METHOD, APPARATUS AND SYS TEM FOR RESERVOIR

SIMULATION USING A MULTI-SCALE FINITE VOLUME

METBOD INCLUDING BLACK OIL MODELING

RELATED AFPLICATION

This nonproviskmal application claims the benefit of co-pending, provisional patent application United States Serial No. 60--'S 14,748, filed on June IS, 2006, and United States Serial No, 60/814,747, filed on J une 18, 2006, which nre both incorporated by reference in Their entireties.

TECHNICλI _f .F(ELD

lite present invention relates generally to the simulation of fluid ilovv in subterranean formations, and more particularly, to reservoir simulations which utilize black oil models and to reservoir simulations which use multi-scale methods.

BACKGROUND Of THE INVENTION

Reservoir simulation is used to predict the flow of lluids in an underground reservoir. The fluid flow may include oil. gas and water. Such reservoir forecasting is important in reservoir management and estimating the potential recovery rrπm a reservoir.

Reservoir simulation is well known throughout the oil industry and in the scientific literature, A good primer on the principles behind reservoir simulation is K. Aziz and

A, Settari. Pαtroleum Reservoir Simulation, Elsevier Applied Science Publishers,

London ( 1979). Another description of huw reservoir simulation is generally

performed Ss described in U .S. Patent No. 6,052.520 to Watts Hi et. al These references are hereby incorporated by reference in their entireties.

The following are general steps taken in a conventional reservoir simulation. First, a S reservoir is selected for which the rock and fluid properties are Jo be modeled and simulated. The leservorr is modeled and discretized into a plurality of cells. Nonlinear governing equations are constructed for each cell, generally in the form of finite difference equations, which are representative of properties of rocks and fluids in the reservoir. Examples of rock properties include porosity, capillary pressure, and

10 relative permeability for each phase of fluid (oil, water, gas.j Examples of fluid properties include oil viscosity, oil formation factor (Bo), and pressure, temperature, and. saturation in each of the cells. Nonlinear terms in these equations are linearized to arrive at a set of linear equations for each timestεp of the simulation. These linear equations can then be solved to estimate solutions for unknowns such as pressure and

I S saturation in the cells, from these values of pressure and saturation other properties can be estimated including the overall production of oil _} gas and water from the reservoir m Ά timestep, The aforementioned steps are repeated over many such timesteps to simulate fluid flow over time in the reservoir.

0 The permeability of natural formations of reservoirs displays high levels of variability and complex structures of spatial heterogeneity which spans a wide range of length scales. In order to improve the reliability of petroleum reservoir simulation models, much effort has been directed towards characterizing the reservoir properties in great detail. Consequently, efficient numerical algorithms are required to tackle the

25 resulting high resolution simulation models {with several million grid cells). Recent advances m muiti-scaie methods have enhanced the ability to make predictions of flow and transport in highly detailed heterogeneous reserve ii models. U.S. Patent No. 6,823,297 to Jenny et al,, U.S. Patent Applπ. 20050203725 to Jenny et al. and papers by Jenny, p.. S. I i Lee, and 11 A. Tchelepi: 2003, Multi-Scale Fmiie- Volume Method

30 for Elliptic Problems m Subsurface Flow Simulation, j. Comp, Phys. 187, 47-67; and Jenny, P., S. H. Lee., and IL A. Tehelepi: 2004, Adaptive Multi-scύle finite Volume Method for Multi -Phase Flow and Transport. Multi-scale Model Sirnuϊ. 3, ( !) 2004,

50-64, describe a multi-scale finite volume (MSPV) method for single phase How and adaptive IMPES (i.e., implicit pressure, explicit saturaiions. The MSFV method solves How and transport sequentially. To compute a fine-scale flow OeId. dual basis functions, which are solutions ol local problems un a dual grid, axe used to construct a global coarse scale system ot ^: equations, which car* be solved to obtain global coarse- grid pressures. A velocity field is reeomiiueted on the fine- scale using primal basis functions. In the MSFV approach, a block based Sehwarz overlapping technique is used to solve hyperbolic saturation equations (i.e., the transport problem). This MSFV method has been proven to be accurate for strongly heterogeneous problems.

Black oil formulation is often used to perform simulation studies in practice. In a black oil model, three components, which are defined as fluid phases at standard (surface; conditions, axe used to represent a fluid system. The hydrocarbon system is described using two pseudo components, namely oil and gas, and the third component represents water. The phast* behavior of the fluid system is described using solubilities and formation volume factors.

The conservation equations of the three pseudo components are nonlinear. They are parabolic due to capillarity (three immiscible fluid phases) and compressibility (rock and fluids). However, in most, settings, reservoir displacement processes are dominated by convection, so that the pressure field is nearly elliptic, while the saturation equations display nearly hyperbolic behavior.

in the .MSFV ^" method of Jenny et a!., dual-grid basis functions, which are soh.rt.ions to local problems on the dual grid, are used to assemble an upsealed transmissibility field on the coarse grid. Solution of tins global system yields the coarse scale pressure field. A fine scale pressure field can be reconstructed using the dual-grid basis functions. Fine scale saturation equations are then solved using the reconstructed

One scale velocity field, In the saturation equation, a Schwar,ϊ overlap method is applied directly for a primal coarse cell with local boundary conditions. h\κ moie. details of the calculation of the MSFV method and the use of dual-grid basis functions, see Appendix A below.

Prior multi-scale methods typically deal with single- or two-phs.se flow and only consider very simplified physics. They neglect capillary pressure, gravity, and dissolution. They also assume that both the porous medium and the fluids are 5 incompressible. Accordingly, there is a need for a multi-scale finite volume method which addresses; the effects of coinpK'SSibiliiy, capillary piessure and gravity,

SUMMARY 0EJHK _. MY!3£πQN

0 The present invention utilizes a multi-scale finite-volume (MSFV) method to deal with nonlinear immiscible three phase compressible flow in the presence of gravity and capillary forces (Le., black oil model). Consistent with the MSl-V framework, flow and transport are treated separately and differently, preferably using a fully implicit sequential algorithm. The pressure Held is solved using an operator splitting :5 algorithm. The general solution of the pressure is decomposed into an elliptic pan, a buoyancy'capjllary force dominant part, and an snhomogeneous paxt with source/sink (wells) and accumulation, A MSFV method is used to compute ihe basis functions of the elliptic component, which captures long range interactions in the pressure field, Direct construction; of the velocity field and solution of the transport problem on the 0 primal coarse grid allows for wide flexibility in accommodating physical mechanisms, such as compressibility, capillary pressure, and buoyancy.

It is an object of the present invention to provide an efficient algorithm tor a black oil model using a multi-scale fmite volume method in reservoir simulation.

An operator splitting algorithm is devised to compute the pressure field, 'Hie general solution of the pressure is decomposed into elliptic, buoyancy dominated, and capillary dominated parts. ^' The effect of wells (sources/sinks) on the ilυw field is represented as a particular solution. 0

A method for simulating the properties of a fluid-containing subterranean reservoir includes ihe step of partitioning a reservoir model of a subterranean reservoir into a

grid of line cells and respective grids of primal coarse cells and dual coarse cells which overlie the grid of fine cells, A first pressure equation is constructed. The first pressure equation includes a homogenous equation component p,^ & gravity and eapillaπty equation component /λ>, and an inhomogenous equation component p _P associated with at least one of sources and sinks and accαmuiation. The homogenous equation, component p _h is solved to obtain dual basis functions utilizing local boundary conditions on the dual coarse cells. The gravity and capillarity equation component ρ _s is solved to arrive at a summation of products of primal coarse cell pressures and dual basis functions and a correction term. The inhomogenous equation component p _P is solved to arrive at primal coarse ceil pressure equations utilizing the basis functions in conjunction with (he correction term. The primal coarse cell pressure equations are solved for primal coarse cell pressures. Then pressures in the fine ceils are constructed, which are a combination of products of primal coarse cell pressures and basis functions augmented by the correction teπn from the p _{ , equation.

The method can. also include the step of constructing Neumann boundary conditions on the primal coarse, cells utilizing the fine eel! pressures previously calculated. In the method, the conventional pressure equations on the fine cells contained in each primal coarse cell can be calculated with the Neumann boundary conditions applied at the boundaries of the primal coarse eel! to arrive at improved fine cell pressures. In the method, transport equations for each phase in the fine cells can be solved utilizing the iπφioved fine ceil pressures tυ arrive at fine ceil saturations for each of the phases. In the method, the mobility and mobility changes and pressure changes can be calculated in each fine cell. If mobility changes and pressure changes are not within a predetermined tolerance, then the method steps can be repeated repeat until the tolerance is satisfied. In the method, the pressure and saturations in fee fine cells can be outputted, and then proceed to the next time step by repeating the previous steps of the method.

ϊn the method, the solving of the homogeneous equation component py, can include employing the basis functions and a correction function. The correction function can be used to account for gravity effects and correct a course scale operator.

λn aspect of tlie present invention includes a tangible media carrying corrspmer- readable instructions for carrying the steps of the method. Another aspect υf the present invention includes a system comprising computer haidware and tangible media carrying computer instructions for carrying out the steps of the method.

According to another aspect of the invention, a method for simulating the properties of a fluid-containing subterranean reservoir includes the step of partitioning a reservoir model of a subterranean reservoir into a grid of line cells and respective grids of primal coarse cells and dual coarse cells which overlie the grid of fme cells An approximate pressure field is computed, which including a solution of a course- scale pressure equation. Fine-scale fluxes are constructed. A phase-transport equation is computed,

In the method, the step of computing the approximate pressure field can include a juxtaposition of iocai solutions computed on the dual course cells, ϊn the method, the step of computing the approximate pressure Held cart also include computing a fine- scale pressure approximation responsive to a set. of basis function? and a correction function, in the method, the course-scale pressure equation can be an elliptic homogenous equation ph, and the basis functions comprise local solutions to the elliptic homogenous equation, in the method, the step of computing the approximate pressure .field can include using the basis equations to relate a course- grid pressure to a finite scale pressure distribution, and extract course-scale transmissibiiities, In the method, the correction function can be independent of the course-scale pressure. The correction function cao account for gravitational effects on the fluid in the reservoir and can correct a course-scale operator.

