a mixed finite element--finite volume formulation of the black-oil model

A MIXED FINITE ELEMENT–FINITE VOLUME FORMULATION

OF THE BLACK-OIL MODEL∗

LUCA BERGAMASCHI† , STEFANO MANTICA‡ , AND GIANMARCO MANZINI§

SIAM J. SCI. COMPUT. c© 1998 Society for Industrial and Applied MathematicsVol. 20, No. 3, pp. 970–997

Abstract. In this paper a sequential coupling of mixed finite element and shock-capturingfinite volume techniques is proposed, in order to numerically solve the system of partial differentialequations arising from the Black-Oil model. The Brezzi–Douglas–Marini space of degree one (BDM1)is used to approximate the Darcy’s velocity in the parabolic-type pressure equation, while the systemof mass conservation laws is solved by a higher order Godunov-type scheme, here extended to triangle-based unstructured grids. Numerical results on 1-D and 2-D test cases prove the effectiveness andthe robustness of the coupling, which seems particularly suited to handling high heterogeneities andat the same time accurately resolving steep gradients without spurious oscillations.

Key words. Black-Oil, mixed finite elements, TVD finite volumes, shock-capturing schemes

AMS subject classification. 65M60

PII. S1064827595289303

1. Introduction. The accurate prediction of the performance of a given reser-voir under a particular recovery strategy is a crucial factor for its economical exploita-tion. In the last decade several mathematical models describing physical phenomenaoccurring in reservoirs have been proposed, and a variety of numerical methods fortheir approximate solution has been investigated rather extensively.

The mathematical models usually considered describe the fluid flowing in a reser-voir as composed by many chemical components which combine in different phases.

In this paper we mainly address the compressible model where three independentcomponents (oil, gas, and water) form the three phases (liquid, vapor, and aqua)present in the reservoir. This model is usually known under the name of “Black-Oil,”and its main feature is the description of mass-exchange processes between all phasesallowing phase-missing cases.

The two chemical components, oil and gas, represent ideal mean hydrocarbons.At standard pressure and temperature (“stock-tank” conditions or STC), the “oil”hydrocarbon will be present in the liquid phase, while the “gas” hydrocarbon is in thevapor phase. Within the reservoir, pressure and temperature are different than STC:oil and gas combine in a different mass density ratio according to a thermodynamicequilibrium constraint.

The Black-Oil model consists of a set of partial differential equations describingthe conservation of mass for each component and the time evolution of the phase pres-sures and velocities. These equations are strongly nonlinearly coupled. Nevertheless,as shown by Trangenstein and Bell (TB) [35], they can be separated into two sets

∗Received by the editors July 20, 1995; accepted for publication (in revised form) October 31,1997; published electronically December 23, 1998.

http://www.siam.org/journals/sisc/20-3/28930.html†Department of Mathematical Models for Scientific Applications, University of Padua, via Belzoni

7, 35131 Padua, Italy ([email protected]). The work of this author was supported in part by ECcontract IC15 CT96-211.

‡Agip S.p.A. Reservoir R&D (RIGR), via Foliani 1, 20097 S. Donato Milanese, Milan, Italy([email protected]). The work of this author was supported in part by joint project (DSP 174)Agip-CRS4.

§CRS4, Applied Mathematics and Simulation Group, Via N. Sauro 10, Cagliari, Italy([email protected]). Current address: IAN-CNR, via Abbiategrasso 209, 27100 Pavia, Italy.

970

NUMERICAL FORMULATION OF THE BLACK-OIL MODEL 971

with a very different nature. The conservation of mass equations form a nonlinear hy-perbolic system if dispersion effects are neglected, or a strongly advection-dominatedparabolic system if these latter ones are included. Instead, in the volume discrepancyformulation adopted by TB, phase pressures obey a parabolic equation which tendsto be elliptic in the limit of null total (fluid plus rock) compressibility.

A reliable numerical model must be able to take into account the complexityof a real reservoir. The irregularity in the geological and geometrical morphologyaffects the computational domain and empirical data such as the permeability andthe porosity fields involved in the calculations. In the present approach, unstructuredtriangle-based grids have been preferred because of their inherent flexibility and ofthe possibility of introducing in a straightforward manner local refinements aroundcomplex geometrical patterns.

The reservoir permeability field is a full tensor quantity that presents very largelocal variations, up to 8 or 10 orders of magnitude [23]. This results in highly dis-continuous terms in the discretized form of the equation that may lead to inaccuratesolutions if not appropriately treated. Moreover, the strong nonlinear hyperbolic na-ture of the mass conservation equations is responsible for the formation of sharpfronts moving through all the reservoir. Therefore, an effective numerical resolutionof the Black-Oil model deserves special attention, and appropriate methods should bedevised to ensure both space and time accuracy.

The experience acquired so far in numerically treating flow models in porous mediademonstrates the effectiveness of blending mixed finite elements for the parabolicequations and shock-capturing finite volume methods for the hyperbolic equations [21,15]. These numerical strategies directly exploit the mathematical properties of themodel, taking proper care of the completely different nature of the equations in thesystem. In particular, this approach is capable of accurate approximation of pressureand velocity flow fields in the presence of strong reservoir heterogeneities as well as ofcapturing unsteady discontinuous fronts.

Mixed finite element methods [14] for the simultaneous solution of fluid pressureand velocity yield a very accurate numerical discretization and have been extensivelystudied; see [17, 26]. In our framework, velocities are approximated using the low-est order Brezzi–Douglas–Marini elements (BDM1) with piecewise-constant pressure.Their accuracy in treating heterogeneous permeability situations has been evaluatedin a previous work [11].

Shock-capturing finite volume methods exhibit the property of accurately resolv-ing steep gradients without spurious numerical oscillations while taking numerical dis-sipation effects at a very low level. A conservative formulation is usually considered,allowing an accurate numerical prediction of the moving discontinuities appearing inthe physical solution. The discretization is based on an evaluation of the fluxes by gen-eralizing the Engquist–Osher numerical flux for systems [24, 25] and on a second-ordertriangle-based TVD reconstruction of the mass concentration variables.

The solution of the model equations is advanced in time via a sequential method.First, pressure and velocity are calculated via an implicit Euler scheme assuming thatmass concentrations are known. Then, an explicit predictor–corrector scheme withcharacteristic tracing updates the hyperbolic mass conservation equations using thevelocity field computed in the previous stage.

As yet, the idea of splitting the equations into two subsystems to be solved sep-arately in an alternate way has been exploited by researchers in porous media flowsand is documented in literature (see, e.g., [9]). It is the purpose of this work to de-

972 L. BERGAMASCHI, S. MANTICA, AND G. MANZINI

velop a new sequential approach for the Black-Oil formulation, considering in thisframework both mixed finite element and unstructured shock-capturing finite volumediscretizations.

The major issue of this paper, from the numerical point of view, relies on illus-trating by direct numerical experiences the effectiveness of this methodology whenapplied to the complex physical phenomena predicted by the Black-Oil model.

Our discretization model generalizes the Durlofsky approach proposed for slightlycompressible two-phase flows [22].

