Sequential Upscaling Method

Transport in Porous Media 46: 179–212, 2002.c© 2002 Kluwer Academic Publishers. Printed in the Netherlands. 179

Sequential Upscaling Method

KOICHI SUZUKI and THOMAS A. HEWETTJapan Oil Development Co., Ltd. e-mail: [email protected] Center for Reservoir Forecasting, Stanford University USA.e-mail: [email protected]

Abstract. This paper presents a new method for scaling up multiphase flow properties which prop-erly accounts for boundary conditions on the upscaled cell. The scale-up proposed does not requirethe simulation of a complete finely-gridded model, instead it calls for assumptions allowing thecalculation of the boundary conditions related to each block being scaled up. To upscale a coarseblock, we have to assume or determine the proper boundary conditions for that coarse block. Todate, most scale-up methods have been based on the assumption of steady-state flow associated withuniform fractional flows over all the boundaries of the coarse block. However, such an assumptionis not strictly valid when we consider heterogeneities. The concept of injection tubes is introduced:these are hypothetical streamtubes connecting the injection wellbore to all inlet faces of the fine gridcells constituting the block to be scaled up. Injection tubes allow the capturing of the fine-scale flowbehavior of a finely-gridded model at the inlet face of the coarse block without having to simulatethat fine grid. We describe how to scale up an entire finely-gridded model sequentially using injectiontubes to determine the boundary conditions for two-phase flow. This new scale-up method is able tocapture almost exactly the fine-scale two-phase flow behavior, such as saturation distributions, insideeach isolated coarse-grid domain. Further, the resultant scaled-up relative permeabilities reproduceaccurately the spatially-averaged performance of the finely-gridded model throughout the simulationperiod. The method has been shown to be applicable not only to viscous-dominated flow but also toflow affected by gravity for reasonable viscous-to-gravity ratios.

Key words: upscaling, relative permeability.

Nomenclature

fw fractional flow of water.g acceleration constant due to gravity.k absolute permeability.kηrp relative permeability of phase p in direction η.M0 endpoint mobility ratio.Ng gravity number.nxc number of coarsely-gridded blocks.nxcf number of gridblocks per coarse-grid domain in the x-direction.nzf number of gridblocks per coarse-grid domain in the z-direction.P pressure.

PI,Kc,img imaginary capillary pressure.

PV I pore volumes injected upstream.q̄pI+(1/2),K volumetric flow rate from coarse block (I,K) to coarse block (I + 1,K) of phase p.

180 KOICHI SUZUKI AND THOMAS A. HEWETT

Qi,kp,tb

(t) cumulative fluid outflow volume of phase p out of injection tube passingblock (i, k) up to time t .

Sw water saturation.S∗

w reduced water saturation, (Sw − Swir)/(1 − Sor − Swir).

S̄I,Kw average water saturation of coarse block (I,K).Swir irreducible water saturation.Sor residual oil saturation.t time.tD pore volumes injected into an entire model.tmax total simulation time from start of water injection.T transmissibility.Tηgpi,k transmissibility between the face η of fine block (i, k) to the global producer.T̄I+(1/2),K transmissibility between block (I,K) and block (I + 1, K).

Ti,kw linear-flow-based well-to-block transmissibility.

Vi,kpm,tb

movable pore volume of injection tube passing fine block (i, k).V̄I,K pore volume of coarse block (I,K).

λ̄I+(1/2),Kp interblock mobility of phase p between coarse block (I, K)

and coarse block (I + 1, K).µp viscosity of phase p.�gp potential of the global producer.

�̄pI,K potential of phase p in coarse block (I,K).

1. Introduction

1.1. MOTIVATION TO DEVELOP A NEW UPSCALING METHOD

Recently, there has been growing interest in the development of a rigorous scale-upprocess. Finely gridded geostatistical models are made up of more gridblocks thancan be handled by present day finite difference reservoir simulators. Of particularinterest are coarser-scale relative permeabilities which can reproduce the averageperformance of the local domains of these finely gridded models with a minimumloss of fine-scale variabilities when coarsely gridded models are considered.

Although considerable research has been devoted to the development of such ascale-up process, rather less attention has been paid to the development of a scale-up process using isolated gridblocks exposed to the proper boundary conditions.A process of upscaling geostatistical gridblocks to a reservoir simulation gridblockshould be able to account for the effect of the flow-path length from an injection po-sition to each coarse gridblock, since resultant coarser-scale relative permeabilitiesare known to be flow-path dependent or boundary-dependent.

1.2. LITERATURE REVIEW

Scale-up is a classic research subject which results in pseudofunction gener-ation. When introduced, pseudofunctions were aimed at reducing the dimensionof simulation models. Hearn [14] presented a method of analytically calculating

SEQUENTIAL UPSCALING METHOD 181

pseudorelative permeabilities to represent a layered two-dimensional (2D) modelby a one-dimensional (1D) model. Coats et. al. [6] introduced the concept of ver-tical equilibrium to calculate the performance of a 3D reservoir with a 2D model.Jacks et. al. [18] proposed the concept of dynamic pseudofunctions in order toreduce the dimension of a model from 3D to 2D.

Ekrann and Dale defined and distinguished two major approaches in averagingflow properties [12]: dynamic-pseudos approaches and effective-properties ap-proaches. In addition to these approaches, there exists a multistep upscaling pro-cess.

1.2.1. Dynamic Pseudos Approach

Dynamic pseudos are generated based on the simulation results of multiphase flowin the whole or a representative portion of a finely-gridded model. Hence, no scalerestrictions apply to this type of approach [12]. The resultant coarser model issupposed to replicate the spatially averaged performance of the representative por-tion under unsteady-state multiphase flow. However, it is beyond the ability of theexisting dynamic-pseudos approaches to isolate and upscale gridblocks away fromwells since the proper boundary conditions are unknown prior to the multiphaseflow simulation of each coarse-grid domain.

Several averaging methods have been proposed to generate dynamic pseudo-functions [13]. Kyte and Berry [21] used individual flow rate and mobility weightedpotentials described later. Stone [27] and Hewett and Berhens [16] developed meth-ods based on average total mobility. Hewett and Archer [15] proposed that poten-tials should be taken at the pressure nodes of coarse gridblocks.

The Kyte and Berry [21] pseudofunctions are probably the most widely useddynamic pseudofunctions in the industry. They presented a method of generat-ing pseudorelative permeabilities using mobility-weighted average phase-pressuresalong the central column of each coarse gridblock. Knowing the phase flow rateacross a coarse grid boundary, Darcy’s law is applied to obtain pseudorelative per-meabilities with the mobility-weighted average phase-pressure difference betweenthe central columns of two neighboring coarse gridblocks. With their method, wecan implicitly incorporate the mobility variation between the pressure nodes of twoneighboring coarse gridblocks.

Stone [27] argued that pseudorelative permeabilities should be calculated withthe transmissibility-weighted average total mobilities along the exit-face grid-blocks of each coarse gridblock. Such averaged total mobilities and the flowrate-averaged fractional flow yield Stone’s pseudorelative permeabilities. AlthoughStone claimed that his method is rigorous, it is obvious that his method cannot ac-count for the interblock mobility variation, especially when a sharp front is formedduring fluid displacement.

