous simulationsto obtain a history matched solution. Towards this end,the ensemble Kalman filter (EnKF) has recently gener-ated significant attention as an efficient approach for as-similating dynamic data. Although the EnKF has manyadvantages such as ease of implementation and efficient un-certainty quantification, it is technically appropriate onlyfor random fields (e.g., permeability) characterized by two-point geostatistics (multi-Gaussian random fields). Real-istic systems however are much better described by non-Gaussian random fields characterized by multipoint geo-statistics, which is capable of representing key geologicalstructures such as channels. History matching algorithmsthat are able to reproduce realistic geology provide en-hanced predictive capacity and can therefore lead to betterreservoir management. In this work, we apply kernel prin-cipal component analysis (KPCA) to parameterize non-Gaussian random fields characterized by multipoint geo-statistics. By using high order polynomial kernels, ker-nel PCA enables preserving arbitrarily high-order statis-tics of non-Gaussian random fields, thereby providing thecapability to reproduce complex geology. The KPCA pa-rameterization in then applied in conjunction with EnKF,which allows dynamic data assimilation while ensuring thatthe prior geological characteristics of the updated randomfields (such as channels) are retained during the Kalmanupdate. Furthermore, the KPCA parameterization dra-matically reduces the number of state variables on whichthe EnKF is applied, thereby improving the efficiency ofthe EnKF. The overall procedure is then applied to severalsynthetic examples. The approach is shown to better re-produce complex geology, which leads to improved historymatches and better predictions, while retaining reasonablecomputational requirements.
Pallav SarmaChevron Energy Technology CompanySan Ramon [email protected]
A new response surface method called Wavelet KernelANOVA, is proposed for uncertainty propagation and sen-sitivity analysis of an expensive multiphase flow reservoirsimulator. The method is based on a wavelet Reproduc-ing Kernel Hilbert Space (RKHS) technique. An adaptiveexperimental design method is also proposed to iterativelyimprove the accuracy of the response surface. The perfor-mance of the method are compared with a similar approachbased on kriging on a realistic reservoir model.
THEREDA AThermodynamic Reference Database for NuclearWaste Disposal in Germany
THEREDA provides consistent thermodynamic datasetsto assist the long-term safety assessment of nuclear wasterepositories. It meets specific German requirements (highsalinities and temperatures). THEREDA is build on a re-lational SQL-database. The application of referential in-tegrity, thermodynamic dependencies, alternative datasetsand uncertainty estimates allows to deliver tailor-made pa-rameter files for use in geochemical modelling software likeEQ3/6, GWB and ChemApp covering law-of-mass-actionand Gibbs-energy-minimization codes.
Mathematical Issues in Subsoil Bioventing Mod-elling and Optimal Design
Oxygen is required in bioventing to improve the activityof bacteria to biodegrade contaminants in the subsoil andthus air is inflated through wells. The mathematical modeldescribes the bacteria population dynamics and the dy-namics of a multiphase, multicomponent fluid in porousmedia. A critical point of the design problem is to choosewell positions and air flow rates to optimise the biodegra-dation process. Several approaches are possible: minimis-ing the costs, maximising the instantaneous biodegrada-tion rate, maximising oxygen concentration, subsoil air flowevaluation and so on.
Application of a Coupled VF/EF Multi-ScaleMethod to Cement Media Homogenization
We present here the results of a work on a multiscale res-olution method using both Finite Volums and Elements inthe field of numerical homogenization. Our method relyupon the coupling of two grid scales: a coarse one and afine one. The trick is to build a Finite Element basis onthe coarse grid from problems solved on the fine one. Inprevious work on this subject, the Finite Element methodwas used in both the fine and coarse scales, whereas, inour approch, the fine scale simulations are made via FiniteVolums. This way, we hope to increase the stability of themethod in view of strong discontinuities and anisotropy ofthe studied media. Our application exemple is a model ofcement media, where very important variations of diffusiv-ity occur.
Subdomain Time Stepping for Transport in PorousMedia
In many problems concerning transport in heterogeneousporous media the simulation domain can be divided intoseveral subdomains with different hydrogeological proper-ties. These different properties imply different time scalesthat we would like to take into account by using differ-ent time steps in the different subdomains. This leads tospace time domain decomposition. We discuss various for-mulations of this domain decomposition method using theSchur complement formulation or the Schwarz waveformrelaxation method.
Jerome JaffreINRIA-Roquencourt78153 Le Chesnay cedex [email protected]
We analyze the irreversible bimolecular reaction A+B-¿Cin a stratified random porous medium. In case of fastreaction, the problem can be solved in terms of a singleadvection-dispersion equation for the mixing ratio (Z). Wecompute the reactants concentrations expected values byusing the probability density function (pdf) of Z. We as-sume the pdf of Z to be beta-distributed with mean andvariance computed by semi-analytical solutions
albert j valocchi, Albert J. valocchiDepartment of Civil & Environmental EngineeringUniversity of Illinois at [email protected], [email protected]
CP20
Modern Techniques of Software Development forSimulation of Complex Applications – A Selectionof Recent Development in the Contextof the DuneFramwork
In this paper we present software techniques as well as re-cent development of the software package DUNE [1], [2]and in particular of the module DUNE-FEM [3]. For sev-eral test problems as well as for more complex applica-tions, such as the simulation of fuel cells, we present nu-merical results. These results have been obtained usingmodern simulation techniques such as higher order contin-uous and discontinuous Galerkin methods (also in a stabi-lized version [4]) in combination with local grid adaptiv-ity. The parallelization of the code has been taken intoaccount which is in combination with local grid adaptivitya non-trivial part since dynamic load balancing has to beendone. References [1] P. Bastian, M. Blatt, A. Dedner,C. Engwer, R. Klofkorn, M. Ohlberger, and O. Sander. Ageneric grid interface for parallel and adaptive scientificcomputing. I: Abstract framework. Computing, 82(2-3):103–119, 2008. [2] P. Bastian, M. Blatt, A. Dedner,C. Engwer, R. Klofkorn, R. Kornhuber, M. Ohlberger, andO. Sander. A generic grid interface for parallel and adap-tive scientific computing. II: Implementation and tests inDUNE. Computing, 82(2-3):121–138, 2008. [3] A. Ded-ner, A. Klofkorn, M. Nolte, and M. Ohlberger, DUNE-FEM – the fem module, http://www.mathematik.uni-freiburg.de/iam/research/projectskr/dune/, Mathematis-ches Institut, Universitat Freiburg, 2008. [4] A. Ded-ner and R. Klofkorn. A generic stabilization approachfor higher order discontinuous galerkin methods for con-vection dominated problems. Preprint Nr. 8 (sub-mitted to SIAM Sci. Comp.), Mathematisches Insti-tut, Universitat Freiburg, http://www.mathematik.uni-freiburg.de/IAM/homepages/robertk/postscript/dedner kloefkorn limiter.p2008.
Nonlinear Solution Strategies for CompositionalSimulation
We present a simulation framework based on automatic dif-ferentiation that provides wide flexibility in choosing thenonlinear formulation and selecting primary and secondaryvariables. The behavior of new and existing nonlinear for-mulations for compositional simulation is analyzed. Fullyimplicit formulations are considered, where the full Jaco-bian in terms of both the primary and secondary variablesis constructed. The behaviors of the Newton iterationsare analyzed for different variable sets, equation alignment,and nonlinear updating strategies.
Both accurate and computationally efficient simulationsof turbulent flows are needed to understand and predictoceanic flows. Reduced-order models represent naturalchoices in these applications. The fundamental challenge isto retain the physics of the underlying turbulent flow whilekeeping the computational cost at a minimum. In this talk,we will present reduced-order modeling strategies synthe-sizing ideas originating from proper orthogonal decomposi-tion and large eddy simulation of turbulent flows. In partic-ular, we will present approaches based on the variationalmultiscale and dynamic subgrid-scale methods. Analysisand numerical illustrations of our methodology will also bepresented.
Numerical Simulation of Population Balance Sys-tems
The talk will describe a numerical approach for simulatingpopulation balance systems which is mainly based on finiteelement methods. It will address the following topics:
• the simulation of the turbulent background flow,
• the simulation of transport–dominated equationswithout spurious oscillations in the computed solu-tions,
• the impact of using different schemes for solving thehigher–dimensional equation for the population bal-ance.
The numerical simulations will consider a precipitationprocess.
Large Eddy Simulation of Mixing in the Lock-exchange Problem
LES of a 3D lock-exchange problem are presented, whichcontains shear-driven mixing, internal waves, interactionswith boundaries and convective motions, while having asimple domain, initial and boundary conditions, and forc-ing. Two general classes of LES models are tested, namelyeddy viscosity models based on constant-coefficient and dy-namic Smagorinsky models, and an approximate deconvo-lution (AD) model. It is found that constant-coefficient
Smagorinsky models can only provide a marginal improve-ment over under-resolved simulations, while both dynamicSmagorinsky and AD models lead to significant improve-ments in mixing accuracy. The primary accomplishmentof this study is that it is shown that the hybrid approachattains the best agreement with the mixing curve fromDNS, while being computationally approximately a thou-sand times faster.
Resolution and Scale Dependence of Relative Dis-persion in a Hierarchy of Ocean Models
We examine a hierarchy of ocean models, ranging from sim-ple 2D turbulence simulations to North Atlantic HYCOMoutput, to determine the effect of Eulerian spatial modelresolution on the two particle statistics of synthetic driftertrajectories. In each case, particle dispersion at large timeand space scales is found to be controlled by hyperbolicstructures produced by identifiable meso-scale features ofthe flow. In all cases, time-distance graphs given in termsof computed Finite Size Lyapunov Exponents show an in-crease in the extent of exponential scaling with increasingspatial smoothing of the velocity field and scaling of thelimiting exponent with resolved hyperbolicity.
Modeling of Multi-Phase Flow Processes in HeatedFracture-Matrix Systems
Analysis of the proposed underground repository for ra-dioactive waste at Yucca Mountain, Nevada, requires pre-diction of water-vapor flow processes in unsaturated frac-tured rock at elevated temperatures near and above theboiling temperature of water. This constitutes a challeng-ing modeling problem. We elaborate on these challengesand present possible conceptual and solution approaches,with specific focus on the respective roles of fractures andmatrix and their interaction.
Jens T. BirkholzerLawrence Berkeley National Laboratory
Massively Parallel Discrete Fracture and MatrixSimulations - A Route to Faster and More Real-istic Predictions
We present a new massively parallel algorithm for discretefracture and matrix simulations which is based on finite ele-ment finite volume discretisations and hierarchical solvers.We observe linear scaling for up to 64 processors on mod-els containing several million degrees of freedom. This nowallows us to resolve the non-linear coupling of small scalecapillary – viscous and large scale gravitational – viscousprocesses adequately for realistic high-resolution represen-tations of fractured reservoirs.
Stephan MatthaiMontan University of LeobenDepartment of Minearl Resources and [email protected]
Dim CoumouPotsdam Institute for Climate Impact ResearchEarth System [email protected]
MS2
Complex Gas–Water Processes in DiscreteFracture–Matrix Systems
Degassing effects may occur in fractures in the vicinity ofdeep radioactive-waste-disposal sites as a result of a pres-sure drop. These effects play an important role e.g. in theinvestigation of the hydraulic conditions in the near fieldof the disposal sites. The aim of this presentation is tocontribute to the understanding of non-isothermal behav-ior of water–gas systems in the near field of atomic wastedisposal sites in fractured porous media. For the simula-tion on the laboratory scale we use a percolation model.To transfer the information from the laboratory scale tothe field scale we use a renormalization scheme.
Rainer HelmigIWS, University of Stuttgart, GermanyInstitut fur [email protected]
Insa NeuweilerUniversity of Hannoverneuweiler @hydromech.uni-hannover.de
Jennifer NiessnerInstitut fuer Wasserbau (IWS)Universitaet Stuttgart, [email protected]
MS2
An Interfacial-area Based Approach for ModelingDegassing Processes in Fractured Porous Media
In the vicinity of radioactive waste disposal sites, degassingmight occur in fractures which is to be avoided as it allowsradioactive substances to reach the ground surface rela-tively quickly. The degassing in the fracture is highly de-pendent on the interfacial area separating the fluid phaseswhich cannot be taken into account by classical models.Therefore, we present a new approach based on interfacialareas and derive constitutive equations for fractures froma micro-scale model for single fractures.
Jennifer NiessnerInstitut fuer Wasserbau (IWS)Universitaet Stuttgart, [email protected]
Production Optimization and Reservoir InverseModeling Using Principal Component Analysis
Principal Component Analysis (PCA) is the basis of pow-erful model order reduction procedures for optimizationwithin the oil industry. By PCA a high-dimensional linearspace is optimally projected onto another space of muchlower dimension. These spaces can represent state vari-ables (pressure and saturation) and reservoir properties(permeability and porosity). In this talk I will explain thefundamentals of PCA and illustrate its use in optimizationby examples from production optimization and reservoirmodel inversion.
Identifiability, Controllability and Observability inHydrocarbon Reservoir Models
Over the past few years systems and control concepts havebeen applied in reservoir engineering, e.g. optimal control,Kalman filtering, model reduction. The success of theseapplications is determined by the identifiability, controlla-
64 GS09 Abstracts
bility and observability properties of the reservoir modelat hand. In this presentation the controllability and ob-servability properties of single-phase and multi-phase flowreservoir models are analyzed and interpreted. Addition-ally, it is shown how to determine and use approximatemodel structures with identifiable parameterizations.
Jorn F.M. van-DorenShell International Exploration & [email protected]
Model Reduction via Multiscale Methods: Appli-cation to Water Flooding Optimization
In this talk we discuss the use of a multiscale mixed finiteelement method (MsMFEM) for model order reduction inreservoir engineering optimization problems. We illustratethe use of the methodology for finding optimal rates andwell-placement in water-flooding examples. We also discusspros and cons for the MsMFEM approach compared to e.g.,PCA (principal component analysis)-based techniques.
Reducing Model Order by the Definition of a Ran-dom Function From a Given Set of Model Realiza-tions
In this talk we provide a novel framework for geostatis-tics/spatial modeling. We establish a method for defin-ing a random function from a set of generated/measuredrealizations, typically associated with uncertainty analy-sis processes. This function reduces drastically the orderof the model. Once the random function is available, weshow how consistent sets of new realizations, with struc-tural properties similar to the initial set, can be generatedwithout re-running the algorithm or consulting the gener-ating source.
Gas Migration in a Nuclear Waste Repository; aUnified Modelling of Fully and Partially SaturatedPorous Materials
Motivated by modelling the gas migration in an under-
ground nuclear waste repository, we derive a new compo-sitional model of compressible multiphase flow and trans-port in porous media, with interphase mass transfer.Thisnew unified modeling of fully and partially water saturatedporous materials makes possible a unified numerical treat-ment of both fully and partially water saturated situations.Existence of solutions, is proved under adequate assump-tions,and numerical simulations showing the efficiency ofthis modeling are presented.
Phase Exchange in Two-phase Flow in Porous Me-dia and Complementary Problems
Two-phase flow with phase exchange is modeled as a setof time dependant nonlinear partial differential equationswith nonlinear complementary conditions. Such a formula-tion allows appearance and disappearance of a phase as wellas the use of some laws which are widely used as Henry’slaw. We discretize the problem with cell-centered finitevolumes method or mixed finite elements and discuss ap-propriate solvers for the solution of the resulting nonlinearcomplementary problems.
Jerome JaffreINRIA-Roquencourt78153 Le Chesnay cedex [email protected]
Fully Coupled HMFE Approximation for Two-phase Flows in Porous Media
For modeling CO2 underground storage, two-phase, two-component flow with possible diappearance of each phaseand phase-exchange is considered. By neglecting tem-perature variations the model is reduced to a system oftwo time-dependent partial differtial equations. The un-knowns are quantities derived from saturations, composi-tions and pressures,with relations independent of time andspace. The discretization ansatz is a mixed finite elementmethod, not necessarily of lowest order. For the chosendiscretization fluxes are explicitely eliminated and conden-sation leads to a nonlinear equation in terms of Lagrangemultipliers at each time step.
Peter KnabnerUniversitat Erlangen-NurenbergDepartment of [email protected]
Principle of Equivalence and Method of NegativeSaturations for Multicomponent Flow with PhaseTransitions Through Porous Media
We analyse the flow with phase transitions when the two-phase fluid can be alternated with the zones occupied bysingle-phase non-equilibrium fluid. The principle of equiv-alence is poved which says that a non-equilibrium single-phase fluid is hydrodynamically equivalent to an imagi-nary two-phase equilibrium fluid having specificic physicalproperties called the equivalence conditions. One of theseconditions necessarily implies that the saturation of oneimaginary phase is negative. We prove the consistence the-orem which shows that such equivalence conditions do notcontradict to fundamental thermodynamic principles. Theequivalence theorem allows for developing an efficient nu-merical method of modeling these processes, by applyingthe uniform two-phase flow equations for all the flow do-main, but different thermodynamic constitutive conditionsfor single-phase and two-phase zones. The moving inter-face between these zones is detected automatically oncethe saturation becomes negative. The method is tested onone-dimensional analytical solutions. The numerical ex-ample of solving two-dimensional flow problems with ir-regular shape of the interface are obtained. The advancedversion of the method is developed for the cases of diffu-sion, adsorption and chemical reactions, when the inter-face between zones is detected as the surface of the weekdiscontinuity of the saturation field. The simulated exam-ples concerns oil displacement by partially miscible gas andCO2 injection in an aquifer.
Mikhail PanfilovLEMTA-UMR 7563Institut National Polytechnique de Lorraine - [email protected]
We explore the application of Gaussian process emula-tors to uncertainty analysis of groundwater flow models.Emulators statistically approximate the output of com-puter models and can be used as cheap substitutes for themodel. A model of groundwater flow through the CulebraDolomite at the Waste Isolation Pilot Plant site is chosenas a test case to illustrate the methods. The uncertainty inthe output of the model given the uncertainty in the inputsis quantified.
In this talk, we present a Generalized Spectral Decomposi-tion method for probabilistic uncertainty quanti?cation inelliptic equations. The de- composition aims at approach-ing the stochastic solution on reduced bases (stochasticand deterministic), similar to the Polynomial Chaos ex-pansions, except that the expansion bases are not selecteda priori but constructed and adapted to the problem equa-tions and parameters uncertainty. Di?erent algorithmsare considered for the construction of the reduced bases(Power-type and Arnoldi-type iterations), and comparedon a test problem related to steady linear and non-linearground water ?ow simu- lations (Darcy equations). Per-formance and implementation aspects are discussed withregard to the classical stochastic ?nite element method,and we discuss the extension of the Generalized SpectralDecomposition method to large unsteady problems arisingin nuclear waste disposal simulations.
Olivier Le MatreLaboratoire d’Informatique pour la Mecanique et [email protected]
Anthony NouyUniversite de NantesInstitut de Recherche en Genie Civil et Mecanique (GeM)[email protected]
Lionel MathelinLaboratoire d’Informatique pour la Mecanique et [email protected]
MS5
Efficient Iterative Solvers for Stochastic Ground-water Flow Problems
We consider finite element discretizations of PDEs with astochastic differential operator arising in groundwater flowsimulations. We report on the design of efficient itera-tive solvers for the resulting large coupled linear systemof equations. In particular we present a new approach forpreconditioning such linear systems which takes advantageof the special Kronecker product structure of the Galerkinmatrix. We demonstrate the efficiency of our techniqueson selected model problems.
Oliver G. ErnstTU Bergakademie FreibergFakultaet Mathematik und [email protected]
MS5
Efficient Numerical Methods for Stochastic Com-putations
Numerical algorithms for effective incorporation of uncer-tainty into differential equations are discussed. The un-certainty is modeled as functions of random variables, andthe governing equations are treated as stochastic. By usinggeneralized polynomial chaos (gPC) expansion, the solu-
66 GS09 Abstracts
tions are expressed as convergent series of orthogonal poly-nomials in terms of the random variables. Based on gPC,various numerical schemes can be designed, via either astochastic Galerkin (SG) approach or a stochastic colloca-tion (SC) approach. In this talk we discuss and comparethe strength and weakness of gPC based SG and SC meth-ods. The focus is on their accuracy and efficiency. Wewill also demonstrate the clear connection between thesemethods and the classical deterministic spectral methodsand their applications to inverse problems such as param-eter estimation.
Interfacial Momentum Balance Between Non-equilibrium Phases
The net stress on a flat interface is not zero during a phasechange. A net stress is needed to balance the jump in mo-mentum experienced by the molecules as they cross theinterface; a stress-free interface introduces an error in thesolution for the velocity field. This work presents an ex-pression for interfacial momentum balance between non-equilibrium phases and investigates the equilibration pro-cess in the presence of natural convection.
Numerical and Mathematical Aspects of Multi-phase/Component Solubility in Water
The numerical and mathematical aspects of multi-phase/component solubility in water are considered where ther-modynamically consistent three-phase behavior is coupledwith nonlinear transport in porous media. In this model,several hydrocarbons are allowed to dissolve in the aque-ous phase while all other components expect water exist inthe oil and gas phases. The proposed computation of anygas solubility in the aqueous phase is based on EOS frame-work. The numerical simulation results are presented anddiscussed.
Tie-simplex Based Parameterization for Thermo-dynamical Equilibrium Computations of Multi-component Systems with Arbitrary Number ofPhases
We present a general framework for the computation ofthermodynamic equilibrium of multi-component systemsthat form an arbitrary number of phase. We parame-terize the high-dimensional compositional space using tie-simplexes (tie-triangles for three-phase) to represent themulti-phase regions. The tie-simplex computation andinterpolation procedure complement the parameterizationand form a complete mathematical framework. We demon-strate the efficiency of the method for several multi-phaseequilibrium problems of practical interest that includeboth equilibrium flash calculations and multi-phase, multi-component flow problems.
On the Use of Streamlines and Front Tracking forCompressible Flow
We investigate the use of streamline methods and fronttracking for compressible flow. Operator splitting is usedto incorporate gravity effects. We apply this method forthe injection phase of CO2 storage and point out in whichsituations the method has advantages and when other ap-proaches should be considered.
