SPE-173262-MS

A Unified Finite Difference Model for The Simulation of Transient Flow inNaturally Fractured Carbonate Karst Reservoirs

Jie He, and John E. Killough, Texas A&M University; Mohamed M. Fadlelmula F., and Michael Fraim, TexasA&M University at Qatar

Copyright 2015, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE Reservoir Simulation Symposium held in Houston, Texas, USA, 23–25 February 2015.

This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contentsof the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflectany position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the writtenconsent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations maynot be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract

The presence of cavities connected by fracture networks at multiple levels make the simulation of fluidflow in naturally fractured carbonate karst reservoirs a challenging problem. The challenge arises inproperly treating the Darcy and non-Darcy flow in the different areas of fractured medium. In this paper,we present a single-phase transient flow model which is based on the Stokes-Brinkman equation and ageneralized material balance equation. The generalized material balance equation proves to be exact inboth cavities and porous media, and the Stokes-Brinkman equation mathematically combines Darcy andStokes flow, thus allowing a seamless transition between the cavities and porous media with only minoramounts of perturbation introduced into the solutions. Finite differences are implemented for the solutionof the proposed transient flow model. This solution method provides a smooth transition from standardmultiple-porosity/permeability reservoir simulators and moreover, it is physically more straightforward,mathematically easier to derive and implement, and more apt to generalization from two-dimensional tothree-dimensional cases than alternative techniques.

Application of the derived transient flow model is shown by examples of three fine-scale 2-Dgeological models. The first two models, although simple, provide verification of the proposed transientflow model. The third example presents a more complex and realistic geological model derived frommultiple-point statistics simulation technique with the second model used as the training image. Theresults of the third model form the foundation for future study of multi-phase and 3-D reservoir cases.

IntroductionNaturally fractured carbonate karst reservoirs are commonly found all over the world (Dabbouk et al.2002; Khvatova et al. 2012; Peng et al. 2009). The modeling and numerical simulation of such reservoirsare a challenging problem because of the presence of vugs (cavities from small to medium size) and caves(large size cavities) which are usually interconnected by natural fractures. The vugs and caves are foundto be of sizes ranging from one centimeter to several meters (Huang et al. 2010; Peng et al. 2009; Zhanget al. 2004). The presence of these cavities and fractures at multiple levels makes the geological structureof such carbonate reservoirs highly heterogeneous and complex, and the coexistence of free flow in the

cavities and fractures (Tuncay et al. 1998) and Darcy flow in the porous media makes the coupled solutionof fluid transport in these reservoirs very difficult.

Various continuum approaches have been developed for the modeling of fluid flow in naturallyfractured carbonate karst reservoirs. Methods based on the multiple-continuum concept model fracturesand vugs as porous media with high permeability values (Abdassah and Ershaghi 1986; Bai et al. 1993;Camacho Velazquez et al. 2005; Kang et al. 2006; Wu et al. 2007; Wu et al. 2006). These methods arealso widely applied in the simulation of unconventional reservoirs (AlfiYanCaoAnWang and Killough2014; Yan et al. 2013). Another continuum approach combines the porous media, fractures, and cavitiestogether as a single effective porous medium (Oda 1986; Sitharam et al. 2001; Wu 2013), and use effectiveporosity and permeability to approximate the fluid storage and transport behavior in the fractured vuggyreservoirs. The continuum approaches have simple formulations and are computationally efficient, but thedifficulty in estimating the cavity permeability or effective porosity/permeability values limit theirapplications in fractured carbonate karst reservoirs (Yao et al. 2010).

Another approach for modeling fluid flow in fractured carbonate karst reservoirs is based on theDarcy-Stokes system (Arbogast and Brunson 2007; Arbogast and Lehr 2006; Peng et al. 2009; Yao et al.2010). The Darcy-Stokes system consists of free flow in cavities and fractures characterized by Stokesequation and fluid flow in porous media by Darcy’s Law. The coupled Darcy and Stokes equations aremore difficult to solve, and additional boundary conditions (Beavers and Joseph 1967; Jäger and Mikelic2000; Saffman 1971) need to be specified at the interface between cavities and porous media to guaranteecontinuity of mass and momentum across the interface. The specification of such boundary conditionsrequires a fairly detailed knowledge of the location and extent of the interface which in turn makes theapplication of the Darcy-Stokes approach complicated (PopovEfendiev et al. 2009).