Even if mixed finite elements are usually accepted for solving this kind of problem,the elements that have been adopted (i.e., BDM1 [13]) in the computation of thevelocity field guarantee a higher level of accuracy at a given computational cost [11]than usual lower order Raviart–Thomas elements [32]. However, the idea of usingfinite volume methods in oil modeling problems is relatively new. Bell, Colella, andTrangenstein [7] and Bell, Dawson, and Shubin [8] did some pioneering work in thisdirection though their discretizations rely upon a finite difference formulation of themodel equations. It is worth pointing out that the mixed finite element formulationfor pressure and velocity is particularly well suited as it directly generates up toan optimal order of accuracy the interelement fluxes which in turn can be easilyincorporated into the finite volume solver. More importantly, we stress again thatthis blended approach allows one to exploit the different nature of the two governingequations by two different methods, each of them being the best suited for the equationat hand.

The paper proceeds as follows. In section 2 we describe the mathematical formu-lation of the Black-Oil model as proposed by TB. Mass density conservation equationsand Darcy’s law for phase velocities and pressures are introduced and a total velocityis defined. In section 3 we present the sequential approach and derive the parabolicpressure equation via the discrepancy error formulation. In section 4 we report thenumerical discretization and time evolution scheme of this equation using mixed finiteelements. In section 5 we illustrate the discretization of the mass conservation equa-tions using an explicit predictor–corrector time marching scheme with characteristictracing and a triangle-based TVD finite volume representation of the flow variables.In section 6 we document the effectiveness and the capability of the proposed ap-proach by a set of test cases describing realistic situations. A final discussion on theperformances of the present method and its future development closes the paper insection 7.

2. The Black-Oil formulation. In this section we formulate the equationsdescribing the Black-Oil model for reservoir simulation. A reservoir fluid is consideredcomposed of three different chemical components, called “oil,” “gas,” and “water,”whose related quantities will be (resp.) indicated by the subscripts o, g, and w.

Oil and gas represent ideal mean hydrocarbons and can combine together andwith water to form three different phases. These phases, called “liquid,” “vapor,” and“aqueous,” and indicated by the subscripts l, v, and a, simultaneously flow throughthe reservoir.

At STC, the liquid and vapor phases are, respectively, composed of oil and gashydrocarbon, so we can identify each phase with its principal components. At reservoirpressure and temperature, oil can also be present in the vapor phase, gas in theliquid and aqueous phase. This mass redistribution is governed by thermodynamicsconstraints, and situations of undersaturation, in which some of the phases are notformed, must be appropriately treated by the simulator.


One of the major features of the Black-Oil model is its capability of predictingmass transfer effects between phases when simulating primary—pressure depletion—and secondary—water injection—recovery. These capabilities have contributed to im-posing the Black-Oil method as the mathematical model mainly used in petroleumreservoir engineering.

The Black-Oil mathematical formulation introduced by TB is based on the fol-lowing assumptions:

• isothermal thermodynamic phase-equilibrium: how components combine toform phases;

• Darcy’s law: how the phase volumetric flow rates are related to the corre-sponding phase pressures;

• equation of state: the fluid completely fills the pore volume;• mass conservation: the total mass of each chemical component is conserved

through phases.

The complex thermodynamic phase-equilibrium occurring in the Black-Oil modelis taken into account by a set of thermodynamic relations which express how anychemical component redistributes over the phases and how the phase fluid occupiesthe porous volume. These relations are usually given as a function of the liquid pressurein the reservoir. Each phase is described by a velocity and a pressure field, relatedby Darcy’s law. Moreover, the phase pressures can be related to a total velocityfield—introduced as the sum of the phase velocities. A further relation among pressure,phase velocities, and chemical mass densities is provided by the equation of state. Theequation of state is usually formulated in terms of the phase saturations—the ratiobetween the volume occupied by the phase and the total pore volume. This equationstates that the total pore volume, where phases flow, is entirely filled by the fluids.However, a sequential approach cannot satisfy all of the flow equations exactly ateach step of the computation and one of the constitutive relations of the model hasto be relaxed. Following TB, we linearize in time the equation of state so that it isonly satisfied approximately, thus introducing a volume discrepancy error, and imposeexactly the thermodynamic equilibrium, Darcy’s law, and the component conservationequations.

2.1. Primary and secondary variables. When mathematically formulatingthe Black-Oil model, it is useful to distinguish between a set of “primary” and “sec-ondary” variables. Time evolution of the previous ones is directly governed by theequations of the model, while the secondary variables depend on the first ones bysome functional relations. Primary and secondary variables can again be subdividedfor sake of presentation in “component unknowns” and “phase unknowns,” due to thephysical nature of the quantities they are related to.

Component primary variables are the mass concentration densities (mass per unitpore volume) indicated by z = [zo, zg, zw]T . These unknowns express the density ofeach chemical component integrated over the different phases in which it is distributed.Let us also introduce as “secondary variables” the generic element zij giving theconcentration density of the component i in phase j, and the component densitymatrix Z:

Z = (zij) =

zol zov 0zgl zgv zga0 0 zwa

.(2.1)


The matrix Z and the vector z satisfy the following condition:∑

j=l,v,a

zij = zi, i = o, g, w, or Ze = z,(2.2)

with e = [1, 1, 1]T , which simply casts the component conservation integrated overthe phases.

In our mixed formulation, phase primary variables are the pressure p and the totalvelocity ~vt given as the sum of the phase velocities: ~vt = ~vl + ~vv + ~va. These latterones, collected into a phase velocity vector ~v = (~vl, ~vv, ~va)

T, are “secondary” variables

and can be obtained at any time step once the total velocity has been updated. Phasesecondary variables are also the saturations u, defined as the ratio between the porevolume occupied by the phase and the total pore volume. As a consequence of thesequential formulation, real saturations no longer satisfy the equation of state and aresubstituted by a vector of “effective saturations” s—which will be referred to simplyas “saturations” in the following. The saturations s depend on the saturations u by

sj =uj

ul + uv + uaj = l, v, a .(2.3)

The equation of state is finally replaced by an obvious normalization condition, withno real physical meaning.

2.2. Thermodynamics and saturation equations. At surface conditions,the liquid phase is only composed by the oil hydrocarbon, the vapor phase by the gashydrocarbon, and the aqueous phase by water. Hence, it is possible to identify for aphase a principal component, i.e., the component present at STC. Let us now intro-duce a dimensionless matrix R whose generic term rij represents the ratio betweenthe mass of the component i and the mass of the principal component of the phase j.The density component matrix Z is then written as

Z = RDz =

1 Rv 0Rl 1 Ra

0 0 1

zol 0 00 zgv 00 0 zwa

,(2.4)

where

Rl =zglzol, Rv =

zovzgv

, Ra =zgazwa

,(2.5)

and Dz = diag[zol, zgv, zwa]. In this way we write the density component matrix Zas a function of the primary variable z and the entry of matrix R, which are givenas functions of the pressure p, either explicitly or (more usually) tabulated in anempirical way. In what follows it will be useful to define the matrix T as the leftinverse of the matrix R (i.e., I = TR). Let us also introduce three formation volumefactors, defined as the ratio between phase saturations and densities of the principalcomponents. These terms are also known as empirical functions of the pressure

Bl =ulzol, Bv =

uvzgv

, Ba =uazwa

.(2.6)

Or, in matrix form:

B = diag(Bl, Bv, Ba) = DuD−1z .(2.7)

The definition of the matrices R and B allows us to express an explicit dependenceof the two variables z and u as

u = BTz.(2.8)


2.3. Undersaturation. Although the Black-Oil model involves three compo-nents, this does not necessarily mean that three phases are always formed in the wholereservoir. When the three components are present, two cases of undersaturation arepossible:

• all gas dissolved into liquid and/or aqua;• all oil dissolved into vapor.