The main challenge in modeling fluid flow throughnaturally-fractured carbonate karst formation is how toaddress various flow physics in complex geological archi-tectures due to the presence of caves which are connectedvia fracture networks at multiple scales. In this paper,we present an efficient upscaling process that is based ona unified fine-scale multi-physics model which adaptivelycouples Stokes-Brinkman with discrete fracture networkmodels. The underlying idea is to use Stokes-Brinkmanmodel to represent flow through rock matrix, void cavesand intermediate flows in high permeability regions and touse discrete fracture network model to represent flow infracture network. This unified approach adaptively treatsfractures as lower dimensional geometries with permeabil-ities assigned according to their apertures. Consequently,various numerical solution strategies can be efficiently ap-plied to greatly improve the computational efficiency inflow simulations.
On Eulerian-Lagrangian Formulations and TheirAnalyses
We present optimal-order error estimates for a class ofEulerian-Lagrangian methods for advection-diffusion equa-tions, which are uniform with respect to the vanishing pa-rameter epsilon. The estimates depend on the given ini-tial, boundary, and right side data but not on the truesolution. We also discuss optimal-order error estimatesfor Eulerian-Lagrangian methods for the coupled system inporous medium flow. These results justify the strength ofEulerian-Lagrangian methods. Numerical results are pre-sented to verify the analysis.
Hong WangUniversity of South CarolinaDepartment of [email protected]
KaiXing WangSchool of Mathematics, Shandong UniversityJinan, Shandong 250100, [email protected]
MS8
Analysis of a Scale Similarity LES Model Designedfor Certain Stratified Flows
We propose and give some mathematical analysis resultsfor a Large Eddy Simulation model which involves regu-larization only in the horizontal variables. The method weconsider, which fits into the class of scale similarity models,is similar to the “Simplified Bardina model’ introduced byLayton and Lewandowski (2006). We are able prove thatour new model has good mathematical properties (exis-tence and uniqueness in appropriate Sobolev space). Themathematical foundation is one of the first steps for thevalidation, even if numerical experiments on realistic prob-lems will be the necessary further step.
The idea of considering extra-viscosity (sub-grid-scaleterms) acting only in the horizontal variables is well dif-fused in the community, especially for the mixing dam
breaking problem in the context of Boussinesq approxi-mation, see Ozgokmen et al. (2007) and this is one of themotivations for our work. Moreover, the analysis of fluidswith very different viscosity coefficients in the horizontaland vertical direction is motivated also by the study of Ek-man layers, see for instance the introduction of Chemin etal. (2000), Desjardins and Grenier (2000) and Pedlosky(1979).
Luigi C. BerselliDipartimento di Matematica Applicata ”U.Universita’ di [email protected]
MS8
A Goal-Orientated Mesh Adaptive and Dual-Weighted Pod for Data Assimilation into OceanModelling
A novel dual weighted POD method for adapting mesh,order reduction and 4D-Var data assimilation is presentedhere. The aim is to (1) optimise the reduced bases , thusimproving the quality of reduced ocean models representa-tion of the goal; (2) design an error measure to guide anadaptive meshing algorithm. The goal functional for opti-mising reduced bases and meshes has been designed to beconsistent with that for 4-D Var data assimilation.
Fangxin FangDepartment of Earth Science and EngineeringImperial College London, [email protected]
MS8
On a LES-deconvolution Model for the Ocean witha Fixed Wind
We model the oceanic flow by using the Navier-Stokesequations together with the rigid lid assumption, whichyields flux conditions for the velocity at the surface, thatcouple the ocean to the atmosphere. We stay here in thecase of a fixed atmospheric wind. We start by showinghow to adapt the periodic Leray-Alpha model to the caseof those realistic boundary conditions. We next turn tothe Layton-Lewandowski deconvolution model, based onthe Van Cittert algorithm. This model cannot be adaptedlike this to the case of the ocean. We observe that the VanCittert algorithm is in fact nothing but a finite differencescheme of a certain evolution equation. This yields a newLES deconvolution model adjusted to the ocean, for whichwe prove the existence and the uniqueness of a smooth so-lution. We also prove the convergence of those solutionsto a dissipative solution to the Navier-Stokes equationswhen the deconvolution parameter goes to infinity. Thisresearch work has been initially performed by A.-C. Ben-nis, R. Lewandowski and E.-S. Titi. R. Lewandowski issupported by the ANR project 08FA300-01.
Three-Dimensional Dynamics in Non-ParallelShear Stratified Flows
The instabilities of non-parallel flows such as those in-duced by polarized inertia-gravity waves embedded in astably stratified environment are analyzed in the contextof the 3D Euler-Boussinesq equations. We derive a suf-ficient condition for shear stability and a necessary con-dition for instability in the case of non-parallel velocityfields. Three dimensional numerical simulations of the fullnonlinear equations are conducted to characterize the re-spective modes of instability, their topology and dynamics,and subsequent breakdown into turbulence. We investi-gate three-dimensional characteristics and present compu-tational results on Lagrangian particle dynamics.
Alex MahalovDepartment of MathematicsArizona State [email protected]
Mohamed Moustaouidept. of mathematics and statisticsArizona State University, Tempe, AZ [email protected]
MS9
Asymptotic and Numerical Modelling of Flows inFractured Porous Media with Finite Volume Meth-ods
We study some asymptotic models used to compute theflow outside and inside fractures in 2-D heterogeneousporous media. The flow is governed by the Darcy law withlarge discontinuities in the permeability tensor. The frac-tures, supposed to have a small aperture with respect to themacroscopic length scale, are then asymptotically reducedto immersed fault interfaces. A cell-centered finite volumescheme on general polygonal meshes fitting the interfacesis derived to solve the set of 2-D/1-D equations with theadditional differential transmission conditions linking bothpressure and normal velocity jumps through the interfaces.We prove the convergence of the FV scheme and derive ex-istence and uniqueness of the solution to the asymptoticmodels proposed. Various numerical results are reportedshowing different kinds of flows in the case of impermeableor partially/highly permeable immersed fractures. Thesenumerical solutions of the asymptotic models are validatedby comparing them to the solutions of the global Darcymodel or to some analytic solutions.
Philippe AngotLATP Universite de MarseilleMarseille, [email protected]
Florence HubertLATPUniversite de Provence, Marseille, [email protected]
MS9
Development of Model Concepts for Flow inMacro-porous Media on Different Scales
The fast infiltration of heavy rainfalls in macro-porous hill-slopes is one of the key processes which triggers fast infil-tration, saturuation and pressure increase and thus defor-mation processes. A so-called cascade model concept hasbeen developed for macropore infiltration based on con-trolled laboratory experiments and it is implemented in theframework of a two-phase flow model concept for porousmedia. The exchange parameters which are determined bythis tool will be transfered to a double continuum modelconcept which is then applied on the field scale.
Kai Germer, Juergen BraunUniversity of Stuttgartkai.germer @iws.uni-stuttgart.de, [email protected]
MS9
Deformation Band Populations in Fault DamageZones - Impact on Fluid Flow
Fault damage zones in highly porous reservoirs aredominated by deformation bands that generally havepermeability-reducing properties. Due to an absence ofsufficiently detailed measurements and the irregular distri-bution of deformation bands, a statistical approach is ap-plied to study their influence on flow. A stochastic modelof their distribution is constructed, and band density, dis-tribution, orientation and flow properties are chosen basedon available field observations. The sensitivity of these dif-ferent parameters on the upscaled flow is analyzed. Theinfluence of a heterogeneous permeability distribution wasalso studied by assuming the presence of high permeabil-ity holes within bands. The fragmentation and position ofthese holes affects significantly the block effective perme-ability. Results of 1D and 2D local upscaling are comparedand qualitatively similar results for the flow characteristicsare obtained. Further, the procedure of iterative local-global up scaling is applied to the problem.
Sylvie SchuellerCIRP, Bergen, Norwaynow at IFP, Rueil-Malmaison, [email protected]
Magne S. EspedalDepartment of Mathematics,University of Bergen, [email protected]
Haakon FossenCentre for Integrated Petroleum ResearchUniversity of Bergen, [email protected]
GS09 Abstracts 69
Alvar BraathenCentre for Integrated Petroleum ResearchBergen, [email protected]
MS9
Preferential Flow in Fissure Systems
The asymptotic analysis of Darcy flow near a very thinhighly-permeable fissure is revisited. The limiting problemis Darcy flow in the region coupled to tangential Darcy flowthrough the lower-dimensional interface approximation ofthe fissure. Stokes flow in the fissure leads to Brinkmannflow in the interface model. For a fine-scale periodic struc-ture of highly-permeable fissures of width decreasing attwice the rate of the periodic scale, we obtain the homog-enized limit and establish the two-scale convergence. (pre-liminary report)
Particle Swarm Optimization with Surrogate Func-tions for Water Management Problems
We provide a framework to improve the efficiency of theparticle swarm optimization method by using surrogatefunctions. For water resources management problems, thishelps alleviate the burden of costly simulation calls re-quired to model groundwater flow and transport that areembedded in the objective function. We present resultson a hydraulic capture problem posed as a mixed-integernonlinear problem in determining the number of wells, wellrates, and locations to minimize clean up costs.
Tailoring Hybrid Optimization Methods for WaterResources Management
Hybridizing optimization methods has emerged as highlypromising. However, the effectiveness of this approach maybe compromised if the methods combined are not suited toone other or to the application of interest. In this talk,we will discuss hybrid optimization in the context of waterresources management. We will focus on the characteristicsof the problem domain and of some optimization techniquesand present numerical results demonstrating the efficacy oftailoring hybrids.
Hybrid Optimization Methods for SimulationBased Problems
Simulation based optimization is increasingly important forhydraulic application problems, especially the need to han-dle real-valued as well as integer-valued variables is emerg-ing. Hybrid optimization methods applied in a well suitedframework can help to solve arising problems with less sim-ulations calls if they are well adapted for the underlyingproblems. In this talk we will discuss mixed-integer non-linear simulation based optimization using surrogate func-tions for efficient derivative free optimization.
We demonstrate a POD (proper orthogonal decomposi-tion) tool for calibration of models for saturated flow inporous media. POD was developed for flow control andbuilds a global basis from “snapshots’ taken from a dy-namic simulation. In our steady-state context we build thebasis from the sensitivities. The advantages of POD arethat matrix assembly and factorization for the finite ele-ment simulation are moved out of the optimization loop.We show three-dimensional results that indicate that theperformance of the optimization using POD is essentiallythe same has that for the much more expensive optimiza-tion using the full finite element simulator. We may alsoshow results for global optimization using POD as the sur-rogate model.
Stacy HowingtonUS Army Engineer Research and Development [email protected]
MS11
Coupling Discontinuous Galerkin DiscretizationsUsing Mortar Finite Elements for Advection-Diffusion-Reaction Problems
We investigate DG-DG domain decomposition couplingusing mortar finite elements to approximate the solu-tion to general second-order partial differential equations.This class of equation includes second-order elliptic andparabolic equations, advection-reaction equations, as wellas problems of (mixed) hyperbolic-elliptic-parabolic type.In the formulation, we consider the upwinded flux forthe advective flux and provide the matching condition byweakly imposing continuity of the total flux on the inflowboundary part of the interface and continuity of the total
70 GS09 Abstracts
flux and the solution on the characteristic boundary partof the interface via mortar finite elements. The subdomaingrids need not match and the mortar grid may be muchcoarser, giving a two-scale method. The diffusion coeffi-cient is allowed to be degenerate. Convergence results interms of the fine subdomain scale h and the coarse mortarscale H are then established. If the interface lies in theadvection-reaction regime, with proper choice of h and H,optimal convergence rates are achieved. A non-overlappingparallel domain decomposition algorithm reduces the cou-pled system to an interface mortar problem. The propertiesof the interface operator are discussed.
Analysis of MPFA Convergence on General Ge-ometries
The similarities between the multi-point flux approxima-tion (MPFA) methods and the mimetic finite differencemethod can be used to prove convergence of the MPFAmethods on general grids in 2D and 3D. We examine theassumptions needed for convergence of the MPFA meth-ods, with particular attention to the limitations posed bythe analysis and how the theory compares to the numericalconvergence results.
Annette StephansenUnifob PetroleumCentre for Integrated Petroleum [email protected]
A Multipoint Flux Mixed Finite Element Methodon Hexahedra
We present a mixed finite element (MFE) method forDarcy flow on hexahedral elements, which reduces to cell-centered finite differences and performs well for discontin-uous full tensor coefficients. Motivated by the multipointflux approximation (MPFA) method where sub-face fluxesare introduced, we introduce an enhanced version of thelowest order Brezzi-Douglas-Marini (BDM) MFE space in-volving four velocity degrees of freedom per face. A specialquadrature rule is employed that allows for local velocityelimination and leads to a symmetric and positive definitecell-centered system for the pressures. Theoretical and nu-
merical results indicate second-order convergence for pres-sures at the cell centers and first-order convergence for sub-face fluxes if the grids are sufficiently regular. Second-orderconvergence for face fluxes is also observed numerically.
Homogenization of Porous Media by the PeriodicUnfolding Method
We consider the Stokes and the Navier-Stokes problems ina perforated domain. The holes are periodically distributedwith a period e, their size is of the order de. Both d ande are small parameters. Our aim is to give the asymptoticbehavior of the velocity and of the pressure of the fluid,as these two parameters go to zero. We first discuss thecase d = 1 corresponding to the classical homogenization(cf. [3] and [4]). For the the general case d = ea (i.e., thecase of small holes), we derive several limit problems, corre-sponding to different values of a. These limit problems areeither the Darcys law, or a Stokes problem or the Bringmanlaw. Finally, we study the case of partially perforated do-mains. The proofs use the periodic unfolding method (see,for instance [1] and [2]) which avoiding the introduction ofextension operators, allows us to treat complex geometriesof domains. References [1] D. Cioranescu, A. Damlamian,G. Griso, Periodic unfolding and ho- mogenization, C. R.Acad. Sci. Paris S er. I Math. 335 (2002), 99-104. [2] A.Damlamian, An Elementary Introduction to Periodic Un-folding, GAKUTO International Series Math. Sci. Appl.24 (2005), 119-136. [3] E. Sanchez - Palencia, Non ho-mogeneous Media and Vibration Theory, Lecture Notes inPhysics 127, Springer Verlag (1980). [4] L. Tartar, Incom-pressible fluid flow in a porous medium- convergence of thehomogenization process. Appendix in [3].
Doina CioranescuLaboratoire Jacques-Louis LionsUniversite Pierre et Marie [email protected]
MS12
Multiscale Models for Processes with Evolution ofMicrostructures
Many problems in science and engineering involve time-varying microstructures. Important examples are phasetransitions with microstructures of dendritic or eutec-tic type, or processes in porous media, where deposi-tion/resolution may change the geometry of pores. In suchcases the application of standard homogenization does notlead to a purely macroscopic model as in more traditionalexamples, but to multiscale models. We address the deriva-tion and analysis of such models and their use in simula-
GS09 Abstracts 71
tions.
Christof EckInstitut fur Angewandte Analysis und NumerischeSimulationUniversity of [email protected]
MS12
Upscaling of Differential Equations Modelling theReactive Flow Through a Deformable System ofBiological Cells
We consider a system of model equations coupling fluidflow, deformation of solid structure and chemical reactionsin biological tissue. Our goal is to obtain the upscaledsystem modeling reactive flow through biological tissue onthe macroscopic scale, starting from a system on the celllevel. The novelty of the upscaled model is that it includesBiot’s equations from the soil consolidation theory, coupledwith reactive transport. (Joint work with W. Jaeger andM. Neuss-Radu)
Andro MikelicInstitut Camille Jordan, UFR MathematiquesUniversite Claude Bernard Lyon [email protected]
MS12
Crystal Dissolution and Precipitation in PorousMedia
We propose a pore scale model for crystal dissolution andprecipitation in a porous medium. As a result of the precip-itation and dissolution the pore geometry may change. Us-ing the proposed model we derive upscaled effective equa-tions on the Darcy scale via a formal homogenization pro-cedure. Numerical simulations show that solutions of theupscaled model match the averaged solutions of the pore-scale model very well.
Tycho L. van NoordenDepartment of Mathematics & Computer ScienceTU [email protected]
MS13
Osmotic, Swelling, and Disjoining Pressures atMultiple Scales
In swelling porous media, pressures commonly used includedisjoining pressure, osmotic pressure, and swelling pres-sure. These are attributed to microscopic forces such aselectrostatic forces, Van der Waals forces, and surface hy-dration. Here we discuss the definitions of these differentpressures and examine the relationship between them andmicroscale forces within the framework of two upscalingapproaches for swelling porous materials: hybrid mixturetheory and homogenization.
Lynn BennethumDept Mathematical and StatSci. Univ Colorado [email protected]
MS13
Electro-chemical and Swelling Behaviour of Active
Hydrated Materials
Charged hydrated materials exhibit internal couplingmechanisms stemming from the intrinsic characteristics ofthe constituents. In this content, the model under consid-eration consists of a fluid-saturated solid matrix carryingvolume-free fixed negative charges, while the pore fluid isgiven by a mixture of a liquid solvent and the cations andanions of a dissolved salt. Based on the well-founded The-ory of Porous Media (TPM), use is made of the assump-tion of quasi-static processes. The governing equations aregiven by the volume balance of the fluid mixture governedby the hydraulic pressure, the concentration balance gov-erned by the cation concentration, the overall momentumbalance governed by the solid displacement and the electri-cal continuity equation governed by the electrostatic force.Furthermore, the mechanical solid extra stress is describedby an extended neo-Hookean material law, while the vis-cous fluid flow follows an extended Darcy’s law, which in-cludes the gradients of the ion concentrations and the elec-trical potential. Furthermore, the ion diffusion is describedby an extended Nernst-Planck equation. Finally, the modelis implemented into the FE tool PANDAS by use of a mixedfinite element scheme. The presented examples proceedfrom boundary conditions depending on internal variablessuch that certain stabilisation techniques are needed.
Wolfgang EhlersInstitute for MechanicsUniversity of [email protected]
MS13
Ionic Multi-Species Transport in Porous Materialswith Account to Surface Charge Phenomena
A set of macro level Nernst Planck Poisson type of equa-tions are derived, describing the diffusion of ionic speciesin the pore water of a porous material with charged porewalls, using the electroquasistatic hybrid mixture theory.The effect of the surface charge on the global ionic diffusionis studied by a separate microscopic tentative approachbased on ionic double layer formations. A numerical so-lution technique based on the finite element approach isformulated.
Bjorn JohannessonDept Civil EngineeringUniversity of [email protected]
MS13
Passive Measurements of Electrical Potential as aNon-intrusive Method to Determine and LocateHydromechanical Disturbances
A pore scale model of transport of ions in a charged porousmaterial is developed using a volume averaging operatorapplied to the Nernst-Planck and Stokes equations thatcouple Maxwell and Biot equations. The macroscopicequations are solved with a finite element method. Wepresent results of the forward and inverse modeling. Theinverse modeling is developed inside a Bayesian frameworkusing MCMC samplers. Different applications are shownin hydrogeophysics and volcanology.
Corrected Operator Splitting for Two-phase Flowwith Gravity Forces in Porous Media
In this talk, we apply the corrected operator splitting(COS) algorithm for two-phase flow including gravity inporous media, combining the characteristics method withthe finite volume method. The motivation is to gain thecomputational efficiency from the hyperbolic solver and agood handling of the non-linear flux term with a more accu-rate shock width using large time step. The methodologyof the COS algorithm is described and numerical resultsare given.
Mixed Multiscale Finite Element Methods forPorous Media Flows Using Limited Global Infor-mation
In this talk, I will describe multiscale methods for fluid-structure interaction. This is a highly nonlinear problemthat involves coupling Stokes equations with elasticity. Wepropose several multiscale methods and their analysis. Nu-merical results are presented. This is a joint work withPeter Popov and Yuliya Gorb.
A major advantage of multi-scale methods compared toclassical upscaling consists in their ability to reconstructfine-scale velosity fields, which can be used to solve forphase transport. Here, the stability of an implicit fractionalflow formulation is investigated and an analysis of the clas-sical Newton-Raphson scheme explains the well known, se-vere time step size restriction for S-shaped flux functions.Then, a simple, unconditionally stable modification is de-vised.
Multi-scale Multi-physics Models for Flow andTransport in Porous Media
Physical processes generally take place on different tem-poral and spatial scales. Also, on the same scale, theymay vary in space. A standard approach would solve forthe most complex processes occurring in the domain of in-terest on the finest relevant scale. This is unnecessarilyexpensive. Therefore, we discuss multi-scale multi-physicsapproaches which allow to take the space and time depen-dence of processes into account, thus reducing the requiredamount of data and computing time.
Jennifer NiessnerInstitut fuer Wasserbau (IWS)Universitaet Stuttgart, [email protected]
Rainer HelmigIWS, University of Stuttgart, GermanyInstitut fur [email protected]
MS15
Reduced-Order Models for Time-Periodic Dynam-ics in Ocean Flows
A number of phenomena in ocean flows exhibit time-periodic dynamics. These may arise due to diurnal forcingor are inherent flow features such as breaking waves orLangmuir cells. In this talk, we discuss the adaptation ofreduced-order model development based upon proper or-thogonal decomposition (also known as empirical orthogo-nal functions) to include sensitivity variables. The result-ing dynamical systems exhibit better long time integrationbehavior. This improvement is preserved as parameters inthe model are varied.
Chemical Data Assimilation: Computational Toolsand Applications
The task of providing an optimal analysis of the state ofthe atmosphere requires the development of novel compu-tational tools that facilitate an efficient integration of ob-servational data into models. We discuss several new com-putational tools developed for the assimilation of chemicaldata into atmospheric models. The distinguishing featureof these models is the presence of stiff chemical interac-tions. The variational tools presented in this talk includeautomatic code generation of chemical adjoints, propertiesof adjoints for advection numerical schemes, calculation ofenergy singular vectors and their use in placing adaptiveobservations. Data assimilation results using the 4D-Varmethod are shown for several real test problems to illus-
GS09 Abstracts 73
trate the power of the proposed methods.
Adrian SanduVirginia Polytechnic Institute andState [email protected]
MS15
Some Mathematical Problems in Geophysical FluidDynamics
Abstract not available at time of publication.