A critical review of dynamic pseudorelative permeabilities can be found inBarker and Thiebeau [2], who highlighted practical problems which may occurin the course of dynamic pseudofunction generation. Barker and Thiebeau also


pointed out a weak point of each of the dynamic pseudofunction approaches. Atypical problem such as negative pseudorelative permeabilities is caused by thenet flow in the opposite direction of the pressure gradient. Such problems will bediscussed more in detail in Section 6.

1.2.2. Effective Properties Approach

Effective properties are defined to be process independent or boundary independentand universally valid when a coarse-grid domain is much smaller than the lengthof the large-scale heterogeneities [12]. Thus, there exists a scale restriction to thistype of approach.

It is known that most effective properties approaches assume the extreme flowregime such as a viscous limit and a capillary limit. Otherwise a fine-scale simula-tion would be needed for flow regimes affected by multiple forces, i.e, both viscousand capillary forces. The extreme flow regimes above require a coarse-grid domainto be exposed to steady-state while actual reservoirs are not strictly steady-state[25].

For the viscous limit, the steady-state assumption above requires the averagefractional flow of water to be constant normal to the main direction of flow [11].As far as this condition is satisfied, the fractional flow of water can be varied alongthe inlet boundary of an upscaled cell as adopted in the work by Durlofsky [9].

A special assumption is the assumption of uniform-fractional-flow boundaries.Such an assumption results in recovering the rock curves as effective relative per-meabilities when a single set of rock curves are assumed in an entire detailedmodel [9, 11]. This means that no heterogeneity can be incorporated in the re-covered effective relative permeabilities (the rock curves). Accordingly, a detailedmodel should be coarsened with a relatively small cell to satisfy the assumption ofuniform-fractional-flow boundary by incorporating heterogeneities with upscaledabsolute permeabilities only. For example, Durlofsky et al. [10] showed a methodof upscaling absolute permeabilities of non-uniform coarse blocks defined basedon the local flowrate distribution where those blocks were exposed to periodicboundary conditions [8].

For the capillary limit, a given capillary pressure fully determines the saturationdistribution inside an upscaled cell. Under such saturation distribution, a steady-state pressure solution is obtained with the corresponding mobility distribution ofa phase assuming no flow boundaries or periodic boundaries [25]. Based on thesteady-state pressure solution above, a phase permeability can be calculated at agiven average saturation.

Primarily due to the steady-state assumption described above, most effectiveproperties approaches [5, 20, 26] have no control on numerical diffusion, unlikemethods such as that of Kyte and Berry [21]. Li et al. [23] incorporated the shock-front velocity into pseudo generation, but the global boundary conditions are notproperly accounted for.


Some researchers showed a method of calculating effective relative permeabilit-ies based on multiphase flow simulations on a limited region including an upscaledcell. In the work by Alabert and Corre [1], they first ran a two-phase flow simulationin a small region. Then, at several instants during the two-phase flow simulation,they took a snapshot of the fine-scale saturation distribution. On this saturationdistribution, they calculated the fluid phase permeabilities (the product of relativeand absolute permeabilities). Then, they ran a steady-state simulation to obtain theeffective permeability for each phase at the instant they considered. In this simula-tion, they considered all the directions and assumed constant pressure boundariesto cause flow in a certain direction and no flow boundaries in the rest of the direc-tions. Pickup and Sorbie [24] showed a similar method described above, but theyincluded the concept of phase permeability tensors. This kind of approach adoptsa limited region of an entire model, but the length effect caused by the globalboundary conditions is not properly accounted for.

Durlofsky [9] showed an effective properties approach which does not requirean assumption of uniform fractional flow. He varied the inlet conditions randomlyand ran a simulation. Then, he collected a sufficient amount of the correspond-ing effective relative permeabilities of an upscaled cell. Finally, a least-square fitwas done against those effective relative permeabilities in conjunction with satura-tion and higher moments (variance or covariance) of relevant quantities such aspressure, velocity, saturation, etc.

1.2.3. Multistep Pseudofunction Generation

As a method intermediate between dynamic pseudos approaches and effectiveproperties approaches, a multistep pseudofunction generation process was intro-duced by Lasseter et al. [22] and was discussed extensively by Kossack et al.[19]. Multistep pseudofunction generation is a process of generating final-steppseudofunctions for the largest-scale gridblocks by successively generating thepseudofunctions of one scale and substituting these pseudofunctions into the nextlarger-scale gridblocks.

In order to synthesize the boundary conditions of the upscaled cell, this processrequires an entrance section made up of several homogeneous gridblocks upstreamof the fine gridblocks being scaled up. Lasseter et al. [22] and Kossack et al. [19]found that the resultant pseudofunctions changed with the length of the entrancesection and became length-insensitive for entrance lengths longer than a specificlength. They determined the pseudofunctions at this length-insensitive limit.

Ekrann and Dale [12] and Hewett et al. [17] derived coarse-grid pseudofunc-tions for the Buckley-Leverett flow [4]. These authors found that the coarse-gridpseudofunctions were length dependent. Further, Hewett et al. [17] showed thatthe coarse-grid pseudofunctions converged to an asymptotic form at several tens ofcoarse blocks downstream of the injection boundary. Thus, it can be saidthat the multistep pseudofunction generation techniques described above adoptasymptotic pseudorelative permeabilities instead of pseudorelative permeabilities


properly distance-adjusted. It is questionable that such a multistep process canappropriately consider the proper boundary conditions.

1.3. OBJECTIVE OF THIS WORK

In this paper, we present a new scale-up process in which a local domain is isol-ated from an entire model and then the corresponding coarser-scale pseudorelat-ive permeabilities are generated with appropriate boundary conditions. This newscale-up process consists of two novel techniques: the sequential scale-up methodand the concept of injection tubes. These two techniques will be described indetail in the subsequent sections. The new process is tested against oil–water two-phase viscous-dominated flow in a heterogeneous 2D permeability field. We willshow that the new process reproduces fine-scale water saturation distributions andthe spatially-averaged performance of the finely-gridded model. Gravity-affectedflow is also considered. The quantities above were reproduced well for a reason-able range of gravity numbers. The scale-up method developed is available for2D-to-1D upscaling and 2D-to-2D upscaling.

2. Main Feature of the Sequential Upscaling Method

In this section, we show the sequential method using isolated coarse-grid domainsassociated with injection tubes at the inlet face. The sequential method is a methodof scaling up a whole finely-gridded model sequentially with each coarse-griddomain being scaled up to its next downstream domain. It is widely known thatdynamic pseudofunctions are flow-path dependent or boundary-condition sensi-tive. Thus, it is a difficult problem to determine what boundary conditions to applyon the inlet face of isolated blocks. To overcome this, we adopt the concept ofinjection tubes which enables us to capture the fine-scale flow behavior of theglobal finely-gridded model at the inlet face of the isolated coarse-grid domainbeing scaled up. The injection tubes will be described later. A detailed descriptionof sequential scale-up and the concept of injection tubes for scaling-up 2D to 1Dhas been previously presented [28, 29, 30].

2.1. GENERAL ASSUMPTIONS

For the moment, we consider a 2D-to-1D scale-up problem using a two-dimensional finely-gridded model, as shown in Figure 1, with an injection wellat the left side, a production well at the right side, and no flow boundaries at thetop and bottom sides. This makes the determination of upstream and downstreamblocks quite simple since the main direction of flow is from the injection wellto the production well. We assume the oil–water two-phase flow of incompressiblefluids in which each fluid has a constant viscosity. Initially, water is distributed with


Figure 1. Schematic of the sequential scale-up method.

the irreducible saturation. We also assume that the two-dimensional finely-griddedmodel above has uniform properties except for permeabilities and porosities.