The Stokes-Brinkman equation (Brinkman 1949) provides a unified approach which avoids some of theproblems encountered in the Darcy-Stokes system. This approach is unified in the sense that it uses asingle equation rather than coupled ones to describe fluid flow in the entire fractured carbonate karstreservoir. The Stokes-Brinkman equation can be shown to be equivalent to the Darcy and Stokes equations(Brinkman 1949; PopovBi et al. 2007) once appropriate parameters are selected in the corresponding flowregions, respectively. Therefore, explicit modeling of the interface is avoided, allowing a seamlesstransition between the vugs and porous media (Gulbransen et al. 2009; PopovBi et al. 2007).

Many papers are devoted to the numerical formulation and solution of the Stokes-Brinkman equation(Bi et al. 2009; Gulbransen et al. 2009, 2010; Ingeborg et al. 2010; Laptev 2003; PopovBi et al. 2007;PopovEfendiev et al. 2009; PopovQin et al. 2007; PopovQin et al. 2009; Qin et al. 2010; Qin et al. 2011),but to the best of our knowledge, all of them consider only steady-state flow, and are applied towards 2-Dstreamline-based type of simulations. In this paper, we propose a single-phase transient flow model forfluid transport in naturally fractured carbonate karst reservoirs. This transient flow model consists of theStokes-Brinkman equation, and a generalized material balance equation which is unsteady state and exactin the entire reservoir. Finite differences are implemented for the solution of the proposed transient flowmodel, which provides a smooth transition from standard multiple-porosity/permeability reservoir simu-lators. This solution method is physically more straightforward, easier to derive and implement, andproves more apt to generalization from 2-D to 3-D cases than alternative techniques. The derived transientflow model is applied to three examples of fine-scale 2-D geological models. The first two geologicalmodels are simple and used to verify the proposed transient flow model. The third application presents amore complex and realistic geological model derived from multiple-point statistics (MPS) simulationtechnique with the second model being used as the training image. The results of this model form thefoundation for future study of multi-phase and 3-D reservoir cases.

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Mathematical ModelsThe solution of fluid flow problems, whether in porous media or not, mostly starts from the formulationof various conservation laws. A brief review of the material and momentum balance equations is givenfor fluid flow in fractured carbonate karst reservoirs. These equations form the basis for all the approachesdescribed in the previous section.

Flow in Porous MediaSingle-phase fluid flow in porous media is completely described by the material balance equation

(1)

and Darcy’s law (Hubbert 1956)

(2)

where � and k are the porosity value and permeability tensor of the porous media, � and � are fluiddensity and viscosity, u� is the Darcy velocity vector, t is time, p is pressure, is mass injection (�) /production (-) rate per unit volume, g is earth gravity, and Z is vertical depth. As a convention, k isassumed to be diagonal. It is worth pointing out that in Equation 2 we have implicitly assumed no-slipflow boundaries. If gas slippage effect is significant, then Klinkenberg factor (Klinkenberg 1941) orKnudsen’s number based methods (AlfiYanCaoAnWangHe et al. 2014; Beskok and Karniadakis 1999)can be used to correct the permeability tensor k in Equation 2.

If we further assume constant fluid density and steady state flow (in the Eulerian specification of theflow field, �/�t � 0), then Equations 1 and 2 reduce to

(3)

(4)

where is the volumetric injection (�) / production (�) rate per unit volume, and g � g Z is thegravity vector. Multi-continuum and effective-continuum methods are primarily based on Equation 1 andvariations of Equation 4.