Roger TemamInst. f. Scientific Comput. and Appl. Math.Indiana [email protected]
Simulation of Flow and Transport Processes in aDiscrete Fracture Matrix System I.GeostatisticalGeneration of Fractures on an Aquifer AnalogueScale
The critical step for the discrete fracture model approachis the generation of a representative fracture network. Inthis study, we will show a geostatistical fracture generatorwhich integrates statistical geometries and spatial charac-teristics in terms of a standardized variogram, neighbor-hoods and a fracture cell density. Later the flow and trans-port behavior of a fracture matrix system is investigated,where the fracture matrix system is represented by the gen-erated fracture network embedded in a porous rock matrix.
Hakon HaeglandUniversity of Bergen, NorwayDepartment of [email protected]
Andras BardossyIWS, University of Stuttgart, GermanyInstitut fur [email protected]
Helge K. DahleUniversity of Bergen, NorwayDepartment of [email protected]
MS16
Automated Characterization of Fractured Reser-
voirs from Well Tests
Geologically-realistic models of fractured reservoirs areconstructed via a workflow involving the integration ofvarious data. However these data are often incompleteand uncertain, such that the model parameters cannot befully characterized. The Covariance Matrix Adaptation-Evolution Strategy has been used to estimate identifiablefracture sets conductivities from well tests data. The globalsensitivities of the conductivities are computed during op-timization from a response surface methodology and theSobol decomposition technique.
A Linear Finite-element Node-centered Finite-volume Method Including Jump DiscontinuitiesApplied to 2-phase Fracture / Matrix Flow
Using operator-splitting, we compute pressure / velocitywith the FEM and 2-phase transport with the FVM. Atmaterial interfaces, where saturation can become discon-tinuous, we enrich this discretisation with as many degreesof freedom as materials are joined. Additional functional-ity computes sector-to-sector fluxes, retaining higher-orderaccuracy, and pressure on the enriched mesh. An implicitcalculation of nonlinear diffusive transfer processes acrossfracture-matrix interfaces complements our new method.
The simulation of flow in discrete fractured media leads tosolve very large linear systems. Those systems are sparseand with a specific shape due to the underlying physicalproblem. In order to solve efficiently this type of systems,different solvers have been tested. We used both directand iterative ones. The iterative solvers were based on amultigrid method or a conjugate gradient with various pre-conditionings. Sequential and parallel versions were testedon clusters.
Jean-Raynald de DreuzyGeosciences Rennes, UMR6118 CNRSUniversite de Rennes [email protected]
74 GS09 Abstracts
MS17
An Inverse Prob-lem for Coupled Heat-Groundwater Transport inthe Shallow Subsurface
In an effort to improve synthetic thermal imagery for re-mote sensing technologies, a suite of closely coupled numer-ical simulators has been developed. This computationaltestbed includes thermal and moisture transport finite el-ement models, coupled with solar and vegetation models.It is well suited for simulations of specific scenarios, whichotherwise might be difficult and time consuming to repro-duce in the field. This talk will focus on the inverse problemrelated to determining material properties.
Owen J. EslingerUS Army Corps of EngineersInformation Technology [email protected]
MS17
Forecasting Subsurface Model Performance DuringSimulation-based Optimization
In subsurface transport models, constraint violations mayoccur before complete simulation of the remediation time-frame. A new approach exploits this behaviour by monitor-ing remediation constraints during simulation. For a givenfunction evaluation, simulation is pre-empted (terminatedearly) if infeasible results are forecasted. A demonstrationproblem is presented involving the design of a barrier sys-tem constructed from sorptive materials. Results suggestthat model pre-emption can significantly reduce computa-tional cost without affecting solution quality.
Inverse Modeling for Non Linear Groundwater Sys-tems with Local RBF and Global OptimizationAlgorithms with Application to a Large ChineseAquifer
We present new optimization algorithms for identifying pa-rameters in nonlinear partial differential equation modelsof groundwater aquifers. The approach incorporates bothlocal and global optimization methods. It includes the useof response surfaces of the objective function in order toreduce the number of times the computationally expensivePDE model must be computed. The method is applied todata from a large (456 sq km) aquifer that supplies waterto Beijing, China.
Stefan M. WildCornell UniversitySchool of Operations Research &[email protected]
MS17
Simulation-based Nonlinear Least Squares Fitting
In this talk we will discuss the development of a trustregion-based derivative-free optimization solver for nonlin-ear least squares problems. Our emphasis is on achievingrapid decrease of the residuals since the simulation-basedobjectives of interest to us are computationally expensiveand rarely allow for many evaluations. The central issueis managing the set of models as the evaluation historygrows. We will illustrate the algorithm on a variety of sci-entific applications.
Jorge J. MoreArgonne National LaboratoryDiv of Math & Computer [email protected]
MS18
Quasi-Positive Families of Continuous Darcy-FluxFinite Volume Schemes
Families of continuous Darcy-flux control-volume dis-tributed (Multi-point flux Approximations) CVD(MPFA)finite volume schemes are presented for the general full-tensor porous media pressure equation. These schemesmaintain flux and pressure continuity while only dependingon a single degree of freedom per control-volume. The newfamilies of schemes are compared with earlier formulationsin terms of quadrature range, numerical convergence andstability for challenging problems involving high full-tensoranisotropy.
Fully Conservative Streamline Methods for Two-phase Flow
Linear transport of a tracer in an ambient fluid can beapproximated by a fully conservative characteristic or La-grangian method in which both fluids are conserved locally.Nonlinear two-phase flow is a similar transport problem,except that streamlines and characteristics no longer coin-cide. We extend linear characteristic methods to two-phasestreamline methods in one space dimension, preserving themasses of both fluids. Our method is effective with time-steps that are several times the CFL step.
Todd ArbogastDept of Math; C1200University of Texas, [email protected]
Chieh-Sen Huangational Sun Yat-sen UniversityKaohsiung, [email protected]
Control Volume Mixed Methods and Relationshipsto MPFA
The control volume mixed finite element method (CVM-FEM), applied to flow equations in porous media, is a vari-ant of the RT0 mixed method in which finite-volume vectorvelocity test functions yield a discrete Darcy law on controlvolumes, in addition to the usual cell-by-cell conservation.The talk will discuss new results on superconvergence forCVMFEM on distorted quadrilaterals, analysis of CVM-FEM with mortar couplings, and schemes that combinefeatures of CVMFEM and multipoint flux approximation(MPFA).
A Mortar Multiscale Solver for Stochastic Approx-imations of Flow in Random Porous Media
Recent work has shown that the construction of a multi-scale basis for the mortar mixed finite element method cangreatly reduce the computational cost in solving the coarsescale interface problem. We show that this multiscale ba-sis only needs to be computed for the mean of the randomfield and can be reused as an efficient preconditioner foreach of the subsequent stochastic realizations.
Modeling the Pore Space in Carbonate Reservoirsby Using Acids
Carbonate reservoirs can contain large fractures that cancause short circuit flows leaving much oil unproduced. Thispaper investigates the idea of permeability reduction in thefracture by injection of a mixture of acids. A 1-D sim-ulation shows that upstream dissolution and downstreamprecipitation of CaSO4 are the dominant processes. A 2-Dsimulation shows that downstream fractures are clogged,whereas continuously wormholes are created upstream. A
Modelling Microbial Clogging in Porous Media onthe Darcy Scale
A concept is presented to model biomass accumulation inthe subsurface on the Darcy scale. It accounts for bacterialgrowth, deposition of bacteria on the porous medium, de-tachment, decay, and the effects of biofilm growth on thehydraulic properties of the porous medium. The intendedapplication is the simulation of the plugging of damagedor fractured cap-rock in injection well vicinity in a CO2
storage reservoir and the protection of well cement againstcorrosion using biofilms.
Al CunninghamCenter for Biofilm EngineeringMontana State Universityal [email protected]
MS19
Fast-reaction Asymptotics for a Reaction-diffusionSystem with Distributed Microstructure
We investigate a reaction-diffusion process in a two-phasemedium with microscopic length scale ε. The diffusion co-efficients in the two phases are highly different (d1/D = ε2)and the reaction constant k is large. First, the homogenisa-tion limit ε → 0 is taken, which leads to a two-scale model.Afterwards, we pass to the fast-reaction limit k → ∞ andobtain a two-scale reaction-diffusion system with a movingboundary traveling within the microstructure. This resultsare the outcome of a joint work with Sebastian A. Meier(Bremen, Germany).
I will discuss efforts, developed over the last 10 years, toformally upscale the transport and reaction processes asso-ciated with biofilms in porous media. Our results to dateare multi-scale, starting from the cell scale, and currentlyending (after a sequence of upscaling efforts) at the Darcyscale. Both local mass equilibrium and non-equilibriummodels have been examined. I will also discuss recent suc-cesses in imaging biofilms in porous media using x-ray to-mography.
Brian D. WoodSchool of Chemical, Biological, & EnvironmentalEngineeringOregon State [email protected]
MS20
Coupled Hydro-Geomechanical Analysis of FaultReactivation and Water Induced Compaction
Injection and production of fluids cause changes of pres-sure, temperature and saturation that affect the stateof stress in oil reservoirs, resulting in porosity and per-meability variations. It is therefore a coupled hydro-geomechanical problem. Reservoir simulation with geome-chanical coupling, considering the appropriate constitutivebehavior of the materials, allows realistic modeling of waterinduced compaction and fault reactivation, that can influ-ence in the production of oil due to pressure and fluid lossin the reservoir.
Leonardo GuimaraesDept Civil Engineering - Federal University [email protected]
MS20
Macroscopic Effects of Dissolution of IntergranularContacts: A Multi-scale Analysis
The paper investigates dissolution/diffusion phenomenathat occur within the intergranular contact and their in-fluence on macroscopic behavior of chalk rocks. The in-tergranular interface is modelled as an evolving structurein which the contact between grains takes place along aset of isolated islands surrounded by fluid. The mathe-matical description of dissolution and diffusion processesis obtained by incorporating molecular and volume aver-aging. Extensive numerical studies of interface structureevolution are carried out. The macroscopic description ofchemo-mechanical coupling is formulated by invoking anevolution law that employed the notion of a dual time scaleassociated with the intergranular contact evolution.
Dariusz LydzbaInstitute of Geotechnics and HydrotechnicsWroclaw University of [email protected]
Reactive Transport in Porous Media: Multiple-Scale Analysis, Instability Phenomena
Reactive transport in porous media is of paramount impor-tance in many applications. Finding the correct chemicalreaction scheme is one of the major modelling complexity.However, coupling with transport brings additional diffi-culties. Two-types of difficulties are considered in this re-view of recent results: (i) multiple-scale problems, with thedevelopment of effective surface properties and ”homoge-nized” models taking into account the various scales of thepore and media geometry, as well as surface roughness andheterogeneity, (ii) hydrodynamics instabilities during dis-solution processes.
Michel QuintardInstitut de Mecanique des Fluides de [email protected]
MS20
Numerical Solution to an Integro-differential Equa-tion Modeling Swelling Porous Materials
We demonstrate the results of a discrete scheme used tosolve a nonlinear model of a swelling porous material. Themodel is cast as a Volterra partial integrodifferential equa-tion (VPIDE) of the second kind where the dependent vari-able is the liquid phase volume fraction. A pseudospectraldifferentiation matrix is constructed to compute the spatialderivatives and a Volterra Runge-Kutta scheme is used tosolve the VPIDE at each time step.
Keith WojciechowskiDepartment of Mathematical SciencesUniversity of Colorado [email protected]
MS21
Setup of the MoMaS Reactive Transport Bench-mark
We will present here the definition of the reactive trans-port benchmark of MoMaS. According to the interests ofMoMaS, the reactive transport problem should be repre-sentative of the problems encountered in nuclear waste dis-posal simulations. Moreover, we want to interest a com-munity as large as possible: geochemistry, hydro geology,numerical methods, applied mathematics... Nevertheless,the high complexity of both transport and chemical phe-nomena occurring in such a system may be an obstaclefor some researcher who may not be familiar with hydro-geological and geochemical concepts. The problems pro-posed here are built on the same mathematical concepts asreal hydro-geochemical problems, but their description hasbeen simplified. The difficulty for building this benchmarkwas also to provide a sufficient simple problem without lossof mathematical and numerical difficulties. The objectivesof this benchmark are to propose a challenging test for nu-merical methods used on reactive transport modelling inporous media. In order to focus on numerical methods,the problem presented here is on quite small size, fromhydrodynamic and from geochemical point of view. Toobtain a really challenging test, the chemical coefficientspresented in this benchmark are not real one; but they arestill realistic This benchmark consists in three independentparts, ranked by complexity: Easy, Medium, Hard. Eachpart consists of a 1D and a 2D reactive transport problem.The flow and transport phenomena are the same for the
GS09 Abstracts 77
three parts. From one part to the other, some chemicalphenomena are added increasing the difficulties.
Results of the GdR MoMaS Reactive TransportBenchmark with RICHY2D
In this talk the software RICHY2D for the simulation of re-active transport in porous media and computational resultsfor nine of the twelve MoMaS benchmark problems are pre-sented. The software is based on the one-step method. Inorder to reduce the size of the resulting discrete problems,it uses a reformulation of the system. RICHY2D is basedon M++ a platform for solving systems of nonlinear partialdifferential equations with Finite Elements in parallel.
Joachim HoffmannUniversity of Erlangen-NurembergDepartment of [email protected]
MS21
Coupling Transport and Chemistry in Porous Me-dia with a Newton-Krylov Method: Application tothe MoMaS Benchmark
Reactive transport modelling leads to the coupling betweena set of advection-diffusion PDE’s, and a set of algebraicequations. The resulting nonlinear system is solved by aNewton-Krylov method (which does not require storing thejacobian matrix), using as main unkowns the total mobileand total immobile concentrations. The main advantageis to keep chemistry distinct from transport, while stillproviding a globally coupled algorithm. We validate themethod on the 1D and 2D MoMaS benchmark.
We have developed a global numerical method for reactivetransport problems based on a robust and efficient Dif-ferential Algebraic Equations (DAE) solver. The coupledPartial Differential Algebraic Equations (PDAE) are firstdiscretised in space by a finite difference method. The re-sulting DAE are then discretised in time by a BDF methodin association with a modified Newton-LU method. Wehave used our method to simulate 2D problems, in partic-ular some of the Geochemistry MoMaS benchmarks.
Caroline de DieuleveultANDRAINRIA RennesCaroline.de [email protected]
Data Assimilation for a Viscous IncompressibleFluid
We study the inverse problem of determining the initialstate, and possibly the forcing, of a viscous incompress-ible fluid observed directly or indirectly over a period oftime. We will formulate this as a Bayesian inverse prob-lem, giving rise to a probability measure on function spacefor the initial vector field, and the forcing (or model er-ror). We will describe effective MCMC methods that allowus to sample from such a distribution, and present somenumerical results.
Quantifying Robustness of Mixing Diagnostics In-ferred from Satellite Altimetry
Several recent studies make use of satellite altimetry datato infer mixing diagnostics in the global surface ocean. Thereliability of these diagnostics is unclear, however: in par-ticular, the effect of unresolved scales on turbulent trans-port cannot be quantified. We examine a range of mixingdiagnostics in simulations of quasigeostrophic and surfacequasigeostrophic turbulence and directly probe their de-pendence on sampling resolution. In this way, we aim toquantify the robustness of altimetry-inferred mixing diag-nostics.
Shane R. KeatingNew York UniversityCourant Institute of Mathematical [email protected]
Shafer SmithCourant InstituteCenter for Atmosphere Ocean [email protected]
78 GS09 Abstracts
MS22
Relative Dispersion in the Atmosphere and Ocean
A signature of Lagrangian chaos is that the separation be-tween two particles deployed at slightly different positionsin a flow will grow exponentially in time. In this talk weexamine the statistics of particle pairs from in situ exper-iments in the atmosphere and ocean. There are indeedindications of exponential growth at smaller spatial scales,but the behavior at larger scales differs, apparently due tothe large scale circulation.
Effective Diffusivity: A Tool to Quantify Inhomo-geneous, Instantaneous, and Irreversible Transport
When advection-diffusion of a passive tracer is describedwith respect to the level set of the tracer itself, it becomespure diffusion with a spatially and temporally varying dif-fusion coefficient. I will review this effective diffusivity for-malism, its relationship to other metrics of statistical me-chanics, its applications to quantify inhomogeneous mixingin the ocean and atmosphere, and some ideas for parame-terization.
Noburu NakamuraUniversity of ChicagoDepartment of Geophysical [email protected]
MS23
An Experimental Setup of Ground-based and Air-borne Systems to Study Spatio-temporal Struc-tures in Atmosphere-Land Surface Energy, Waterand CO2 Exchange
Exchange processes between land surface, vegetation andatmosphere over structured, inhomogeneous regions are in-vestigated in an experimental approach. For that purposeground-based long-term monitoring and dedicated cam-paigns are combined within the Rur catchment, Germany.Four campaigns covering different vegetation periods havebeen performed with instrumentation ranging from leaflevel gas exchange, eddy correlation stations, scanning re-mote sensors to aircraftobservations. Synergistic data anal-ysis shall reveal spatio-temporal structures in the exchangeprocesses and their relation to external parameters.
Christian SelbachInstitute for Geophysics and MeteorologyCologne [email protected]
MS23
Patterns in Soil-Vegetation-Atmosphere Systems:Theory, Modeling and Data Assimilation
The Transregio 32 is an interdisciplinary project basedon the hypothesis that explicit representation of patternsin the soil-vegetation-atmosphere system in experimentaland theoretical studies will improve our understandingand predictions of the pertinent mass, energy, and mo-mentum fluxes and their interactions. We will providean overview of the research activities at the Universitiesof Bonn, Cologne and Aachen and the Research CenterJlich including the application of coupled simulation toolsin conjunction with measured data and data assimilationtechniques
Non-invasive Methods and Modelling Approachesto Study the Impact of Subsurface Structures onFlow and Transport
Geophysical methods may play an important role inmanaging our terrestrial environment and in maintainingecosystem functioning and services. Especially, the appli-cation and further development of hydrogeophysical meth-ods combined with mathematical models seem very promis-ing to maintain and protect soil and groundwater quality.Hydrogeophsical methods may help to improve our controlon storage, filter and buffer functions of soils and ground-water systems. Moreover, methods are needed that willhelp us to bridge the gap between the scale of measure-ments and observations and the scale at which manage-ment of terrestrial systems takes place. In this presenta-tion several examples will be presented showing how hydro-geophysical research can contribute in meeting these chal-lenges and may be used to characterize subsurface waterflow and transport.
Atmospheric-land Feedbacks in Clear and CloudyBoundary Layers
We investigated the effect of land surface heterogeneity oncloud formation using a large eddy simulation model. Ourstudy showed that by altering the turbulent structure ofthe atmospheric boundary layer, heterogeneity may createconditions that are favourable for cloud formation. How-ever, the results of the model are sensitive to the numerics.The choice of the advection scheme has significant influenceon the transport over the interface between the turbulentboundary layer and the laminar free atmosphere, which isat the height at which clouds form.
Jordi VilaMeteorology and Air QualityWageningen University and Research Center
Chiel van HeerwaardenWageningen University and Research Center Wageningen,The [email protected]
MS24
Analysis and Algorithms for a Regularized CauchyProblem arising from a Non-Linear Elliptic PDEfor Seismic Velocity Estimation
We derive and study nonlinear elliptic PDE’s in 2D and 3Dconnecting the Dix velocity and geometrical spreading ofthe image rays, and hence the true seismic velocity which isa product of the Dix velocity and the geometrical spread-ing. The physical setting allows us to pose only a Cauchyproblem, and hence is ill-posed. However we are still ableto solve it numerically on a long enough interval of time tobe of practical use.
Ray-theoretical Aspects of Seismic Time Migrationand Demigration
This work is devoted to a ray-theoretical analysis of ele-mentary wavefields inherent to time migration and demi-gration of seismic data. For such processes we base our-selves on the standard hyperbolic traveltime approxima-tion with respect to source-receiver offset and migrationaperture. Essential is also the construction of two-waysurface-to-surface ray propagator matrices correspondingto normal rays, for which the slowness vector is normalto the subsurface reflector, and image rays, for which theslowness vector is normal to the measurement surface inthe time-migration domain. The combination of such ma-trices provides useful insight into the processes of seismicmigration and demigration.
Time Migration Velocity Analysis by Image-wavePropagation in the Common-image Gathers
Image-wave propagation or velocity continuation describesthe repositioning of a migrated seismic event as a func-tion of migration velocity. In the common-image gather(CIG) domain, it can be used for iterative migration veloc-ity analysis. Continuation of CIGs allows to detect thosevelocities at which events flatten. A correction formulatranslates constant flattening velocities into varying time-migration velocities. Thus, the migration velocity modelcan be improved iteratively until a satisfactory result isreached.
Joerg SchleicherUniversity of CampinasDept. Applied [email protected]
An Hybrid Finite Volume Method for Two PhaseFlow Problems: Formulation, and Numerical Re-sults
We will present extension of Sushi finite volume method inthe case of two phase flow in porous media. The schemeuses unknowns at the centre of the cells and unknowns onthe interfaces. In relation with theses last unknowns, wehave to ensure continuity of some fluxes : different choicescan be retained and we will discuss this point. We willcompare our results with results obtained by other methodsas mixed finite elements.
Clement ChavantEDF (Electricite de France) R&DDepartement [email protected]
In this talk, we study the near well discretization of CO2storage models using cell centered multipoint flux approx-imation schemes for the Darcy fluxes. The finite volumediscretization uses an hybrid mesh connecting a near wellradial mesh to the reservoir mesh for a deviated well. Thesystem of equations is a two phase two components Darcyflow model with dissolution of the CO2 component into theaqueous phase.
Discretisation Schemes for Anisotropic Heteroge-neous Problems on Near-Well Grids
In this work, we construct analytical solutions for near-well flow which is not aligned with a radial inflow pattern.These solutions resemble strongly heterogeneous, possibleanisotropic media, which is less accounted for in existingnear-well models. We compare different control volumediscretisation schemes and radial-type grids for such cases,and give their convergence behavior. The results of thesesinglephase flow test cases are supported by multiphaseflow simulations.