2.2. COARSE-GRID ISOLATION

In the sequential scale-up method, we isolate a coarse-grid domain for scale-upand its next coarse-grid domain downstream from a global finely-gridded modelas shown in Figure 1. We define these two coarse-grid domains as an isolatedcoarse-grid connection or an ICGC as abbreviated. In this approach, we can ob-tain the representative potentials of the upstream and downstream domains in onesimulation run. This allows us to generate scaled-up relative permeabilities of theupstream domain of each connection.

The isolation described above requires us to synthesize the inlet and outletboundary conditions of an ICGC model. For this purpose, the sequential scale-


up process adopts injection tubes for the inlet boundary and a synthetic producerfor the outlet boundary as shown in Figure 1.

2.2.1. Synthetic Inlet Boundary Conditions

Based on the results of a two-phase flow simulation in the ICGC model, we modelthe fine-scale performance at the exit face of the upstream coarse-grid domainusing as many injection tubes as all the fine gridblocks on the exit face. We as-sign variable pore volumes, transmissibilities, and saturation functions to theseinjection tubes such that each injection tube can capture the fine-scale inlet-facebehavior of the next coarse-grid domain downstream as if we included all thefine gridblocks upstream. As we proceed to scale up the next ICGC downstream,we update injection tubes using the simulation results of this isolated connectionmodel.

For instance, we wish to sequentially upscale a two-dimensional finely-griddedmodel to a one-dimensional coarse model made up of nc coarse blocks with eachcoarse-grid domain upscaled from nxcf ×nzf fine gridblocks. When we were at thestep of upscaling coarse-grid domain ic, the use of injection tubes would allow usto replace all the fine gridblocks upstream [(ic − 1)× (nxcf × nzf)] by nzf injectiontubes.

2.2.2. Synthetic Outlet Boundary Conditions

As shown in Figure 1, we assume a fully-penetrating synthetic producer at thedownstream edge of the ICGC model. We preserve the properties of fine gridblockson the downstream edge including saturation functions of the rock curves exceptfor grid-to-well transmissibilities. We can use two different kinds of transmis-sibilities between the synthetic producer described above and each of all the finegridblocks on the downstream edge: global-model-based well transmissibilitiesand gridblock-based well transmissibilities.

2.2.2.1. Global-model-based well transmissibilities. When a single-phase-flowsolution of a global finely-gridded model is available, we can determine the trans-missibilities so that the ICGC model can preserve the flowrate variation along thedownstream edge observed in the global model. We can calculate a global-model-based well transmissibility for each exit-face gridblock using the single-phase-flowsolution described above, i.e., the inter-fine-block flowrate across the downstreamedge, the gridblock pressure, and the producing pressure of the global producer.In other words, use of global-model-based well transmissibilities allows us to im-plicitly account for heterogeneities contained in fine gridblocks downstream of theICGC model. The detailed description of a well model is given in Ref. [28].

2.2.2.2. Gridblock-based well transmissibilities. When a single-phase-flow solu-tion of a global finely-gridded model is unavailable, we use the permeabilities of


the exit-face gridblocks to calculate the well transmissibilities for the syntheticproducer.

3. Sequential Method for 2D-to-2D Upscaling

In this section, we extend the 2D-to-1D sequential upscaling method described inthe previous section to be capable of a 2D-to-2D upscaling problem. The conceptof injection tubes is continued to be adopted so that the appropriate inlet boundaryconditions are provided to an upscaled cell. A synthetic producer is also assumedat the outlet boundary of an upscaled cell.

3.1. CONSIDERATION OF MULTIDIMENSIONAL UPSCALING

3.1.1. Additional Technical Aspects Beyond 2D-to-1D Upscaling

Modern reservoir simulation practice requires us to scale-up a detailed multidimen-sional model to a more coarsely-gridded model in the same dimensions. However,most dynamic pseudos approaches were developed to reduce the simulation-modeldimensionality. Thus, it is important for us to discuss additional technical aspectswhich are likely to appear in a multidimensional upscaling problem.

Multidimensional upscaling introduces difficulty in modeling flow behavior atthe coarse block boundaries in the transverse flow direction. We considered thesequential 2D-to-1D upscaling method in the previous section. In such a dimensionreducing method, we only have to upscale the longitudinal flow behavior and wecan ignore the transverse flow. However, multidimensional upscaling forces us totake into account the transverse flow in addition to the longitudinal flow.

3.1.2. Known Problems in Multidimensional Upscaling

The difficulty in multidimensional upscaling can be found not only in the mul-tiphase flow condition but also in the single-phase flow condition when we calcu-late interblock flow properties. The most undesirable difficulty is the inconsistencybetween the interblock potential gradient and the net flow of a fluid. In this section,such difficulty is referred to as a negative transmissibility problem in the single-phase flow condition and a negative pseudorelative permeability problem in themultiphase flow condition.

3.1.2.1. Negative transmissibility problem. A negative transmissibility causes adata check error with most commercial simulators. Mathematically, any simulation-based transmissibilities, irrespective of their signs, positive or negative, reproduceall the coarse-scale interblock flowrates and potentials of a given single-phase in-compressible fluid system. Simulation-based transmissibilities are referred to astransmissibilities derived by applying Darcy’s law to coarse-scale interblock flow,i.e.,

Tsim = q̄

λs

�L

(�u −�d), (1)


where Tsim is the simulation-based transmissibility; q̄ is the interblock flowrate; λsis the single-phase fluid mobility; �u − �d is the interblock potential differencebetween the upstream block and the downstream block where the potentials arerepresented by those of the pressure nodes; �L is the inter-node distance.

To ensure a system of positive transmissibilities, each of the coarse block bound-aries is allowed to be intersected by a certain streamline only once [32]. This meansthat all the inter-fine-block flow on each of the coarse block boundaries should beuni-directional.

In this section, we establish a system of positive transmissibilities as follows atthe stage prior to two-phase upscaling:

1. Run a single-phase flow simulation over an entire finely-gridded model.2. Determine a coarse grid so that each of coarse block boundaries is a unidirec-

tional flow boundary as much as possible.3. Choose a pressure node in each coarse block so that the interblock flow is

consistent with the interblock potential gradient. A group of all such pressurenodes are referred to as a consistent set of pressure nodes.

4. Calculate interblock transmissibilities using Equation (1).

3.1.2.2. Negative pseudorelative permeability problem. Difficulty in multi di-mensional upscaling for multiphase flow was extensively discussed in the reviewof pseudofunctions by Barker and Thiebeau [2].

They pointed out that negative pseudorelative permeabilities occur when the netflow of a phase is in the opposite direction to the interblock pressure gradient.

Even when non-negative transmissibilities are ensured in a single-phase coarsesystem, it cannot be guaranteed that a negative pseudorelative permeability prob-lem never occurs in upscaling the corresponding multiphase coarse system. Sincefluid displacement causes the mobility distribution to change with time, the non-negativeness condition established in the single-phase flow may no longer hold inthe multiphase flow. In addition, counter flow of oil and water at the coarse blockboundaries can occur due to gravity forces. Such counter flow may cause a negativepseudorelative permeability problem.