Flow in Vugs and FracturesSingle-phase laminar flow in nonporous regions (cativites and fractures) is governed by the continuityequation

(5)

and Navier-Stokes equation

(6)

where v is true velocity vector, and all the other variables are as previously defined. In Equation 6 wehave applied Stokes hypothesis (Gad-el-Hak 1995), and assumed that the fluid is Newtonian with constantdensity and viscosity. Detailed derivations and extended discussions of Equations 5 and 6 can be foundin the classical textbook (Bird et al. 2006).

If we further assume incompressible flow (D�/Dt � 0) and steady state in Lagrangian’s specificationof the flow field (Dv/Dt � 0), then Equations 5 and 6 reduce to

(7)

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(8)

Equation 8 is the Stokes equation. Coupling Equations 3 and 4 with Equations 7 and 8 yields theDarcy-Stokes system, the solution of which requires additional boundary conditions (Beavers and Joseph1967; Jäger and Mikelic 2000; Saffman 1971) at the interface between nonporous regions and porousmedia.

Stokes-Brinkman EquationObserving that Equations 3 and 7 have the same form

(9)

Brinkman (Brinkman 1949) combines Equations 4 and 8 into the Stokes-Brinkman equation

(10)

where u is fluid velocity, �* is called the effective fluid viscosity, and the other variables are aspreviously defined. Equations 9 and 10 constitute the Stokes-Brinkman system, which unifies fluid flowin nonporous regions and porous media in the sense that Equation 10 can be made mathematicallyequivalent to both Equations 4 and 8 by appropriate assignments of k and �* values to different flowregions: it reduces to Equation 4 if we set �* � 0 in porous media, and approximates Equation 8 if wechoose large k values (ideally let k ¡ �) and set �* � � in the nonporous regions. In practice, insteadof assigning different �* values in different flow regions, we set �* equal to � throughout the reservoir(Brinkman 1949; Gulbransen et al. 2009; PopovBi et al. 2007), which only introduces a small perturbationinto the solutions compared with setting �* � 0 in porous media (PopovBi et al. 2007), since �*�u isseveral orders of magnitude smaller than the other terms on the left-hand side of Equation 10 in typicalporous media (Gulbransen et al. 2009).

The Stokes-Brinkman system has very limited applications since the material balance equation(Equation 9) assumes steady-state flow in the Eulerian specification. However, neither Darcy’s law norStokes equation makes such assumption, so the Stokes-Brinkman equation itself is applicable to bothsteady-state and unsteady-state flow. Therefore, to obtain a transient flow model, we only need to replaceEquation 9 with a generalized material balance equation

(11)

Equation 11 is identical to Equation 1 in the porous media, and reduces to Equation 5 once we set �� 1 in the nonporous regions. In other words, Equation 11 is exact in the whole reservoir given properporosity values in different flow regions. The Stokes-Brinkman equation (Equation 10) and the general-ized material balance equation (Equation 11) constitute our proposed single-phase transient flow model.

Numerical FormulationA reservoir simulator has been developed to solve the proposed transient flow model, which partiallyutilizes the structure of FTSim, a simulator based on the TOUGH� family of codes (Moridis and Freeman2014). In this section, we present the numerical formulation underlying our reservoir simulator, anddiscuss some challenges that arise in the computational solution of the transient flow model.

Finite Difference DiscretizationThe standard blocked-centered finite difference method is employed in the discretization of our transientflow model with Cartesian coordinates and uniform grids. The generalized material balance equation(Equation 11) is a scalar equation, and for a grid block with indices (i,j, k), it can be discretized into

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(12)

where ux, uy, uz are velocities in the corresponding directions, m is the mass injection (-) / production(�) rate, V � �x�y�z, Ax � �y�z, Ay � �x�z, Az � �x�y are the volume and face areas of the cuboidgrid blocks, and �x, �y, �z are the spacing in the x-, y-, z- directions, respectively. The subscript n denotesthe nth time step, and the subscripts i�1/2, j �1/2, k � 1/2 are the indices of the variables in the x-, y-,z- directions, respectively. In these subscripts, �1/2 means that the velocities are defined at the blockinterfaces rather than block centers. It is worth noting that we should always define pressure at blockcenters and velocities at block interfaces in order to maintain mass conservation in the discretizedequations and stability of the numerical solutions. Moreover, each variable in Equation 12 has threesubscript indices corresponding to the 3-D Cartesian coordinates (e.g. ux,i�1/2 is actually ux,i�1/2,j,k), butwe have omitted any of the three subscripts i,j,k wherever possible to make the appearance of the equationmore concise. We will adopt this convention in all discretized equations throughout this paper.