Local-global Two-phase Upscaling of Flow andTransport in Heterogeneous Formations
We present a local-global two-phase (LG2P) upscaling ap-proach to generate upscaled transport functions. TheLG2P upscaling directly incorporates global coarse-scaletwo-phase solutions into local upscaling. It effectively cap-tures the impact of global flow, while avoiding global two-phase fine-scale simulations. Local boundary conditionsare updated with time-dependent coarse solutions, there-fore capturing the global flow both spatially and tempo-rally. Through various examples, we show that the methodconsistently provides accurate coarse models for both flowand transport predictions.
Dynamic Upscaling of Multiphase Flow in PorousMedia Via Adaptive Reconstruction of Fine ScaleVariables
We propose an upscaling method that is based on dynamicsimulation of a given model in which the accuracy of the up-scaled model is continuously monitored via indirect error-measures. If the indirect error measures are bigger thana specified tolerance, the upscaled model is dynamicallyupdated with approximate fine scale information that isreconstructed by a multi-scale finite volume method. Thenew upscaling algorithm is validated for two-phase, incom-pressible flow in heterogeneous porous media.
Generic Global Scale-up: Advantages and Chal-lenges
Generic global scale-up is based on global flow solutionsobtained using generic boundary conditions as opposed towell-driven boundary conditions tailored to specific flowscenarios as in other global scale-up methods. This paperreview some of the advantages of generic global scale-up, itsrelation with global multiscale finite element methods, andthe challenges in its practical applications. Some practicalresolutions to address the challenges are also discussed.
Numerical Upscaled Model of Transport with Non-separated Scales
We show numerical results for a new model for advection-diffusion-dispersion in porous media in the presence of mul-tiple scales which are not necessarily well separated. Themodel, developed theoretically in prior work and motivatedby experiments of Haggerty et. al., includes as specialcases the classical homogenized model as well as the doubleporosity models, but is characterized by presence of addi-tional memory terms. We discuss various discretizationsfor the model with special attention paid to handling thenew memory terms. Most significant issue is how to detectwhich terms are important in what regimes of flow andtransport and how the model compares with experimental
A Unified Mixed Formulation Naturally CouplingStokes and Darcy Flows
Solving Stokes-Darcy coupled flow using stable Galerkinformulations for both subproblems usually leads to unbal-anced rates of convergence. We present a stabilized mixedFEM for Darcy flow, compatible with Galerkin stable el-ements for Stokes, with balanced rates of convergence forvelocity and pressure. The discontinuities of the solutionson the interface of the free fluid and the porous medium areincorporated in the finite element approximation by lineartransformations derived from the interface conditions.
Maicon CorreaNational Laboratory of Scientific ComputingLNCC, [email protected]
Abimael LoulaNational Laboratory of Scientific Computing, [email protected]
MS27
The Chaotic Dynamics of Anomalous Dispersion asModels by a Nonstationaryextension of BrownianMotion
Given an arbitrary mean square displacement, we showhow to construct a family of stochastic processes with thisdisplacement and with independent, nonstationary incre-ments. The resultant process is used to model anomalousand classical diffusion. By computing the fractal dimensionit is shown that the complexity of the family of processesis the same as that of Brownian motion. An analytical ex-pression is developed for the finite-size Lyapunov exponentand numerical examples presented.
John CushmanDept Earth Atmospheric SciencesPurdue [email protected].
MS27
Domain Decomposition for Poroelasticity and Elas-ticity with DG Jumps and Mortars
We couple a time-dependent poroelastic model in a regionwith an elastic model in adjacent regions. We discretizeeach model independently on non-matching grids and werealize a domain decomposition on the interface by intro-ducing DG jumps and mortars. The unknowns are con-densed on the interface allowing the computation in eachsubdomain to be performed in parallel. We establish errorestimates for an algorithm where the computation of the
Tim M. WildeyThe University of Texas at AustinAustin, [email protected]
MS27
Preferential Flow and Geomechanics
Dynamic capillary pressure effects alter classical parabolicmodels of filtration flow to those of pseudoparabolic type.These effects become substantial in fast processes or atsmall scales. We consider these issues in the filtrationthrough elastic porous media with such pressure-saturationrelationships.
Ralph ShowalterDepartment of MathematicsOregon State [email protected]
MS28
Multi-D Upwinding for Multi-Phase Transport inPorous Media
Multidimensional methods for hyperbolic equations canlessen the correlation of numerical errors with the com-putational grid reducing bias in inherently unstable flowproblems. We present a family of monotone multidimen-sional methods for two-phase flow with gravity, which isextendable to more general scalar hyperbolic equations. Alocal coupling in the flux calculation is introduced throughthe use of interaction regions resulting in a compact stenciland extension to any grid topology.
Margot GerritsenDept of Petroleum EngineeringStanford [email protected]
MS28
Multidimensional Upwind Schemes for Flow inPorous Media on General Quadrilateral and Tri-angular Grids
Standard reservoir simulation schemes employ single-point
82 GS09 Abstracts
upstream weighting for convective flux approximation.These schemes introduce both coordinate-line numericaldiffusion and cross-wind diffusion into the solution thatis grid and geometry dependent. New locally conserva-tive higher-order multi-dimensional upwind schemes thatminimize both directional and cross-wind diffusion are pre-sented for convective flow approximation. The schemes arecoupled with full-tensor Darcy flux approximations. Thenew schemes are comprised of two steps; (a) Truly multi-dimensional upwind approximation, which involves fluxapproximation using upwind information obtained by up-stream tracing along wave-vector paths where wave infor-mation travels in multiple dimensions. Multi-dimensionalformulations using edge-based and cell-based approxima-tions that reduce cross-wind diffusion are presented. Con-ditions on tracing direction and CFL number lead to alocal maximum principle that ensures stable solutions freeof spurious oscillations. (b) Higher-order approximationthat corrects the directional diffusion of the approxima-tion. Benefits of the resulting schemes are demonstratedfor convective reservoir simulation test cases involving arange of unstructured grids and permeability fields. Whilestandard simulation methods yield relatively poor resultsfor such cases, the new methods prove to be particularlyeffective.
Eulerian-Lagrangian Methods for Multiphase Mul-ticomponent Transport
Transport in porous media is often advection-dominated.This leads to efforts to incorporate Lagrangian techniquesinto numerical schemes, in order to overcome CFL limita-tions, numerical dispersion, and non-physical oscillations.In multiphase transport, these efforts are made easier byworking with an adjoint system, whose natural interpre-tation is in terms of mass movement rather than wavepropagation. The talk will explain this in the context ofEulerian-Lagrangian methods for multiphase multicompo-nent transport and will outline some recent developments.
Discontinuous Galerkin Methods for Transport inPorous Media
Two-phase flow in porous media has important applica-tions for petroleum reservoir engineering and groundwa-ter processes. Both applications may involve multipletime and spatial scales, long simulation time periods, andmany coupled nonlinear components. In particular, theadvection-dominated component and the nonlinear cou-pling of compressibility, capillary pressure and relative per-meabilities often result in sharp saturation fronts, whichdemands steep gradients to be preserved with minimal os-cillation and numerical diffusion. In this talk, we considerthe combined method of discontinuous Galerkin (DG) andmixed finite element (MFE) for simulating two-phase flow
in porous media. A number of numerical examples are pre-sented to illustrate computational advantages of DG meth-ods for porous media flow, with emphasis on the treatmentof capillary pressure heterogeneity and the dynamic meshmodification.
Stochastic Velocity Field Models for Eddy-RichFlows
Recent high-resolution radar observations of surface veloc-ity have revealed submesoscale eddies in the coastal areas.By an objective estimation method, eddies are detectedand their parameters such as center coordinates, size, andintensity are estimated. The obtained statistics are usedto parametrically represent the birthdeath process of ed-dies via a model stochastic velocity field known as inlarflow. The model is developed further to represent the eddyamplitude decay more accurately.
Solving the Pressure Poisson Equation in Large-aspect Ratio Ocean-shaped Domains
A key challenge in the development of non-hydrostaticocean models is that the Poisson equation for pressure mustbe solved in a domain with extremely large aspect ratio ε.This leads to a matrix with a condition number which isbounded from below by cε−2 for some positive constant cas ε → 0, in the case in which Neumann boundary condi-tions are applied everywhere. Since the convergence rate(required number of iterations for a given tolerance) foriterative solvers scales with condition number, this makesstandard iterative solvers extremely slow. However, in thecase in which Dirichlet boundary conditions are specified,the condition number is bounded from above by the con-dition number of the matrix which one must solve for ahydrostatic model. This motivates the preconditioner ap-proach used by the MITgcm ocean model. However, as for-mulated, that approach can not easily be extended to fullyunstructured meshes such as those used by the ImperialCollege Ocean Model (ICOM). In this talk, we prove someestimates on the condition number for finite element dis-cretisations of the Poisson equation in the large aspect ratiolimit, and suggest a new approach for preconditioning thematrix system based integrating an approximated reducedsystem for the top surface boundary condition in a multi-grid solver. These solvers enable previously unachievablesimulations in large aspect ratio domains (such as three-dimensional gyres) and we will present some of our latestresults from these simulations.
Stephan KramerImperial CollegeApplied Modeling and Computational [email protected]
MS29
CABARET in the Ocean Gyres
A new high-resolution numerical method is proposed formodelling quasigeostrophic mesoscale ocean dynamics ineddying regimes. The method is based on a novel, second-order non-dissipative and low-dispersive conservative ad-vection CABARET scheme. The properties of the newmethod are applied to the classical model of the double-gyre ocean circulation. It is demonstrated that, in turbu-lent regimes, the new method leads to a significant acceler-ation of the numerical solution convergence, in comparisonto the conventional second-order method.
Sergey KarabasovUniversity of CambridgeDepartment of [email protected]
Pavel BerloffPhysical Oceanography Dept.Woods Hole Oceanographic [email protected]
Vasily GolovizninMoscow Institute of Nuclear Safety (IBRAE)[email protected]
MS29
Large Eddy Simulation of Sub-mesoscale MotionDue to Surface Frontal Instability
The behavior of ocean flows at the sub-mesoscale range ispoorly understood despite its importance for many prac-tical problems, such as the short-term dispersion of pollu-tants and oils spills. Most ocean models rely on hydrostaticdynamics, mixed-layer and subgrid-scale parameterizationsto represent these processes. In this study, large eddy sim-ulations of frontal instability are carried out and mixingacross the front is quantified using background potentialenergy, drifter releases to compute finite-scale Lyapunovexponents and relative dispersion.
Improving Time Integration by Conservative Reg-ularization of Atmospheric Equations
We present a conservative regularization approach to thecompressible Euler equations that replaces the pressuregradient in the momentum equations by the gradient of asmoothed pressure field. The smoothing selectively slowsdown the fastest sound wave components only, without al-tering the slow dynamics and does not introduce additionalviscosity. We will present numerical results from a 2D ver-
tical slice model.
Tobias HundertmarkUniversitaet PotsdamInstitute for [email protected]
Peer Methods for the Compressible Euler Equa-tions.
A new time-splitting method for the integration of the com-pressible Euler equations is presented. It is based on a two-step peer method which is a general linear method withsecond-order accuracy in every stage. The scheme usesa computationally very efficient forward-backward schemefor the integration of the high-frequency acoustic modes.With this splitting approach it is possible to integrate sta-bly the compressible Euler equations without any artificialdamping. The peer method is tested with the dry Eulerequations and a comparison with the common split-explicitsecond-order three-stage Runge-Kutta method by Wickerand Skamarock shows the potential of the class of peermethods with respect to computational efficiency, stabilityand accuracy.
Stefan JebensLeibniz Institute for Tropospheric Research,Permoserstrae 15, D-04318 Leipzig, [email protected]
MS30
Auxiliary Variables and Deferred Corrections forDivergence Constrained Flows
A class of methods for the numerical solution of incom-pressible and low-Mach number flows based on a novelcombination of deferred corrections and auxiliary variablesis presented. Temporal integration is done using spectraldeferred corrections which allow multiple terms in an equa-tion to be treated either implicitly or explicitly and withdifferent time steps and can easily be constructed to attainhigh formal order of accuracy. These methods are com-bined with a finite volume discretization of an auxiliaryvariable formulation of the equations of motion to producehigher-order accurate flow solvers. The key idea in auxil-iary variable methods is to formulate an equation for a vari-able which does not have an explicit divergence constraintbut from which the desired constrained velocity can be nu-merically extracted by the analog to the procedure usedin projection methods. Examples from incompressible, 0-Mach, and low-Mach number flows will be presented.
Michael MinionUniversity of North Carolina-Chapel HillMathematics [email protected]
MS30
A Discontinuous Galerkin Hp-Adaptive and TimeImplicit Mesoscale Model
Moore’s law predicts a doubling in available computingpower every 18 months: at this rate a O(1km2) global
84 GS09 Abstracts
climate simulation will only be achievable 30 years fromnow with the numerical methods currently employed in at-mospheric models. We explore a class of linearly implicittime-stepping methods: Rosenbrock W-methods. Withpreconditioning techniques, they offer an efficient way ofintegrating the compressible Euler equations. We also di-cuss a new way to reduce the cost of evaluating Jacobians.
Amik St-CyrNational Center for Atmospheric ResearchInstitute for Mathematics Applied to the [email protected]
MS31
Velocity Model Determination by Double Path-integral Imaging
Path-integral imaging sums over the migrated images fora set of migration velocity models. Those velocities wherecommon-image gathers align horizontally are stationary,thus favoring these images in the sum. Thus, the imageforms without knowledge of the velocity model. By exe-cuting the path-integral imaging a second time using thevelocity as a weight factor in the sum, the stationary ve-locities can be extracted by a division of the two images.
Joerg SchleicherUniversity of CampinasDept. Applied [email protected]
In this paper, we consider the hyperbolic, the shifted hy-perbola, the rational and the generalized traveltime ap-proximation for the qP- and qSV-waves in a homogeneousVTI medium. We also consider the qP-wave in acoustic ap-proximation. Fomel and Stovas (2009) proposed the gener-alized moveout approximation with other approximationsbeing the special cases. Being compared with the hyper-bolic, shifted hyperbola and rational approximations in fewexamples, the generalized approximation gives the best re-sults.
In statistics, semblance is related to the second momentand in optimization, to the least-squares solution of max-imum signal energy as a characterization of reflectionevents. We define extensions of semblance by replacingsecond-order by higher-order quantities. These semblancemeasures behave differently in the estimation of Common-Reflection-Surface parameters by means of the hyperbolictraveltime approximation applied to multicoverage data.We find improved parameter estimates using a fourth-ordersemblance.
Commercial Simulation Software for Porous Media:Capabilities and Developments
In structural engineering, automotive, oil-gas or aircraft in-dustries the commercialization of computational mechanicssoftware is fairly advanced. Today, a powerful finite ele-ment/finite volume technology has been established therein hands of only a few big players worldwide. In contrast,however, looking to porous media, where we mainly focuson subsurface water resources, environmental, geothermaland industrial porous material problems, the situation israther different. Surprisingly, the water resources marketis still widely dominated by a classic finite difference pro-gram, the USGS code MODFLOW, available for free (ex-cept GUI), which stems from the eighties and possessesin the meantime a number of extensions and adaptations.It has similarities to an open-source development project.On the other hand, there is an increasing number of re-search codes at universities funded by public authorities,which cover a large disciplinary spectrum ranging frommulti-phase flow via chemical reaction systems, fractureflow modeling, deformation processes to different numer-ical approaches. It suggests that most of the problemsseem now solvable and those software products could sat-isfy most desires in research and practice. Nevertheless,there are also commercial simulation software systems, forexample FEFLOW, which has shown a further growing po-tential in the porous media market. In the present paper wewill discuss the scope and requirements for such a commer-cial porous media modeling system. We critically ask: whoneeds commercial software, for what and what are the basicrequirements, advantages and challenges? Where are thedeficiencies and technological constraints? We characterizethe status quo regarding the capabilities available, numer-ical features and inherent software technology. In develop-ing innovative and globally competitive software productsthe following guiding principles become more important:depth and breath of capabilities, completeness (modeling
GS09 Abstracts 85
that works), availability and usefulness (GUI, interfaces,service, teaching), scalability as well as adaptive architec-ture. From the current stage we give an outlook to futureneeds and progressing software developments. Examplesare given for geothermal applications and large-swelling ab-sorbing porous materials.
Applications of Software Tools D3F and R3T forCoupled Problems of Variable Density Flow andTransport in Porous Media
The software tools D3F and R3T can solve numerically sev-eral complex applications of subsurface flow and transportproblems. The tools are based on library UG (Unstruc-tured Grids) that enables the computations on complexgeological domains discretized by an unstructured, locallyadapted, multilevel grid that can be redistributed on pro-cessors of parallel computer. The tool D3F (DistributedDensity Driven Flow) solves nonlinear flow and transportequation that are fully coupled due to the dependence offluid density on transported concentration. The tool R3T(Retardation, Reaction, Radionuclides, Transport) solvesmany nonlinear transport equations that are coupled dueto reactions (e.g., decay, slow and fast sorption, etc.) andwhere the transport is dominated by the advection deter-mined from the D3F velocity field. In this talk we intro-duce not only the most interesting features of these soft-ware tools (implicit/explicit in time discretizations, New-ton solvers with analytic linearization, level set methods,etc.), but also the experiences of a general interest con-cerning the development and the usage of complex softwaretools for this type of problems.
Peter FrolkovicDepartment of MathematicsSlovak University of [email protected]
MS32
Discontinuous Galerkin Method for ConvectionDominated Transport in Porous Media UsingDUNE
In this paper we present software techniques as well asrecent development of the software package DUNE andin particular of the module DUNE-FEM. For several testproblems as well as for more complex applications, such asthe simulation of fuel cells, we present numerical results.These results have been obtained using modern simulationtechniques such as higher order continuous and discontinu-ous Galerkin methods (also in a stabilized version) in com-bination with local grid adaptivity. Since we are interestedin the simulation of complex problems the parallelizationof the code has been taken into account. In adaptive simu-lations this becomes a non-trivial part since dynamic loadbalancing needs to be done. Applied to different problemsthe code shows a very good scalability even on a high num-ber of processors.
Robert KlofkornMathematisches Institut, Abt. fur AngewandteMathematikUniversitat Freiburg
Simulation of Density and Temperature DrivenFlow and Contaminant Transport in FracturedPorous Media Using d3f and r3t
In this talk, we discuss the mathematical models and thenumerical methods for the simulation of the density andtemperature driven flow and the contaminant transport infractured porous media. The fractures are represented bymanifolds with the reduced dimensionality. In the frac-tures, we consider the same phenomena as in the bulkmedium. The discretized PDEs for the bulk medium andthe fractures are solved as a coupled system by the Newtonmethod with the multigrid linear solver. We present nu-merical results obtained using the simulation programmesd3f and r3t extended to the case of the fractured media.
Michael LampeGoethe Center for Scientific ComputingUniversity of Frankfurt a.M., [email protected]
MS33
Mixed Multiscale Finite Element Methods usingLimited Global Information
I will describe mixed multiscale finite element methods us-ing limited global information. In particular, I will stresshow multiple global information can be incorporated intomultiscale basis functions. This is a joint work with J.Aarnes and L. Jiang.
A Loosely Coupled Hierarchical Fracture Model forthe Iterative Multi- Scale Finite-Volume Method
Recently, the multi-scale finite volume (MSFV) method forflow in heterogeneous porous media was combined with anefficient iterative procedure, which allows to converge so-lutions to the corresponding fine-scale references. Here,this iterative MSFV (iMSFV) method is extended by a hi-erarchical fracture model, where the flow in the resolvedfracture network is represented on lower dimensional man-ifolds. Therefore, a similar datastructure as for the well
In recent years, many different (but related) methods thatattempt to capture the multiscale behavior of flow inheterogeneous reservoirs have been developed. Here, weextend our own previous models (variational multiscalemixed methods) for the simulation of flow in fracturedreservoirs, in which fractures are viewed as entities of lowerdimensionality than the domain. We will conclude the pre-sentation with results for realistic problems.
A General Iterative Scheme for the Multiscale Fi-nite Volume Method
In the Multiscale Finite-Volume (MSFV) method a con-servative velocity field is constructed from an approximatepressure field, which is obtained by superimposition of lo-cal solutions coupled by a global problem. Due to the lo-calization assumption, the MSFV solution differs from theexact solution of the problem. By estimating the localiza-tion error, an iterative algorithm can be constructed thatconverges to the exact solution and conserves mass at anyiteration.
Deformation of Porous Media on the Pore Scaleand Induced Variations
The deformation of porous media is calculated by solv-ing the elastostatic equations discretized on unstructuredmeshes made of irregular tetrahedra. The macroscopic con-ductivity and permeability necessitate the resolution of theLaplace and the Stokes equations on the initial and de-formed meshes. Applications will be provided for a numberof media either reconstructed or measured by microtomog-raphy. Results relative to the evolution of the media and ofthe macroscopic properties of the media will be presented.
Bounds of Dynamic Permeability and RelatedProblems
Transfer coefficients within porous media are assessed bycombining the physical principles of the homogenization ofperiodic media and the geometrical simplifications of theself-consistent method. This approach provides two staticand dynamic permeability assessments that enable to buildbounds for a wide class of media. Similarly the Klinken-berg effect for rarefied fluids is estimated. As for diffusionproblems, estimates of trapping coefficient and of thermalpermeability involved in the dynamic gas compressibilityare given.
How to Model Dispersion in Unsaturated DoublePorosity Medium? An Integrated Approach
Multi-scale, multi-components, multi-phases are the keywords that characterize the double porosity media subjectto geo-environmental conditions. In relation to this con-text we present an integrated upscaling approach whichcombines three issues: theoretical, numerical and exper-imental. A physical model was designed to imitate thedouble-porosity media and to enable performing the disper-sion experiments in fully controlled conditions. The mod-elling is based on the observations and follows the logic:from micro towards macro.