Thus, single-phase based gridding may not be able to completely prevent un-desirable problems like negative pseudorelative permeabilities under multiphaseflow with a low mobility ratio and gravity-affected flow. In this section, we willobtain positive coarser-scale relative permeabilities as follows:

1. Incorporate (imaginary) capillary pressures to make up negative fluid potentialdifferences if observed.

2. Use a pseudoization technique like Stone’s pseudos in which the exit-faceproperties represent interblock flow properties.


3.2. SEQUENTIAL METHOD FOR 2D-TO-2D UPSCALING

We developed three approaches of sequential 2D-to-2D upscaling methods. It isassumed that the longitudinal direction flow occurs from the left to the right, orin the x+ direction, by placing an injector on the leftmost edge and a produceron the rightmost edge of the model. These approaches isolate different extentsof a simulation domain out of an entire detailed model to sequentially generatepseudofunctions. The common features of the approaches are given below:

– the concept of injection tubes explained in the previous section is adopted;– all the resultant coarser scale relative permeabilities are made directional, i.e.,

each coarse block can have different sets of coarser scale relative permeabilitiesfor the x+, z+, and z− directions;

– non-uniform coarse gridding is available.

The first approach, the isolated coarse grid connection (ICGC 2D) approach,uses two neighboring coarse columns to scale up coarse blocks which verticallysubdivide the upstream coarse column. The second approach, the isolated coarsegrid domain (ICGD 2D) approach, uses only a single coarse column being up-scaled into vertically aligned coarse blocks. The third approach uses only the coarseblock being upscaled. We abbreviate the third approach by SEQ 2D.

All three of the approaches assume that a single-phase pressure solutionof an entire model is available. This single-phase pressure solution allows us todetermine all coarse block boundaries, pressure nodes, and thus interblock andwell-to-block transmissibilities prior to two-phase upscaling. This process will bedescribed in detail later.

3.2.1. ICGC 2D Approach

Figure 2 shows a schematic of the ICGC 2D approach. This approach uses twoneighboring coarse columns similar to the 2D-to-1D version of the ICGC approachdescribed in the previous section. Each coarse column is subdivided into nzc coarseblocks. Injection tubes are assumed in one fine-block column upstream of the inletface of an ICGC 2D model. A synthetic producer is attached at the exit face of thedownstream coarse column.

Fine blocks on the exit face of the upstream coarse column provide the informa-tion necessary to generate the properties of injection tubes for the next down-stream ICGC 2D model. The relevant equations can be found in earlier references[28–30].

The potential of each of these coarse blocks is represented by a potentialobserved at the corresponding predetermined pressure node. For coarse block(I,K), the corresponding potential is given as �̄p

I,K(t) at time t for p = o,w.The average water saturation, S̄I,Kw (t), is obtained by pore-volume-weighted aver-aging of all the fine blocks contained in the coarse block. Coarse-scale interblock


Figure 2. Schematic of the sequential scale-up method using two neighboring coarse columns(ICGC 2D).

flowrates are obtained by summing up a fine-scale interblock flowrate across thepredetermined coarse block boundaries, i.e.,

q̄p

I+(1/2),K(t) =k2∑

k=k1

qp

i2+(1/2),k(t) for p = o,w, (2)

for the x-direction and

q̄p

I,K+(1/2)(t) =i2∑

i=i1qp

i,k2+(1/2)(t) for p = o,w, (3)

for the z-direction where coordinates i1, i2, k1, and k2 denote the boundary coordi-nate of the coarse block in each direction.

There are two options available to generate interblock coarser scale relativepermeabilities as shown in Figure 3.

3.2.1.1. Dynamic-based option. The first option adopts the Hewett and Archermethod [15] not only for the longitudinal flow but also for the transverse flow,since interblock potential differences are available in both directions. This is calleda dynamic-based option.Derivation of dynamic pseudos. For each coarse block, different sets of pseudo-functions are assigned to longitudinal flow across face x+ and transverse flow


Figure 3. Schematic of two options of calculating interblock pseudofunctions in the ICGC 2Dapproach.

across face z+ and face z−. The longitudinal directional pseudorelative permeab-ility, k̄xrp, is given by

k̄xrp(S̄I,Kw (t)) = µpq̄

p

I+(1/2),K(t)

T̄I+(1/2),K(�̄p

I,K(t)− �̄p

I+1,K(t))for p = o,w. (4)

The transverse directional pseudorelative permeability, k̄z+rp , for flow from coarseblock (I,K) to coarse block (I,K + 1) is given by

k̄z+rp (S̄I,Kw (t)) = µpq̄

p

I,K+(1/2)(t)

T̄I,K+(1/2)(�̄p

I,K(t)− �̄p

I,K+1(t))for p = o,w. (5)

For flow from coarse block (I,K) to coarse block (I,K − 1), the transversedirectional pseudorelative permeability, k̄z−rp , is given by

k̄z−rp (S̄I,Kw (t)) = µpq̄

p

I,K−(1/2)(t)

T̄I,K−(1/2)(�̄p

I,K−1(t)− �̄p

I,K(t))for p = o,w. (6)

Practical solution to negative pseudorelative permeabilities. At this point, weshould recall that dynamic pseudos using the potential difference as shown inEquation (4)–(6) can cause unwanted results such as the negative pseudorelativepermeabilities pointed out by Barker and Thiebeau [2]. This usually occurs in the


transverse flow since the transverse flow is likely to be affected by a fluctuationsuch as the mobility change and is likely to cause the inconsistency of the net flowwith interblock potential gradient. We show a practical solution when we encounternegative pseudorelative permeabilities.

– Oil phase pseudorelative permeabilities: When the oil phase causes negativepseudorelative permeabilities, we can skip the corresponding saturation inter-val if it is short compared to the final saturation increment from the initialcondition. However, when the interval is long, we go back to the coarseningstage, since most simulators work on an oil-phase potential basis.

– Water phase pseudorelative permeabilities: For flow from block (I,K − 1) to(I,K) at time t , we assume a situation when the oil phase results in reasonablepseudos but the water phase shows negative pseudorelative permeabilities, i.e.,

q̄wI,K−(1/2)(t) > 0 (7)

and

�̄wI,K−1(t)− �̄w

I,K(t) < 0. (8)

To overcome this inconsistency between the water flowrate and the potentialdifference shown above, we use capillary pressures to make the correspondingwater potential difference to be positive and reasonable. For a saturation in-terval causing water-phase negative pseudorelative permeabilities, we assumeimaginary capillary pressures, P̄ I,K−1

c,img (S̄I,K−1w (t)), so that the modified water-

phase pressure difference, ��̄w,modI,K−(1/2)(t), is made equivalent to the potential

difference observed in the single-phase flow simulation, ��̄oI,K−(1/2)(t = 0),

i.e.,

��̄w,modI,K−(1/2)(t) = ��̄o

I,K−(1/2)(t = 0), (9)

where

��̄w,modI,K−(1/2)(t) = �̄w

I,K−1(t)− P̄I,K−1c,img (S̄I,K−1

w (t))− �̄wI,K(t). (10)

Substituting Equation (10) into Equation (9) yields

P̄I,K−1c,img (S̄I,K−1

w (t)) = �̄wI,K−1(t)− �̄w

I,K(t)−��̄oI,K−(1/2)(t = 0). (11)

Thus, the imaginary capillary pressures become negative due to the inequalityshown in Equation (8).