There are no other cases of undersaturation because water is not allowed to appear ineither liquid or vapor. Furthermore, only one undersaturated situation is allowed ata time, because oil must be present in liquid or in vapor phase and it cannot dissolveinto the aqueous phase.

We say that a phase is missing if the corresponding component of the diagonalmatrix Dz (or equivalently, by (2.4), of the vector Tz) is negative.

Let us analyze briefly the two possible cases of undersaturation.

2.3.1. Vapor phase missing. The physical meaning of the negative vapor com-ponent of Tz is that the fluid pressure p is higher than the bubble-point pressure pb,at which vapor phase forms. This bubble-point pressure can be found by solving forp the relation

zgv ≡ (T (p)z)v ≡ −Rl(p)zo + zg −Ra(p)zw = 0.(2.9)

We require that both Rl and Ra are nondecreasing functions of pressure so that (2.9)can always be solved.

When the vapor phase is missing we would also like to have zov = 0 and, conse-quently, Rv = 0. It is therefore useful to define a new dimensionless matrix R

R =

1 0Rl Ra

0 1

,(2.10)

where Rl ≡ Rl(pb) and Ra ≡ Ra(pb). Now matrix R is no longer nonsingular, so weneed to define a new matrix T which plays the role of T in the saturated case yieldingT R = I:

T =

(

1 0 00 0 1

)

.(2.11)

Since there is no vapor phase, all of the oil must be in liquid and all of the water mustbe in aqua, giving

u = [ul, ua]T , Dz = diag(zozw), B = diag(Bl, Ba),(2.12)

and

Z = RDz =

zo 0Rlzo Razw

0 zw

.(2.13)

2.3.2. Liquid phase missing. When the liquid component of Tz is negativewe have that the liquid phase is missing. To force this component to be zero we needto solve

zgl ≡ (Tz)l ≡ Rvzg −RvRazw + zo = 0,(2.14)


which gives

Rv(p) =zo

zg −Ra(p)zw.

As in the vapor missing case we can define R and T as follows:

R =

Rv 01 Ra

0 1

, T =

(

0 1 −Ra

0 0 1

)

.(2.15)

Since there is no liquid phase, the amount of gas in vapor zgv must be equal to zg−zgaand all the water must be in aqua; that is,

u = [uv, ua]T , Dz = diag(zg −Razw, zw), B = diag(Bv, Ba),(2.16)

and

Z = RDz =

Rv(zg −Razw) 0zg −Razw Razw

0 zw

.(2.17)

2.4. The equation of state. The basic physical principle defined by the equa-tion of state is that the fluid which flows in the reservoir completely fills the rock porevolume. This fact is formally expressed by the constraint that the total sum of thephase volume is equal to the total porous volume and given by

ul + uv + ua = 1.(2.18)

The saturation u depends on the pressure and the component mass densities. Sincethese latter fields are time dependent, the equation of state can be directly exploitedto get an equation for the time evolution of the pressure. In a truly coupled model,the equation of state should be solved simultaneously with the other equations of themodel. We adopted here the approach proposed by TB, where a linearized form of theequation of state is considered. This approach yields an equation for the time variationof the pressure at a price of introducing a “volume discrepancy error” because theequation of state is not exactly satisfied. The linearization of the equation of statealso allows to separate the system of equations of the Black-Oil model in two subsetswhich can be treated sequentially, using ad hoc numerical methodologies.

2.5. Darcy’s law for the total velocity. The three phase velocities ~vj arerelated to phase pressures by Darcy’s law:

~vj = −Mj

[

~∇(p+ pjl) − ρjg~∇h]

, j = l, v, a,(2.19)

where p is the pressure for the liquid phase, pjl is the difference between the pressurein the phase j and the liquid phase, g is the gravity, h is the depth, and ρj is thedensity in phase j. In what follows, in order to simplify the notation, we will neglectthe capillary pressure, which is usually modelized as a function of the saturations.

Let us define the phase mobility Mj and the fractional flow fj as

Mj = Kkrjµj, fj =

krj/µj∑

s krs/µs=Mj

Mt, Mt =

∑

j

Mj ,(2.20)


where K is the absolute permeability tensor, krj the relative permeability of the phasej, and µj the phase viscosity. We can introduce a total velocity ~vt as a function of theliquid phase pressure:

~vt =∑

j

~vj = −∑

j

Mj

[

~∇p− ρjg~∇h]

.(2.21)

For subsequent convenience we define the phase vector f

fj = fjρj , j = l, v, a, and f = [fl, fv, fa]T ,

and we rewrite (2.21) in the compact format:

M−1t ~vt = −~∇p+ eT fg~∇h .(2.22)

Inversely, the velocity of phase j can be written as a function of the total velocity(2.21):

~vj = fj

~vt −∑

s 6=j

Ms

[

(ρs − ρj)g~∇h]

, j, s = l, v, a.(2.23)

Moreover, we set

tj = fj∑

s 6=j

Ms(ρj − ρs) g, j, s = l, v, a,

and

f = [fl, fv, fa]T , t = [tl, tv, ta]

T

to obtain a compact expression for the phase velocity vector

~v = f~vt + t~∇h.(2.24)

The quantities µj and krj are functions of the pressure p and of the saturationss, respectively. Usually, the relative permeabilities are given as tabulated values inconjunction with special models for three-phase flow (see the Appendix).

2.6. Mass conservation equations. The conservation of the mass for the threedifferent components is given by the balance of the contribution of each phase to themass fluxes. In the saturated case they read

∂(zoφ)

∂t+ ~∇ ·

(

zol~vlul

+zov~vvuv

)

= 0,

∂(zgφ)

∂t+ ~∇ ·

(

zgl~vlul

+zgv~vvuv

+zga~vaua

)

= 0,(2.25)

∂(zwφ)

∂t+ ~∇ ·

(

zwa~vaua

)

= 0 .

Using the definitions of Z and Du previously introduced, these relations can be ex-pressed in a more compact vector form, which is valid also in the case of undersatu-ration, provided that ~v is the vector of existing phase velocities such as

∂(zφ)

∂t+ ~∇ · (ZD−1

u ~v) = 0.(2.26)


It is convenient to reformulate the mass conservation laws given by (2.26), in terms ofthe matrices R and B. These matrices are known functions of the pressure p, which isthe solution at each time step, together with ~vt, of the mixed pressure-velocity system.We have

∂(zφ)

∂t+ ~∇ · (RB−1~v) = 0,(2.27)

which holds even in the general case of missing phases, once the proper definition ofundersaturated B and R is employed.