Tien Dung Tran NgocBRGM, Departement Geothermie, 3 avenue ClaudeGuillemin, [email protected]
MS34
A Three-Scale Model of pH-Dependent Flows In-cluding Ion Adsorption in Kaolinite Clays
A new three-scale model to describe the coupling betweenpH-dependent flows and ion transport in kaolinite clays isproposed. The poorus medium is characterized by threeseparate nano-micro and macroscopic length scales. Thepore (micro)-scale is characterized by micro-pores satu-rated by an aqueous solution containing four monovalentionic species and charged solid particles surrounded by thinelectrical double layers. The two-scale nano/micro modelis homogenized to the macroscale leading to the derivationof effective equations.
Marcio A. MuradNational Laboratory of Scientific Computation
Statistical Distributions of Conservative and Re-active Species Concentrations in HeterogeneousPorous Media
We present a semi-analytical method of deriving the fullprobability distribution of concentration at observationpoints or volumes, which is based on first-order approxima-tions of one- and two-particle displacements. The resultingconcentration pdf approximately follows a beta distribu-tion. This pdf can be mapped to the concentration pdf s ofspecies that react upon mixing. Estimates of mean valuesand standard deviations based on second-order perturba-tion analysis rather than mapping entire pdf s are biased.
Combined Deterministic and Stochastic SensitivityAnalysis - Application to Uncertainty Analysis
For a model y=f(x), x uncertain vector, we want to 1)resume the influence of components of x on componentsof y, 2) use it for uncertainty analysis. We first choosesimple o,p so, that the singular value decomposition (SVD)of (p(F(o))’(x) is almost independent of x (deterministicanalysis). The SVD of [cov(p(y),o(x))]inv([cov(o(x),o(x)])fairly solves 1). We can precede the analysis with a properorthogonal decomposition of o(x) to deal with correlatedinput components.
Estelle MarchandINRIA RocquencourtProjet Estime BP [email protected]
MS35
Movement of Fluid Interfaces and Their AveragedProperties During Two-phase Flow in Heteroge-
neous Porous Media
The fluid-fluid interface during immiscible displacementhas a strong influence on mass transfer during the flowprocess in heterogeneous porous media. Important criteriaare the front roughness and the averaged saturation at thefront. Similar to the concept of dilution and dispersion insolute transport theory, these two measures differ in theirrelation to processes at the fluid interfaces. This presenta-tion discusses predictions of large scale front behavior ob-tained theoretical methods and experimental observations.
Insa NeuweilerLeibniz University HannoverInstitute for Fluid [email protected]
Global Random Walk Simulations for Sensitivityand Uncertainty Analysis of Transport Models
The global random walk algorithm (GRW) performs thesimultaneous tracking on a fixed grid of huge numbers ofparticles at costs comparable to those of a single-trajectorysimulation by the traditional particle tracking approach.Ensembles of GRW simulations of a typical advection-dispersion process in aquifers are used to obtain reliableestimations of the input parameters for the upscaled trans-port model and of their correlations, input-output correla-tions, probability distributions, and relations between in-put and output uncertainty.
Finite Volume Schemes for Simulating Meso- andMicro-scale Atmospheric Flows
Two different finite volume schemes for simulating meso-and micro-scale atmospheric flows are discussed in detail.First an adaptation of Smolerkiewicz’s MPDATA schemeon unstructured grids is presented. The second finite vol-ume scheme is based on flux-based wave decompositionsuggested by LeVeque. The f-waves scheme is describedin detail and Euler solutions for different benchmark prob-lems are presented. The scheme is also compared with theNational Center for Atmospheric Research’s state-of-the-art WRF model.
Nash’at AhmadScience Applications International Corporation1710 SAIC Drive, M/S 2-3-1, McLean, VA 22102, [email protected]
MS36
A Runge-Kutta Discontinuous Galerkin Methodwith Time Accurate Local Time Stepping
Our research objectives are the construction and applica-tion of accurate and efficient schemes for unsteady flow
88 GS09 Abstracts
problems. In this talk, a recently developed explicit Runge-Kutta based discontinuous Galerkin discretization is pre-sented. The focus is set on the time accurate local timestepping algorithm and an efficient implementation usinga mixed modal/nodal approach. To demonstrate the ac-curacy and efficiency of the method, several test problemsfor the compressible Navier-Stokes equations are shown.
Theory and Numerics of Sound-proof Models forAtmospheric Motions
The classical Ogura and Phillips’ (1962) anelastic model,subsequent extensions by Dutton and Fichtl (1969), Lippsand Hemler (1982), or Bannon (1995/96), and the pseudo-incompressible model by Durran (1988) will be revisited.We demonstrate that only the Ogura-Phillips model hasa systematic single-scale asymptotic derivation, but thateven multiple scales asymptotic techniques are incapableof yielding the extended anelastic or pseudo-incompressiblemodels. In turn, we demonstrate that these models reduce,at small scales, to the incompressible Boussinesq approx-imation (anelastic) and to the zero Mach number, vari-able density flow equations (pseudo-incompressible), re-spectively. Thus these models are compatible with par-ticular low Mach number limits of the compressible Eulerequations at least on small scales. To show that these mod-els do provide valid approximations to low Mach numberflows in the atmosphere, more advance techniques of math-ematical analysis are needed. The outline of a proof that iswork in progress at the time of submission of this abstractwill be provided. A Godunov-type projection method thatwas motivated by these considerations will be summarizedand tests involving small-scale flow with large density vari-ation as well as breaking topographic internal waves willbe presented.
Rupert KleinPotsdam Institute for Climate Impact ResearchFree University of [email protected]
Implementation of Pressure and Velocity Regular-ized Equations
Numerical modeling of atmospheric flows is a multiscaleproblem for which the treatment of turbulence and scaleseparated phenomena is essential. Our approach consistsin studying both phenomena by means of regularizationsof the underlying model equations. We have in particu-lar studied velocity (α-Euler models) and pressure regu-larizations. While the concept of velocity regularization iswell-known from incompressible Navier-Stokes equations,the extension of velocity regularizations to compressibleflows is non-trivial. Pressure regularizations arise naturally
from semi-implicit time-stepping methods as well as fromLagrangian particle methods. We have implemented andstudied both types of regularizations for the shallow-waterequations and the ICON spherical spatial discretization ap-proach. After a survey of the basic regularization conceptswe will present numerical results. This is joint work withMarco Restelli, Marco Giorgetta, Tobias Hundertmark andPeter Korn. The work has been funded by the German Sci-ence Foundation (DFG) under the SPP Metstroem.
Sebastian ReichInstitut fur Mathematik, Universitat PotsdamAm Neuen Palais 10, 14469 Potsdam, Germany, Tel.: +49331 [email protected]
MS37
A Multilayer Saint-Venant System : Derivationand Numerical Validation
We are interested in this talk in the derivation and analysisof a multilayer model for shallow flows. The model allowsthe fluid to circulate from one layer to the connected ones.The main properties (energy, hyperbolicity) of the modelare exhibited. A kinetic interpretation and some numericalsimulations including a recirculation case with wind forcingare also given.
Emmanuel AudusseLaboratoire d’Analyse, Geometrie et ApplicationsUniversite Paris 13, [email protected]
MS37
Finite Volume Simulation of the Geostrophic Ad-justment in a Rotating Shallow Water System
We focus on the simulation of the geostrophic adjustmentphenomenon for rotating shallow water models by means offinite volume methods. The well-balanced properties andthe discrete dispersion laws of the numerical schemes playa fundamental role. Here we consider spatial discretizationbased on a first order Roe- type method and some higherorder extensions based on WENO reconstructions. Thetime discretization is designed in order to provide suitableapproximations of inertial oscillations, taking into accountthe Hamiltonian structure of the system for these solutions.Some numerical experiments for 1d and 2d problems willbe shown.
A Roe-type Scheme for Two-phase Shallow Gran-ular Flows over Variable Topography
We consider a depth-averaged two-phase model for gravity-driven flows made of solid grains and fluid, moving overvariable basal surface. In particular, we are interested inapplications to geophysical flows such as avalanches anddebris flows. We numerically solve the model equations bya high-resolution finite volume scheme based on a Roe-typeRiemann solver. Well-balancing of topography terms isobtained via a technique that includes these contributionsinto the wave structure of the Riemann solution.
Anne MangeneyInstitut de Physique du Globe de ParisUniversite Paris [email protected]
MS37
High Order Well-balanced Finite Volume Schemesfor Systems of Balance Laws
A new family of high-order well-balanced schemes for thenumerical solution of hyperbolic systems of balance laws isproposed. The schemes are designed with use of two setsof variables, conservative and equilibrium ones. We dis-cretize the equations in the conservative variables, while forreconstruction we use the equilibrium ones. We constructwell-balanced schemes up to the fourth order and apply ourtechnique to the shallow water equation and nozzle flow.
Alexander KheLavrentyev Institute of Hydrodynamics630090 Novosibirsk, [email protected]
MS38
The Solidification of An Ideal Ternary Alloy in aMushy Layer
We examine a model for the solidification of a ternary alloyin a mushy layer. The effects of species diffusion are in-cluded along with heat diffusion in order to investigate thepossibility of double-diffusive and other modes of convec-tion in this system common in many geophysical systems.We investigate the properties of non-convecting base statesolutions for this ternary system and then present linearstability results that reveal convective modes of instability.
Daniel AndersonDepartment of Mathematical SciencesGeorge Mason [email protected]
Geoffrey McFaddenNational Institute of Standards and [email protected]
Bruce T. MurrayDepartment of Mechanical EngineeringSUNY at [email protected]
MS38
Stability of Upwelling Mantle in a Solubility Gra-
dient
Channelized melt flow due to reactive infiltration in aporous medium has been proposed as a mechanism for meltextraction in the Earths mantle. We present analytical andnumerical results from an extended mathematical formula-tion for melt migration in a deforming mantle column thatundergoes steady upwelling. To study the spatial distribu-tion of the channels, we explicitly track the abundance of adissolving mineral in the otherwise nonreactive solid. Wepresent approximate analytic solutions for steady upwellingcolumns, and study their stability to small perturbationsusing linear stability analysis. The height of the upwellingcolumn is larger that the compaction length of the sys-tem, and therefore compaction is stabilizing the systemmore than previously acknowledged. Upwelling, porositydependence of the bulk porosity, and increasing disequi-librium also stabilize the system. These linear results arein good agreement with high-resolution numerical simula-tions at early times. Although the system studied is stableto initial perturbations in the range of parameters of geo-logical interest, sustained perturbations of the porosity atthe inflow boundary lead to localization. Melt flow intothe channel is limited by the formation of a compactingboundary layer, not previously observed. If boundary per-turbations are sustained long enough the dissolving mineralis exhausted in regions of high melt flow and the channelsplits due to compaction. Upward branching of channels isexpected in the upper part of the upwelling column, in con-trast to the inverted cascade obtained in previous studies.In summary, it is more difficult to localize melt flow thanpreviously thought and sustained perturbations are nec-essary to force melt localization. The presence of mantleheterogeneity in the upwelling mantle might provide suchsustained perturbations. This may provide a link betweensource heterogeneity and the distribution of dunite bodiesin the mantle section of ophiolites.
Alan SchiemenzDepartment of Applied MathematicsBrown Universityalan [email protected]
Marc A. HesseDept. of Geological SciencesBrown Universitymarc [email protected]
MS38
Open Melt Conduits in Mantle Undergoing Decom-pression Melting
We examine the dominating physicial processes for an openchannel of melt within a partially molten viscous matrix.Melting of the residual rock is fundamentally driven bythe rate that heat is transported from the hotter mantlebeneath; in an open conduit this is mostly provided by themotion of melt rather than matrix. Large melting rateskeep channel walls open against viscous closure due to thereduced channel pressure, which is essentially magmostatic.
Reactive Channelization in Mushy Layers and theMantle
The formation of chimneys in a mushy layer and the for-mation of dunite channels in the mantle both occur by thereactive infiltration instability. In both cases, fluid flowis driven by buoyancy through a permeable, reactive ma-trix up a solubility gradient, leading to matrix dissolutionand channelization of flux. The theory and experimentsused to study these two systems have been developed inde-pendently and yet much can be learned by a comparativestudy. For example, theory for the reactive-convective in-stability in a mushy layer assumes chemical equilibriumbetween crystals and brine, while the theory for magmaticsystems is based on disequilibrium and linear reaction ki-netics. New simulations of magmatic flow beneath a mid-ocean ridge demonstrate the power of a computational im-plementation based on equilibrium thermodynamics andthe Enthalpy Method. New experiments on mushy layerssuggest that disequilibrium and kinetics are important inexplaining new phenomena. This talk will examine simi-larities and differences between reactive channelization inmushy layers and the mantle. It will weigh the motivationsand benefits of assuming thermodynamic (dis)equilibriumin theoretical descriptions of the two systems.
Simulation of a Benchmark - Solutions and ErrorEstimates
Geological storage of carbon dioxide in deep saline aquifersis considered as a means to reduce greenhouse gas emis-sions. Different storage mechanisms play a role in this con-text. In our contribution, we are concerned with the mech-anisms of residual- and dissolution trapping. We present2D- and 3D numerical results related to a model of an ex-tended sloping aquifer and investigate the effect of the men-tioned trapping mechanisms and their proper time scales.The results of the 2D- and the 3D simulations are com-pared, as well as various types of boundary conditions.The results are also compared to semi-analytical solutionsby Hesse et al (2008) where a sharp interface and no dis-solution is assumed.
Maria EleniusBergen Center of Computational ScienceUniversity of [email protected]
Jan M. NordbottenDepartment of MathematicsUniversity of [email protected]
MS39
Vertically Integrated Approaches to Large ScaleCO2 Storage
Models of CO2 injection into geological formations mustcapture both the large-scale plume and small-scale leakage
along wells. Traditional numerical methods are computa-tionally expensive for large complex geological systems andwith large numbers of wells. The VESA model combinesvertically-averaged governing equations with a subscale an-alytical model for wellbore flow. CO2 injection is solved nu-merically on a coarse grid, capturing the large-scale injec-tion problem, while the embedded analytical model elimi-nates expensive grid refinement around wells.
Investigation of Numerical Methods for CO2 Injec-tion
This talk will focus on numerical methods for the bench-mark problem. We investigate methods utilizing methodsspecially suited for advection dominated flow and grav-ity segregation and use operator splitting to combine thismethods. In the investigation of several methods for thegiven benchmark problem, we focus on methods which eas-ily can be extended to more complicated geometry. In par-ticular we investigate streamline methods and reorderingmethods.
This talk will give an overview of the proposed benchmarkproblem for this minisymposium. Recognizing that despitethe simplicity of the benchmark, it poses severe computa-tional challenges. The talk will discuss available modelingchoices that may be applied to reduce the computationalload, and assess these modeling choices in the context ofthe benchmark.
Jan M. NordbottenDepartment of MathematicsUniversity of [email protected]
MS40
Homogenization-based Multiscale Finite Elementsfor Heterogeneous Porous Media
Multiscale methods are used to solve flow problems withheterogeneous permeability. We show that a popular mixedmultiscale finite element fails to reproduce constant flowfields, and so fails to converge in any meaningful way. Theproblem arises for anisotropic permeability. We expect thesame for isotropic permeability when the microstructure
GS09 Abstracts 91
leads under upscaling to a tensorial homogenized perme-ability. A modified method based on homogenization isshown to converge with respect to this microstructure.
Todd ArbogastDept of Math; C1200University of Texas, [email protected]
MS40
Sparse Finite Element Method for Periodic Multi-scale Nonlinear Monotone Problems
In this talk, I will present a sparse tensor product Finite El-ement (FE) method for the high-dimensional limiting prob-lem obtained by applying the multiscale convergence to amultiscale nonlinear monotone problem in Rd. The limit-ing problem is posed in a product space, so tensor prod-uct FE spaces are used for discretization. This sparse FEmethod requires essentially the same number of degrees offreedom to achieve essentially equal accuracy to that of astandard FE scheme for a partial differential equation inRd. It is proved that for the Euler-Lagrange equation ofa two scale convex variational problem in a smooth andconvex domain, the solution to the high-dimensional limit-ing equation is smooth. An analytic homogenization erroris then established, which together with the FE error pro-vides an explicit error estimate for an approximation to thesolution of the original multiscale problem. Without thisregularity, such an approximation always exists when themeshsize and the micro scale converge to 0, but without arate of convergence.
A Multiscale Mixed Finite-element Method for theStokes-Brinkman Equations
We present a multiscale mixed finite-element method fordetailed modeling of free-flow and porous regions in vuggyand naturally-fractured reservoirs. The method uses astandard Darcy model to approximate flow on a coarsegrid and captures fine-scale effects through basis functionscomputed numerically by solving local Stokes-Brinkmanflow problems on the underlying fine-scale grid.
We discuss upscaling of inertia effects in heterogeneousporous media. From lab scale to reservoir scale we con-sider non-Darcy flow model; we present recent analyticaland numerical results extending work with C. Garibotti.A separate project with Trykozko is on upscaling fromporescale to lab scale. Connecting these scales, explain-
ing why heterogeneity and inertia seem to diminish eachother’s impact, as well as handling anisotropy emergingfrom upscaling are the main issues of this talk.
Malgorzata PeszynskaDepartment of MathematicsOregon State [email protected]
MS41
Analysis of Immiscible Two-phase Flows in PorousMedia with Discontinuous Capillarities
We consider a simplified model for two-phase flows inporous media made of different rocks. We focus on theeffects of the discontinuity of the capillarity field. We firstconsider a model with capillarity within the rocks. Thenwe look for the asymptotic problem for capillarity forcesremaining only at the interface. We propose a simple mod-eling of oil-trapping by the mean of non-classical shocks atthe interface.
A Phase-field Model of Unsaturated Flow – Stabil-ity Analysis and the Development of Gravity Fin-gers
Infiltration of water into homogeneous dry soil often leadsto preferential flow in the form of fingers. The canonicalmodel for unsaturated flow, known as Richards equation,is totally stable and therefore unable to reproduce this be-havior. We use this physical problem to introduce a newclass of models of multiphase flow in porous media, whichaccount for the presence of a macroscopic interface (thewetting front), and explain why and how fingering occurs.
Outflow Boundary Conditions in Porous MediaFlow Equations
To model the flow in porous media one typically uses ei-ther the Richards equation if the flow can be describedeffectively by one phase, or the two-phase flow equationsif two phases must be modeled. Due to the wide rangeof applications, the interest in these two systems is enor-mous. Rigorous analytical results became available in the80ies, when strong nonlinearities in time-dependent prob-lems were treated systematically. In most contributionson the subject, the analysis is simplified by restricting toDirichlet boundary conditions, even though a physicallymore appropriate boundary condition is the outflow con-dition. After motivating and explaining this condition wepresent the corresponding qualitative analytical problems,a regularization technique and existence results.
Non-linear Interface Models for Multi-phase FlowProblems in Heterogeneous Media
We consider two extensions to standard capillary pressurerelationships for two phase problems. Firstly, to correctthe non-physical behavior, we use a recently establishedsaturation-dependent retardation term. Secondly, in thecase of heterogeneous porous media, we apply a model witha capillary threshold pressure that controls the penetrationprocess. Mathematically, we rewrite this model as inequal-ity with possible discontinuities in the saturation and pres-sure at the interfaces. Numerical examples in 2D and 3Dshow the influence of the modifications.
Monotonicity for Multi Point Flux ApproximationMethods on Triangular Grids
We study the monotonicity behaviour of MPFA methodson triangular grid. For single phase flow, we find sufficientconditions for the MPFA O- and L-methods. The foundmonotonicity regions for the methods are also tested nu-merically. The tests are done for cases corresponding toboth homogeneous and heterogeneous media. We also in-vestigate the robustness of the methods with respect tomonotonicity for two-phase flow. The results obtained inthis work may be utilised in grid generation.
Ivar AavatsmarkCenter for Integrated Petroleum ResearchUniversity of [email protected]
MS42
A Finite Volume Method for Solving the SteadyConvection-Diffusion Equation
We discuss a cell-centered finite volume method to approx-imate the steady convection-diffusion equation on meshesof triangles and tetrahedra. The method is second-orderaccurate through a piecewise linear reconstruction withineach cell and at mesh vertices. A non-linear definition ofthe face gradients for the numerical diffusive fluxes allowsus to demonstrate the existence of a Maximum Principle.We also consider a reformulation for meshes of convex poly-gons of any shape, and the connection with the DDFVmethod and the MFD method.
Gianmarco ManziniIstituto di Matematica Applicata e TecnologieInformatiche
Monotone Finite Volume Discretization ofthe Convection-Diffusion Equation on PolyhedralMeshes
We consider the cell-centered finite volume discretizationof the steady convection-diffusion equation. The diffusiontensor may be full and anisotropic and the operator mayhave the dominated convection part. The computationalmesh (conformal or non-conformal) is assumed to consistof convex polyhedral cells. The proposed finite volumemethod is monotone, i.e. it preserves non-negativity ofthe differential solution. Monotonicity of the method isprovided by nonlinear two-point diffusion and advectionfluxes derived on faces of mesh cells. The method is the 3Dextension of the 2D finite volume discretization [1]. 1.Lip-nikov K., Svyatskiy D., Vassilevski Yu. Interpolation-freemonotone finite volume method for diffusion equations onpolygonal meshes.J.Comp.Phys., 2008
Kirill NikitinInstitute of Numerical MathematicsRAS, Moscow, [email protected]
Nonlinear Finite Volume Methods for Convection-Diffusion Problems on Unstructured PolygonalMeshes
Predictive numerical simulations of subsurface processesrequire not only more sophisticated physical models butalso more accurate and reliable discretization methods.The discretization methods used in existing simulations failto preserve positivity of a continuum solution of the ellipticequation when the media is anisotropic and heterogeneousand/or the mesh is strongly perturbed. We present thenonlinear finite volume approach that guarantees positiv-ity of the discrete solution on unstructured meshes andfor strongly anisotropic diffusion tensors. Recently severalvariations of this approach have been proposed. We com-pare their numerical and computational properties.