3.2.1.2. Exit-face-based option. The second option adopts the Hewett and Archermethod [15] only for the longitudinal flow while coarse-scale relative permeabilit-ies for the transverse flow are generated based on the exit-face fine-blockproperties. This is called an exit-face-based option. Exit-face-based pseudorel-


ative permeabilities are given by fine-scale transmissibility weighted averagingfine-scale relative permeabilities, i.e.,

k̄z+rp (S̄I,Kw (t)) =

∑i2i=i1 Ti,k2+(1/2)krp(Si,k2

w (t))∑i2

i=i1 Ti,k2+(1/2)for p = o,w, (12)

k̄z−rp (S̄I,Kw (t)) =

∑i2i=i1 Ti,k1−(1/2)krp(Si,k1

w (t))∑i2

i=i1 Ti,k1−(1/2)for p = o,w, (13)

These exit-face-based pseudofunctions look similar to Stone’s pseudos [27].Stone’s pseudos are based on transmissibility-weighted total mobilities and simu-lation-based average fractional flows. However, we excluded use of simulation-based average fractional flows, since average fractional flows can be negative in thetransverse flow. Thus, we average fine-scale relative permeabilities along the trans-verse boundary to obtain non-negative relative permeabilities irrespective of theflow direction. It is impossible for this kind of exit-face-based pseudo to explicitlyaccount for the mobility variation between the pressure nodes as the Kyte and Berrymethod [21] does. As shown in Ref. [28], Stone’s pseudos are unable to reproduceanalytical pressure drops but they are able to capture frontal advance. Later, wewill show that exit-face-based pseudos can work in upscaling a heterogeneousmedium.

3.2.2. ICGD 2D Approach

Figure 4 shows the schematic of the ICGD 2D approach. This approach uses asingle coarse-column to generate pseudofunctions of isolated blocks vertically sub-dividing the coarse-column. Similar to the ICGC 2D approach, injection tubes arelocated at the inlet face of the coarse-column being scaled-up. On the other hand, asynthetic producer is assumed along one fine-column downstream of the exit faceof the coarse-column being scaled-up. We generate the properties of injection tubesfor the next downstream ICGD 2D model based on fine-scale interblock flowratesand potentials at fine gridblocks on the exit face.

Use of only a single coarse-column makes the interblock potential differencesunavailable in the x-direction. Thus, we need to use a method like Stone’s pseudos[27] which requires no downstream potentials as shown in Figure 5.

Contrary to the transverse flow, negative fractional flows are rarely seen acrossthe coarse block boundaries in the longitudinal flow. Thus, we adopt Stone’spseudos [27] for the x-direction pseudorelative permeabilities given by

k̄xrp(S̄I,Kw (t)) = f̄ I+(1/2),Kp (t)λ̄I+(1/2),K

T(t) for p = o,w, (14)

where

f̄ I+(1/2),Kp (t) = q̄p

I+(1/2),K(t)∑

p′=o,w q̄p′I+(1/2),K(t)

for p = o,w, (15)


Figure 4. Schematic of the sequential scale-up method using a single coarse column(ICGD 2D).

Figure 5. Location of exit face blocks to calculate exit-face-based pseudos in the ICGD 2Dapproach.

and

λ̄I+(1/2),KT (t) =

∑k2k=k1

Ti2+(1/2),kλT (Si2,kw (t))∑k2

i=k1Ti2+(1/2),k

. (16)

For the transverse direction, exit-face-based pseudofunctions are adopted as de-scribed earlier in Equations (12) and (13).


3.2.3. SEQ 2D approach

3.2.3.1. Main feature. The SEQ 2D approach uses a single isolated coarse block,not a coarse-column consisting of multiple coarse blocks. This requires us to pro-vide proper boundary conditions at the inlet and exit faces of such a coarse blocknot only in the x-direction but also in the z-direction. Thus, we have to model allthe boundaries of a coarse block being scaled up using either injection tubes or asynthetic producer depending on the flow direction across each boundary.

We desire to consistently define the upstream–downstream relationship amongall the coarse blocks so that we can apply the concept of sequential scale-up. Wehave to avoid a mixed boundary condition in which an influx fine-block face andan outflux fine-block face coexist on the same coarse block boundary. Hence, weset up all the coarse block boundaries such that all of the exit face fine blocks alongeach boundary exhibit uni-directional flow in a fine scale as much as possible. Inother words, the summation of fine-scale flowrates in a certain direction along theboundary has to largely exceed the summation in the opposite direction so thateither an influx boundary condition or an outflux boundary condition yields a goodapproximation of a mixed boundary condition.

This approach consists of the following three major steps:

1. Determination of the coarse grid: We define the coarse grid over an entiremodel based on a single phase flow simulation result such that the assump-tion of uni-directional flow boundaries described above is satisfied as much aspossible. We will explain this step later using test data.

2. Ordering coarse blocks according to the flow directions. The coarser-scale in-terblock flow directions allow us to order all the coarse blocks from upstream todownstream. Then, all the coarse blocks are numbered in this order. Figure 6shows a schematic of this ordering with example interblock flow directions.It should be noted that each of the outflux boundaries of a certain coarseblock acts as an influx boundary to the next downstream block. Hence, weconstruct injection tubes based on the information obtained at the exit-face

Figure 6. Schematic of numbered coarse blocks in the order from upstream to downstream.


Figure 7. Schematic of the sequential scale-up method using an isolated coarse block(SEQ 2D).

fine blocks along such outflux boundaries. Those injection tubes are added tothe corresponding inlet face of the next downstream block. This allows us tosequentially isolate and upscale all the coarse blocks as if each of the SEQ 2Dmodels included all the fine blocks located upstream.

3. Sequential scale-up and construction of injection tubes: We sequentially visitall the coarse blocks according to the order defined above. Figure 7 shows aschematic of this step. At each coarse block, we construct a SEQ 2D model,conduct two-phase flow simulation, scale up the relative permeabilities of thecoarse block, and construct injection tubes for the next downstream blocks. Wewill provide a detailed explanation of this step from the most upstream coarseblock to the subsequent downstream blocks.

3.2.3.2. Detailed explanation for the most upstream coarse block. The most up-stream coarse block is surrounded by only outflux boundaries in terms of inter-coarse-block flow.

Construction of a SEQ 2D model

1. Local boundary condition: The most upstream coarse block is supposed tocontain a global injector at the inlet face in the x-direction. The remaining facesof such a block must act as outflux boundaries. This means that we incorporateno injection tubes in such a block. It should be noted that we have to considerinjection tubes for the subsequent downstream blocks.


2. Model domain: The corresponding SEQ 2D model consists of two domains:the internal domain, 'in and the external domain, 'ex:

(a) The internal domain, 'in, includes all of the fine blocks inside the coarseblock being scaled up.

(b) The external domain, 'ex, consists of a column and/or a row of fine blocksat one block downstream of each boundary. This domain is further sub-divided into two subdomains below:

(i) The influx-external subdomain, 'inex, is made up of a row or column

of fine gridblocks on the influx boundaries. This subdomain containsinjection tubes.

(ii) The outflux-external subdomain, 'outex , is made up of a row or column

of fine gridblocks on the outflux boundaries. This subdomain containsa synthetic producer.

(c) The outermost fine blocks in the internal domain facing the outflux-externaldomain are grouped into the boundary, (ob. Such fine blocks are used toconstruct the properties of injection tubes for the next downstream block.