3. The sequential approach. The sequential method is one of the approachesused in literature to solve the equations of the Black-Oil model and relies upon alinearization of the equation of state. Such a method assumes that the state equationis not exactly satisfied and introduces in this way a volume discrepancy error. However,it yields a correct split of the equations into a parabolic and a hyperbolic set.

Numerically, the mixed equations provide an expression for the pressure and thetotal velocity, taking at the previous time step the values for the saturations and thedensity z. Then, the hyperbolic system is solved as a function of z. The nonlinearcharacter of this equation depends on the phase velocities, which are functions oftotal velocity and saturations. The mass component densities are evolved in time byfreezing pressure and total velocity as solution of the mixed equation at the currenttime step.

3.1. The pressure equation. Following TB, we develop a differential equationfor the pressure assuming that the state equation is not exactly satisfied, and intro-ducing in this way a volume discrepancy error. We start by writing the state equationas a function of the pressure at the previous time step and of the component densityvector z. By the chain rule, we can write the time derivative of u as

∂u

∂t=∂u

∂p

∂p

∂t+∂u

∂z

∂z

∂t.(3.1)

Introducing a linear approximation of ∂u∂t in the time interval ∆t

∂u

∂t=

un+1 − un

∆t,(3.2)

we impose that at time tn+1 the state equation is exactly satisfied, or eTun+1 = 1.Equation (3.1) can be rewritten, after multiplication by eT , as

1 − eTu

∆t= eT

∂u

∂p

∂p

∂t+ eT

∂u

∂z

∂z

∂t,(3.3)

where, for simplicity, we have dropped the superscript n. Actually, the pressure p iscomputed in order to correct the errors introduced in the state equation. Substituting∂z∂t from (2.27) in (3.3), we get

eTu− 1

∆tφ =

(

−φ eT∂u

∂p+ eT

∂u

∂zz∂φ

∂p

)

∂p

∂t+ eT

∂u

∂z~∇ ·

(

RB−1~v)

.(3.4)

Observe now that, by (2.8), the saturation u is a homogeneous function of the firstdegree in z. In fact,

∂u

∂z= BT, which implies

∂u

∂zz = u.(3.5)


Equation (3.4) can then be written as

eTu− 1

∆tφ =

(

−φ eT∂u

∂p+ eTu

∂φ

∂p

)

∂p

∂t+ eTBT ~∇ ·

(

RB−1~v)

,(3.6)

where the phase velocity vector ~v is related to the total velocity ~vt by (2.24).

3.2. Splitting errors. The splitting technique we consider in this paper intro-duces an O(∆t) error in the numerical method which is usually referred to as “volumediscrepancy error.” This essentially indicates the accuracy by which the equation ofstate eTu = 1 is solved. Hence, the pressure computed from the parabolic equationsolves only approximately the volume balance equation. However, this O(∆t) errorcan be removed by correcting the pressure in order to exactly satisfy the equationof state. This technique is based on an algebraic iterative solution (by, e.g., New-ton’s method) of the equation of state, regarded as a nonlinear function of pressureonly f(p) ≡ eTu(p) − 1 = 0. The application of this technique results in a relevantreduction of the volume discrepancy error, as investigated by Dicks in [18].

4. Mixed finite element formulation. The mixed finite element approachstarts from a weak formulation of the pressure equation (3.6) and Darcy’s law (2.21),considering the liquid phase pressure p and the total velocity ~vt as main unknowns.

Let Th = T be a regular partition (in the sense of Johnson [28]) of the domainΩ into triangular elements whose maximum diameter is h, and let Wh, Ph be twofamilies of finite dimensional subspaces dense in H(div; Ω) and L2(Ω), respectively.We look for a function ph ∈ Ph approximating the pressure p and a function ~qh ∈Wh approximating the total velocity ~vt. The two discrete spaces Wh and Phshould satisfy the discrete inf-sup condition [12, 4]. A proper choice for Ph and Wh isrepresented by the space of piecewise constant functions P0(T ), and the BDM1 spacedefined as

Wh = ~q ∈ H(div; Ω) | ∀T ∈ Th, ~q|T ∈ (P1(T ))2,

respectively. This space coupling ensures first-order spatial accuracy on the approxi-mation of pressures and second-order accuracy for velocities, while second-order ac-curacy for pressure can be found at the element centroids [20]. A proper choice ofdegrees of freedom yields a flux conservative scheme.

The discrete weak formulation of the mixed problem is easily obtainable by theprojection of (3.6) and (2.21) onto the spaces Ph and Wh under their usual scalarproducts, while the discretization in time is performed by a simple forward difference:

∫

Ω

eTu− 1

∆tφ ψ dA =

∫

Ω

(

−φ eT∂u

∂pn+ eTu

∂φ

∂pn

)

pn+1 − pn

∆tψ dA(4.1a)

+

∫

Ω

eTBT ~∇ ·(

RB−1~v)

ψ dA ∀ψ ∈ Ph,

∫

Ω

M−1t ~vt · ~ω dA = −

∫

Ω

~∇p · ~ω dA+

∫

Ω

eT fg~∇h · ~ω dA ∀ω ∈Wh.(4.1b)

We designate, now, the total number of elements in the discretized domain as Neand the total number of edges as Nf . The total velocity and pressure fields can be


represented in terms of degrees of freedom pk and qm and basis functions ψk and ~ωm

as

~vt =

2Nf∑

m=1

qm~ωm ; p =

Ne∑

k=1

pkψk .(4.2)

Here, we choose the basis functions in order to satisfy ψk = 1 over the element Tkand zero elsewhere and, for j = 0, 1,

∫

ei

~ωm · ~ni lj dl =

1 if j Nf < m ≤ (j + 1)Nf and mod (m,Nf) = i,0 otherwise,

where ei is the edge of index i and ~ni is the respective normal.The matrices R, T (R, T in the undersaturated case) and B appearing in (4.1a)

are the ratio matrices defined in section 2.2, and their elements, known functions ofthe pressure at the previous time level, can be considered constant over each elementTj of the discretization mesh. Choosing the test function ψ in (4.1a) to be the pressurebasis function ψj characteristic of element Tj , it is possible to simplify the last integraltherein as

∫

Ω

eTBT ~∇ ·RB−1~v ψjdA =

∫

Tj

eT ~∇ · ~v dA =

∫

Tj

~∇ · ~vt dA .(4.3)

We can thus rewrite (4.1a) and (4.1b) expanding the two unknowns ~vt and pn+1 interms of the basis functions ~ωm and ψj :

2Nf∑

m=1

qm

∫

Tj

~∇ · ~ωm dA+ pj hnj =

∫

Tj

eTu− 1

∆tφ dA+ hnj p

nj , j = 1, . . . , Ne,(4.4a)