According to yet another aspect of the invention, a method for simulating the properties of a fluid-containing subterranean reservoir includes the step ol partitioning a reservoir model of a subterranean reservoir into a grid of fine cefe and respective grids of primal coarse ceils and dual coarse cells which overlie the grid υf fine cells.

An approximate pressure (it-Id is computed. The computing of the approximate pressure filed includes computing a solution of a course-scale pressure equation. The computing of the approximate pressure field includes a juxtaposition of local

solutions computed on the dual course cells. The computing of the approximate pressure OcId includes computing a fine-scale pressure approximation responsive to a set of basis functions and a correction function. Fine-scale fluxes are constructed. A phase-tvarsspoπ equation is computed.

in the method, the course-scale pressure equation can be an elliptic homogenous equation /?«,, and the basis functions can comprise local solutions to rhe elliptic homogenous equation. In the method, the step of computing the approximate pressure field can also iπdικk- using the basis equations to relate a course-grid pressure to a finite scale pressure distribution, and extract eourse-seak transmissibiliϋes. in the method, tht correction, function can be independent of the course-scale pressure, in the method, the correction function can be utilized to account for gravitational effects on the fluid in the reservoir, and correct a course-scale operator.

These and other objects, features and advantages of the present invention will become better understood Vvith regard to the following description, pending c laims and accompanying drawings when-:

FIG. 1 is a flow-chart of a method, practiced in accordance with the present invention, for simulating fluid flmv in a reservoir;

FlG, 2 shuws (a) a global problem domain partitioned imo a primal (solid linest and dual course grid (dashed line) and Cb? dual control volume blown up depicting an underlying fine grid structure:

PlG. 3 shows a coarse grid with nine adjacent coarse volumes (suhd lines), λlsυ shown is a corresponding dual grid (dashed lines), f our adjacent coarse dual control vohrroes comprise the boundary conditions for the coarse grid of a cell 5;

FIGS 4<a.K ^" 0 show ^, tor Example 1 ;

FIGS. 5(aj-(f) show depletion with constant pressure boundary conditions at time " ^: 50 days with (a) logarithm of permeability; (b) pressure MSFV: (c ) S ₁ ., from MSFV; (d) ,.V _S from MSf V: (c) 6V from fine scale simulation; and (f) S _% from fine scale simulation;

FlG. 6 shows contour fines of water saturation for Example 2 (a) 0.21 FVi; (b) 0.42 PVI; and (c) 0.64PV1 with solid contours represent multi-soak, dashed contours represent fine scak results;

FfG. 7 is a heterogenous model with two wells (Example 3} vvith(a) log-permeability distributions; and Cb) oil saturation; just alter breakthrough;

FlG 8 is a comparison of production rates for Example 3 with raultiscale (solid Hues}; and fine scale references (dashed);

FlG. ^Q is a tϊeld-seate heterogenous model with five wells (Example 4} with (a) log permeability distribution; (b) oil saturation just after breakthrough;

FIGS. 10 shows production rates fur Example 4;

FIG. 1 1 is a flowchart of a srsethod, practiced in accordance with the another embodiment of present invention, for simulating llusd How m a subterranean reservoir;

FlG. 12 is a global problem domain partitioned into a primal or course grid (solid lines) together wϊtk its dual coarse grid (dashed lines)

FIGS, 13λ- 13D illustrate water-phase saturation at three different times ( 13A- BC) and the natural logarithm of the heterogeneous permeability field U 3D).

HG. Ai _> associated with Appendix A ₅ illustrates a coarse 20 grid of coarse cells with an. overlying dual coarse grid including a dual coarse control volume and an underlying ^" uie of tint- cells;

FiG. A2 illustrates a coaxse grid including nine adjacent coarse cells (bold solid lines) with a corresponding overlying dual coarse grid (bold dashed lines) including dual coarse control volumes and an underlying fine grid (thin dotted lines) of floe cells; and

FIG. A3 shows flux contribution and qf ^} due to the pressure in a particular coarse ce.U 2.

1 , Overview

Most ^" practical reservoir simulation studies are performed using the so-called black oil model, in which phase behavior is represented using solubilities and formation volume iactois. The present invention extends the multi-scale finite-volume TMSFV) method, as referenced above, io deal with nonlinear immiscible three phase compressible flow in the presence of gravity and capillary forces (i.e., black oil model). Consistent with the MSl-V framework, flow and transport arc treated separately arui differently, preferably using a fully implicit sequential algorithm. The pressure field is solved using an operator splitting algorithm. The genera! solution, of the pressure is decomposed into an elliptic part, a buoyancy/capillary force dominant part, and an inhoϊoogeneous part with source/sink {we Hs) and accumulation. A MSFV method, similar to that described in U.S. Pas.. No. 6,823,297, is used to compute the basis functions of the elliptic component, which captures long range interactions in the pressure field. The methodology is demonstrated using black oil examples of πorsHπtiax tompressibie multiphase (low in ,stroπ,yiy heterogeneous formations. Direct construction of the velocity ikki and solution of the transport problem on the primal

coarse grid allows lot wide flexibility in accommodating phvsical mechanisms, such as cαmpiessibiltey, capillary pressure, and buoyancy,

Pl.G. ! is a How chart depicting steps taken Sn a preferred embodiment, of the present invention: which are as follows: partitioning a reservoir model of a subterranean reservoir into a grid of fine cells and respective grids of primal coarse cells and dual coarse tells v.-hkh overlie the. grid of fine eelis; 110

constructing a first pressure equation comprising i) a homogenous equation component p^ ii) a gravity and capillarity equation component p _κ ; and iii) an

^.homogenous equation component p., associated with at least one of sources and sinks and accumulation; 120

(i) solving the homogenous equation component to obtain dual basis functions utilizing local boundary conditions on the dual coarse cells; IM

(U) solving the gravity and capillarity equation component p_ _s to arrive at a summation of products of primal coarse eel! pressures and dual basis functions and a correction term; |40

(in) solving the- tnhoraogenous equation component /.^ to arrive at primal coarse eel! pressure equations utilizing the basis functions in conjunction with the correction term; 150

solving the primal coarse cell pressure equations for primal coarse cell pressures;_ϊ 60

constructing pressures in the fine cells which are a combination of products of pπmal coarse cell, pressures and basis functions augmented by the correction term from the pg equation, 170

constructing Neumann boundary conditions on the primal coarse cells utilizing the tine ceil pressures from step (ε); 180

solving conventional pressure equations on the fine cells contained in each primal coarse cell with the Neumann boundary conditions applied at the boundaries of the primal coarse ceil to arrive at improved line cell pressures:

$ 120

solving transport equations for each phase in the line cells utilizing the improved fine ceil pressures of step (f} to arrive at Ii tie cell saturations for each of the phases; _. 200

S O calculating the mobility and mobility changes and pressure changes in each fine cell; _. 21θ

if mobility changes and pressure changes are not within a predetermined tolerance, then repeating steps 120 to 210 until the tolerance is satisfied; _. 220 and 15 output the pressure and saturations in the fme cells and proceed to the next time step, 230

The detailed description of a preferred embodiment of using the present invention is 0 organized as follows. IB section 2, ihe governing equations for three phase transport arc described and a discretized formulation is derived Transport equations are rearranged to yield a single pressure equation and two phase transport equations In section 3, an operator splitting scheme is introduced that decomposes the pressure equation into three components. Corresponding multi-scale algorithms are derived for 5 each the three components. Section 4 describes a sequentially fully implicit scheme for solving saturation equations. Also, the use of adaptivity in the MSFV method is discussed. The novel algorithm of FKJ, 3 for a picfeπcd embodiment of this invention is summarized in Section 5. Finally, numerical examples are presented in Section 6,

0

^■ 1

2. Governing Equations and Discretixed Formulation

The standard black oil model includes two hydrocarbon phases (oil and gas) and one aqueous phase (water) with rock and fluid compressibility, gravity effects, and capillary pressure The thermodynamic equilibrium between the hydrocarbon phases is described by the solubility of gas in oil. The conservation equations are nonlinear due to the strong nonlinear character of the relative permeability anά capillary pressure relations, high compressibility (especially of gas} ₅ large density and viscosity differences, and phase appearance and disappearance effects.

The governing equations for the black oil formulation are:

on the domain ω , svith boundary ¹ conditions on iX2 , Hers;. It ₁ - I Z B ₁ and

A _/ ■■" kk _! //.'. tor phases ! - o, M\ q (i.e., oil, water and gas). B ₁ denotes the formation volume factor of phase I , i.e., the volume at reservoir conditions to volume at standard conditions. Similarly, S ₁ . A _f ,/j _; p ₍ . denote, respectively, the saturation, relative permeability, viscosity, and density of phase / , The well volurnetiic How rate is q.. . The tensor k describes the permeability field, which is usually represented as a complex multi-seaie function of space thai tends to be highly discontinuous in discrete representations. Porosity is denoted by $,/? _< • is the phase pressure, g is gravitational acceleration, z denotes reservoir depth, and K _^ is the solubility of gas in oil. In general, μ, _^ p ₁ , B^ ₁ R. aj)d φ are functions of pressure. The relative permeabilities. £,, , are strong functions of saturation.