Yuri VassilevskiInstitute of Numerical MathematicsRussian Academy of [email protected]
MS43
Atmospheric Simulation Using High Order Meth-ods on Locally Adapted Unstructed Parallel Grids
We address the issue of simulating atmospheric motionin 2d using a high order conservative scheme (DG) onunstructured, locally adaptive grids. Testcases includeclassical mountain overflow problems and warm non-
GS09 Abstracts 93
precipitating cloud model with 3 components of air. Thelatter being an intermediate step towards an implementa-tion of a simplified COSMO model using high order, con-servative scheme. The advantage of such an approach is ineffective resolving of the impact of orography and dealingwith boundary problems in more natural manner.
High-Order Semi-implicit Time-integrators for theEuler and Navier-Stokes Equations
In this work, we describe the construction of high-orderaccurate in time semi-implicit methods for the Euler andNavier-Stokes equations used in mesoscale nonhydrostaticatmospheric modeling. The goal of this research is to com-pare and contrast various forms of the governing equations(e.g., should one use Exner or density for the mass andshould one choose potential temperature, density potentialtemperature, or total energy). The form of the continuousgoverning equations used in the models can restrict oneto only certain classes of time-integration methods (e.g.,if the equations are in flux form, then it is not possibleto use classical semi-Lagrangian methods). In addition,some forms of the governing equations can be more expen-sive to solve computationally than other forms. Anothertopic of discussion concerns the inclusion of adaptive time-stepping machinery into these models. While not impor-tant to the outline of the talk, for the moment we are as-suming that an element-based continuous Galerkin methodis used for approximating the spatial derivatives. In the fu-ture, discontinuous Galerkin methods will be included intothis generalized approach. We will describe the challengesfacing us with discontinuous Galerkin methods and howto implement them in conjunction with semi-implicit time-integrators.
Frank GiraldoNaval Postgraduate School, Department of AppliedMathematics833 Dyer Road, Bldg. 232, Spanagel 262, Monterey, [email protected]
A 2d Discontinuous Galerkin Model for the (Non)-hydrostatic Atmosphere
The vertical structure of the averaged free atmosphere isdominated by the hydrostatic balance. Large scale waves,with horizontal spatial scales larger than 10km, are presentin a non-hydrostatic model and develop similar for hydro-static equations. In the hydrostatic case, phenomena ofsmaller scales either destroy the balance or do not givecorrect wave speeds and dispersion properties. Wave prop-agation will be studied within a hydrostatic and a non-hydrostatic 2-dimensional model. Both models are dis-cretized with a high order discontinuous Galerkin methodand a semi-implicit time stepping.
Matthias LaeuterAlfred Wegener Institute for Polar and Marine [email protected]
Francis X. GiraldoAssociate Professor of Applied Mathematics [email protected]
Marco RestelliMax-Planck-Institut fur MeteorologieHamburg, [email protected]
The logically rectangular finite volume grids for two-dimensional PDEs on the sphere and for three-dimensionalproblems in a spherical shell introduced in [SIAM Review50(2008) pp. 723-752] have nearly uniform cell size, avoid-ing severe Courant number restrictions. We present recentresults with adaptive mesh refinement using the GeoClawsoftware and demonstrate well-balanced methods that ex-actly maintain equilibrium solutions, such as shallow waterequations for an ocean at rest over arbitrary bathymetryor a stratified atmosphere in three dimensions.
Randall J. LeVequeApplied MathematicsUniversity of Washington (Seattle)[email protected]
Marsha BergerCourant Institute of Mathematical SciencesNew York [email protected]
94 GS09 Abstracts
MS44
Approaches to Coupled Land-energy Flux Model-ing
We will discuss benefits and associated difficulties of con-structing coupled models to simulate interactions of thegroundwater-landsurface-atmosphere system. These mod-els potentially provide understanding of two-way feedbacksand are, thus, a promising tool for process understandingand prognosis. However, related physical processes cannotbe described fully using first-principles approaches leadingto parameterizations that are intrinsically tied to partic-ular spatiotemporal scales. Therefore, understanding ofprocess scalability is an important component in coupledmodel construction requiring careful consideration.
Stefan KolletMeteorological InstituteUniversity of [email protected]
MS44
Introduction to the Minisymposium: Coupled Ap-proaches to Groundwater-atmospheric Modeling
The linkage between groundwater and the lower atmo-sphere occurs via shallow soil moisture and energy trans-port processes. Complex, coupled nonlinear physiochemi-cal processes and a wide range of spatiotemporal scales ne-cessitates numerical simulation of these interactions. Thesecomputationally challenging solutions must utilize efficientalgorithms and preconditioners resulting in a balance ofstability, efficiency and accuracy. This talk will introducethese issues and demonstrate some numerical approachesto bridging these two systems.
Reed M. MaxwellDepartment of Geology and Geologic EngineeringColorado School of [email protected]
MS44
Surface-groundwater Flow Coupling Based onBoundary Condition Switching
We develop a model of surface-groundwater flow based onboundary condition switching between surface-subsurfacedomains. The model is shown to be able to reproducerelevant processes at different spatial and temporal scales.Explicit finite differences for surface routing and implicitfinite elements for Richards equation allow for differenttime-stepping strategies to capture different time scales.Model performance is analyzed numerically on sample andreal-world test cases and theoretically within the contextof domain decomposition methods.
Mario PuttiDept. Mathematical Methods and Models for AppliedScienecesUniversity of [email protected]
MS44
Strategies for Coupling Subsurface Flow, Land Sur-face, and Atmospheric Models
Fully coupled bedrock to atmosphere models require astrategy for time stepping and passing information be-tween the subsurface, land surface, and atmospheric mod-
els. We will present results of investigations into time cy-cling and operator splitting methods for a coupled modelincluding the ParFlow variably saturated subsurface flowcode, the CLM land surface flow code, and the WRF atmo-spheric simulation code. This work performed under theauspices of the U.S. Department of Energy by LawrenceLivermore National Laboratory under Contract DE-AC52-07NA27344. This work was supported by the LLNL Cli-mate Change Initiative. LLNL-ABS-409527.
Initiating Large Slope Movements by Rain-inducedShear Bands
Against the background of the global climate change, thecomputer aided prediction of landslides induced by heavyrain events is of increasing importance in order to protectlife and property in alpine regions. This makes it necessaryto accurately model the complex inelastic deformation be-haviour of partially or fully saturated soil on the basis ofsophisticated multi-phase continuum theories. In this re-gard, we present a triphasic porous media model capableof describing all prominent hydraulic and solid mechani-cal effects triggering the evolution of shear zones towardscomplete slope failure.
Wolfgang EhlersInstitute for MechanicsUniversity of [email protected]
A Double Continuum Approach for Two-phaseFlow Simulation in Natural Slopes
The fast infiltration of heavy rainfalls in macro-porous soilsis one of the key processes which triggers the movementof natural slopes as it leads to a ’fast’ pressure increaseas well as to a fast saturation of a hillslope. The macro-porous medium is characterized by a complex channel-likenetwork which cannot be discretely taken into account. Tosimulate flow processes in such media, a double-continuummodel concept for two-phase flow has been developed andshould be presented showing results of plausibility testsas well as of a field experiment. Special emphasis is puton the exchange parameters which are determined fromlaboratory experiments.
Theoretical and Experimental Aspects on Trigger-ing Rapid Mass Movements of the Flow-Type
Many of the world’s most devastating landslide disasterscan be attributed to landslides of flow type involving noncohesive soils. The flow-type characteristics of the post-failure stage are linked to the initial acceleration of thefailed mass, determined by the soils’ mechanical instabil-ity. In the presentation, experimental evidences of partiallysaturated non cohesive soils leading to flow-type behaviourare illustrated and interpreted within a geomechanical con-ceptual framework.
Internal Erosion in Soils: A Multi-phase Model onMultiple Scales
Internal erosion, i.e. the debonding and movement of finesthrough the porous soil by hydraulic forces, could affectthe mechanical and hydraulical stability of geostructuresdramatically. Internal erosion is causative for many natu-ral and technical disasters in connection with flood eventsand collapse-like slope failures where small triggers leadto a liquefaction of the soil. To simulate internal erosionprocesses, we propose a mixture theory-based multi-phasemodel which we compare with laboratory experiments.
Holger SteebAssistant Professor, U Twente, The NetherlandsProfessor, U Bochum, [email protected]
Computing Geological Storage of CO2: A Grid En-vironment for Massively Parallel Simulation, Stor-age and Statistics
In this presentation we describe a grid environment(GE) for massively parallel simulations related CO2-sequestration in deep aquifers. In addition to the computefunctionality, the environment provides data-storage facili-ties and tools for statistical data-exploration. The compute
part of the GE has access to a number of supercomput-ers located in the Nordic countries Denmark, Finland andNorway. Storage and data-evaluation is provided by theNorway and partners in UK and Ireland. The infrastruc-ture is based on the middlewares ARC (Nordugrid) andDiGS (EPCC). It supports the simultaneous execution ofvarious instances of massively parallel simulations and thesubsequent evaluation of the generated data. The currentlysupported software is MUFTE-UG, a simulation platformfor Multi-Phase-Multi-Component flow simulations. Simu-lated time-series are stored in XML-annotated sets of filesand statistical investigations can be carried out on the baseof the complex meta-data. The capability of the environ-ment will be illustrated by selected models.
Numerical Schemes with Low Dissipation for Mod-elling Contaminant Transport in Porous Media onSimplicial Meshes
We present two numerical schemes with low dissipation formodeling contaminant transport in porous media. Bothschemes use operator splitting and cell-centered finite vol-ume paradigm. However, the schemes differ in the con-struction of the diffusive flux and the advection substeps.Also, the schemes are derived for different classes of 3Ddynamic meshes, i.e. conformal tetrahedral and non-conformal hexahedral (octrees). We present the basic prop-erties of the schemes such as order of convergence, mono-tonicity, computational efficiency, and compare them onthe solution of several test problems.
High Performance Simulation of CO2 GeologicalStorage
The objective of the French ANR project SHPCO2 is to de-velop and study advanced numerical methods for the simu-lation of CO2 reactive transport on massively parallel com-puters. We have identified four challenging subjects thatwe will discuss during the session. 1. Coupling multiphaseflow and reactive transport models. 2. Reactive transportnonlinear solvers. 3. Time space domain decompositionand local time stepping. 4. Dynamic load balancing for amulti-physics and multi-domain application.
Modelling the Transport of Particulate Suspen-sions and Formation Damage During the Deep In-jection of Supercritical Carbon Dioxide in Sand-stone Formations
Prediction of CO2 injection performance in deep subsurfaceaquifers and reservoirs rely on the well ability to maintainhigh flow rates of carbon dioxide during several decadeswithout significantly impairing the host formation. Dy-namics of solid particulate suspensions in permeable mediaare recognized as one major factor leading to injection wellplugging in sandstones. The invading supercritical liquid-like fluid can contain variable concentrations of exogenousfine suspensions or endogenous particles generated in situby colloidal or hydrodynamic release mechanisms. Sus-pended solids can plug the pores leading to possible forma-tion damage and permeability reduction in the vicinity ofthe injector. As such, models which can predict well injec-tivity decline, are useful in the operations of planning, de-sign, and maintenance, related to carbon dioxide injection.In this study we developed a finite element based simula-tor to predict the injectivity decline nearby CO2 injectionwells and also for production wells in the context of EOR.The numerical model solves implicitly a system of two cou-pled sets of finite element equations corresponding to thepressure-saturation two-phase flow, then a system of so-lute and particles convection-diffusion equations. Particleequations are subject to mechanistic rate laws of colloidal,hydrodynamic release from pores bodies, blocking in poresbodies and pores throats, and interphase particles trans-fer. The model was validated against available laboratoryexperiments at the core scale. At the field scale, challengesstill exist for an accurate assessment of the permeabilitychange due to limited current knowledge of SupercriticalCO2 and water phases micro-interactions at pore surfaces,but also to the multiscale nature of the numerical problem.Numerical demonstration examples in a saline sandstoneaquifer reveal that formation damage during CO2 injectionwill primarily depend on the injected particles wettability,the injection flow rate, and the medium tortuosity. Othersimulation examples are provided for a CO2 injection ina five spots pattern EOR oil field for performance assess-ment of the production with occurence of in-situ sandingin a poorly consolidated sandstone reservoir.
Using Energy Minimizing Basis Functions as aMultiscale Method, With Applications to Multi-grid and Domain Decomposition Solvers
We demonstrate the applicability of energy minimizing ba-sis functions for two-phase flow simulations, and we high-light their numerous benefits. We show how they can beimplemented to obtain efficient serial (algebraic multigrid)
and parallel (Additive Schwarz with coarse space correc-tion) linear solvers for large-scale heterogeneous problems.Moreover, they possess the same advantages of other mul-tiscale techniques, but without the need of constructing acoarse mesh, and they are well-suited for adaptive algebraiccoarsening.
Ilya D. MishevExxonMobil Upstream Research CompanyDepartment Technical Software [email protected]
Ludmil ZikatanovDepartment of Mathematics,PennState [email protected]
MS47
A Multiscale Finite Volume Method for UnsteadyStokes-Darcy Equations
The talk focuses on the application of the iterative-multiscale-finite-volume (IMSFV) procedure to the Stokes-Darcy system, describing flow in porous media. The stan-dard IMSFV method is extended for solving the momen-tum and pressure correction equations in a fractional timestepping algorithm framework. The multiscale method al-lows to simulate flow and transport in porous media withgeometries that can be too complicated to be resolved bya feasible computational grid. The coarse scale ensuresglobal coupling and the fine scale ensures appropriate res-olution. The method is illustrated with examples and iscompared with fine scale solutions.
A Multilevel Multiscale Mimetic (M3) Method forTwo-Phase Flows in Porous Media
The M3 method builds recursively a problem-dependentmultilevel hierarchy of models for flow in porous media.Each model is locally mass conservative. The methodsupports full diffusion tensors on unstructured polyhedralmeshes and accommodates general coarsening strategies.We describe well modeling and adaptive strategies for up-dating the multilevel hierarchy. Numerical simulations forpermeability fields with long correlation lengths show thateven with large coarsening factors, 50 and more in each co-ordinate direction, the multiscale solution remains within
Multilevel Solvers and Upscaling Via Explicit En-ergy Minimisation
In this talk we highlight the strong link between robustmultilevel iterative solvers and upscaling/multiscale tech-niques for subsurface flow. Motivated by recent theoreticalresults for two-level overlapping Schwarz domain decom-position methods, we investigate the construction of up-scaling techniques based on energy minimisation. Theseinclude the Multiscale Finite Element Method as a specialcase, but in contrast to MsFEs do not require any artifi-cial boundary conditions on coarse elements. We presentand analyse a preconditioner for the arising constrainedminimisation problem, that is robust to mesh refinementand arbitrary coefficient variation (requiring no scale sep-aration). The resulting method is O(h−d) where h is thesubgrid size.
Robert ScheichlDepartment of Mathematical SciencesUniversity of [email protected]
Generalized Forchheimer Equation for Two-phaseFlow Based on Hybrid Mixture Theory
We discuss the derivation of a Forchheimer-type equationfor two-phase flow through an isotropic porous medium us-ing hybrid mixture theory. Hybrid mixture theory consistsof upscaling field (conservation) equations by averaging,and then exploiting the entropy inequality to obtain consti-tutive equations. Isotropic function theory is then used tosimplify the equation for application to an isotropic porousmedium. As time permits, we will discuss the extension ofthis theory to multiphase swelling materials.
Tiziana GiorgiDepartment of MathematicsNew Mexico State [email protected]
MS48
Multiscale Adaptive Modeling of Inertia and Non-equilibrium Processes
New experimental and modeling techniques make possi-ble to identify and quantify inertia, dynamic effects, andnon-equilibrium processes that until recently by necessitywere deemed next-order or lower-scale only. The new termsand couplings that arise bring challenges for analysis andnumerical modeling. We are concerned with how to deter-mine their significance adaptively the way one handles gridadaptivity. The talk will be partly based on joint researchwith Showalter and Klein.
Malgorzata PeszynskaDepartment of MathematicsOregon State [email protected]
MS48
Two-scale Pore Network Modeling for MultiphaseFlow in Heterogeneous Geometries
Most of the transport phenomena of interest in subsurfaceflow modeling occur on multiple scales, and the complexsubsurface pore spaces geometry on micro scale is notoriousfor dictating macroscopic flow of interest. We present an al-gorithm to geometrically match pore throat networks fromtwo separate scales (that can either originate from availablemeasurements or be constructed according to generally ac-cepted knowledge) and show preliminary results for flow innetworks based on realistic carbonate geometry.
Masa ProdanovicUniversity of Texas at AustinCenter for Petroleum and Geosystems [email protected]
Adrian SheppardAustralian National UniversityDepartment of Applied [email protected]
MS48
Experimental Analysis of Non-equilibrium Pro-cesses for Multiphase Flow in Porous Media
The effect of flow rate on interfacial configurationsin porous media was investigated using computed x-ray microtomography and image analysis. One fulldrainage/imbibition cycle was measured and subse-quently residual configurations were assessed followingdrainage/imbibition at flow rates varying over an order ofmagnitude. Results indicate that both flow rate and sys-tem history are important, especially for the imbibitionprocess.
Analysis of Numerical Methods for CouplingTransport Andgeochemistry Equations
Reactive transport equations involve PDE for advection-diffusion, algebraic equations for chemistry at equilibriumand ODE for kinetic chemistry. Classical methods arebased on a method of lines where spatial discretization isfollowed by temporal discretization. We analyze here thenumerical properties and convergence of four such meth-ods, namely SNIA, SIA, DSA and DAE. We illustrate thecomparison by some numerical examples.
Caroline de DieuleveultANDRAINRIA Rennescaroline.de [email protected]
MS49
Efficient Solution of Large Reactive-TransportProblems Including Mineral Reactions
The global-implicit approach applied to large multicompo-nent reactive transport problems in general leads to largenonlinear systems. A method is proposed to reduce the sizeof the system using equivalence reformulations, leading toan elimination or a decoupling of many of the unknowns.If precipitations/dissolution reactions are involved, a for-mulation is given which can be solved by the semismoothNewton method. Numerical results of the proposed algo-rithms are presented.
Serge KraeutleInstitute for Applied MathematicsUniversity of [email protected]
MS49
Adaptive, Selective Coupling of MulticomponentTransport and Kinetic Reactions
The efficiency of the process-preserving, globally implicitapproach to solve reactive multicomponent transport prob-lems with Newton’s method can be enhanced by modifiy-ing the Jacobian. Therefore the reaction network, whichdefines the connectivity pattern of the system matrix, isanalysed to neglect terms whith the aim to decouple speciesequations on the level of the linear solver without deteri-orating the quadratic performance of the global, process-preserving iteration. The fully adaptive algorithm can beused potentially in every time step, and works with directas well as iterative linear solvers for reactions of kinetictype in arbitrary dimensions.
Alexander PrechtelMathematics DepartmentUniversity of Erlangen-Nuremberg, [email protected]
MS49
Mixed Finite Elements and Newton Method forReactive Solute Transport in Porous Media
We present a mass conservative finite element scheme for
reactive solute transport in porous media. The transportis modelled by a convection-diffusion-reaction equation, in-cluding equilibrium sorption. We especially considered thecase of a Freundlich type isotherm, when the equation be-comes degenerate. The algorithmic aspects of our schemeare presented in [F. A. Radu, M. Bause, A. Prechtel and S.Attinger, Analysis of an Euler implicit - mixed finite ele-ment scheme for reactive solute transport in porous media,Numerical Mathematics and Advanced Applications, K.Kunisch, G. Of and O. Steinbach (editors), Springer Ver-lag, 2008, pp. 513-520.] and the analysis of the discretiza-tion error in [F. A. Radu, I. S. Pop and S. Attinger, Anal-ysis of an Euler implicit - mixed finite element scheme forreactive solute transport in porous media, Numerical meth-ods for partial differential equations, 2008, to appear.]. Wefocus here on the applicability of the Newton method onsolving the nonlinear problems arising at each time step.For the degenerate case a regularization step is necessary.An explicit condition for the quadratic convergence of theNewton method in terms of the discretization parametersand the regularization number is derived.
Florin A. RaduUFZ-Helmholtz Center for Environmental Research,[email protected]
Iuliu Sorin PopDept of Mathematics and Computer ScienceTU [email protected]
MS50
Balance-Preserving Finite Element Methods onUnstructured Meshes
The dynamical equations of motion for the atmosphereand ocean support oscillations a vast range of spatial andtemporal scales. However, large-scale flows in the at-mosphere and ocean remain in a state of slow evolutioncalled geostrophic balance in which fast oscillations arevery weak; these slowly-evolving dynamical states repre-sent our weather and the global circulation patterns. Tomake predictions on these timescales, special care must betaken to design numerical methods which reflect the prop-erties of this underlying balance. Recently there has beenmuch interest in modelling the atmosphere and ocean onadaptive unstructured meshes; this calls for new discretisa-tion methods which reflect geostrophic balance. We intro-duce a new family of mixed finite element methods whichprovide the best possible representation of geostrophic bal-ance on unstructured meshes. The proofs of these proper-ties are very simple since they are based on underlying ge-ometric structure: the discretisations have exact sequenceswhich mimic the div-curl and curl-grad relations of vectoranalysis.
Discretised Poisson and Nambu Brackets in theContext of an Atmospheric General CirculationModel
We want to establish a coupled atmosphere and oceanmodel for climate simulations and numerical weather pre-diction. As model properties are foreseen (i) conservationof mass, (ii) conservation of tracer mass and its consis-tency with the continuity equation, (iii) conservation ofenergy and a vortex quantity; all to be achieved on triangu-lar/hexagonal meshes. Exploring the algebraic structure offluid equations in Poisson/Nambu bracket form facilitatesto obey these properties in the numerical realization.
Hamiltonian Particle-mesh Simulations for a Non-hydrostatic Vertical Slice Model
We develop a non-hydrostatic vertical slice model in thecontext of the Hamiltonian Particle-Mesh Method (HPM)for the dry adiabatic atmosphere. The slice model is testedwith the bubble-experiments described in Robert (1992)and the gravity wave experiment in Skamarock and Klemp(1994). The solutions are maintained smooth largely dueto a ”regularization” in the absence of the artificial diffu-sion. The regularization is implemented in harmony with aconservative force field and does not interfere with the hy-drostatically balanced reference state. The accuracy of theHPM simulation is comparable to those in these referencesand the model performances show that the HPM method ispotentially applicable to non-hydrostatic atmospheric flowregimes.