2Nf∑

m=1

qm

∫

Ω

M−1t ~ωm · ~ωqdA−

Ne∑

k=1

pk

∫

Ω

~∇ · ~ωqψkdA =

∫

Ω

eT fg~∇h · ~ωq dA,

q = 1, . . . , 2Nf,

(4.4b)

where we have, for simplicity, considered homogeneous Dirichlet boundary conditionsin the application of the divergence theorem to (4.1b), and set (in view of (2.8))

hnj =1

∆t

∫

Tj

(

−φeT∂BT

∂pnz + eTu

∂φ

∂pn

)

dA .(4.5)

The mixed system, built with (4.4a) and (4.4b), can be written in compact matrixform as follows:

Mx = b ,(4.6)

with

M =

(

A CCT −D

)

, x =

(

q

p

)

, b =

(

r1

r2

)

.(4.7)


The system matrix M is symmetric but not positive definite. In fact, the pressurestiffness matrix D has positive entries due to the parabolic character of the pressureequation (see [35]). Nevertheless, the simple block structure of M suggests its Schurdecomposition with respect to D, which leads to a single equation in the variable q:

Sq = c , with S = A+ CD−1CT , c = r1 + CD−1r2 .(4.8)

This formulation reduces the problem to the solution of a symmetric positive definite(SPD) linear system with a lower number of degrees of freedom. Furthermore thesparsity pattern of matrix S is the same as that of A—with at most ten nonzeroentries per row. The condition number of S is strictly related to the time step, whichis at the denominator of the elements of matrix D. If the time step is very small, thenthe condition number of S is very close to that of A. By contrast, if ∆t has a largevalue (of the order of 10 days in the units used in the Black-Oil model), then theill-conditioning of the divergence matrix CCT can influence the condition number ofS. In such a case the solution of system (4.8) is not recommended: the mixed problem(4.4a) and (4.4b) can be reformulated yielding the well-known hybrid formulation,which gives rise to an SPD system matrix [14].

Another approach [2] starts from the Schur decomposition of system (4.6) withrespect to the matrix A and consists of the iterative solution of the, once again, SPDsystem

(

CTA−1C +D)

p = r2 + CTA−1r1

for the special preconditioning techniques, which are required in the solution of thissystem; see, e.g., [27, 31, 10] and the references therein.

4.1. Second order in time pressure correction. In section 4 we developeda parabolic equation which is second-order accurate in space and only first-order ac-curate in time. A globally second-order algorithm would require a more accurate timediscretization of the pressure equation resulting in a nonlinear iterative scheme andadding considerable computational complexity to the algorithm. For sake of complete-ness we mention that second-order methods, which do not require the solution of anonlinear equation, are reported in literature. An example of such procedures (basedon a predictor–corrector approach) is described in [19]. Nevertheless, Trangenstein [1]justifies the reduced time accuracy by the fact that the Black-Oil model was devel-oped to describe primary and secondary oil recovery processes—characterized by aslow variation of the pressure field—and hence the time truncation errors are expectedto be small.

5. Shock-capturing finite volume formulation. The sequential approachintroduced in section 3 separates the pressure and velocity equations from the massbalance equations. Assuming pressure and velocity fields given by the solution atthe previous time step, mass balance equations form a hyperbolic system which isdiscretized on 2-D unstructured meshes in a finite volume framework via an explicithigher order Godunov-type method. This method allows an accurate approximationof the moving discontinuities which appear in the physical solution. This is possiblebecause, in the conservation form of our shock-capturing finite volume approach,discontinuities are computed as part of the numerical solution [29]. The main drawbackof most of these schemes is that steep gradients are approximated by continuoustransitions which usually spread the discontinuity over many cell points. A higher


order shock-capturing scheme yields a solution which is at least second-order accuratein smooth regions and provides nonoscillating sharp shock profiles.

The multidimensional scheme developed in the present work takes origin fromthe class of multidimensional methods historically proposed by Colella in [16]. Thesemethods were originally developed to numerically approximate the solution of strictlyhyperbolic systems of conservation laws. They were subsequently extended to generalnonstrictly hyperbolic systems with local linear degeneracies in [7] and applied to oilrecovery situations.

In the 1-D version, a Taylor series extrapolation in space and time along char-acteristic curves gives two approximations of the solution at each cell-interface, onefrom the cell on the left (zL) and the other from the cell on the right (zR). Then, alocal Riemann problem is solved and the Godunov flux, required to update the cell-averaged solution, is defined as the physical flux calculated at the Riemann solutionstate. If zL and zR are simply taken to be the cell-averaged values, considering inthis way a constant representation of the spatial dependence of the solution withinany cell, a first-order accurate scheme is obtained. Second-order accuracy is insteadgiven by a more complicated preprocessing calculation (the so-called predictor step,in Colella’s terminology) where initial states of the Riemann problem at the sides ofthe cell i are defined as approximations of

zni ±

∆x

2

(

∂z

∂x

)

+∆t

2

(

∂z

∂t

)

,(5.1)

zni being the ith cell-averaged state at time tn. This approach requires a monotonized

slope computation with some form of limiting to avoid numerical oscillations at dis-continuities. The resulting piecewise linear solution is traced along the characteristicfamilies approximated by the eigenstructure computed at some mean state. In thepresent work, a cell-averaged approximation of the solution logically associated to thecentroid of each cell is advanced in time via an explicit predictor–corrector time step-ping scheme using a conservative flux balance estimation. The higher order Godunovmethod follows through three main steps:

1. trace along characteristic curves to define Riemann problems at cell interfaces;2. estimate numerical fluxes via approximate Riemann problem solution proce-

dures;3. update the cell-averaged solution within any cell.

Two main variants have been introduced, with respect to the original scheme of Colellato make possible the calculation on a triangle-based unstructured mesh. First, thecentral difference monotonized slope considered in [7] for the piecewise linear recon-struction has been substituted by the triangle-based TVD estimation of gradients ofBarth and Jespersen [6]. Then, tracing forward along characteristics is done in thedirection normal to each cell-interface.

This approach differs qualitatively from the unsplit approach on structured gridsof [7], whose multidimensional method considers contributions to fluxes both in thedirection normal to the cell-interface as well as in the transversal direction. However,the numerical experiments we report in section 6 account for the capabilities and therobustness of our approach.

5.1. The finite volume conservative framework. Integrating (2.26) over theelement Tk yields

∫

Tk

z∂φ

∂t+ φ

∂z

∂tdA =

∫

Tk

~∇ ·(

RB−1~v)

dA .(5.2)


The application of the divergence theorem allows a finite volume formulation of themass conservation laws:

∫

Tk

z∂φ

∂t+ φ

∂z

∂tdA = −

3∑

i=1

∫

ei

RB−1~v · ~ndl,(5.3)

where the summation is extended over all the edges of the element Tk. The vector ~vis the phase velocity vector at time step tn+1 from the mixed system.