Saturations arc constrained by;

and the three phase pressures p, _t „ />, and p are related by two indepeRtieBi capillary pressure functions:

The oil phase pressure is chosen as the primary variable, p - ρ _B and substituted for the others in Fqs. { } ) to (3). The following semi -discrete forms are obtained ¹

I \ ^" in,, \, - i \ ^" ρ - gfl^ ^;■ - _. ! - q., \ ' ^" ' ^■ ' .

,./.-1 l »,« + ! ^. -1- S ... , .J- f.f > ψ-

^' ϊ ^' .he nme step srzε is δ/ and the superscript n denotes the state at the old time level ( n -!- 1 is the current time kve!). ϊiqs. (7) to (9) are multiplied with factors a, , respectively :

A subsequent summation step removes the dependency of the current level of saturations in the accumulation term of the resulting equation. If the resulting equation is iurthet linearized around pressure p° for p"" , then the we) i -known (parabolic) pressure equation is obtained: v/ ^{i !} )

with:

- I i -

ⁱ JC ■ ^■ ? ^' [: -f &ϊb" $?) )

RH ?

and:

For a celS (J . i\ k } in a Cartesian grid the advection terms in Kq. ( S l ) is modeled by

^{~h r} - .%!,**:< ^!' ^!.>i: + ] -I α'w

where The iυcal line scale transmissibilitics tor C ^' artcsifin grids arc given by:

i\ , ■ i A.r.-λ-v, l i lV:

For a Cartesian grid w!teκ the grid lines are aligned with the principal directions of the hdcrυyeiKύv. the interface, permeability becomes the harmonic mean of the vakurs of i\vo adjacent c«lLs. For instance, the interface permeability in x direction becomes:

Far a more general ease with nononhogonai grids and tensor permeabilities, the transmirisibilities become muitipoint stencils, in place of the Uvo-point approximation of Gq, (15).

Eq. (1 1 ) is solved for /?" ⁺¹ at the beginning of each iteration of a Newton Raphson loop and the updated pressure Is then used to derive the- linearized transport equations.:

r • i ;T ^{< J} I Af -ϊ- V" ~~i- ^' (5." ^{: !} - .ST i I ■ ^{! !} - ψ.>> V : ϊ 1

where / e |o, g|- stands for OiJ ₁ water, and gas. The third phase can be readily computed from the simple saturation constraint (4) at the end of each iteration if Eq. s 1 8} Js soived for S ₁ , and 5 _}S .. Note that the states denoted by the superscripts Sλ ϊ' 4 L and n + 1 arc the same alter convergence is achieved.

3» Multi-scaie Finite-Volume Formαlaiion

The teachings of U.S. Patent No. 6,823.29? to Jenny et a!., U.S. Patent Appln. 20050203725 to Jenny et al., Je«ny ₅ P., S. II. Lee. and H. A. Tchekpi : 2003, Muhi- Scaie fimtc-Vυlume Method for Elliptic Problems in Subsurface Flow Simulation. J. Corap. Phys. 187, 47-67. Jenny, P., S, H, Lee, arid H, A, Tcheϊcpi: 2004, Adaptive Mulii-scale Finite Volume Method for Multi-Phase Flow and Transport. Multi-scale Model. Simul. 3. SO 64, are hereby incorporated by reference m their entireties into this specification. See .Appendix λ below for a condensed summary of methods described in U.S. Fat, No. 6.823,297.

IB the sfjulti-.seak iiuήe-vohssrse (JVISFV) algorithms described in the references, the global (fine scale) problem domain is partitioned into prtmaJ and dual coarse volumes as similarly illustrated in FϊG. 2. A set of pressure basis functions is computed for each dual volume and the coarse grid pressure is solved using upscale^

vransmissibiiitjes. The same pressure basis functions allow for reconstruction of the fins scale pressure from the coarse solution.

The original MSFV algorithm of these references was oπyinalϊy designed to solve the elliptic pressure equation for incompressible fiovy in highly heterogeneous formations.

The black oil mode!, which accounts for compressibility and capillarity effects, yields a nonlinear paraboHc pressure equation. However,, compressibility and capillary effects are, in general, quite small. As a result, the pressure equation exhibits a nearly elliptic behavior. A multi-scale algorithm is designed for pressure based on this observation.

In a preferred embodiment of this invention, the pressure is decomposed into a homogeneous part and two inhomogeneαus parts. The homogeneous (elliptic) component is denoted by /?„ : the inhomogeneous part is composed of two contributions: gravity and capillarity, ,>>., , and additional source/sink due to well and πonliπearity of physical parameters, p _r . Following this procedure, the governing equations for the different pressure components can be written as'

and the pressuie equation becomes:

where,

;

The pressure due to capillary force is m ^~ general much smaller and localized than the pressure due to viscous force and gravity. ^' The first tenn bi the right hand side of Eq,

122) vanishes as the pressure .solution converges \p ^v>l ~ /? ^r -> 0) and the second term becomes only the source/sink as the fluid becomes incompressible {a _t , ~-> l). (n the subsequent sections, the modifications to the original (incompressible) MSFV framework are discussed thai can accommodate the operator splitting approach of Eq. 5 (19).

3.1 Homogenecrm Pressure Solution

The original MSFV method of U. S Pat ^" No. 6,823,29? yields an accurate fine scale pressure solution by employing a coarse grid. The reconstructed fine scale Held is obtained using fbe coarse scale pressure solution and the locally computed basis 10 functions. A general MSFV algorithm for the black oil formulation will now be derived. A conforming coarse grid with N nodes and M cells is constructed on the original fine grid (see FiCs, 2), Each coarse cell, ω' (/ <= {I,,., λ/}) , is composed of multiple fine ceils. A dual coarse grid, conforming to the i i ae grid, is constructed such thai each dual coarse cell ω" ^' (/ e ;L.,,,λ ^f f), contains exactly one node rsf t.be coarse

I S grid in its interior. The coarse dual grid has M nodes, λ\(i e |l, M \), each in the interior of a coarse cell ω" ^' . Each dual grid has N. comers (Ibur in two dimensions and eight in three dimensions), λ set of dual basis functions, θ' , is constructed; one for each comer / of each dual coarse cei! ω" .

The dual basis functions are used to assemble a coarse scale transmissϊbiiuy field, 20 which is used to compute the coarse scale pressure, /?' . The dual basis function θ', , for example, is the local solution of Eq. (20);

::>,λ ^" i v _> ^■ v ^" ^} + <:t,,. \ ^" t λ ^' ^. \ ^" θ* j -j- c\ _λ T ^; - S i ^{" ■} S ^' Wj ) — ^ ^n S i ^* ,\ t ^' 2ό)

where the uπdetlying tins gfid, the fine scale total mobility field, and the reduced prtiblciϊi boundary condition:

1 ?

- ( λ rθ"j), ~ O t-» CJγS ¹ ? .

6'x _;

are used. Here, the subscript t denotes the tangential vector eomponervr to and;

λ ^::: λ., -f- λ,, H- X ₄₁ ^• • ^• --- <. _\ λ\. 4- ■ »„ . *, ^' _v + ii,; v ^' ,; ! - ^" '

The value at the node x _k of ω" is given by:

where S _{1 1} is the Kroπecker delta. By definition &, (x)∞ 0 fox A' e ω ^' ^ . Effective coaree scale transπvissibilities are extracted from the dual basis functions θ', . A set of fine scale basis functions, θ| , is constructed; lor each coarse coil ω;\ one basis function for each adjacent coarse cell ω\ , which includes ω: ^' . The boundary conditions for the local computations of the fine scale basis functions are extracted from the dual basis functions. Finally, given a coarse scale solution p\ , the phase flux across the interface;

is approximated by:

where T J ₅ , are phase transmissibiiities obtained as follows:

( A; ro'j B ,->τ

,/,-s. ^s T.

The vector n is the unit norma! of 5ω^ _{] ;:} pointing in the direction from ω;, .to £>.!, Note that the homogeneous solution is later used to compute a coarse grid pressure operator (restriction operator) and the coarse grid solution is interpolated to fine grid solution via the basis functions (prolongation operator).

3.2. Itihwftoggϊieous Solution with Gravity and Capillary Pressure

As shown in Hq. ( _. 21), the inhornogeoeons solution p,, accounts for gravity and capillary forces. Due to the complexity of the fiaetjona! iiow Function in the presence of gravity, the potential field cannot be represented by a simple superposition of the basis functions. λn efficient algorithm is proposed to compute p _g by splitting it into two ports. The first part is represented by the original basis functions; the second part is a correction term based on solving a local problem that accounts for buoyancy effects. Following this idea, p _g is decomposed into two parts:

^ - ιζ + />;; - > + pl w <γ* f32)

Note that in the dual coarse grid Q ^' % pζ is represented by a weighted linear combination of the coarse grid pressure, pi'' , where the weights are given by the basis functions. The correction term, p*h obtained from Eq, (21 ):

where solutions of reduced problems consistent with Eq. (33) serve as boundary conditions:

λ - V/i*'. ) ---- ™- { G i , O:i S'j/' . ! λ4 ;

<:7X _; \ ^{■ 'J} J i :jχ,

where:

G — t ^; _v / ⁾ ,,,V., í O,, /> _i( A ;,. t ' _{' S} (i^',, i« v ^" ; - C. _N ..\{,. ^■ \ ^' />. ..... - M ,,.\' ^■ S ^' p _{t !K} , 1 ^' iδ ;

As discussed in the previous subsection, the reduced system cau be applied along the boundary (edge in two dimensions and interfaces in three dimensions) to specify the boundary conditions for each dual coarse grid. For instance, the reduced systems from Ecj. (34) may be solved along the boundary lines with the boundary conditions:

"9 s < ^~~ 0 tor i L

Note that the correction term /?,'' is computed with the simple boundary conditions for the reduced system that is independent of the global pressure distribution. This iucalizaiiυn assumption to compute />* is analogous so the one used to construct the dual basis iimctkm in the absence of gravity effects. Consequently, this localization assumption is effective in resolving the fine scale structures of complex heterogeneous problems where buoyancy plays an important role. Substituting Eq. (32} into the governing equation Eq. (21 } and applying the Green's theorem for me coarse grid operation, the solution from μζ becomes an additional source/sink for the coarse grid operator:

<t\

where:

^ ^{:=r ■•■} ILI / C ^λ ; - Wλ, - ^ i ^{is J;λ} K

Once />!: is computed, the coarse grid pressure p _f ' ean be computed by the coarse grid transmissibiiity described in the prtvious section and the additional source term from p _g ° .. ^' The fή, , as a result, does not have to be calculated separately. The coarse grid

block pressure caτι be calculated together with other sυυrce''siuk and accumulation terms, which are Ueated in the next section.