3. Global injector: A global injector is connected to each injection fine blocklocated on the inlet face of the block. We assign a linear-flow-based well-to-block transmissibility, T i,kw , given by

T i,kw = 2ki,kx �y�z

�x, (17)

and the rock curves to the completion data of a global injector.4. Construction of a synthetic producer: We model a synthetic producer such that

a SEQ 2D model preserves the flowrate variability along the outflux boundar-ies obtained in an entire detailed model. Thus, to each of all the fine blocks(i, k) contained in the outflux-external domain described earlier, we assign ablock-to-producer transmissibility, based on a single-phase pressure solutionof an entire model. We calculate global-based block-to-producer transmissib-ilities, T xgpi,k for face x+, T z−gpi,k for face z−, and T z+gpi,k for face z+. We considerface x+ only for the x-direction, since a cross-sectional model being assumedrarely sees backward flow in the x direction. Using a directional flowrate acrossthe block boundary and a potential difference between the fine block and aglobal producer, those transmissibilities above are expressed by, i.e.,

– when face x+ acts as an outflux boundary,

T xgpi,k = µoqoi+(1/2),k

k0ro(�

oi,k −�gp)

, (18)

– when face z− acts as an outflux boundary,


T z−gpi,k = µoqoi,k−(1/2)

k0ro(�

oi,k −�gp)

, (19)

– and when face z+ acts as an outflux boundary,

T z+gpi,k = µoqoi,k+(1/2)

k0ro(�

oi,k −�gp)

. (20)

It should be noted that the fine-scale interblock flowrates used in Equa-tions (18)–(20) are outward-positive against the exit face of the outflux-externaldomain. Although we will define a coarse block boundary formed by fineblocks with unidirectional flow as much as possible, such a boundary maycontain a minor number of fine blocks with flow in the opposite direction. Onsuch occasions, the interblock flowrates in Equations (18)–(20) are made allpositive assuming that flow contributed by the fine blocks with flow in theopposite direction is small.

Generation of coarser-scale interblock relative permeabilities. Following two-phase fine-scale simulation using a SEQ 2D model, we generate coarse-scale inter-block relative permeabilities. The SEQ 2D approach allows no interblock potentialdifferences to be available from the simulation above. Thus, we adopt Stone’spseudos [27] to generate the x-direction and the z-direction pseudorelative per-meabilities as illustrated in Figure 8. The x-direction pseudorelative permeabilitiesare given by Equations (14)–(16). The z-direction pseudorelative permeabilities forflow from coarse block (I,K) to coarse block (I,K + 1) are given by

k̄z+rp (S̄I,Kw (t)) = f̄ I,K+(1/2)

p (t)λ̄I,K+(1/2)T

(t) for p = o,w, (21)

Figure 8. Location of exit face blocks facing outflux boundaries to calculate Stone’s pseudosin the SEQ 2D approach.


where

f̄ I,K+(1/2)p (t) = q̄

p

I,K+(1/2)(t)∑

p′=o,w q̄p′I,K+(1/2)(t)

for p = o,w, (22)

and

λ̄I,K+(1/2)T (t) =

∑i2i=i1 Ti,k2+(1/2)λT (Si,k2

w (t))∑i2

i=i1 Ti,k2+(1/2). (23)

For flow from coarse block (I,K) to coarse block (I,K − 1), the z-directionpseudorelative permeabilities are given by

k̄z−rp (S̄I,Kw (t)) = f̄ I,K−(1/2)

p (t)λ̄I,K−(1/2)T (t) for p = o,w, (24)

where

f̄ I,K−(1/2)p (t) = q̄

p

I,K−(1/2)(t)∑

p′=o,w q̄p′I,K−(1/2)(t)

for p = o,w, (25)

and

λ̄I,K−(1/2)T (t) =

∑i2i=i1 Ti,k1−(1/2)λT (Si,k1

w (t))∑i2

i=i1 Ti,k1−(1/2). (26)

Construction of injection tubes for the next downstream block. Basically, the cal-culation of injection tube properties described in earlier references [28–30] is ap-plicable to the SEQ 2D approach except for the determination of pore volumesupstream. We use the inter-fine-block flowrates and fine-block potentials obtainedat the outermost fine blocks in the internal domain if those blocks face the outflux-external subdomain.

In the 2D-to-1D sequential scale-up problems described in the previous section,we assumed that the global boundary condition propagates only in the x-direction.This made the total pore volumes of injection tubes to be simply a summation ofthe pore volumes of all the fine blocks from a global injection face to the exit faceof a coarse column being scaled up.

However, the SEQ 2D approach propagates the global boundary conditions ina two-dimensional manner as described earlier. Accordingly, we have to determinethe pore volumes upstream of each of the coarse block boundaries in the x andz-directions.

For the most upstream block, or block (I,K), the x+ face and the z+ faceare assumed to be outflux boundaries while the x− face acts as a global injectionface. The z− face is assumed to be located on the global no-flow boundary. Thepore volumes relevant to injection tubes for the next downstream block, PV II,K ,is given by the pore volume of the internal domain, V̄I,K ,

PV II,K = V̄I,K . (27)


As described in earlier references [28–30] we determine the pore volume of eachinjection tube, V i,k

pm,tbfor (i, k) ∈ (ob, proportionally to the cumulative oil outflux

from the internal domain to the external domain through each fine block (i, k) onthe outflux boundary, Qi,k

o,tb(tmax),

V i,kpm,tb

= Qi,ko,tb(tmax)

∑(i′,k′)∈(ob

Qi′,k′o,tb (tmax)

PV II,K. (28)

Summing up Vi,kpm,tb

along each of the outflux boundaries yields the total pore

volumes of injection tubes, V̄ I+(1/2),Kpm,tb

, and V̄ I,K−(1/2)pm,tb

, connecting a global injectorand such boundaries, i.e.,

V̄I+(1/2),Kpm,tb

=k2∑

k=k1

Vi2,kpm,tb

, (29)

and

V̄I,K+(1/2)pm,tb

=i2∑

i=i1Vi,k2pm,tb

, (30)

respectively.

3.2.3.3. The subsequent downstream blocks. All the steps described above aresequentially applied to the subsequent downstream blocks. But, modifications be-low are required.

1. Instead of a global injector, injection tubes generated in the previous upstreamblocks are added to one fine block upstream of the influx boundaries of thecoarse block being scaled up.

2. For coarse blocks containing a global producer, a global producer replaces asynthetic producer. The corresponding SEQ 2D models of such coarse blocksexactly reproduce any completion intervals of the global producer.

3. The determination of the pore volumes upstream should consider the total porevolumes of injection tubes associated with the influx boundaries. For instance,let us assume coarse block (I,K) with the influx boundaries at the x- andz-boundaries and the outflux boundaries at the x+ and z+ boundaries. In thisexample, the pore volumes upstream, PV II,K , include the total pore volume ofthe internal domain and the total pore volumes of the injection tubes associatedwith those influx boundaries described above, i.e.,

PV II,K = V̄I,K + V̄I−(1/2),Kpm,tb

+ V̄I,K−(1/2)pm,tb

. (31)

For the blocks further downstream, the total pore volumes upstream givenabove is decomposed to each of the outflux boundaries as shown by Equa-tions (28)–(30).


3.3. TEST RESULTS OF SEQUENTIAL 2D-TO-2D UPSCALING

We tested the three approaches of the sequential 2D-to-2D upscaling method de-scribed earlier against three different cross-sectional permeability fields. Permeab-ility Fields 1, 2, and 3, were all generated with the sequential Gaussian simulationprogram sgsim of Gslib [7]. In this paper, we describe the results for permeabilityfield 1. The results for permeability fields 2 and 3 can be found elsewhere [28].