Let us call zk,n the value of the density vector over the element Tk at time n. Using(2.24) for the phase velocities, the finite volume formulation of the mass conservationlaws is given by

φkzk,n+1 − zk,n

∆t+ zk,n+1 ∂φk

∂p

∆pk∆t

= −1

Ak

∫

∂Tk

RB−1(

f~vt + t~∇h)

· ~n dl,(5.4)

where f is the vector of fractional flows expressed as a function of mass densitiesthrough the saturations by (2.20), known at time step n. The pressure field p and thetotal velocity field ~vt are given by the initial “parabolic” step of the sequential methodand considered constant in the time interval (tn, tn+1]. The porosity φk(p) as a functionof p can also be assumed constant within each cell. If, in addition, φk is consideredconstant in time, we can view (5.4) as the finite volume formulation corresponding tothe following hyperbolic system of conservation laws written in differential form:

φ∂z

∂t= −~∇ · ~g(z) = −

∂~g

∂z· ~∇z,(5.5)

where ~g = RB−1~v = RB−1(f~vt + t~∇h) is the physical flux function.

5.2. Predictor–corrector method with characteristic tracing. Let us fo-cus on the Taylor extrapolation used to calculate left and right states at cell interfaces.In any cell we start from a Taylor expansion up to first-order terms of z(~x, t) aroundthe centroid position ~xc, which produces a linear correction in space and time of thecell-averaged solution z at time tn. We use this linear representation of the solutionin Tk × (tn, tn+1] to compute the initial states of the local Riemann problems at themidpoint of each cell-interface ~xf , as required by the algorithm. The first-order Taylorexpansion yields

z

(

~xf , tn +

∆t

2

)

= z(~xc, tn) + (~xf − ~xc) · ~∇z +

∆t

2

∂z

∂t(5.6)

and, after substitution of (5.5) into (5.6),

z

(

~xf , tn +

∆t

2

)

= z(~xc, tn) + (~xf − ~xc) · ~∇z −

∆t

2

1

φ

∂~g

∂z· ~∇z .(5.7)

Let us define ∆~x2 = ~xf − ~xc, where the factor 1/2 is introduced only for convenience

reasons, in order to have the same final form of the analogous relations in [7]. We

decompose ∆~x and∂~g∂z onto the directions ~η and ~τ , normal and tangent to the face f

at ~xf , respectively, as shown in Figure 5.1:

∆~x = (∆~x · ~η)~η + (∆~x · ~τ)~τ ,(5.8)


xx

c

ηf

A B

C

τ

Fig. 5.1. The geometry of triangle-based decomposition for the local characteristic tracing.

∂~g

∂z=

(

∂~g

∂z· ~η

)

~η +

(

∂~g

∂z· ~τ

)

~τ .(5.9)

Substituting them in (5.7), we have

z

(

~xf , tn +

∆t

2

)

= z(~xc, tn) +

1

2

[

∆~x · ~η − ∆t1

φ

∂~g

∂z· ~η

]

~η · ~∇z(5.10)

+1

2

[

∆~x · ~τ − ∆t1

φ

∂~g

∂z· ~τ

]

~τ · ~∇z.

For nonlinear problems, as suggested in [7], the estimate provided by (5.10) mustbe modified in order to disregard the wave components which do not correspond towaves traveling towards the cell-interface. This procedure is coherent with the upwindnature of the scheme and has been reported in literature to yield a more robust schemeeven in strongly nonlinear cases.

Let Hη and Hτ be the normal and the tangent component of the Jacobian, re-spectively; i.e., Hη = (∂~g/∂z) · ~η, and Hτ = (∂~g/∂z) · ~τ . The subtraction of wavestraveling in wrong directions is accomplished via a local projection operator, definedas P = XΛ+X−1, where X is the matrix of right eigenvectors of the Jacobian inthe η direction; i.e., X−1HηX = Λ, Λ = diag(λk) is the diagonal matrix of the Hη

eigenvalues and Λ+ the diagonal matrix defined as Λ+ii = [1 + sign(λi)]/2. Applying

the projection operator P , we have the following as a final expression:

z

(

~xf , tn +

∆t

2

)

= z(~xc, tn) + P

1

2

[

∆~x · ~η − ∆t1

φHη

]

~η · ~∇z(5.11)

+1

2

[

∆~x · ~τ − ∆t1

φHτ

]

~τ · ~∇z.

The gradient ~∇z which appears in (5.11) must satisfy a TVD stability condition toensure that numerical oscillations do not appear whenever a discontinuity forms in thesolution. This is directly imposed as a monotonicity constraint in the triangle-basedreconstruction algorithm which we present in the next section.

5.3. The TVD triangle-based linear reconstruction. A linear reconstruc-tion of the flow variables within a generic triangular cell Tk can be written in thefollowing way:

z(~x, t) = z(~xk, t) + (~xk − ~x) · ~∇z|k,(5.12)


(i)

k i p

(j)

n

Fig. 5.2. The triangle-based stencil for the linear TVD reconstruction.

where ~xk is the centroid of the triangle Tk and zk = z(~xk, t) is the value logicallyassociated to it at time t. All the relations involving quantities like zk are to beinterpreted componentwisely. An estimate of a “mean” gradient ~∇z|k inside Tk mustbe somehow provided, which implies an estimate of the two terms ∂z

∂x |k and ∂z∂y |k. The

application of the divergence theorem can easily yield these derivatives which can betaken with second-order accuracy in space as the values at centroids. Let us considerthe divergence theorem applied to a generic vector field ~V :

∫

Ω

~∇ · ~V dΩ =

∮

∂Ω

~V · ~nds,(5.13)

where ∂Ω is the close boundary which defines a generic domain of integration Ω and~n is the outward normal to this boundary.

Let us take for Ω the triangular cell Tk and consider two vector fields ~V = (z, 0)

and ~V = (0,z). Hence, applying the relation (5.13), we obtain the required estimatesfor ∂z

∂x and ∂z∂y .

The satisfaction of a TVD condition is obtained by introducing a limiting functionθ on the slopes given by the gradient in the piecewise linear reconstruction. Hence,the reconstruction takes the form

z(~x, t) = z(~xk, t) + θ(~xk − ~x) · ~∇z|k .(5.14)

The limiter θ takes values in the range between 0 and 1. Any limiter introduced inthe literature can be used; see [34] for a review. However, these limiters are essentiallymonodimensional. A really 2-D limiter has been introduced by Barth and Jespersen(see, e.g., [6]) and is reported here.

Let us indicate with zLi and zR

i , respectively, the left and right reconstructedvalues on the ith edge ei of the mesh. Referring to Figure 5.2, the piecewise linearreconstruction inside Tk yields the left state zL

i while the piecewise linear reconstruc-tion inside Tp yields the right state zR

i . “Left” and “right” are defined according tothe orientation of the normal vector at the edge ei which points outward to Tk andinward to Tp.

The limiter proposed by Barth and Jespersen is designed in order to ensure thatthe following two properties are satisfied by the reconstructed values:

(A) the reconstruction must not exceed the minimum and maximum of the neigh-boring cell averages;

(B) the difference in the interpolated values zRi − zL

i at the ith edge and thedifference in the corresponding cell averages z(~xk, t) − z(~xp, t) should havethe same sign.


These two properties can be summarized as

1 ≥zRi − zL

i

z(~xk, t) − z(~xp, t)≥ 0 .(5.15)

Barth and Jespersen showed that to satisfy property (A) it is sufficient to design thelimiter as follows:

• Compute zmaxk and zmin

k as the maximum and minimum of the cell averagesin the neighbors of cell Tk.