.1.3 , Particular Solution with Weils and Accumulation Terms

In addition so capturing ihe effects of She presence of sourcε-iλmks (wells ^' ) in the domain, the particular solution ;?,, , as given by Hq. (22). accounts for compressibility effects on the accumulation terms. ^' Vo simplify the derivation of the inho-mogeneous solution here, wells are represented as distributed fine scale source terras in she coarse grid containing the well.

The coarse grid transmissibility is obtained from the homogeneous (elliptic) component. If the coarse grid tiansmissibtiitv field does not change for the particular solution with souτcεs ^; 'sinks and accumulation terms, the coarse grid pressure can be computed directly. With the coarse grid traπsmissibility approximation, a mass balance equation can be written:

wherv

O. V f < < ;,... ! ?/, -f /?. < ^■ /„ )' ^' ),

)

% ' σ-! .'λ. ^. (4 !

for each coarse eel! ω' {/ ;~ {l,..., .<V/}) and the entire system can be solved for the coarse pressure , -s- /^ , ) . Note that a coarse grid s ^' factor is employed to roinhϊuz.e the pressure dependency on the new coarse saturation. The following coarse arid a ¹ factors are used;

T' " ^' . <.Xb'; .SV

Once the coarse grid pressure is computed the fine grid pressure in the dual grid cars be obtained using the basis functions:

_3* _fJ { χ ) -T ^j (X j — ^) ^ ( ^ KK _M * lC: i + /£ (30- > ⁽ ' ^γ ..-. <">' ^; t -J -t

.• — 1

The pressure from the particular solution and the linear gravity part is interpolated by die dual grid basis functions and then the correction gravity pressure is added.

3,4. Conservative Velociiy fidd and Improved Pressure Field

The velocity field as readily obtained from She reconstructed fine pressure (i.e., using coarse grid solution p _t ' ^' and the dual basis functions θ ^! ,- } suffers from local mass balance errors at the dual coarse cell boundaries, As a remedy, a second set of (primal) basis functions may be used for the ccmstruction of a conservative velocity field. Using the fine scale fluxes obtained from the dual basis functions as boundary conditions when computing the fine scale basis functions ensures that the rceonsirucu'd fine scale velocity field is conservative (See FlG. 3). Even though an adaptive method can be employed to eliminate unnecessary computation of the second set of basis functions, the large number of such basis functions does not guarantee that this approach is numerically more efficient than direct computation of the velocity and saturation fields every lime siep, or iteration.

As a lesuii. a dixecs approach is used for obtaining velocities instead for this preferred embodiment. This is, in fonsirucfing ihe velociiy field on local domains, the

KCøniλruUed line &cale picture suluiiuu ii> suinpk'd as the local boundaucs inr fluxes λϊIδ local Neumann probk-mt- arc suhetl. l he detect vdoeii} coustiuchon allows <*ll nonlinear cffeds Iu be included (i.e., full black oil sies), however, adapύvή> he lost. As a secondary benefit, the fine pressure -solution obtained from the Veumann. solution is likeh to he of higher quality ύ e , it is locally consistent vuth the \ field) as compared to the proline solution from the conrse solution and the basis lunelnms (a? in ihe oπginal nieihod of T S. Pat No δ,82?,29 ^' ? This direct approach now be detailed.

hvivs the coarse grid ρr«.-suje boiution and tls« do<J giid basis iuuctioπs, the line grid. pressure OeId in a prim a! coarse pjkL f~_T , as show n in Bq. (44), is constructed The Neumann boundary condπi>ms art- (.onstxueted along the primal coart-e grid, fi"t

and Hq. (19; is solved ior the fine scale velocity, in ordci to obtain a conservative velocity iidd U s\ hich conforms to u' , the rctjiόreincnt:

i i ϊi -- u ^■ U I l ^! > ^" i

S^. imposed at the interlaces between coarse ee!K. vsiicfc n is the interface unit norma] vcctct l he loe-U fine seak- solution is the soiutiun of Kq. (1 1 ) with the local boundary cortditusiiS ni ^" {4b i. i;q, (1 1 J cun be rearranged the iocαl boundai s conditions κ» obtain ¹

w 3 th.

;<' Iλ> .,;;: \ Vt,. ^! -' , ' '/ ( > ^•

λi

with the local boundary conditions:

\) — u" ^{" ■} u <m OiY < U?M

The resulting pressure solution is of good quality and replaces the rnuiU-seale reconstructed pressure solution i« the black oil case. The local pressure solution is readily converted ϊo velocities, at the boundary the velocities Eq. (49) will be enforced.

.3.5. Saturation Solution

From she conservative pressure field in Hq. U l ). the fluxes can be readily computed K) a cell and the fluxes .substituted in Eq. ( 18):

. ι ->0.»

which arc solved lor S ₁ , and <S _^ at the end oϊ each ϊteraiioti. Notice that the states denoted by (he superscripts v, κ í l _; and w + l axe the same after convergence is achieved.

4, Algorithm Summary

The above steps can be summarized into the form of an algorithm as described below.

1. Partition problem space into coarse and dual coarse volumes (i.e., coarse grid and dual grid).

2. Compute dual basis functions {/>/,) iϊπrH Eq. (25) and local boundary conditions Eq, (26) and Eq. {28)- This can be done adaptively.

3. Compute the correction terra of gravity-capiϋaiy piesstuc solution {p^ } ftorn

Eq. (33 } and local boundary condiϋons Eq. (34) and Eq. (35). Compute the coarse pressure values for i nhomogeneous equation with source/sink and gravity, /.>f , located at X ₁ , as shown in Eq. (39).

5 4. The coarse pressure field p ^c together with the dual scab basis functions and the gravity-caps Oary pressure correction terra p.'~ are used to construct fine scale pressure for the inhomogeneous equations ij?, _; (x)} in Eq. (44),

5. Fϊom the fine soak ¹ pressure solution p{x} ~ p _k + ρ _u -f- p., construct the Neumann boundary conditions for the primal dual grid. Eq. (47). This pressure

H) solution will yield fine scale phase velocity Held U ₁₅ , u , ,,U,, .

6. The phase velocities are used to solve phase transport equations on the firse scale. Here an explicit or an implicit discretization scheme can be used.

7. tJsina the new saturation distribution, the mobility field K is updated and the basis functions are recomputed where it is necessary (which includes updating

15 the effective coarse scale transmissifoiiitUss). Here an adaptive scheme is preferably applied.

8. If an implicit solution, method is applied, one proceeds with the next Newton Raphso.0 iteration by repeating steps 2-7, until convergence is achieved.

9. The next time step is performed by repeating steps 2-8.

20 5. Sequential Fully Implicit Coupiiag And Adaptive Computation

The algorithm, summarized in the previous section, comprises two major steps. First the phase velocities sre obtained from solving the pressure equation with the MSFV method and rhcn the transport equation is solved on the fine grid by using the reconstructed fine scale phase velocity field U - A Schvvarz overlap method is applied 2 S for the saturation equations. The transport problem is solved locally on each coarse

volume with an implicit upwind scheme, where the saturation values from the neighboring coarse volumes at the previous iteration level are used for the boundary conditions. Once the transport equation is converged (.for hyperbolic systems the Schsvarz overlap method is very efficient), the new saturation distribution determines the new total mobility field tor the elliptic problem of the next Newton iteration. Note that,, in general, some of the basis functions have to be recomputed each time step. These steps can be iterated until convergence of all variables at the current time level.

The MSFV approach can be easily adapted to a sequential fully implicit treatment. The KiSFV implementation allows fur performing an IMPES, traditional sequential, or a fully implicit scheme. Here, the full nonlinear transniissibility terms at the new time step level are retained so that stability is guaranteed. The converged solution using this sequential approach should be generally identical to the solution obtained using the simultaneous solution strategy, which is usually used to deal with the coupled fully implicit system,

Aziz, K, and A Settari: 1979, Petroleum Reservoir Simulation. Elsevier Applied Scientific Pub. and Watts, J. W.: 1983. ¹ A Compositional Formulation of the Pressure and Saturation Equations', In: Proceedings of 7th SPE Symposium on Reservoir Simulation: SPE 122) , San Francisco, CA, pp. 1 13 — 1.22» investigated a so-called sequential method in which the pressure is solved exactly the same way as tor the IMPES method; and then the saturation is obtained with the linearized implicit transniissihilities. The serai-implicit treatment of transπus&ibility in the second step improves the numerical stability. Since the transrnissibilities in the pressure and saturation equations are at different tinse levels, this formulation will not satisfy the material balance. Nevertheless, the material balance error is very small in a black oil system. Watts derived a sequential algorithm of compositional formulation and also noted that the material balance error was not a problem in applications. A typical time step needs several iterations for full convergence.

ϊn the conventional implicit scheme is xescrvoir simulation, a Newton method is used to solve the nonlinear transport equations with variables (e.g., relative permeabilities, pressure and saturation) at a new time level The linearized equations are solved

simultaneously for pressure and saturation changes. Note that this implicit scheme of may be a sequential, iterative implicit scheme that is somewhat different from She conventional fully implicit method. Nevertheless, the solution should converge to the folly implicit solution. The sequential implicit MSFV method has been tested ibr large and stiff problems and the coupling scheme did not fail., even for very large time steps.