The Stokes-Brinkman equation is a vector equation, and can be expanded into three scalar equationsin the Cartesian coordinates:

(13)

(14)

(15)

Discretization and proper rearrangement of Equations 13~15 for a grid block with indices (i,j, k) yields

(16)

(17)

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(18)

where kx, ky, kz are the permeability values in the corresponding directions, and all the other variablesare as previously defined. At each block interface, the permeability value is estimated by harmonicaverage of those of the neighboring two grid blocks, while � and � take values from the upstream side,i.e. the grid block that has a higher fluid potential than the other. Such estimations apply to all of Equations12 and 16~18, and assure mass conservation across the grid blocks.

Linear System and Solver SelectionUnlike Darcy’s equation (Equation 4), we cannot obtain an explicit expression of the velocity vector u asa function of the pressure p from the Stokes-Brinkman equation (Equation 10). Therefore, we have tosolve both p and u explicitly. If we divide the reservoir into Nx, Ny, and Nz grid blocks in the correspondingdirections, respectively, and impose no-flow boundary conditions on all the boundaries, then we have4NxNyNz — NxNy — NXNZ — NyNz unknown variables and the same number of discretized equations, soa unique solution is guaranteed. The unknown variable vector X is arranged as follows:

(19)

where p, ux, uy, and uz are column vectors listed in a consistent manner, i.e. all of them go throughthe indices i,j, k in the same order. The discretized equations can then be recast into a system of nonlinearresidual functions, and solved with the Newton’s method (Nocedal and Wright 2006) by constructing thenumerical Jacobian matrix and iteratively solve for pressure and velocities at each time step. We prefernumerical Jacobian to the analytical one because the numerical Jacobian can be written in a very genericway, and is easy to code and parallelize, which is crucial in large-scale simulations.

Figure 1 shows that the Jacobian matrix for our transient flow model has a sparse, multi-diagonal, andsymmetric pattern. However, the Jacobian matrix is indeed nonsymmetric and very ill-conditioned, so wecan no longer use the conjugate gradient (CG) method (Hestenes and Stiefel 1952) as the matrix equationsolver. This problem can be circumvented in steady-state systems by applying the Gaussian eliminationmethod, since the matrix equation is only solved once. In transient flow problems, however, the matrixequation needs to be solved at each time step and hundreds of times in total, so direct methods likeGaussian elimination are generally not considered because they are highly time-consuming. Fortunately,in our case, the Jacobian matrix remains positive definite, so the generalized minimal residual (GMRES)method (Sadd 2003) can be employed to solve it. GMRES generally converges more slowly than CG, andconsumes more computer memory. In our simulator, a restarted GMRES solver (Ju and Burkardt 2012)in the compressed row storage (CRS) form is implemented to reduce memory cost, and LU factorizationis used as a preconditioner to speed up the convergence.

Model SpecificationsLiquid n-octane is chosen as the single-phase fluid in our simulation. The density and viscosity of liquidn-octane are computed from the following Yaws’ equations (Yaws 1998):

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(20)

(21)

where a, b, Tc, n, A, B, C and D are coefficientsthat pertain to the the specific liquid, c0 is the oilcompressibility, and pref is the reference pressure. Itis worth noting that the original Yaws’ liquid den-sity equation has the form � � a· b�(1�T/Tc)n, i.e.Yaws assume that liquid density is only a functionof temperature. We remedy this by assuming aconstant oil compressibility c0 in the pressure rangeof our simulation and that the original Yaws’ equa-tion are for liquid densities at the reference pressure.These assumptions lead to the multiplication of theterm ec0(p�pref) on the right-hand-side of Equation20. The values of these coefficients, the propertiesof the porous medium, and the grid parameters arelisted in Table 1, where k and � are the permeabilityand porosity of the porous medium, Nx, Ny, Nz arethe number of grid blocks in the x-, y-, z- directions,and �x, �y, �z are the spacing in the correspondingdirections, respectively. Two wells are placed in thereservoir to form a quarter five-spot pattern. Thelocation (indices of the grid block where the well islocated) and mass rate of each well are shown inTable 2. The values in Tables 1 and 2 are used asstandard inputs in all the applications.