Seoleun ShinInstitute for Mathematics, University of PotsdamAm Neuen Palais 10, 14469 Potsdam, [email protected]
Conservation and Wave Propagation on Hexagonaland Geodesic C-grids
A ‘geodesic’ grid, comprising hexagonal and pentagonalcells, is attractive for global atmospheric modelling be-cause it gives a nearly uniform and isotropic coverage of thesphere, avoiding the pole problems of a latitude-longitudegrid. A C-grid staggering, in which the mass variable isstored in cells and normal velocity components are storedat cell edges, is also considered attractive because it givesa relatively good representation of fast waves and hencethe geostrophic adjustment process. However, early stud-ies of wave propagation for the shallow water equations ona regular hexagonal C-grid showed spurious large frequen-cies for near-grid-scale Rossby waves - a serious obstacle tothe use of such a scheme for global atmospheric modelling.We show here that these spurious large frequencies can beavoided by a suitable discretization of the Coriolis terms. Ageodesic grid comprises somewhat distorted hexagons andalso pentagons. By careful consideration of the discretevorticity budget, the regular hexagonal grid scheme is ex-
tended to the case of distorted hexagons and pentagons(indeed, any polygonal voronoi grid), ensuring both satis-factory Rossby wave propagation and energy conservation.Some sample numerical results will be shown.
John ThuburnSchool of Engineering, Computing and MathematicsHarrison Building, University of Exeter, North ParkRoad, [email protected]
We solve the coupling of Richards equation in hetero-geneous soil with non-moving surface water or depth-averaged shallow water equations. The Richards equationis treated without linearization by Kirchhoff transforma-tion, convex minimization and monotone multigrid meth-ods in homogeneous soil and non-overlapping domain de-composition methods to address heterogeneity. Gravity isincluded by upwinding. The coupling is provided by hy-drostatic pressure and mass conservation leading to fixedpoint problems by different time scales for ground and sur-face water.
Discontinuous Galerkin Methods for CouplingDepth-integrated Shallow Water Models withRichards Equation
We describe a discontinuous Galerkin method for couplingshallow water flow and unsaturated ground water flow.The local discontinuous Galerkin method is used to ap-proximate both the shallow water equations and Richards’equation describing flow in the vadose zone. Coupling be-tween the two models is done weakly. Error estimates andnumerical results are presented.
Clint DawsonInstitute for Computational Engineering and SciencesUniversity of Texas at [email protected]
MS51
Hierarchic Modelling of Free Surface Shallow Wa-ter Flow
Mass Conservative DG/FV Schemes for Couplingthe Richards Equation with Surface Flows
We propose and analyze numerical schemes to couple sub-surface and overland flows. The governing equations areRichards’s equation in the subsurface, the kinematic waveequation on the surface, and the matching of pressure andnormal velocity at the interface. Space-time discretiza-tion uses Discontinuous Galerkin/Finite Volume schemescombined with a multi-step implicit scheme in the subsur-face. Multi-step coupling algorithms are proposed to en-sure overall mass conservation. Numerical results illustratethe performances of the proposed algorithms.
Tomographic Analysis of Reactive Fluid InducedPore Structure Changes
We have employed synchrotron microtomography withflow-column experiments to capture and quantify snap-shots in time of dissolution and secondary precipitationchanges in the microstructure of sediments exposed to sim-ulated caustic waste. Dissolution induced changes includedan increase in the number of larger pores. Precipitation in-duced changes included reduction in the number of smallpores and closure of small throats, with accompanying re-duction in pore coordination numbers and reduction in thenumber of pore pathways.
W. Brent LindquistState University of New York, Stony [email protected]
Interfacial Energy, Surface Tension and ContactAngle
Capillary phenomena can be equivalently described interms of surface energies necessary to create the interfaces,or in terms of surface tensions that tend to contract theinterfaces. We discuss the issues arising in presence of asolid phase and the tension balance at the triple contactline, including the component perpendicular to the solidwhen Youngs law applies. Also implications for porous
Detailed Pore Scale Fluid Displacement ModelingVia the Level Set Method
An accurate description of pore level immiscible fluid dis-placement could significantly improve macroscopic param-eter predictions in real porous media. We present a simplebut robust quasi-static displacement model for drainageand imbibition based on the level set method that resultsin geometrically and topologically correct interfaces, andis independent of the pore space complexity. Many appli-cations include investigating fracture matrix transfer, andcoupling with sediment mechanics. The code, LSMPQS, ispublicly available.
Masa ProdanovicUniversity of Texas at AustinCenter for Petroleum and Geosystems [email protected]
Steven L. BryantPetroleum and Geosystems Engineering DepartmentUniversity of Texas at Austinsteven [email protected]
MS52
Pore Scale Modeling of Moving Crystal-fluid Inter-faces Using Level Sets and Phase Field Models
We present level-set and phase-field formulations of crys-tal dissolution/precipitation processes on the pore scale.These formulations are then compared to each other andused to simulate the evolution of crystalline solid-fluid in-terfaces inside a pore. Furthermore we will briefly dis-cuss how these formulations can be used to upscale disso-lution/precipitation processes to the Darcy scale.
Tycho L. van NoordenDepartment of Mathematics & Computer ScienceTU [email protected]
MS53
Simulation of Core-Scale Multiphase Flow Experi-ments with CO2 and Brine
Sequestration in Stanford General Purpose Re-search Simulator
We present the capabilities of Stanford’s General PurposeResearch Simulator (GPRS) to simulate CO2 sequestrationproblems. The key known mechanisms, including hystere-sis, diffusion and dispersion, and chemical equilibria have
GS09 Abstracts 101
been incorporated in GPRS. Comparison between GPRSand other simulators show close agreement and competitiveefficiency for non-mineralization cases. A CO2 mineraliza-tion case in the Johansen formation is presented, where wedemonstrate that the amount of mineralized CO2 dependsstrongly on the in-situ mineral composition.
Two-phase Flow in Deformable Porous Media: Ap-plication to CO2 Storage
Recently, CO2 sequestration as concluding long-term stageof closed Carbon Dioxide Capture and Storage (CCS) pro-cesses considered for reducing carbon dioxide emissionsinto the atmosphere is subject of worldwide intensive in-vestigations. To take into account all the relevant phe-nomena during spreading and storage of the injected CO2in the subsurface (flow and transport of multiple phases,rock deformation, non-isothermal conditions, geochemicalreactions) physically founded complex mathematical mod-els are required. The multiple coupled problems cannot besolved analytically. Rather, sophisticated numerical meth-ods have to be applied.
In this presentation, a series of models will be developedto describe the dispersal of CO2 plumes through a sub-surface aquifer, including the effects of capillary retention,drainage through fault zones or leakage into a lower per-meability overlying seal rock. The models will be analysedto develop approximate similarity solutions to describe thedispersal of the plume with time, and the ensuing shapeof the residual trapped CO2 as the continuing current mi-grates through the stata.
Andrew WoodsBP-Institute for Multiphase FlowUniversity of [email protected]
MS54
Constrained Adaptive Voronoi Gridding for Reser-voir Modeling
We present our recently developed techniques for adaptiveconstrained 2.5D Voronoi grid generation based on a rigor-ous procedure of constructing Voronoi grid conforming to
piecewise linear constraints. Our approach addresses reser-voir modeling challenges associated with accurate model-ing of complex reservoir geometry and heterogeneous reser-voir properties. Together with an accurate scale-up of fine-scale geologic properties, our gridding approach improvesthe consistency between geologic descriptions and reservoirsimulation models, leading to more accurate simulation re-sults.
A New Multiscale Approach for the Simulation ofMultiphase Flow Processes in Porous Media
The contamination of the unsaturated zone with a lightnon-aqueous phase liquid is studied, corresponding to adomain with randomly distributed heterogeneities wherecomplex three-phasethree-component processes are rele-vant only in a small(local) subdomain. This subdomainneeds fine resolution as the complex processes are governedby small-scale effects. For a comprehensive fine-scale modeltaking into account three-phasethree-component processesas well as heterogeneities in the whole (global) model do-main, data collection is expensive and computational timeis long. Therefore, we developed a general multi-scale con-cept where on the one hand, the global flow field influ-ences the local three-phasethree-component processes onthe fine-scale. On the other hand, a coarse-scale satura-tion equation is solved where the effects of the fine-scalemulti-phasemulti-component processes in the subdomainare captured by source/sink terms and the effects of fine-scale heterogeneities by a macrodispersion term.
Rainer HelmigIWS, University of Stuttgart, GermanyInstitut fur [email protected]
Jennifer NiessnerInstitut fuer Wasserbau (IWS)Universitaet Stuttgart, [email protected]
In this talk we present a methodology for coarse-scale mod-eling of multiphase flow in 3-D channelized domains suchas those encountered when simulating gas injection pro-cesses. It integrates techniques for transmissibility upscal-ing, adaptive mesh refinement, and MPFA finite-volumediscretization in order to balance accuracy, efficiency, androbustness. We focus on aspects that are specific to multi-phase flow, including saturation-based adaptivity criteria,dynamic adaptation during transport, and streamline trac-ing.
James V. LambersStanford UniversityDepartment of Energy Resources [email protected]
Margot GerritsenDept of Petroleum EngineeringStanford [email protected]
MS54
Downscaling: the Inverse to Upscaling
Determination of conductivities from measured head-flowrate pairs is generally called inverse modelling. We presenttwo inversion methods: the Double Constraint methodand Constrained Back Projection. We do so for two dis-cretization methods: node-based finite elements and block-centered finite differences. Downscaling is inverse mod-elling to determine fine-scale conductivities in coarse-scalegrid blocks. The examples show that downscaling is a prac-tical complement to homogenization if the porous mediumis not periodic.
Pore-network Modelling of Non-equilibrium Capil-lary Pressure Effect
Traditional two-phase flow models assume that macro-scopic phase pressures difference is a function of macro-scopic saturation. However, under dynamic conditions, thedifference of macroscopic phase pressures is known to bea function of time rate of saturation change. We have de-veloped a three-dimensional regular lattice dynamic pore-network model in which we solve for the two fluid pressurefields separately. We have simulated both drainage andimbibitions processes for different viscosity ratios.
V. Joekar-NiasarEarth Sciences Department, Utrecht [email protected]
Long Time Behavior for Dynamic Capillarity Mod-els and Relation to Hyperbolic Conservation Laws
We discuss an extended BuckleyLeverett (BL) equation de-scribing two-phase flow in porous media, including a thirdorder mixed derivatives term modeling dynamic effects inthe capillary pressure. We derive existence conditions fortraveling wave solutions, leading to admissible shocks forthe original BL equation violating the Oleinik entropy con-dition. This provides nonmonotone weak solutions of theinitial-boundary value problem for the BL equation con-sisting of constant states separated by shocks, confirmingresults obtained experimentally.
Lattice Boltzmann Modeling of Macro-PorousFlow: Effects of Image Segmentation Algorithmsand Comparisons with Observed Data
In this presentation we simulate saturated flow throughmacroporous soil columns (7.62x18 cm) with a latticeBoltzmann model and compare results with measured sat-urated hydraulic conductivities. Porous geometry was ob-tained with an industrial CT scanner yielding a resolutionof 119 microns (656x656x1482 voxels) and processed withseveral segmentation algorithms to generate pore solid clas-sifications from gray-scale data. In this presentation wewill define the optimal resolution and discuss the merits ofselected segmentation algorithms.
Upscaling Density-dependent Flow and the Role ofGravity Forces in Case of Transversal Dispersion
In the talk we will discuss upscaling density-dependentflow, in particular the role gravity forces in case of transver-sal dispersion. We did some lab experiments, mathematicalanalysis and numerical computations. This is a highly in-teresting topic, with many details not yet fully understood.
Ruud SchottingGeosciencesUniverstiy of Utrecht, The [email protected]
MS56
Solute Transport in Heterogeneous Porous Media
Although being linear advection-dominated transport isa notoriously difficult problem. In strongly variable flowfields even high resolution finite volume methods may ex-hibit a high degree of numerical diffusion. We presentan Eulerian-Langrangian Localized Adjoint method on un-structured grids in two space dimensions that is locallyconservative and monotone. The scheme is compared to ahigh-resolution discontinuous Galerkin scheme.
Convergent Numerical Schemes for Reactive Flowin Unsaturated Porous Media
Reactive porous media flow models are coupled systemsof nonlinear degenerate parabolic equations. Due to thedegeneracy, solutions of such models are lacking regularity,and an efficient numerical simulation needs requires appro-priate discretization schemes. In this talk we consider Inthis talk we analyze the convergence of a mixed finite ele-ment scheme for reactive porous media flows. We focus onthe coupling between the unsaturated flow component andthe reactive component of the model.
Florin A. RaduUFZ-Helmholtz Center for Environmental Research,[email protected]
Iuliu Sorin PopDept of Mathematics and Computer ScienceTU [email protected]
MS56
Numerical Solution of the Transport of Contami-nants in Surface and Subsurface Flows
This work models groundwater contamination throughrivers and lakes. The resulting multiphysics problem cou-ples the Navier-Stokes equations, the Darcy equations and
the transport equation. The flow is numerically solved bythe finite element method in the surface region whereasit is approximated by the primal discontinuous Galerkinmethod of any order in the subsurface. The transportequation is solved with an improved discontinuous Galerkinmethod that minimizes the amount of overshoot and under-shoot in the convection dominated regions. Convergence ofthe scheme and numerical simulations are shown.
Guaranteed and Robust Discontinuous Galerkina Posteriori Error Estimates for Convection-diffusion-reaction Problems
We derive robust a posteriori error estimates for discon-tinuous Galerkin discretizations of stationary convection-diffusion-reaction problems. The estimates do not involveany undetermined constants and can be used for actual er-ror control. They are based on H(div)-conforming diffusiveand convective flux reconstructions. They are also locallyefficient and hence suitable for adaptive mesh refinement.Numerical experiments illustrate their performance.
Martin VohralikUniversite Pierre et Marie CurieParis, [email protected]
Alexandre ErnUniversite Paris-EstCERMICS, Ecole des [email protected]
Annette StephansenUnifob PetroleumCentre for Integrated Petroleum [email protected]
MS57
Linear Fluctuation-dissipation for Low FrequencyClimate Response
Recently, we developed and tested novel computationalalgorithms for predicting the mean linear response of achaotic dynamical system to small changes in external forc-ing via the fluctuation-dissipation theorem (FDT). Thenew linear response algorithms are tested on the T21 trun-cation of the barotropic climate with realistic Earth-like to-pography and two types of forcing which mimic behaviourof the atmosphere at 300 and 500 hPa geopotential height.The new methods yield greater accuracy than classicalFDT methods for the linear response of both mean stateand variance for both dynamical regimes.
Rafail AbramovDepartment of Mathematics, Statistics and ComputerScienceUniversity of Illinois at [email protected]
MS57
Statistical Model for an Incompressible Particle-
104 GS09 Abstracts
mesh Method
In long time simulations with numerical methods, statisti-cal averages converge - under ergodicity assumptions - toensemble averages in an equilibrium measure. Which equi-librium measure depends on the numerical method used.In this talk we describe a model of the dynamics of theHamiltonian particle-mesh method as adapted for quasi-geostrophic potential vorticity flow. The HPM methodpreserves potential vorticity pointwise, and the motion ofdiscrete particles can be embedded in an area-preservingcontinuum flow. Using this knowledge, we propose a modelbased on random permutations of the initial PV distribu-tion on a uniform particle arrangement. This model is val-idated using Monte-Carlo simulations. Next we constructa statistical mechanics theory for the model. Finally, themean field predictions of the statistical mechanics theoryare compared with long time simulations with HPM. Anonlinear stream function-PV profile is observed for bothskewed and flattened distributions.
We will explore the connection between several stochas-tic model reduction techniques in dynamical systems withtime scale separation. In particular, we will explore howthe approach of stochastic centre manifold reduction islinked to the homogenization approach valid on long timescales. In an application we will then use both reductiontechniques for data assimilation taking into account theirvalidity on different temporal scales.
Georg A. GottwaldSchool of Mathematics and StatisticsUniversity of [email protected]
MS57
Parametric Estimation of Effective StochasticModels from Discrete Data
It is often desirable to derive an effective stochastic modelfor the physical process from observational and/or numer-ical data. Various techniques exist for performing estima-tion of drift and diffusion in stochastic differential equa-tions from discrete datasets. In this talk we discuss thequestion of sub-sampling of the data when it is desirableto approximate statistical features of a smooth chaotic tra-jectory by a stochastic differential equation. In this caseestimation of stochastic differential equations would yieldincorrect results if the dataset is too dense in time. There-fore, the dataset has to sub-sampled (i.e. rarefied).
Numerical Analysis of the Navier-Stokes/DarcyCoupling
We present mathematical and numerical models for sim-ulating incompressible fluid flows through porous media.The main applications of our interest are the hydro-logical environmental ones and mass transfer in biome-chanics. We outline the analysis of a coupled Navier-Stokes/Darcy problem. After proving its well-posednessusing the Beavers and Joseph interface conditions, we in-troduce a stable Galerkin finite element approximation andwe study effective iterative schemes based on domain de-composition theory to compute its solution.
Marco DiscacciatiEcole Polytechnique Federale de LausanneInstitute of Analysis and Scientific [email protected]
Lori BadeaInstitute of Mathematics of the Romanian AcademyBucharest, [email protected]
A Fully Mixed Finite Element Method for the Cou-pling of Stokes and Darcy Flows
In this paper we analyze a fully mixed finite elementmethod for the coupling of fluid flow with porous me-dia flow. Flows are governed by the Stokes and Darcyequations, respectively, and the corresponding transmis-sion conditions are given by mass conservation, balance ofnormal forces, and the Beavers-Joseph-Saffman law. Weconsider dual-mixed formulations in both the Stokes do-main and the Darcy region, which yields the introduc-tion of the traces of the porous media pressure and thefluid velocity as suitable Lagrange multipliers. The finiteelement subspaces defining the discrete formulation em-ploy Raviart-Thomas elements for the velocities, piecewiseconstants for the pressures, and continuous piecewise lin-ear elements for the Lagrange multipliers. We apply theBabuska-Brezzi theory together with a classical result onprojection methods for Fredholm operators of index zero,to show stability, convergence, and a priori error estimatesfor the associated Galerkin scheme. In addition, we pro-vide a residual-based a posteriori error estimator for thiscoupled problem. Finally, some numerical results are re-ported.
Gabriel GaticaUniversidad de ConcepcionDepartamento de Ingeniera [email protected]
Optimal Neumann-Neumann Solvers for MortarCoupling of Stokes-Darcy
We consider the coupling across an interface of a fluid flowand a porous media flow. The differential equations in-volve Stokes equations in the fluid region and Darcy equa-tions in the porous region, and coupled through an inter-face with Beaver-Joseph transmission conditions. The dis-cretization consists of Stokes finite elements in the fluid re-gion, Raviart-Thomas finite elements in the porous region,and the mortar Lagrange multipliers on the interface. Weallow nonmatching meshes across the interface. Due to thesmall values of the permeability parameter of the porousmedium and the nonmatching nature of the problem andthe discretization, the resulting symmetric indefinte dis-crete system is ill conditioned and has a nontrivial saddlepoint problem structure. In this talk we discuss these is-sues and show that preconditioners based on the Finite El-ement by Tearing and Interconnecting (FETI) method aremore suitable than preconditioners based on the BalancingDomain Decomposition (BDD) method. We discuss con-dition number estimates for the preconditioners and theirdependence on the permeability and mesh size ratio acrossthe interface. Numerical experiments will be presented toconfirm the sharpness of the theoretical estimates.
Marcus SarkisWorcester Polytechnic InstituteInstituto de Matematica Pura e Aplicada (Brazil)[email protected]
Multiscale Mortar Methods for Coupling of Stokesand Darcy Flows
We discuss numerical modeling of coupled ground waterand surface water flows, based on Beavers-Joseph-Saffmaninterface conditions. The domain is decomposed into aseries of small subdomains (coarse grid) of either Stokesor Darcy type. The solution is resolved locally (on eachcoarse element) on a fine grid, allowing for non-matchinggrids across subdomain interfaces. Coarse scale mortar fi-nite elements are introduced on the interfaces to imposeweakly certain continuity conditions. By eliminating thesubdomain unknowns the global fine scale problem is re-duced to a coarse scale interface problem, which is solvedusing an iterative method. We precompute a multiscaleflux basis, solving a fixed number of fine scale subdomainproblems for each coarse scale mortar degree of freedom,on each subdomain independently. Taking linear combi-nations of the multiscale flux basis functions replaces theneed to solve any subdomain problems during the inter-face iteration. We present theoretical and numerical resultsfor local discretizations based on conforming or discontin-uous elements for Stokes flow and mixed finite elements for
Darcy flow.
Benjamin GanisUniversity of PittsburghDepartment of [email protected]
Experimental Measurement and Modeling of theOscillation of Oil-water Interface Lined with Para-magnetic Nanoparticles
We present multidisciplinary effort in developing tech-niques for determination of oil saturation in reservoir rocksusing paramagnetic nanoparticles. The efforts includepore-scale experiments with nanoparticles designed to pref-erentially absorb to the oil/water interface and that canbe detected remotely as well as numerical modeling. Themodeling assumes capillarity to be a dominant restoringforce when the oil-water interface is exposed to an oscil-lating external magnetic field, resulting in a pressure wavefrom the interface movement.
Steven BryantDepartment of Petroleum and Geosystems EngineeringInstitute for Computational and Engineering Sciencessteven [email protected]
Chun HuhUniversity of Texas at AustinDepartment for Petroleum and Geosystems [email protected]
Masa ProdanovicUniversity of Texas at AustinCenter for Petroleum and Geosystems [email protected]
MS59
Macroscale Modeling of Porous Media Systems:The Devil is in the Deviations
When applying averaging theory to obtain equations forporous media flows, it is well known that deviation termsrelated to the difference between products of averages andaverages of products require careful attention. When us-ing the thermodynamically constrained averaging theory(TCAT), one is also able to show that differences betweenaverages of quantities calculated over different domainscan be important. Here, we illustrate this effect and pro-vide some examples where this issue impacts the equationforms.