In the iteϊatioπ of saturation -.okition, the total velocity between ceils should remain constant in. order to conserve the mass for each ceil It is thus convenient to use a generalized fractional How and the total velocity formulation. In the sequential implicit algorithm, the saturation equations are iterative!)- solved with the total velocity fixed to preserve the conservation of mass (i.e., divergence free). This algorithm can be readiiy implemented by the total velocity and fractional flow formulation. ^" However, for the case of counter-current Sow, the original fractional flow that is dependent on a single saturation has to be modified In urttef tυ maintain mass conservation. The convergence and accuracy of the sequential implicit method with various error tolerances in the outer Newton loop have been examined. The results indicated that this sequential algorithm converges as fast as ihe fully implicit scheme.

The MSFV is well suited for adaptive computation, which can lead to significant gains in computational efficiency. The most expensive part of the algorithm is the computation of ihe dual basis functions, In general, this is performed every iteration due to changes in the saturation (mobility) field. An. adaptive scheme can be used to update the dual basis functions. Since the basis functions are constructed with local

Support, the change of the total mobility is used to decide when and where to update the basis functions m the clomam.

I V- ^'

- ^■ ■. ^' 1 4- * > ■• I ^• ! ]. ,!

1 + *.χ λ"

For a compressible fluid, an effective total mobility change criterion am. be employed for adaptable computation of pressure field:

I -!•

K ^~ '

Mere,

T: ^« λ ! -Xl !

and e λ is a user-defined threshold for mobility and pressure change.

5 6, Numerical Examples

6.1 Cnmputaikm of Gravity Pressure

As the gravity pressure is computed by an operator split, it is instructive to examine the gravity pressure. From numerical experiments, it has been found that the algorithm with local correction is very accurate if the flow is close to one-dimensional

I O vertical motion because she reduced boundary conditions for p* closely represents the real flow field. Tn this example, the case is studied in which the fluids are very much away from a gravitational equilibrium and also have a strong lateral motion in addition to a vertical motion of fluids, A simple two phase two-dimensional model is constructed with 12 s \2 fine grids that is divided by 4 x 4 coarse grids. The oil I S saturation is given by S ^' , - 0.5 for / ~ l - 6 and £ - 5 - 6 and S ₁₁ — 1 in the remainder oi ^* the model. Density and viscosity for fluids are selected as LO g/cc ami 1 ep for water and 0.5 g/ce and 2 Cp for oil, respectively. The quadratic relative permeabilities are employed. In FiG. 4, the original oil saturation is shown in {a} and the gravity pressures computed by the fine scale simulation in (b).

20 ILe uneven water sauuttiϊon ia the left lower earner will induce lateral and vertical motion of fluids doe to the presence of the density difference of the fluids. The countcreuTTcnt flow is expected around the fluid interface and also the water will settle, down inside the left-bottom quarter as the water was uniformly distributed

('V== 0.5) .

in (c) and (d) of FIG. 4, it is shown that the two pans of gravity pressure ( /^ and P _ς i ) computed by the multi-scale algorithm discussed above and the combined pressure /y in FKJ . 4 (e). in addition, the difference of the fine scale simulation and the iirst ran Ui -scale results from the dual grid is depicted in FiG. 4 (f). Due to Use lateral motion of .fluids in this example, the boundary conditions for the reduced system for dual grid in the multi-scale modeling JS less accurate, compared with the fiisi example. Some differences are shown in the gravity pressures from the

- ^" ?9 ~

Table 1. Physical Properties of Black Oi!

inultiscaie calcitiation and the fine scale simuiation. However, as expected the error is very much localised around the boundaries of the dual grids where a reduced system 5 was applied and the error is of a few percentages in most of the domain Clearly, the quality of this multi-scale solution e;io be improved when the solution is calculated WJm improved boundary conditions.

6.2, H ^' aierftood In Linear Geometries

IO

This lest case has a two dimensional geometry with 1 10 x 60 fine cells. A Uniterm coarse grid of 22 x ύ is used for the muili-scale run. The permeability description is taken from the tsrst layer of the Tenth SPE Comparative Solution Project, Christie, M A, and M. J, Blum; 2001, Tenth SfE Comparative Solution Project: A Comparison of I S ϋpsccdmg Techniques. SPORE 4(4), 308 — 1 ?. As shown in HG, 5A, the permeability varies 10 orders of magnitude. A highly correlated area of low permeability is found in the left-hand-side of the model and that of high permeability in the right-hand side of the model,

20 The physical properties of the black oil mode! include three compressible fluid phases, i.e., oil. water and gas, The pressure dependence of the densities is described

using formation volume factors, and the phase equilibrium between the oil and gas phases is described using the solution gas-oil-ratio. A typical physical properties of black oil is tabulated in Table I. For Shis example, the gravity effect is not considered and a higher oil compressibility. 4.8 ^■ 10 ^'" ' for p < p> _; is employed, a stringent test of compressibility. As part of the solution gas in oil moves from oil phase to gas phase as pressure decreases, the oil volume decreases as pressure decreases in p < p _h . The solution gas is constant above the babble point pressure, the oil volume decreases as the pressure increases fur p > p _k ,

The mode] was initialized with oil {S _u ^~ \) and constant pressure (4G(X)p.?/#) . At ,' ~ 0 , water is injected at a constant pressure of SOOO p,y/V/ from the left side; the right boundary is maintained at IQOύpxfύ . This numerical example is a challenging test due to the large pressure drops applied between two boundaries and a large vacation in permeability distribution in the model. A constant timcstep size, 1 day, was employed for this example.

1« FlG. 5 the results ftom the MSFV simulator and fine scale reference simulations are depicted at 50 days. Because gas compressibility is two to three orders of magnitude larger than that of either oil or water, a large pressure drop occurs at the boundary of the oil phase above the bυbbie-pohst pressure and the mixed (oil and gas) phase below the bubble point. This sharp pressure drop is demonstrated clearly in HG, 5 (c). 1« addition, FI ⁽ J. 5 (c)-(d) indicate that the saturation distributions of water and gas obtained from the MShV summation are in excellent agreement with the reference fine scale solutions,

6 3 Two Phase Displacement with Gravity

The accuracy of the black oil MSFV ^" model in the presence of gravity is now assessed with a two phase displacement example. The geometry is a two dimensional 25 x 25 grid with vertical orientation, the multi-scale coarse grid is 5 x 5. The permeability is homogeneous, k ~ 100 md. and the porosity is ^ - 0,2 . The physical properties are

given in Table 1. For a strong gravity effect, we changed the oil density ai S IC to 0.5 g/cc. The mode! was initialized at. A-OOϋp.sia at. the top of the model and water was injected at constant rate from the bottom kit corner displacing the uil towards the producer located at the top right corner. The average pressure was closely maintained 5 at the initial value.

The overall fluid movement is upward, In FIG. δ, water saturation contours are compared between the multi-scale run (solid lines) and the fine scale, reference result (dashed) for three different times (or pore volumes injected, PVI). The contours 10 indicate downward movement of the heavier fluid (water). For ail three snapshots, the saturation contours from the .MSFV approach are in excellent agreement with those from the fine scale reference simulation.

6.4. A Three Dimensional Heterogeneous Model with Two Wails

This example employs a three dimensional model wife two wells at two opposite comers and heterogeneous permeability. The permeability field is generated by the Sequential Gaussian Simulation Method, Deuisch, C. V and A. G. Journel: 1998, GSLIB: Oeostatistica! Software Library and User's Guide. New York: Oxford Univ.

-» 0 Press. The logarithm of permeability has a Gaussian histogram with mean and standard deviation, of 30 md and 1.5, respectively. The variograns is spherical with range of 30 m and 15 m in the directions of 45 degree and 135 degree in the horizontal plane, respectively, and 7.5 m in the vertical direction, The permeability is shown in FKT 7 (a). The mode! comprises 150m x 15m x 48m in physical size and is 25 uniformly discretized using 45 x 45 x 30 fine grids. The multi-seaie coarse grid is 9 x 9 x 6 (uniform).

The permeability distribution is shown in FiG. 7 (a). The fluid properties in Table ϊ are also used for this example. The reservoir is initially at gravitational equilibrium 30 with AOOOpsia at the top oϊ the model Water is injected at constant rate from the bottom left corner (eel! (L Ll ) in coarse grid) displacing the oil toward the producer located at the top right corner (eel! (9,9,6) in coarse grid}. FIG. 7 Cb) shows oil

saturation distribution at the water breakthrough. The pressure around the production weil is below the bubble point, the solution gas becomes free gas and the gas saturation becomes around 0.2. In FlG. 8 die production rates from MSFV art; compared with those from the ttne scale reference simulation. The comparison clearly S shows that MSKV is very accurate in simulating multiphase flow with gravity and compressibility in heterogeneous media.

An upscaled model is computed for this problem. The basis functions were used for l he upscale^ trammissibility and computed pressure and saturation in coarse scale 0 without the reconstruction of fine scale solution. The results are also depicted in FIG.8. Kven though the results from the upseak-d model was qualitatively following those from the fine scale reference simulation, the upscaled model yielded, due to large numerical dispersion, much less accurate production rates than the multi-scale finite volume method. This numerical example clearly illuminated that the 5 reconstruction of tine scale solution in MSFV is an important step in obtaining a high quality solution.