Permeability field 1 is made up of 275 × 21 fine grid blocks. We scaled upthis permeability field to a 10 × 2 coarse model where viscous-dominated flowand gravity-affected flow were considered. We determined the irregular Cartesiancoarse grid so that each of the coarse block boundaries was as much as a uni-directional flow boundary in order to avoid a negative transmissibility problem.

The permeabilities range from 1 mD to 10 000 mD with the standard deviationof logarithmic permeability [σ (log10 k)] at 1.3. This permeability field presents amoderately layered nature. The x-direction semivariogram indicates the relativelylong correlation length of more than 120 grid units, or the relative correlationlength, lx , of 0.44. On the other hand, the z-direction correlation length is madezero so that extremes like high permeability streaks distribute independently ofeach other in the z-direction.

Throughout the test examples being presented, we made the following generalassumptions.

1. uniform porosity;2. a single set of quadratic rock curves;3. uniform size of a fine gridblock (25 ft × 1 ft);4. negligence of capillary forces.

Changes in other properties like an end point mobility ratio and a gravity numberwill be stated when they appear.

3.3.1. Results of 2D-to-2D upscaling for viscous-dominated flow

We upscaled permeability field 1 to a 10 × 2 coarse-grid system using the threedifferent sequential scale-up approaches: the ICGC 2D, ICGD 2D, and SEQ 2D.An end point mobility ratio, M0, of 10 was assumed.

3.3.1.1. Reproduction of the global performance. Figure 9 compares the per-formance of the resultant upscaled coarse models with that of the finely griddedmodel. This figure also includes the performance of the corresponding rock-curvecoarse model which has the same grid-system except for the rock curves insteadof the upscaled relative permeabilities. On this figure, we compare three quantitiesindicative of the accurate reproduction of the global performance as functions ofthe water volumes injected against the total movable pore volume, tD:

1. average fractional flow at the global producer, f̄w;2. cumulative oil recovery relative to the initial movable oil-in-place, NpD;3. interwell pressure drop, �Pi−p.


Figure 9. Global performance under viscous-dominated flow with M0 = 10 calculated withthe resultant coarse model of non-uniform 10 × 2 coarse blocks with ICGC 2D, ICGD 2D,and SEQ 2D approaches compared to the finely-gridded model of Permeability Field 1 madeup of 275 × 21 fine blocks where the results with a rock-curves coarse model are also shown.

It can be seen that the two options of the ICGC 2D approach (dynamic-basedand exit-face-based) and the ICGD 2D approach almost perfectly reproduced thefine-model performance. The SEQ 2D approach also shows a good reproductioncomparable to the two approaches above, except for a small separation observed onthe f̄w curve during a low fractional flow period with f̄w less than 0.2. It should benoted that the SEQ 2D approach indicates a breakthrough time sufficiently close tothat of the finely-gridded model. Thus, we can say that these scale-up approachesenable us to reproduce the global performance of the finely-gridded model.


Figure 10. Coarse block S̄∗w under viscous-dominated flow with M0 = 10 for the top layer

calculated with the resultant coarse model of non-uniform 10×2 coarse blocks with ICGC 2D,ICGD 2D, and SEQ 2D compared to the finely-gridded model of permeability field 1 madeup of 275 × 21 fine blocks where the results with a rock-curves coarse model are also shown.

On the other hand, the rock-curve coarse model shows a clear separation in allthree quantities we compared, despite the same coarse-grid system including inter-block and well-to-grid transmissibilities. This means that upscaling trans-missibilities is insufficient to calculate two-phase flow behavior with the coarsermodel.

3.3.1.2. Reproduction of the local saturation history. In addition to the globalperformance described above, we also evaluated the local saturation history, orthe temporal change of the average reduced water saturation of each coarse block,as a measure of the accurate reproduction of the spatially-averaged oil and waterflowrates obtained using the finely-gridded model. Figures 10 and 11 comparethe average reduced water saturations as a function of tD described earlier amongseveral models: the finely-gridded model, the upscaled models based on the threedifferent approaches, and the rock-curve coarse model.

Figure 10 contains graphs for coarse blocks in the top layer while Figure 11contains graphs for coarse blocks in the bottom layer. These figures show that


Figure 11. Coarse block S̄∗w under viscous-dominated flow withM0 = 10 for the bottom layer

calculated with the resultant coarse model of non-uniform 10×2 coarse blocks with ICGC 2D,ICGD 2D, and SEQ 2D compared to the finely-gridded model of permeability field 1 madeup of 275 × 21 fine blocks where the results with a rock-curves coarse model are also shown.

the upscaled models almost exactly reproduced the local saturation history of thefinely-gridded model.

On the other hand, the rock-curve coarse model shows a discrepancy in mostcoarse blocks. Especially, coarse blocks in the upstream portion, (1, 1), (2, 1),(1, 2), and (2, 2) show a large separation. Blocks (10, 1) and (10, 2) forming themost downstream coarse column seem to reproduce the average saturation profileto a great extent, but we can observe a visible delay in the initial rise compared tothe finely-gridded model.

3.3.2. Results of 2D-to-2D Upscaling for Gravity-Affected Flow

Following the viscous-dominated flow condition described earlier, we upscaledpermeability field 1 to the same 10 × 2 coarse-grid system under gravity-affectedflow with a gravity number, Ng, of 1.8 as defined in Batycky [3]. In this up-scaling, we applied the ICGC 2D and the ICGD 2D approaches excluding theSEQ 2D approach for the moment. This is because the ICGC 2D and the ICGD 2D


Figure 12. Global performance under gravity-affected flow of Ng = 1.8 calculated with theresultant coarse model of non-uniform 10 × 2 coarse blocks with ICGC 2D and ICGD 2Dcompared to the finely-gridded model of permeability field 1 made up of 275 × 21 fine blockswhere the results with a rock-curves coarse model are also shown.

approaches can consider the counter flow of oil and water in the direction ofgravitation, while the SEQ 2D requires uni-directional flow along the boundary.

3.3.2.1. Reproduction of the global performance. Figure 12 compares the globalperformance using the same three quantities as used in the viscous-dominated flowexample, f̄w, NpD, and �Pi−p in the finely-gridded model, the upscaled coarsemodels, and the rock-curve coarse model. The ICGC 2D and ICGD 2D approachesreproduced the global performance of the finely-gridded model quite well. Thebreakthrough time was perfectly reproduced by the upscaled models, though we


Figure 13. Coarse block S̄∗w under gravity-affected flow of Ng = 1.8 for the top layer calcu-

lated with the resultant coarse model of non-uniform 10 × 2 coarse blocks with ICGC 2D andICGD 2D compared to the finely-gridded model of permeability field 1 made up of 275 × 21fine blocks where the results with a rock-curves coarse model are also shown.

can observe a visible separation among the curves of f̄w’s especially from a tDof 0.3 to a tD of 0.4. The curves of NpD and �Pi−p show almost no separationbetween the finely-gridded model and the upscaled models. The rock-curve coarsemodel shows the poorest reproduction of all the three quantities: delayed waterbreakthrough, more optimistic recovery, and insufficient interwell pressure dropcompared to the finely-gridded model.