• Then a local limiting value θj is introduced for each vertex (j) of the cell Tk:

θj =

min(1,zmax

k −zk

zj−zk), if zj − zk > 0,

min(1,zmin

k −zk

zj−zk), if zj − zk < 0,

1, if zj − zk = 0,

(5.16)

where zj is the extrapolated solution variable in the vertex j.• Finally, a global limiting value is estimated as θ = min(θ1, θ2, θ3) .

To satisfy property (B) a check is performed edge by edge on the reconstructedvalues. If these values do not satisfy condition (B), they will be substituted by theirmean.

5.4. Flux estimation. A standard way to define a numerical flux in shock-capturing schemes consists of computing the physical flux in some reference state(typically given by an upwind choice) or considering the mean of the right and theleft fluxes and then in correcting that by a dissipative term. The dissipative correctionis introduced to stabilize the computation and to ensure convergence to the correctentropy weak solution. The procedure used in Godunov-like methods to define thenumerical flux is generally referred to as the Riemann solver, since it considers asa model of the wave propagation of the numerical information an exact or an ap-proximate solution of the Riemann problem locally defined by the left and the rightinterface states. The solution of a Riemann problem for a hyperbolic system of Nequations with initial conditions zL,zR consists of a composition of a finite num-ber of simple waves progressing from zL to zR. A path connecting the two initialstates must be built via a sequence of intermediate states which correspond to theelementary waves—shocks and rarefactions—forming the solution. This procedure isa common approach to provide constructive proofs of existence and uniqueness theo-rems. A general theory is described by Smoller in [33] for systems of conservation lawssatisfying a strict hyperbolicity and a genuine nonlinearity constraint. However, thiscondition is not satisfied in the Black-Oil model considered here and more generallyin Riemann problems arising in oil recovery models. As pointed out by Bell, Colella,and Trangenstein [7], a generalized approach is thus necessary to allow more complexsituations where local linear degeneracies and loss of strict hyperbolicity occur.

Let us call rl the lth right eigenvector of the eigenvalue matrix Λ—i.e., the lthcolumn of the matrix X. In the case of small amplitude jumps zl − zl−1 = αlr

l +O(|zR − zL|2); that is, the right eigenvectors approximate the wave curves in phasespace to the second order in the jump between the left and the right state. This jumpcan thus be expressed by

zR − zL =

N∑

l=1

αlrl +O(|zR − zL|2) .(5.17)


In the general case of finite amplitude jumps we can introduce a set of “generalized”eigenvectors l which represent the net change along Γl and which are normalized tobe of unit length

zR − zL =

N∑

l=1

αll .(5.18)

The generalized eigenvectors l can be numerically approximated by computing themin a mean state z = (zL + zR)/2 as ¯l = rl(z). In the same way, we can introduce ageneralized eigenvalue λl which is an approximation of the wave speed along the linesegments corresponding to the generalized eigenvectors ¯l. An approximation of theexact solution of the Riemann problem [1] can now be estimated at the interface, by alinear combination of the contributions associated with the waves traveling from theleft (positive eigenvalues) and those coming from the right (negative eigenvalues):

zRiemann = z(

zL,zR)

= zL +∑

λl<0

αll .(5.19)

We stress that this scheme still provides us with an approximate solution, which isactually the exact solution of the Riemann problem linearized around the mean statez. The numerical Godunov flux can now be calculated as the physical flux at thisRiemann solution state. This procedure may be affected by a well-known pathology,due to nonuniqueness of the weak solution in the case of transonic expansion. Nev-ertheless, our numerical experiments show that in particular situations, it yields avery effective numerical method, capable of producing high quality results in a largevariety of situations. A more sophisticated approach has been proposed by Engquistand Osher [24, 25], where the flux is calculated at a reference state, chosen in order tosatisfy an upwind criterion, and a dissipative correction term is defined as an integralcontribution of the eigenvalues along the solution path.

Let us first define an upwind reference state by considering the following meanspeed σ:

σ =

(

gη(zL) − gη(z

R))

· (zL − zR)

||zL − zR||2,(5.20)

where gη ≡ ~g · ~η .

If σ ≥ 0, we can choose zL as a reference state and the numerical flux is given by

~gEO(zL,zR) = ~g(zL) +

N∑

l=1

[∫ αl

0

min(λl, 0)dα

]

¯l .(5.21)

Otherwise, if σ < 0, the numerical flux is given by

~gEO(zL,zR) = ~g(zR) −

N∑

l=1

[∫ αl

0

max(λl, 0)dα

]

¯l .(5.22)

The preceding integrals are computed by representing the wavespeeds λl as cubic poly-nomial interpolation along the approximated path Γl. A cubic polynomial is neededto approximate the wavespeeds λl because of inflection points to be represented inthe physical flux function. The cubic interpolation is built with the values of λl in


the two extrema zl−1 and zl and in two interior points on the path Γl. Finally, inorder to avoid computing zeros of cubic functions, which would provide a very expen-sive algorithm, a simplification is adopted and the interpolated cubic is replaced by apiecewise linear interpolation.

When two wavespeeds λl and λm coalesce, a loss of strict hyperbolicity occurs. Inthis case, the expansion coefficients αl and αm in (5.18) are assumed to be unreliable,because the generalized eigenvectors l and m are nearly parallel. Their contributionto the numerical flux calculation needs a special treatment. Following [7], we firstcheck whether the detected eigenvector deficiency is associated with a transonic wave.If it is not, the portion of the jump related to the involved wavespeed is collapsedin a “mean” jump defined by a generalized eigenvector lm and a “mean” expansioncoefficient αlm:

lm =αl

l + αmm

‖αll + αmm‖, αlm = ‖αl

l + αmm‖ .(5.23)

In this case, when the reference state is, for example, zL, the numerical flux is givenby

~gEO(zL,zR) = ~g(zL) +

N∑

k=1k 6=l,m

[∫ αk

0

min(λk, 0)dα

]

¯k +

[∫ αlm

0

min(λlm, 0)dα

]

¯lm ,

where λlm is a linear function such that

λlm(0) = maxλLl , λLm and λlm(αlm) = minλRl , λ

Rm .

If a transonic transition is detected, the corresponding integral corrections terms arereplaced by a Rusanov-like dissipative term −ν αlm ¯lm/2, where

ν = max (|λl(zL)|, |λm(zL)|, |λl(zR)|, |λm(zR)|) .(5.24)

Finally, the scheme can be modified by incorporating a small amount of numericaldissipation in order to ensure convergence to the entropy satisfying solution. The formsuggested in [7] is

~gn+1/2 = ~gEO(zL,zR) − ν(zR − zL),(5.25)

where ν = 0.1 maxl(λRl − λLl ). This viscosity term is different than zero only when

the wavespeed at the left state is greater than the wavespeed at the right state, thatis, the case of a compressive (shock) wave. The form of this artificial viscosity termis quadratic in the jump (zR − zL) and thus maintains second-order accuracy of themethod. A detailed discussion of this issue can be found in [7].