Figure 1—Pattern of Jacobian Matrix for 5�5�5 Grid Blocks

Table 1—Standard Inputs

Property Value Units

Oil Density Coefficients

a 2.2807�10�1 kg/m3

b 2.5476�10�1 -

Tc 2.9568�102 K

n 2.6940�10�1 -

co 1.0�10�8 Pa�1

pref 1.0�105 Pa

Oil Viscosity Coefficients

A -5.9245 -

B 8.8809�102 K

C 1.2955�10�2 K�1

D -1.3596�10�5 K�2

Medium Properties

k 1.5�10�14m2

� 1.5�10�1 -

Grid Parameters

Nx 100 -

Ny 100 -

Nz 1 -

�x 2.0�101 m

�y 2.0�101 m

�z 2.0�101 m

Table 2—Well Properties

Well Number Location (i,j,k) Rate (kg/s)

1 (2,2,1) -3.0

2 (99,99,1) 3.0

Figure 2—Pressure Distribution at 100 days, Homogeneous PorousMedium

SPE-173262-MS 7

ApplicationsThe transient flow model is applied to three examples of fine-scale 2-D geological models. The firstexample is a homogeneous porous medium which is used to test the simulator in the Darcy flow region.The second example is a synthetic model constructed from the one studied by Gulbransen and Bi (Bi etal. 2009; Gulbransen et al. 2009). These two geological models provide verification of the proposedtransient flow. The third application presents a more complex and realistic geological model derived fromMultiple-point Statistics (MPS) simulation technique with the second model used as the training image.

Homogeneous Porous MediumThe transient flow simulator is first applied to a reservoir composed of a homogeneous porous mediumwith properties specified in Table 1. The pressure distribution at 100 days is plotted in Figure 2 and showsclear radial flow patterns around the two wells. The results show that our simulator works well with pureDarcy flow. Note that the natural logarithm of the pressure is plotted in Figure 2 rather than the pressureitself, and we will do so in all the examples.

Synthetic ExampleA 2-D synthetic geological model is binarized from the one studied by Gulbransen and Bi (Bi et al. 2009;Gulbransen et al. 2009) using a developed Visual Basic code TiConverter. The simulation results for thissynthetic model are plotted in Figures 3(b)–(d), which show that the presence of fractures and cavities(labeled in red as “karsts” in Figure 3(a)) significantly alters the shape of the isobars. By comparing Figure

Figure 3—Numerical Results for the Synthetic Example: (a) Reservoir Structure, (b)-(d) Pressure Distribution at 20 days, 50 days, and 100 days,respectively

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3(b) with 3(c), and 3(c) with 3(d), we see thatwhenever a pressure change reaches one end of twointerconnected cavities, it quickly spreads throughthe interconnecting fracture to the other end, whileit takes much longer for the pressure change toextend from the cavities and fractures into the po-rous medium. Consequently, the shapes of the iso-bars are largely distorted from those of radial flow(Figure 2), and point towards the directions inwhich the fractures elongate, indicating significantchange of flow patterns in the reservoir.

Derived ExampleThe geological model in Figure 3(a) is treated as atraining image to generate a more realistic 2-Dmodel with the MPS simulation technique. For thispurpose the Single Normal Equation Simulation(SNESIM) algorithm of Stanford GeostatisticalModeling Software (SGeMS) is used (Remy et al.