6.5. A 3-D Field Scale Mode! with Five. Wells

0 This example employs a three-dimensional field-scale model with four production wells at the upper four corners and one injection well at the center of the bottom layer. The model consists of more than one million ceils (1058851 ^~ 147 x 14? x 40). The permeability field was generated by the Sequential Gaussian Simulation Method Deutsch, C. V. a.nd A. G. Journeh 199%, GSIJB: Geostafistical Soihrare Library and 5 User's Gnkk. New York: Oxford Univ. Press., with the same specifying parameters of example 3i; and the black oil physical properties in Table I were also used for this example. The permeability distribution is shown in FIG. 9 (a), A strong spacial correlation is noted in permeability distribution. Especially the correlated permeability is oriented in the direction along production wells 1 and 3, A uniform coarse grid of 0 21 x 21 x 7 was used that results in a scale-up factor of 343 \~ 7 ^:S } between fine and coarse grids. Water is injected at constant rate from the center of the bottom layer displacing the oil toward the four producers located at the top corners. The model was

simulated for 2500 days with the timcstep size of 30 days. The oil saturation distribution around the end of simulation was depicted in FIG. 9 (b). The preferred water path irom the injection well to the production wells are well established along the correlated orientation of high permeability, in FIG. 10 phase (oil. water and gas) production rates are plotted for the production welis. Due to the strong spaeiai correlation of heterogeneous permeability, the water breakthrough patterns are very different between wells located along the major correlation direction (wells 1 and 3) and the ones in the minor correlation direction (wells 2 and 4), This non-uniform distribution of saturations is also evident in FlG, 9 (b).

In summary, a muiti-scale finite-volume (MSFV) method has been developed for the black oil formulation of multiphase flow and transport in heterogeneous porous media. The black oii formulation, which involves three phase flow with rock and fluid compressibility, gravity, and capillary effects _, , is widely used in practical field-scale simulations. This approach extends the sequential fully implicit MSFV method to noiiUneai three phase compressible ilow with mass transfer expressed using solubilities. An operator splitting algorithm is devised to compote the fine scale pressure field. The black oil MSFV method extends the ability to deal with large scale problems of practical interest. The treatment ensures that the nonljnearity due to .Quid properties, gravity, a:nd capillarity can be resolved by a localization assumption for the boundary coiid itions.

6.6 Three Phase Displacement or Flow Influenced by Gravity

In another embodiment, the Mukiscaie Finite- Volume (MSFV) method 300 (FlG. 1 1) has been developed to solve multiphase flow problems on large and highly heterogeneous domains efficiently. U employs an auxiliary coarse grid (PlG. 12k together with its dual, to define and solve a coarse-scale pressure problem. A set of basis functions, which, are local solutions on dual ceils, is used to interpolate the coarse-grid pressure mid obtain the approximate fine -scale pressure distribution. However, when flow takes place in presence of gravity, it is preferable to utilize a correction function, added to the basis-function interpolated pressure, in order to

obtain accurate fine-scale approximations. Numerical experiments demonstrate excellent agreement between the MSFV solutions for pressure and saturation and the corresponding One-scale reference solutions. 6.6.1 Introduction

To model flow and transport in geological porous media efficiently, a number of upscfilirtg. techniques have been developed m the past 30 years to coarsen the numerical grid used for the simulations. The price paid to reduce the computational costs is the loss of fine-scale information. To overcome this drawback, several multiεcaie methods have been recently proposed to model aquifers and reservoirs; the MuitLscale finite -Element Method (flou arid Wu 1997); the Mixed Muitiscaie Finite - Element Method (Aames et al. 2005; Arbogast 2002; Chan and Hou 2003.); and the .MuJdscaie Finite-Volume (Jenny et al. 2003 ). In contrast to upscaUng, the focus is not simply on capturing the large-scale behavior of the system, but on solving the problem with the original resolution. The goal is to be as accurate as the fine-scale solution keeping the computational costs low - ideally comparable with traditional upsealing methods.

These methods usually deal with simplified physics (incompressible flow, negligible capillary and gravity effects). With this method, the application of a multiscale algorithm to gravity-driven multiphase How Ls shown, considering die MSFV method, which in contrast to other methods is based on a finite-volume discretization, This has technical and practical advantages. Indeed, a conservative flux fieki is obtained by solving a mass balance equation, which facilitates incorporating complex physical processes into the model. Moreover, it follows the main-stream of the currently established simulators in reservoir engineering, which are based on unite- volume discretization These facts increase the appeal of the method for industrial and practical applications.

The M Sl- ^' V method was developed to salve eilipuc (homogeneous) equations on large and highly heterogeneous domains efficiently and has been applied for incompressible multiphase flow problems with negligible capillary pressure and gravity. The MSFV method has been modified to incorporate additional physics and ii has been applied

for compressible flow. The modified algorithm 300 consists of three main steps: computation of an approximate pressure solution., which includes the compulation of the basis functions to extract coarse-scale effective traivsmissibiiities and the solution of the coarse-scale pressure equation 3JJ); construction of conservative fine-scale fluxes 320; and solution of the transport equations 330. Especially for three phase flow with gravity, it is preferable to modify the pressure approximation ύ> order to treat ilow in presence of gravity correctly 340,

6.6.2 Mathematical Model of Multiphase Fimr

10 When considering the flow of three incompressible phases m a rigid porous medium with constant porosity, the mass balance equation of each phase α ^:;: 1 , 2- 3 is

o

(54)

5?

where ^ [L?''L ^J ] is the porosity of the medium; 5 _U [L" ¹ ZLf] the saturation; and q _a , j ! /T] 15 the source terra { ^' positive when extracted), ϊf we neglect capillary pressure and consider an isotropic porous medium, ihe volumetric flux per unit area. Uu [UT]. is given by. ihe generalized Darcv's law, i.e..

Ua = ^■■• vlεrA ^' (yp - pag ) ( 25

where /« is the α-phase relative mobility; k [L"] the intrinsic peπneabϋity, which is K) fluid independent; p [M(LTj the pressure: /λ> (M/ 1/ J the density; and g/ ^' L/T] the gravity acceieiation vector ,

Summing up Eqs (54), defining the total velocity, « "- £< _* « _* , and the total source term, q ^::: £«<:/ _* , and noting that £«5 _« , = /, we obtain

V ^■ u + q ™ 0 (56)

According to Eq, (55), the total velocity can be written as

u - - M { Vp - G } ( 57 }

where

[MfTL ² ], is the modified gravity; / - ∑«/L the total mobility; and fi, ~ λ _v f λ _r the fractional flow function of the a -phase. Once Lq. (56) has been solved for the pressure, /> , the velocities of two phases can he computed and used in the two corresponding transport equations, which have the form of Eq. (54). The saturation of the third phase car. be obtained from the constraint S^ «&J ~ I .

6.6,3 Finite- Volume Discretisation

Given a grid with M nodes and N cells, which defines a partition of the domain, ω , into /V control volumes, Cl _* can derive a set of discrete mass-balance equation by integrating Eq. (56) over each control volume, i.e., ) «JV - H + f/|rf/ ~- 0. Neglecting the source term, using Eq. (57), and applying Gauss' theorem (or divergence theorem), we can write

^■ (59.) where cω. _v ^: - c£L f)dωi is the interface between two cells. ω _f and ω; . the unit vector orthogonal to dCli, pointing from £1 Xo ω,- . In Eq. (6) a choice has to be made in order to relate the continuous pressure gradient, V/? . to the discrete potential drop between the two cells, p, ~ p_ _: . . In the standard cell-centered finite-volume scheme, a piecewise linear interpolation is used and flux continuity is enforced at the interlace; in the MSFV method localized line-scale solutions of the pressure equation are used as pressure i πterpoktors .

6.6.4 The MSfV Method 300

FlG. 12 illustrates a course grid (solid lines) together with its dual grid (dashed lines).

Referring to FIG. 12, in the MSFV method an auxiliary coarse grid with M nodes and

S λ ^; cells is constructed, which defines a partition, f>.^ ,,.. _] of the domain ω . A dual coarse grid is constructed such thai each dual coarse cell, ω" ^" ''" ^' ^, contains exactly one node of the coarse grid in its interior. The ciυal coarse grid .has N nodes, % <- [ ^■ ■ ,v.L exactly one in the interior of each coarse cell,

The modified MSFV method 300. consists of three main steps' H ⁾ computation of an approximate pressure field 320, which includes the solution of a coarse- scale pressure equation; construction of conservative fme-scak fluxes 330: and solution of the phase-transport equations 340. The pressure approximation consists of a. juxtaposition of local solutions computed on the dual cells 3Jj., To compute the fine-scale pressure approximation, a set of basis functions and a 15 correction Junction are employed 3 _. 1 _. 2. The basis functions, which are local solutions of the elliptic homogeneous equation, are used to relate the coarse-grid pressure to the fiue-seale pressure distribution and to extract the coarse-scale lransmissibiliiies 313, Die correction function, which is independent of the coarse-scale pressure, is used to account for the gravity effects and correct the coaxse-scaie operator 314 _. ,

0 Once the approximate pressure is obtained, the fluxes induced at the coarse-cell boundaries are extracted and used as Neumann boundary conditions for local problems (one in each coarse cell), which are solved to compute a conservative fine- scale J] o w field. Therefore, the .(lux approximation is defined as a juxtaposition of local .solutions computed on the coarse cells. Mass conservation is guaranteed by the 5 fact that .(luxes are co.ntintf.ous at the coarse-cell interfaces since adjacent cells share the same flux boundary conditions (with opposite sign).

The conservative total -flux field is used in the phase- transport equations, which are solved by a Schwann's overlapping method using an implicit scheme locally within the coarse cells. Since mobility depends on saturation, basis functions have to be

updated, in order io keep the MSFV method efficient it is crucial that the basis functions are update adaptiveϊv only in regions where mobility changes exceed a given threshold. Like this, moat functions can be reused for the successive step. In the following, we describe how the pressure approximation is constructed for a more 5 comprehensive dbsciiption of the method including the adapt ively criterion to update the basis and correction functions.; the construction of the flux approximation; the solution of the phase transport equations; and the coupling between pressure arid saturation equations.