William G. GrayUniversity of North Carolina - Chapel Hill
Interfacial Area, Capillary Pressure and Saturationat the Pore-scale: Observations and Lattice Boltz-mann Simulations
We present a comparison between interfacial areas ob-tained from microtomographic images of glass bead porousmedia and lattice-Boltzmann simulations for drainage andimbibition processes. There is good agreement betweenthe measured and simulated capillary pressure curves andinterfacial areas and we find a unique capillary pressure-saturation-interfacial area relationship that is useful in de-scribing hysteretic phenomena. Interfacial area per volumeis appears to be dependent upon the dominant flow mech-anism and pore connectedness.
Effect of Pore Space Morphology on InterfacialConfigurations in Porous Media
Fluid configurations in porous media can vary significantlybetween drainage and imbibiton. Using computed x-raymicrotomography, we measure significant differences inwetting-nonwetting interfacial area for different pore spacemorphology, and for drainage and imbibition. The data isanalyzed using pore network characterization and expres-sions based on energy dissipation.
Dorthe WildenschildDepartment of Civil, Constr. and EnvironmentalEngineeringOregon State [email protected]
Masa ProdanovicUniversity of Texas at AustinCenter for Petroleum and Geosystems [email protected]
MS60
High Resolution Parallel Simulations of CO2 Stor-age in Saline Aquifers
Modeling long-term movement and risk of escape of vastamounts of CO2 will require coupled models that capturethe physical and chemical evolution of the system at theappropriate scale. The geological models are often up-scaled for numerical models to reduce the computationaltimes. However, upscaling causes loss of fine grid informa-tion and introduces numerical dispersion. Long term, high-resolution, and large-scale modeling of flow and transportof CO2 will be presented using an in-house simulator withdistributed computing and efficient numerical algorithms
and solver.
Mojdeh DelshadDepartment of Petroleum and Geosystems EngineeringThe University of Texas at [email protected]
Sunil G. ThomasCenter for subsurface modelingThe University of Texas at [email protected]
Balance of Forces During CO2 Injection in Geolog-ical Formations
The CO2 storage capacity of geological formations is ofgreat interest for the selection of potential storage sites incarbon capture and storage (CCS) projects. A detailedanalysis essentially requires a thorough understanding ofthe interaction of forces acting within the system. By defin-ing characteristic quantities for length, time, pressure andvelocity, the governing multiphase flow equations can benon-dimensionalised. This allows for the definition of phys-ically sound dimensionless numbers in conformity with themultiphase flow equations. The dimensionless numbers re-semble the ratios of acting forces like viscous, capillary andgravitational forces. An analysis of the relation of forces inreservoirs with different parameter setups allows their in-tercomparison with respect to their CO2 storage capacitypotential. To back up the analysis, a comprehensive reser-voir parameter database with more than 1200 reservoirs isanalysed and statistical characteristics are derived. Effectsof reservoir parameters like depth, temperature, absoluteand relative permeability, as well as capillary pressure areinvestigated by analytical and numerical 1D and 3D ex-periments. It is shown that dimensionless numbers can beused to qualitatively order reservoirs by their CO2 storagecapacity with respect to the forces acting in the reservoir.Moreover, it is shown that the relative permeability rela-tions together with the residual saturations have a greatinfluence on the balance of forces.
Rainer HelmigIWS, University of Stuttgart, GermanyInstitut fur [email protected]
Analytical Models of CO2 Storage at the BasinScale Including Residual Trapping and Dissolution
Carbon capture and storage will be a viable climate changemitigation technology only if several gigatonnes of carbondioxide are injected every year. In this talk, we will present
GS09 Abstracts 107
mathematical models of CO2 migration at the scale of ageologic basin that, despite their simplicity, account forthe essential flow physics during the injection and post-injection periods. In particular, we present new analyticalresults that include the combined effect of residual trappingand dissolution into the brine.
Probability Density Function Approach for Model-ing Complex Multi-phase Flow in Porous Media
We present a new methodology, which provides a link be-tween Lagrangian statistics of phase particle evolution andDarcy scale dynamics. Each particle has a state vector con-sisting of its position, velocity, fluid phase information andpossibly other properties like phase composition. The ap-proach is applied for modeling CO2 dissolution into brineand resulting non-equilibrium multiphase dynamics. It isshown that opposed to the stochastic formulation, the cor-responding Darcy based deterministic formulation is un-closed.
Darcy-Forchheimer Upscaling for Near-well FlowModeling
The objective of this work is to provide a methodologyto upscale the Darcy-Fochheimer flow equation which ac-counts for non-Darcy effects. These effects are importantin high velocity regions (near-well). Unlike Darcy trans-missibility, the upscaled Darcy-Forchheimer transmissibil-ity depends on flow rate. To overcome this difficulty aniterative local-global upscaling technique is proposed. Themain idea is to use the rate information from global coarsemodel as target rate when performing the local upscaling.
We propose a new method for flow-based coarsening thatadapts to varying requirements with respect to accuracyand simulation time. By adapting the coarse grid to high-flow regions and retaining fine-grid fluxes along each coarseedge, we achieve good accuracy of production curves andsaturation profiles. Incorporating causality of the discretefluxes as an additional measure enables the use of highlyefficient nonlinear block solvers.
The multiscale method is based on using the coarse gridfor pressure equation and fine grid for saturation equa-tions. The essential feature of the method is constructionof the basis functions by solving the one-phase stationaryequations. These basis functions take account of fine gridstructure. It is possible to construct total permeabilitytensor and to up-scale the relative permeability using thedissipative energy integral approximation. The coarse gridequation is solved using support operator method. Someresults of modeling are represented.
MPFA Mortar Multiscale Method for MultiphaseFlow in Porous Media
An iterative coupling method employing a mortar MPFAmethod and a discontinuous Galerkin method for the pres-sure and saturation equations respectively is formulatedfor multiphase flow. This approach can be viewed as amultiscale scheme for the coupled multiphysics problem.
Tim M. WildeyThe University of Texas at AustinAustin, [email protected]
108 GS09 Abstracts
MS62
Stochastic Galerkin Method for Transport Equa-tion
In this work we consider a transport equation with stochas-tic coefficients which model pollution concentration
−div(�(x, ω)) = f(x, ω) in G × ⊗, G ⊂ R�,�(x, ω) = a(x, ω)∇c(x, ω) −−→q (x, ω)c(x, ω),
c = 0 on ∂G,where ω is a random variable, a(x, ω) a diffusion coefficient,modelled by a random field, c(x, ω) is the concentration ofone substance in another substance and the source term ismodelled by f(x, ω). The flow −→q (x, ω) can be computedfrom the equation predicting the groundwater flow throughthe aquifer G. The governing equations are
−div(−→q (x, ω)) = p(x, ω) in G × ⊗, G ⊂ R�,u = g(x) on ∂G,
where −→q (x, ω) := κ(x, ω)∇u(x, ω), the conductivity co-efficient κ(x, ω), the right-hand side p(x, ω) and the so-lution u(x, ω) are random fields. The term ∇u modelsthe pressure gradient. The initial domain G is occupiedby the aquifer. After dicretisation of the deterministicand stochastic operators we apply the stochastic Galerkinmethod to solve this equation. In the conclusion we give anumerical example.
Computational Modeling of Inertia Effects atPorescale
We propose algorithms for computational upscaling offlow from porescale (microscale) to lab scale (mesoscale).We use traditional continuum Navier-Stokes solvers atporescale. Properties of flow in complex pore geometriesare averaged to derive permeability and inertia coefficients.Convergence of solutions and averaging techniques are ma-jor concerns but these can be relaxed if only mesoscopicparameters are needed. For media which are heterogeneousand anisotropic at mesoscale we discuss appropriate non-Darcy models extending Forchheimer model.
Malgorzata PeszynskaDepartment of MathematicsOregon State [email protected]
MS62
Numerical Methods for Unsaturated Flow with Dy-namic Capillary Pressure
Traditional unsaturated flow models use a capillarypressure-saturation relationship determined under staticconditions. Recently it was proposed to extend this re-lationship to include dynamic effects and in particular flowrates, which results in model equations of nonlinear, de-generate pseudo-parabolic type. We study numerical dis-cretizations of such models; we discuss the difficulties as-sociated with the degenerate pseudo-parabolic character of
the equations and multiscale heterogeneity of the medium,and the convergence of schemes.
Multi Point Flux and Mixed Finite Element Ap-proximation in Porous Media: Error Analysis andNumerical Studies
Here, in particular multi point flux and mixed finite el-ement approximations of flow and transport processes inporous media are considered. A new proof of convergencefor a multi point flux approximation scheme on triangu-lar meshes is presented. Multi point flux approximationcontrol volume methods are discretization techniques de-velopped for an accurate and reliable reservoir simulation.They offer explicit discrete fluxes, which allows a wide classof applications. Various aspects of a reliable simulationof simultaneous reactive transport processes are also ad-dressed.
A Stability Criterion for Heterogenoues DensityDriven Flows
Variable density flows can occur due to temperature dif-ferences in deep aquifers and due to salinity differencesin coastal aquifers and refuse dumps. Therefore, theirrelevance cuts across many practical applications like ex-ploitation of geothermal energy resources, oil recovery fromaquifers and remediation of contaminated sites. A typicalfeature of density dependent flow problems is that theycan become unstable (physically or numerically). Variabledensity flow problems are difficult to solve due to the non-linearities and coupling between fluid flow and solute trans-port processes. A big challenge to-date is to derive a gen-eral criterion that states whether flow is physically stable orunstable; and the optimum computational grid resolutionrequired to solve the problem without creating numerical(artificial) instabilities. We present a new stability crite-rion for heterogeneous flow based on the homogenizationtheory. Relevant numerical simulations are presented tosustain our theoretical results.
Error Estimates for the Finite Volume Method fora Copper Heap Leaching Model
This work is motivated by a combined mixed finite element(MFE) - finite volume (FV) scheme of a two phase flowmodel for the heap leaching of copper ores modeled by adegenerate parabolic equation
∂tu−∇ · (∇β(u) + F (u)) = r(u), in QT ≡ (0, T ) × Ω.(1)
Initially we have u(0) = u0 in Ω, whereas u = 0 on ∂Ω.In the above 0 < T < ∞ is fixed, Ω is a bounded do-main in Rd(d ≥ 1) with a Lipschitz continuous boundary.The function β : R → R is non-decreasing and differ-entiable. By degeneracy we mean a vanishing diffusion,namely β′(u) = 0 for some u. We prove error estimatesfor the finite volume discretization for this model. Sev-eral numerical results illustrating the performance of thealgorithm are provided.
A Multiscale Preconditioner with Applications toSolute Transport in Porous Media
The mortar mixed finite element method can be viewed asa multiscale method for the pressure equation, with recentdevelopments showing that the onstruction of a multiscalebasis can greatly reduce the computational cost. We showthat this multiscale basis does not need to be recomputedif used as a preconditioner for a Krylov method. We ap-ply this preconditioner to an IMPEC formulation for thetransport of a solute.
2.5D Finite-Difference Solution of the VTI Acous-tic Wave Equation
2.5D processing is a mathematical formalism of a 3D wavepropagation in a 2D model. Assuming that there is novariation in the earth model in the transverse direction ofthe aquisition line. Using this symmetry, the 3D prob-lem can be reduced to a repeated 2D problem. In thiswork, we apply the 2.5D formalism to describe a VTI wavepropagation in a 2D model that preserves 3D geometrical
spreading effects. This propagation is cheaper than the 3Dwave propagation.
Joerg SchleicherUniversity of CampinasDept. Applied [email protected]
PP0
An Application of Fuzzy Transform in Geophysics
Fuzzy Transform (F-transform) have been introduced asan approximation method which encompasses both classi-cal transforms as well as approximation methods studied infuzzy modelling and fuzzy control. It has been proved that,under some conditions, F-transform can remove a period-ical noise and reduce random noise significantly. In thiswork, we show how to use F-transform in order to solvewave equation with noise and how to use this theory ingeophysics.
Joerg SchleicherUniversity of CampinasDept. Applied [email protected]
PP0
Paraxial Approximations for the VTI AcousticWave Equation
In vertical transversally isotropic (VTI) media the acousticwave equation is
∂4u
∂t4−(1+2η)v2 ∂4u
∂x2∂t2= v2
v∂4u
∂z2∂t2−2ηv2v2
v∂4u
∂x2∂z2(2)
where, u(x, z, t) is the pressure field, η is a anisotropy pa-rameter, and vv and v are, respectively, the vertical qP-wave velocity and NMO velocity. Paraxial wave-equationapproximations are used to describe wave propagation witha preferred direction. In this work, we derive paraxial ap-proximations for equation (1), using higher order Pade ap-proximations, in order to reduce computation cost in mi-gration algorithms with a good accuracy.
Standard real-valued finite-difference (FD) migration can-not handle evanescent waves correctly, which can lead tonumerical instabilities in the presence of strong velocityvariations. A possible solution to these problems is thecomplex Pade approximation, which avoids problems withevanescent waves by a rotation of the branch cut of thecomplex square root. In this work, we apply this approx-imation to the acoustic wave equation for vertical trans-versely anisotropic media to derive more stable FD migra-tion for such media.
Fluid Flow, Transport and Reaction Processes,Monod and Van Genuchten Parametrization, Pa-rameter Identification, Output Least SquaresMethod, Formfree Identification, Adaptive SqpMethod
Recent challenges like bioremediation, longterm under-ground storage of reactive waste or underground carbon-dioxide sequestration require more and more complex mul-ticomponent reactive transport and fluid flow models. Al-though being demanding concerning their efficient numer-ical approximation, the decisive bottleneck in using suchmodels seems to lie in the availability of the increasingrange of reaction and hydraulic flow parameters enteringsuch a model (Monod parameters in multiplicative Monodmodels in conjunction with bioremediation and rate pa-rameters in kinetic mass action law models as well as vanGenuchten parameters for the modelling of hydraulic prop-erties ...). We address the reliable and accurate identifica-tion of such parameters from one of most controlled ex-perimental set ups, namely from soil column breakthroughcurves (letting the upscaling issue aside), but the followingmethology can also applied to field experiments. It is well-known that the (missing) sensitivity and the correlationof parameters prevent a reliable reconstruction from naivehistory matching (output least squares minimization). Fora fixed experimental setup we propose a systematic use ofthe singular values of the sensitivity matrix in the defini-tion of the error functional to design an adaptive approach
in which after each termination in a (local) minimum theerror functional is changed. Applications to the identifica-tion of Monod and van Genuchten parameters show signif-icant improvements in possible accuracy. Furthermore, afavourable form free approach with hierarchical treatmentfor the global parametrization of one- or multidimensionalnonlinearities are used. Thereby, because of the so calledcurse of dimensionality sparse grids are applied in case ofhigher dimensional problems to decrease the degree of free-dom significantly. In addition the presented methods canalso be combined with a hierarchical concept to filter outthe most sensitive parameters and identify them first. Ina further step these approaches can be used also withinexperimental design to find more appropriate sequences ofexperiments which can be taken into account into an mul-tiexperiment identification approach.
Michael BlumeUniversity of Erlangen-NurembergDepartment of [email protected]
Peter KnabnerUniversitat Erlangen-NurenbergDepartment of [email protected]
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Detection and Approximation of Fault Lines fromScattered Data
We propose a method for the detection and accurate ap-proximation of surface fault lines. First, to locate all thedata points close to fault lines, we consider a procedurebased on a local interpolation scheme involving a IDW for-mula. Then, we find further sets of points generally closerto the faults than the fault points. Finally, after applyinga nearest-neighbour searching procedure and a refinementtechnique, we outline some approximation methods for thefault lines.
A Method for Conditioning Stochastic Groundwa-ter Flow Fracture Network Models.
Calculations of groundwater flow through fracture net-works are of great importance for performance assessmentsof potential radioactive waste repositories located in hardrocks. We present a method for conditioning the transmis-sivities in a fracture network model on observed measure-ments of groundwater heads and flows to boreholes. Themethod is illustrated by application to a number of simpletest cases and to a model of a site for which field data areavailable.
A Modified Idw Algorithm for Scattered and TrackData Interpolation
We propose an accurate and efficient interpolation method,and the relative algorithm, for modelling unknown surfaces.It is based on a modified IDW formula (with least squareor radial basis function approximants as nodal functions)and exploits the particular strip structure to optimize thesearching procedure of the nearby data points. The methodworks well from both scattered data and track data, andguarantees a high parallelism.
Soil Reaction Estimates for Laterally Loaded PilesUsing Euler-Bernoulli Beam Theory and a Sim-ple Polynomial Curve Fitting Algorithm on LateralDeflection Measurements
A simple numerical procedure is proposed to estimate soilreactions along laterally loaded piles. The technique in-volves using lateral deflection measurements of the pileobtained from slope inclinometer measurements. The de-flected shape of the laterally loaded pile is estimated usinga simple polynomial curve fitting algorithm that is appliedsequentially to the slope inclinometer data with depth. Theprocedure is used to derive curvature versus depth and mo-ment versus depth. The procedure is calibrated with afull-scale experimental program involving lateral load testto three well-characterized piles where slope inclinometerdata was available as well as curvature and moment dataalong the pile depth. The proposed procedure providesvery good estimates of the measured bending moments andthe soil reactions back-calculated using more conventionalmethods such as the use of a finite difference discretizationof the pile with the soil reaction modeled with p-y curves.
Evolution of the Uralide Orogenic Wedge - a Nu-merical Simulation
The Uralide orogen exhibits a bivergent orogenic wedgewith a foreland thrust belt at the pro-wedge side and
lower crustal material originating from the pro-wedge inthe retro-wedge. In this study a Distinct Element Methodis used to investigate the tectonic evolution of the Uralidewedge and to determine the influence of the neighbourhoodof strong island arc rocks and a serpentinitic melange, iso-static compensation and basal viscous flow on the structureof the orogen.
Parameter Estimation for Water Transport in Het-erogeneous Porous Media
The solution of Richards’ equation requires the estima-tion of the parameters of a parametrisation of the hy-draulic properties (capillary pressure-saturation curve, rel-ative permeability). A typical method is multi-step out-flow, where a sample is placed on a ceramic plate and de-saturated be reducing the pressure stepwise at the lowerboundary. Parameter estimation from multi-step outflowexperiments requires usually the assumption of macro-scopic homogeneity for the sample. If the structure ofthe sample (the arrangement of materials) can be deter-mined independently (e.g. by geophysical measurements)it is possible to take it into account in the inversion pro-cess and estimate the hydraulic properties of the subscalematerials. A very robust forward model massive parallelcomputing is necessary to get a fast solution of the large3D forward problems which need to be solved.
Olaf IppischInterdisziplinary Center for Scientific ComputingUniversity of [email protected]
Information Mining Models for InterdisciplinaryResearch
The conference contribution describes the concept, imple-mentation and application of an innovative informationmining approach in an interdisciplinary research project.Target of the research project is to simulate the long-termdeformation of large mountain sides. One key challengein the interdisciplinary research project is to identify theinteraction of the different physical processes such as me-teorology, hydrology, surface flow, infiltration, sub-surfaceflow, soil mechanics, shear band deformations and hill slopemovements with given geological and topographical het-erogeneity. Key idea of this ongoing research work is tocombine the different well known data mining methods toidentify relationship between the different physical statevariables as interdisciplinary application.
The conference contribution describes the concept, imple-mentation and application of an innovative informationsystem to couple different simulation models. Applicationtarget is an interdisciplinary research project to simulatethe long-term deformation of large mountain sides. Thesystem supports the coupling of models from hydrology,multi-phase groundwater flow and soil mechanics as wellas laboratory experiments and field measurements by gen-eralized sets of objects to manage physical state variablesincluding scaling, mapping and transformation. This ap-proach allows a more flexible coupling of simulation modelsfrom different disciplines on flexible scales and approxima-tion levels.
Colloidal Influence on Fluid Dynamics in PorousMedia
A microscopic model at the pore scale for fluid flow inporous media influenced by colloid dynamics is presented.Special attention is paid to ad-/desorption processes of col-loidal particles at the solid matrix caused by the total in-teraction energy between particles and matrix as well asto thereby influenced evolving microstructure. In orderto achieve some macroscopic description of the concernedphenomena, the derived model is homogenized with themethod of asymptotic two-scale expansions.
A New Method for Assigning ,the Eigenvalues Signin Equation (Ax = λX) and (Ax = λBx)
The inertia of an n × n complex matrix A, is defined tobe an integer triple, In(A) = (π(A), ν(A), δ(A)), whereπ(A) is the number of eigenvalues of A with positive realparts,ν(A) is the number of eigenvalues with negative realparts and δ(A) is the number of eigenvalues with zero realparts. We are interested in computing the Inertia for largeunsymmetric generalized eigenproblem (A,B) for equationAϕ = λBϕ Where A and B are n × n large matrices. Forstandard eigenvalues problem let B = Identity matrix. Anobvious approach for determine Inertia of pair(A,B), is totransform this to a standard eigenproblem by inverting ei-ther A or B. In this paper we show that the eigenvalues signcan be computed by assigning the interval that includingall the eigenvalues and this method is compared by resultsin Matlab.
Nonstandard FDTD Scheme for Computation ofElastic Waves
Finite-difference method in time-domain (FDTD) is one ofthe most popular techniques used for modeling of wavepropagation in many fields. Recently, FDTD schemescalled nonstandard FDTD have been developed in com-putational electromagnetics and acoustics to efficiently re-duce the numerical dispersion and grid anisotropy. In thisstudy we propose a nonstandard FDTD scheme for elasticwave computation, which gives highly accurate solutionsboth for P and S waves.
Efficient Time-Stepping for Advection DominatedReactive transport in Heterogeneous Porous Media
Reactive transport in heterogeneous porous media is com-mon to many applications, including groundwater contam-ination and CO2 storage. Such processes are inherentlymultiscale in nature and highly non-linear,rendering accu-rate and fast numerical solutions challenging. We demon-strate that a finite volume discretization can be integratedin time with exponential integrators where a linear systemis solved exactly. Using a Krylov subspace or Leja pointsto approximate the exponential makes these methods com-petitive compared to standard integrators.
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