Table 3—The parameters used in SNESIM algorithm for the DerivedExample

Parameter Value

Seed 211175

# of Categories 2

Target Marginal Distribution 0.546 0.454

# of Nodes in Search Template 60

Search Template Geometry (rangesand angles)

21, 21, 21 and 0, 0, 0

Hard Data (property) Facies

Min # of Replicates 1

Servosystem Factor 0

Re-simulation Threshold -1

Re-simulation Iteration # 1

# of Multigrids 3

Previously Simulated Nodes 4

Template Expansion Isotropic

Training image The image in Figure 3(a)

Figure 4—Numerical Results for the Derived Example: (a) Reservoir Structure, (b)-(d) Pressure Distribution at 20 days, 50 days, and 100 days,respectively

SPE-173262-MS 9

2009). The generated model is conditioned to the facies data (binary indicators for matrix and karstshaving proportions of 0.546 and 0.454, respectively) of Stanford VI dataset (Castro 2007). The condi-tioned model is shown in Figure 4(a), and The SNESIM parameters used to generate this 2-D model arelisted in Table 3. The cavities in Figure 4(a) are less interconnected than those in Figure 3(a), and moreunevenly distributed in the reservoir. Accordingly, by comparing Figures 4(b)–(d) with Figures 3(b)–(d),respectively, we can see that the pressure change propagates more slowly in the derived geological modelthan it does in the synthetic one. Moreover, the pressure change propagates much faster on the upper leftcorner than on the lower right corner of the derived model, which is clearly shown in Figure 4(d). It isalso noticeable that the shapes of the isobars are quite similar to those of radial flow (Figure 2), whichsuggests that the presence of the cavitites alone does not substantially alter the flow patterns in thereservoir.

ConclusionsIn this paper, we have presented a single-phase transient flow model consisting of the Stokes-Brinkmanequation and a generalized material balance equation. This transient Stokes-Brinkman model properlydescribes fluid flow in both porous media and nonporous regions given appropriate selection of perme-ability and porosity values. The transient flow model has been applied to three examples, of which the firsttwo are used to validate the model. The numerical results of the second and third examples show thatpresence of cavities interconnected by natural fractures significantly changes the pattern of fluid transportin the naturally fractured carbonate karst reservoir, while the cavities alone don’t have such effects. Theseresults form the basis for future study of multi-phase and 3-D flow cases.

The computational efficiency becomes a significant problem in the applications of our transient flowmodel. The discretization of the transient Stokes-Brinkman model results in almost four times as manyunknown variables as there are in a Darcy flow model with the same number of grid blocks, since we haveto solve for the velocities explicitly. On the other hand, the nonsymmetry of the Jacobian matrixdisqualifies the CG method as a matrix solver so that we have to look for less effective substitutes likeGMRES. Therefore, the solution of our transient flow model will quickly become computationallyintractable as the number of grid blocks increases, and parallelization is inavoidable in problems involvedwith large-scale systems.

AcknowledgmentThis publication was made possible by the NPRP award [NPRP 6-485-2-201] from the Qatar NationalResearch Fund (a member of The Qatar Foundation). The statements made herein are solely theresponsibility of the authors.

Nomenclature

Physical quantitiesp � pressure, PaT � temperature, K� � porosity� � fluid density, kg/m3

� � fluid viscosity, Pa·s�* � effective viscosity of the fluid, Pa·su � velocity, m/s

� mass injection (�) / production (-) rate per unit volume, kg/(m3-s)m � mass injection (-) / production (�) rate, kg/sq˙ � volumetric injection (�) / production (-) rate per unit volume, s�1

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Z � depth, mg � earth gravity, m/s2

t � time, s�t � time step, sc � compressibility, Pa�1

Reservoir and GridV � volume of the grid block, m3

A � face area of the grid block perpendicular to a specified direction, m2

�x � spacing in the x-direction, m�y � spacing in the y-direction, m�z � spacing in the z-direction, mN � number of grid blocks in a specified directionVectors and matricesu � velocity vector, m/su¯ � Darcy velocity vector, m/sv � true velocity vector, m/sk � permeability tensor, m2

g � gravity vector m/s2

Superscripts and Subscriptsx, y, z � Cartesian coordinates/directionsi, j, k � indices of the grid blocko � oilref � referencen � number of time steps

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