In the viscous-dominated example described earlier, the ICGC 2D andICGD 2D approaches showed perfect reproduction with no such separations. Thus,this gravity-affected flow example implies that two-dimensional upscaling can beless accurate in the gravity-affected flow condition than in the viscous-dominatedflow condition.

3.3.2.2. Reproduction of the local saturation history. Figures 13 and 14 show theaverage reduced water saturation, S̄∗

w, of each coarse block in the top layer and thebottom layer, respectively. On these figures, the upscaled models show a reasonablereproduction of the local saturation history at most coarse blocks, but to a lesserextent compared to the viscous-dominated flow example. Coarse blocks in the top


Figure 14. Coarse block S̄∗w under gravity-affected flow of Ng = 1.8 for the bottom layer

calculated with the resultant coarse model of non-uniform 10×2 coarse blocks with ICGC 2Dand ICGD 2D compared to the finely-gridded model of permeability field 1 made up of275 × 21 fine blocks where the results with a rock-curves coarse model are also shown.

layer show a worse reproduction than those in the bottom layer, since gravitation ofwater influences the top layer more than the bottom layer. Especially, blocks (7, 1)and (8, 1) show only a qualitative reproduction of the average saturations.

However, the upscaled models still present a better reproduction. The rock-curve coarse model shows a completely different trend for coarse blocks in thetop layer. This implies that use of relative permeabilities without explicitly con-sidering the gravity effect is likely to fail to represent gravity-affected flow, evenqualitatively, in the coarser scale.

4. Conclusions

In this paper, we developed a new upscaling method using isolated blocks ex-posed to appropriate boundary conditions. This new method has been tested againstviscous-dominated flow where the resultant coarsely-gridded model almost per-fectly reproduced the spatially averaged performance of the finely-gridded model.This method also has worked reasonably well for gravity-affected flow. The majorconclusions obtained in this paper are given below.


1. Sequential 2D-to-2D scale-up method

(a) We extended the new upscaling method to be capable of 2D-to-2D upscal-ing. Three different approaches were developed. Similar to the sequential2D-to-1D upscaling method, all three approaches adopt injection tubes atthe inlet face and a synthetic producer at the exit face. To prevent the incon-sistency of interblock flowrates against interblock potential gradient, thecoarse block boundaries are defined to be uni-directional flow boundariesas much as possible using a single-phase flow simulation over an entirefinely-gridded model.

(i) ICGC 2D. This approach uses two coarse columns and vertically di-vides the upstream coarse column into multiple coarse blocks. Twooptions in calculating pseudorelative permeabilities are available.

– Dynamic-based option. This option adopts Hewett and Archerpseudos [15] for interblock flow properties in both x- and z-direc-tions. It should be noted that this option requires interblock flow-rates and the corresponding potential gradients to be consistent.Otherwise, a negative pseudorelative permeability problem canoccur. When we see a negative pseudorelative permeability of aphase, we may be able to make it positive by using an imaginarycapillary pressure if the phase is a water phase. Thus, we needed asafer option described below.

– Exit-face-based option. This option adopts Hewett and Archerpseudos [15] for the x-direction only while the z-direction inter-block flow is upscaled using transmissibility-weighted relative per-meabilities along the corresponding coarse block boundary. Thepositiveness of pseudorelative permeabilities is ensured.

(ii) ICGD 2D. One coarse column is used and vertically divided into mul-tiple coarse blocks. This approach adopts Stone’s pseudos for the x-direction and the exit-face-based pseudos for the z-direction.

(iii) SEQ 2D. This approach works on a coarse block, not on a coarsecolumn. The boundaries of a coarse block are attached with injectiontubes or a synthetic injector, depending on the flow direction deter-mined based on single-phase flow simulation. This approach is theonly one that cannot be applied to upscaling under gravity-affectedflow since no counter flow is allowed on the coarse block boundary.

(b) We tested the three approaches in upscaling three different permeabil-ity fields and two different mobility ratios under viscous-dominated flowand gravity-affected flow (Ng of 1.8). In this paper, the results from oneof those three permeability fields were presented. The endpoint mobilityratio, M0, was 10.


(i) Viscous-dominated flow. In the test runs, all three approaches workedwell where the resultant coarse models reproduced the recovery per-formance and the local saturation history of the finely-gridded modelto a great extent. The ICGC 2D approach and the ICGD 2D approachworked nearly perfectly.

(ii) Gravity-affected flow (Ng of 1.8). The recovery performance wasreproduced well by both the ICGC 2D and ICGD 2D approaches.The breakthrough time was nearly perfectly replicated. On the otherhand, the two approaches above showed the reproduction of the localsaturation history to a lesser extent than the corresponding viscous-dominated flow case. Some blocks showed a good reproduction, butothers showed only qualitative reproduction. However, these upscaledmodels resulted in far better reproduction than the rock-curve coarsemodel.

(c) The test results imply that the three approaches worked very well underviscous-dominated flow in terms of the overall recovery performance andthe local saturation history. Under gravity-affected flow, the ICGC 2D andICGD 2D approaches were quite good for the overall recovery perfor-mance. In terms of the reproduction of the local saturation history, theseapproaches perform reasonably well in some blocks but only qualitativelyin other blocks. This means that gravity forces amplify the complexity inthe z-direction flow since two-phase fluids can move in different directions.

2. Future directions

(a) The new upscaling method described in this paper may be extended tobe capable of 3D-to-3D and 3D-to-2D scale-up problems. We may con-sider upgrading the current 2D-to-2D sequential upscaling methods to athree-dimensional version. The current version requires us to sequentiallysimulate all the coarse-grid domains upstream of an upscaled domain. Thisis done to provide appropriate inlet boundary conditions to the upscaleddomain. Thus, it would not be feasible to directly extend the current ver-sion to a three-dimensional version, since it would be expensive to simulatethe upstream coarse-grid domains in a fine scale, even sequentially. In suchan occasion, we would have to modify the current method so that the loadof fine-scale simulations can be reduced. Consequently, we would have torelax the inlet boundary conditions of the upscaled domain.

(b) However, determination of the inlet boundary conditions is the centraldifficulty in upscaling multiphase flow as stated by Wallstrom et al. [31],since phase flowrates along the domain boundaries are subject to spatialand temporal changes. In this paper, we showed that the distribution ofphase flowrates and fractional flows along the coarse-block boundaries washeterogeneous. Thus, the relaxed version should be able to provide suchheterogeneous phase flowrates and fractional flows along the boundary


as close to those that would be obtained in an entire model as possible.Instead of simulating all the coarse-grid domains upstream in a fine scale,we would have to develop a less expensive method of synthesizing ap-propriate boundary conditions. Use of local absolute permeabilities in thevicinity of the upscaled domain would be one of the alternatives. We wouldalso be able to incorporate the results of a single-phase flow simulationover an entire model if available.

(c) In this paper, we also showed that frontal advance through heterogeneousmedia was sensitive to flow-path lengths. This is the reason why we de-veloped the concept of injection tubes to incorporate boundary effectscaused by flow-path lengths and heterogeneities contained in the domainsupstream. Thus, the modified version of the current sequential upscal-ing method should be able to account for not only heterogeneities butalso boundary effects. The modified version would require us to quantifyboundary effects, for instance, streamline lengths for flow-path lengthsand correlation lengths for heterogeneities. Based on these quantities, afunction would be needed to relate these quantities to the inlet boundaryconditions.

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