Finεseai e pressure approximation _* The MSPV method relies on approximating the

I O fine- scale pressure by a juxtaposition of localizer! solutions computed In the dual ceils. /Ane , and on the representation of these solutions as

v/borc the bassR muedons, gJ*, ar^ local solutions tu

and

0

Note thai the boundary condition assigned on dQ. ^v is equivalent to require thai

V 1. -y ~ 0 (63)

where the subscript i. denotes the vector componenl perpendicular to ciV\ and

p{xή ~ PJ (64)

5 TbLs boundary condition proved to provide accurate results for a large set of numerical test cases (Jenny et al. 2003; Jenny εt ai. 2004; Limaii and Jenny 200(Sa)

Since the velocity is divergence free and V - u ^~ V ■ -u +- V|» , Eq. {6X} is equivalent to assign the solution of the reduce problem V \ a ~ 0 as Dirichlet boundary conditions on do. ^* , where the subscript a denotes the vecioi component parallel to cω ^e .

5 Coarse-scale problem. The coarse-scale pressure equation can he obtained inserting the approximate pressure, Eq, (60), ioio Eq. (59.). This yields the following set of discrete eoarse-cdl mass-balance equations;

Y ^% T«p< - Y" / λkVφl • η ds - I λk G * η d$ t € (i, N\ ( ⁶ ^

' ^ whvxe UJB trausiϊifssibilities, _i

are delϊned as in the case without gravity effects. The terms on the right-hand side of Eq (65) represent two surface source terms on cω.- , Since the coarse- scale operator, 15 Ta ,does not include gravity effects, It. yields fluxes across d£X that are incorrect. The first term on ihe rtghi-hand side of Eq. (65) represents a correction to these inaccurate fluxes and can be regarded as a local correction to the coarse-scale operator, Ti ₁ , independent of the coarse-scale pressure.

0 6.6,5 Numerical Tests:

in this section the accuracy of the MSFV method for density-driven flow is investigated by comparison with a fine-scale reference solution. We consider a I D counter-current How problem in a. homogeneous permeability field and a 2D lock- exchange problem in a heterogeneous permeability field. Io both cases no-flow 5 conditions apply to all boundaries.

TABLE 1 , Properties of the phases used for the .numerical simulations

Counter-current flow. We consider a three-phase eoumer-eurrem flow in an homogeneous permeability field. Initially, the lower part of the domain is filled with SO ⁰ ZC! heavy oil and 50% brine, and She upper part with 100% water. Quadratic mobilities are assumed for all phases; phase viscosities and densities are given in Tabic 1 , At time zero brine and oil begin to separate due so buoyancy effects, When the brine saturation decreases at the water interface, the water becomes heavier than the "mixture" below and starts moving downwards, while oil continues flowing upwards. This numerical experiment allows having in a single lest a stable interface (water/brine ), an unstable interface (water/oil), and phase separation.

The numerical solutions are computed on a ID domain of size L, which is diseretized into 54 fine cells. The coarse grid used for the MSFV method consists of 9 cells, Referring to Charts IA and 1 (B) ₁ the evolution of the vertical profiles of oil and water saturation obtained with the MSFV method are compared with the corresponding fine-scale iefeiencc solutions. H can be observed that solutions are almost identical.

ϊ ock-«\cha«ge problem We. eonsidu a tbte^-phase *-> A ^» m α)'i~.!vhju> oi ^aUT h^ht oil and j durrsrm ot iiv the oil p~t<>.e intuai'j water oeuipjej t'κ ⁵ k*ft hali oi iiu' domain, white the πght part is filled wnb oil Dtk to the oor sitx iliffcieticc, jecuculation n induced Skie m conUd^t u< tht umiitei vuπcnl flθλ ptobkin gtavitv t>igmfivdnll> tola! \ and iheteioit, αi" actuiate gldbαl λe&sui« inlunuatior is needed to cor cctK cαptυic the flow mdutcd

!<> ui the hoπ/ental t'nev-hon ihc situtat OJI I>! ^! he dunπn) phase, whwh t^ imU.Jh not present in tiu sjitt-m, i*- ubtaiueci πoai the cuJibtiaiπcd X i< j» u^ed to cUevk tok'iante ( _^ 'uadxatic lnobilujcs arc ds*-unieJ for al nlv->es phase >S*-CO-ϊ1IC^ and densities ^rc y^ en π 1 v'b.c 1

HK numtπca sol ition^ sire computed υι a 2D doitias i o* st/e I ^■ » /, w^ich t> disercϊ 'ed mto ^v t4 > >4 lint, s ^i >s ilv coarse jrui u^c^ iot ihc MSl-X r\cihod c^ns ^ s oi 0 v 9 ₍ \>sj _s ] ^ ₁ , ^eosretnc ⁱ ne-;n o ^f t ic Vtaoguicous p> uncabihtx JcId (I % PD) is k - 13 10 "'n ⁷ iiul tte t rr,

H ⁽ Ji; 13 V'31, iiUJs'ialv xsαici-pnas^ »λtuiaι _* oπ t,»n»tour itne-, UJ 001 02"> 0"» 0 ^" "> 0 and C 999) at tluct. dirtcϊent ume--, xsnh Oit, unt— caio solution (&ohJ tontouist an J the tontouis) Md HP illustrates tb _κ ηatusal Sogauthni of

- -P

the heterogeneous permeability field. In FIG. 3, the evolution of the saturation distribution obtained with the IVISFV method is compared with the line- scale reference solutions, Shown are water-saturation contour lines at three different times. Note the sharp drainage front (oil invasion) and the expansion wave behind the wetting front (water invasion) due to viscosity difference. The two solutions are in excellent agreement even after large simulation times, which proves that the velocities of the two invading fronts are accurately captured by the MSFV method, The saturation of the dummy phase always remains smaller that the error tolerance of the phase-transport equations.

An accurate treatment of densilv-dπven flow m the IVISFV method has been achieved adding a correction function to the basis-function interpolated pressure. This correction yields very accurate fine-scale pressure fields. The numerical experiments performed for density-driven flow problems (counter-current flow and iockexehange) demonstrate that the MSFV solutions for pressure and saturation are in excellent agreement with the corresponding fine-scale reference solutions.

While in the foregoing specification tills invention has been described in relation to certain preferred embodiments thereof, and many details have been set forth for purpose of illustration, it will be apparent to those skilled in the art thai the invention is susceptible io alteration and that certain other details described herein can vary considerably without departing from the basic principles of the invention,

APFEKDIX A

ϊ. FLOW PROBLEM

A, One Phase Flow Fluid flow in a poxous media can be described bv the elliptic problem:

V - (A - Vp) - f υπ ω (Al )

where p is the pressure, λ is the mobility coefficient (permeability. K ₅ divided by fluid viscosity, μ) and ω is a volume or region of a subsurface which is to be simulated. λ source term / represents λvelis, and in the compressible case, lime derivatives. Permeability heterogeneity is a dominant factor in dictating the flow behavior m natural porous formations. The heterogeneity of permeability K is usually represented as a complex multi-scale function of space. Moreover, permeability K tends to be a highly discontinuous full tensor. Resolving the spatial correlation structures and capturing the variability of permeability requires a highly detailed reservoir description.

rhe velocity u of fluid Row is related to the pressure field through Darcy's law:

u ^ -λ - Vp . <;A2)

On the boundary of a volume. (3ω _; the flux q — u - v is ¹ SPeCiFiCd, where v is the boundary unit normal vector pointing outward. Equations (1) and (2) describe incompressible How in a porous media. These equations apply for both single and multi-phase flows when appropriate interpretations of the mobility coefficient λ and velocity is are made. This elliptic problem is a simple yet representative description of the type of systems that should be handled efficiently by a subsurface flow simulator. Moreover, the ability to .handle this limiting case of incompressible flow ensures that compressible systems car: be treated as a subset,

B. Two ^" Phase Flow The flow of two incompressible phases in a heterogeneous domain may be mathematically described by the following:

φ ^dS « . % op

~ ^~ Q ifo; at ox; Mo ox;

(A3)

ds», ^' K Ar op

φ ^'( :lW at OX; M-w <&/ }

on a volαrøe ω, where p is the pressure, S ₁ . _h are the saturations (the subscripts o

and w stand for oil and water, respectively) with 0 < S ₁ , . _, , ≤ 1 and S, -)- -S\,, ϊϊ 1 , k ϊs the heterogeneous permeability, k _r are the rehuive permeabilities (which are functions of 5',, _l( . )., ,„/.,„ the viscosities and {/,.,, are source terms which represent the welis. The system assumes thai capillary pressure and gravity are negligible. Equivalent!)', system (3) can be written as;

^~ V-ιι^q _o +q _{w {A4)}

φ - ^! - + V-i ^• ' ^■ - u I- -q.

on ^ with

i. = ~^>. _{A6)

and the total mobility

λ^fc^+M. _(A?j

Equation f ^' 4) is known as the "pressure equation" and equation (5) as the "hyperbolic transport equation." Agam, equations (4) aod (S) are a representative description of the type of systems that should hε handled efficiently by a subsurface flow simulator.

Siϊcb How simulators, and techniques employed to simulate flow, are well known to

those skilled in the art and are described in publications such as Petroleum Reservoir Simulation. K. Aziz and A, Settari, Stanford Bookstore Custom Publishing, 1999,

IL MULTI-SCALE FiNLTE-VOLUMK (MSFV) METHOD

A. MSFV Method for One Phase Flow

J , Finite- Volume Method

A cell centered finite-volume method will now be briefly described. To solve the problem of equation (] ), the overall domain or volume ω is partitioned into sraaJkr volumes {ω, ; , A fmite-voluroe solution then satisfies

jV u«'ω - [.. u- vt/r ^ - f fall (AK)

for each control volume ω _; - where v is the unit normal vector of the volume boundary <?ω. pointing outward. The challenge is to find a good approximation for ύ-v at oω, , In general, the Ii ux is expressed as:

Equation (9) is a linear combination of the pressure values, ^ , in the volisines {ω. J- oϊ the domain Q , l he total number of volumes is ⁿ and ^ denotes traπstnissibility between volumes \ω,, \ . By definition, the fhixes of equation { 9} are continuous across the interfaces of the volumes Jω, f and, as a result the finite-volume method is conservative.

2, Construction of. the Effective Transmiϋsibilifies

The MSFV metiiod results in iπuiti-point .steαcϋs for coarse-scaie fluxes. For the folio wing description, an orthogonal 2D grid 20 of grid cells 22 is used, as shown in FIG. Al . An underlying fine grid 24 of tine grid cells 26 contains the fine-scale

permeability K information. To compute the transmissibilities T between coarse grid eelis 22, a dual coarse grid 30 of dual coarse control volumes 32 is used. A control volume 32 of the dual grid 30,ω , is constructed by connecting the- mid-points of four adjacent coarse grid cells 22. To relate the fluxes across the coarse grid cell interfaces 34 which Ms inside a particular control volume 32, or ω , to the finite -volume pressures yJ ^s (& - !,4) in the four adjacent coarse grid cells 22, a local elliptical problem in the preferred embodiment is defined as

For one skilled in the art. the method can easily be adapted to use a local parabolic problem .

Fur an elliptic problem, Dirichlet oi Neumann boundary conditions are Io be specified on boundary DQ . Ideally, the imposed boundary conditions should approximate the true How conditions experienced by the sub-domain in the full system. These boundary conditions can be time and Dow dependent. Since the sub-domain is embedded in the whole system, WaHsirom et al. (T. C. Wallsvrom, T. Y. Hou, M. A Christie, L, J. Duriofsky, and D. H. Sharp, Application of a new two-phase upseahng technique to realistic reservoir cross sections, SPC 51939, presented at the SPE Symposium on Reservoir Simulation, Houston, 1999) found that a constant pressure condition at the sub-domain boundary tends to overestimate flow contributions from high permeability areas. If the correlation length of permeability is not much larger than the grid size, the flow contribution from high permeability areas is not proportional Io the nominal permeability ratio, The transmissibility between two cells is a harmonic mean that is closer to the lower permeability, As a result, uniform flux conditions along die boundary often yield much better numerical results for a sub-domain problem than linear or constant pressure conditions,

Hou and Wu (T. Hou and W, Ii. Wu, A muiii-scate finite element method for elliptic problems in composite materials and porous media, J. Corπp, Fhys, 134: 169-189, I W7) also proposed solving a reduced problem

A = 0, ex. dx

(.Al 1 )

to specify the boundary conditions for the local problem. The subscript t denotes the component parallel to the boundary of the dual coarse control volume 32 QV CX.1 , For equation ( I i) and fox the following part of this specification, Einstein summation convention will be used. The elliptic problem on a control volume C), with boundary conditions of equation (1 1 ) on Xl can be solved by any appropriate numerical method, fn order to obtain a pressure solution thai depends linearly on the pressures p" {j --- ^• IA) _y this preferred embodiment solves four elliptic problems, one for each cell-center pressure. For instance, to get the solution for the pressure /V ^! in the coarse grid cell having node 1 at its center, p ^Ϋ ~ δ _H . is. set. The four solutions provide the dual basis functions φ" (k ■■■ 1.4) in control volume ω , and the pressure solution of the local elliptic problem in a control volume ω is the linear combination

Accordingly, the flux q across J he grid ed! interfaces can be written as a linear combination

where q" {k ~ \λ) are die flux contributions Jroin the corresponding dual basis functions, given all φ* (k - 1,4) from all control volumes ω . The effective transmissibilities T arc commuted., which can he used for finitc-voiume simulations

b> assembling the Ou.* contributions, in the preferred embodiment integral flux contributions across the cell interfaces 34.

Note that the domain ω can have any fine-scale distribution of mobility coefficients λ. . Of course the boundary condition given by equation ( 1 1) is an approximation that allows one to decouple the local problems. The MSFV and global fme-sealε solutions are identical, only if equation (1 1 ) happens to capture the exact fme-scaie pressure solution. However, numerical experiments have been performed which indicate That equation (1 1) is an excellent approximation of the boundary condition

Although the MSFV approach is a finite-volume method, it resembles the multi-scale finite-element method of Wu and HULL briefly mentioned above. The construction of the dual basis functions is similar, though in the present MSFV method they are represented on the disai coarse grid rather than on the boundary of a finite element. A significant difference is thai the present MSFV method is a cell-centered finite-volume method and is conservative. On the other hand, (he mass matrix in the multi-scale finite-element method is constructed based on a variational principle and does not ensure local conservation. In the next section., the importance is illustrated of a fine-scale velocity field that is conservative.

3. Reconstruction of a Conservative Fine-Scale Velocity Field

Fluxes across the coarse cell interfaces 34 can be accurately computed by roulti-seale transmissibilύies T, In some cases, k is interesting to accurately represent the small-scale velocities u (e.g., to predict the distribution of solute transported by a fluid). A straightforward approach might appear to be to use the dual basis functions

φ of equation (12). However, (lieu the reconstructed line-scale velocity field is, in general, discontinuous at the. cell interfaces of the dual grid 30, Therefore, large errors can occur in the divergence field, and local mass balance is violated. Note that mass conservation is always satisfied for the coarse solution using the present MSFV method.

The construction of a second sel of local fine-scale basis> functions φ will now be described which is fully consistent with the fluxes q across the cell interfaces given by the dual basis functions φ . This second set of fine-scale basis functions φ allows a conservative fine-scale velocity field to be .reconstructed

FIG. A2 shows a coarse grid 20 with nine adjacent grid cells 22 and a corresponding duaj grid 30 of dual coarse control volumes 32 or ω , For indexing purposes, these particular cells and corresponding dual volumes shall now be identified with numerals "ϊ-9 ^5' and letters " ¹ A-D" at their respective centers. Also shown is the underlying fine grid 24 of fine grid cells 26. The coarse grid, having the nine adjacent coarse cells 1 - 9, is shown in bold solid lines. The corresponding dual grid 30 of dual coarse control volumes A-D are depicted with bold dashed lines. The underlying fine grid 24 of fine grid ecus 26 is shown with thin dotted lines.

To explain the reconstruction of the fine-scale velocity, the mass balance of the center grid cell 5 is examined. The coarse scale pressure solution, together with the dual basis functions φ , provides the fine-scale fluxes q across the interfaces of coarse cell 5.

To obtain a proper representation of the fmc-scaie velocity field in coarse cell 5, fi) the fϊne-scaie fluxes across an interface of coarse ceil S must match, and (ii) the divergence of the fine-scale velocity field within the coarse volume satisfies

construction of the fine-scale basis functions φ' wiii be described. Each coarse celi pressure ρ{i ~ 1.9) contributes io the fine-scale fhjx q For example, let the contribution of the pressure in cell 2 to the flux q in grid cell 5 be q'" ^; , Note that t) ^Ui is composed of contributions q ^l~ -' and q^' ' coming from She dual basis functions

associated with node 2 of volume A arid volume B, respectively, Io compute the fine-scale basis function φ' associated with the pressure in a coarse cell ^? , p' ■■■ S ₁₁ is set, and the pressure .field is constructed according to the following equation.

The fine-scale iluxes f/ are computed from the pressarε iieicl These iltsxes provide the proper boundary condition for computing ihe fine-scale basis function φ' . To solve the elliptic problem

V - (A -Vp) = /' _oπ ω _{5 (A l 6)}

with the boundary conditions described above, solvability must be ensured. This is achieved by setting

which is an equally distributed source term within Q ₅ . Finally, the solution of the elliptic problem. ( 16) and (J 7), is the fine-scale basis function φ' for coarse cell 5 associated with the pressure in volume ^J" . The small-scale velocity field is extracted from the superposition

P - σ P ^J' φ{

(Al 8)

For incompressible flow, this velocity field is divergence free every where. Computing the line-scale basis { ^' unctions φ ^! requires solving nine small elliptic

problems, which are of the same size as those for the transmissibiliiy calculations, Note that this step is a preprocessing step and has so be done only once. Furthermore, the construction of the fine-scale basis functions φ' is independent and therefore well suited for parallel computation. The reconstruction of the fine-scale velocity field is a simple superposition and is; ideally performed only in regions of interest.

Alternatively, a conservative line-scale velocity field may also be constructed directly in place. This construction may be performed as follows: (i) compute the fine-scale fluxes across the coarse cell interfaces using the dual basis functions with the pressures for the coarse cells; (ii) solve a pressure equation on each of the coarse ceils using the fine-scale fluxes computed in step (f) as boundary conditions to obtain fine- scale pressmes; (iii) compute the line-scale velocity field from Darey ^' s law using the fine-scale pressures obtained in step (ii) with the underlying fine-scale permeability. The pressure solution of step (ii) may be performed on a system with larger support (e.g., by over-sampling around the coarse eellj.

A conservative imc-scaic velocity field may also be constructed directly in place as follows: (1 ) Compute the fluxes across the coarse cell boundaries using the- dual basis functions with the coarse pressures. (2) Solve the pressure equation on each of the coarse cells using the fine-scale fluxes computed in Step I. as boundary conditions. (3) Compute the tine-scale velocity Held from Darcy ^' s law using the fine-scale pressure obtained m Step 2 with the underlying fine-scale permeability. The pressure solution of Step 2 may be performed on a system with larger support (e.g., by over-sampling around the coarse cell}.