6. Numerical results and validation. We dedicate this section to the valida-tion of the MFE-FV code via a 1-D test case and to the presentation of several 2-Dsimulations. Although the standard Black-Oil model could easily be handled (impos-ing Rv ≡ 0 and Ra ≡ 0), the computations are performed in the most general masstransfer framework with volatile oil as well as gas dissolving in both the liquid andthe aqueous phase.

A volume discrepancy error control has been developed following Dicks [18], asexplained in section 3.1, and an automatic time-stepping procedure based on the CFL


Fig. 6.1. Typical mesh employed in 1-D simulations.

[30] stability condition has also been implemented in order to attain reasonable elapsedtimes for executions. A CFL number of 0.9 is used throughout all the computations.

We carried out several numerical experiments in order to evaluate the effective-ness of the proposed numerical approach in terms of accuracy, amount of dissipationintroduced in the discretization, and ability of the scheme to produce oscillation-freeresults in 1-D and 2-D cases.

The eigenvalues and eigenvectors, required in the characteristic tracing and in theEngquist–Osher flux calculation, have been determined via a characteristic analysis,as discussed in TB, and explicitly computed in the code.

The test cases we will present in the continuation of the paper rely on bothsaturated and undersaturated flow and show many of the physical, mathematical,and numerical features of the Black-Oil model.

The change from a saturated situation to an undersaturated one produces a dis-continuity in the eigenvalues and eigenvectors, since the entries of matrices B and Rare not differentiable across a phase change. For this reason the phase space construc-tion for weak waves of section 5 is invalid for part of the jump, and this requires anaddition to the flux of a quadratic viscosity, depending on the maximum jump of thesaturations. We thus modify the viscosity term of (5.24) as

ν = max

ν,0.2 diam ∆s

∆t

,(6.1)

where diam is the diameter of the inner circle in the element and

∆s =

maxj |sRj − sLj | if maxj |s

Rj − sLj | > 0.2,

0 otherwise.(6.2)

6.1. Validation on 1-D test cases. For the validation of the numerical model,we first present a 1-D test case representative of a mixed saturated-undersaturatedflow with phase changes. Our results should be compared to those published by TB,test case 5.

The computations are performed for a homogeneous reservoir 1000 feet in lengthwith constant permeability of 100 millidarcies and no dip. Two wells, an injector, anda producer are located at the two ends of the reservoir (left and right, respectively)and operate at fixed pressures. The pressure-volume properties (B,R, µ, and φ) aswell as the relative permeability data adopted in the simulation can be found in theAppendix.

The referred test case is one dimensional in the original work by TB. Neverthe-less, the code implementing the full 2-D algorithm has been used to produce the resultreported here. The 1-D nature of the reservoir was simulated using the 100 × 1 reg-ular grid shown in Figure 6.1. The number of degrees of freedom necessary for thediscretization of the parabolic equation is over-dimensioned in this case. All the edgesparallel to the flow direction are actually treated as “no flow” boundaries (the physicalflux is set to zero across them). Moreover, as this example does not include any grav-itational effect, the flow direction is unique from left to right. The Engquist–Osher


0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600 700 800 900 1000

VAPOR

LIQUID

AQUA

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600 700 800 900 1000

VAPOR

LIQUID

AQUA

Fig. 6.2. Test case 1. Saturations at 250 (left) and 750 days (right). The lower curve representsthe vapor saturation and the upper curve the sum of the vapor and the liquid saturations.

procedure yields in this case a simple upwind flux determination. Finally, the algebraiceffort of higher order mixed finite elements (BDM1) is not justified here, due to theconstant permeability and porosity fields. However, we stress that the purpose of suchcomputations is just to validate the algorithm in some interesting test problems fromthe physical and mathematical viewpoint by a comparison with results previously re-ported in literature. We do expect the effectiveness of the coupled MFE-FV approachproposed in this work to be evident on truly 2-D problems with highly heterogeneouspermeability and porosity fields, and in presence of counter-current flows.

Test case 1 (TB 5) is representative of flows with sharp interfaces between phasesdue to relevant composition changes. The injector and producer wells operate at 4300and 4100 psi, respectively, and the reservoir initial pressure is 4200 psi. The injectedand initial reservoir component densities are given by

zinj =

.0179178.68.357

, zres =

.668100.19.0668

.

With these definitions, at the beginning of the computation, the reservoir con-ditions are undersaturated with the vapor phase missing, where the gas componentis completely dissolved into the liquid and aqueous phases. The composition of theinjected fluid, a two-phase mixture with the liquid phase missing, is such that a three-phase region suddenly develops and a sharp front of gas, followed by a rarefactionwave, moves from left to right into the reservoir (see Figures 6.2 and 6.4).

In the three-phase region the characteristic wave structure of the hyperbolic prob-lem is similar to that of the three-phase Buckley–Leverett equation (see, e.g., [1]).The gas disturbance moving into the reservoir can thus clearly be identified with aBuckley–Leverett rarefaction-shock compound pattern.

As the simulation proceeds, the liquid is displaced from the region near the in-jection well and thermodynamic equilibrium between the aqueous and vapor phasesis there settled.

At 250 days, as shown in Figure 6.2, three regions separated by sharp fronts arepresent in the reservoir: liquid and aqua coexist near the production well, aqua andvapor near the injector well, and a three-phase zone lies in the middle.

The more mobile vapor phase reaches the production well around 550 days, ascan be seen from the oil rate and gas-oil ratio (GOR) plots of Figure 6.3. In fact,


0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

0 500 1000 1500 2000 2500

WOR

GOR

OIL RATE

Fig. 6.3. Test case 1. Scaled production quantities: oil rate and gas-oil (GOR), and water-oil(WOR) ratios.

the gas production up to 550 days can be related to dissolved gas, while the suddenincrease in the GOR curve slope clearly gives evidence of breakthrough time.

From a numerical viewpoint we remark that, on the linear 200 element meshconsidered here, the proposed method is capable of resolving, with high-order spatialaccuracy, the sharp gradients which form in the solution without being affected byspurious numerical oscillations. Moreover, these results are definitely comparable, bothqualitatively and quantitatively, to that of the TB computations.

6.2. 2-D Simulations without gravity. In this section we present results from2-D test cases with constant and variable permeability fields. We analyze the standardcase of a quarter of five spot: a square domain with an injector and a producer welllocated at two opposite corners (left and right, respectively, in the following figures).In our simulations, we considered a 1000-foot side domain discretized with a regularmesh of 3200 triangles.

The simulations for the test cases of this section are performed extending to 2-Dthe 1-D test case of the previous section. From the saturation plots of Figure 6.5 itcan be seen how steep gradients separating different zones of the reservoir are wellcaptured by our scheme without spurious oscillations even in 2-D test cases.

The next test case (3) is somewhat more representative of the typical situationsoccurring in reservoir simulation. We consider a sand-shale permeability field, depictedin Figure 6.6, with a strong range of variation (10−5 – 103 mD). Standard techniques,whether based on finite elements or on finite differences, lack in accurately renderingthe velocity fields when highly discontinuous geological properties are present. Mixedfinite elements, on the other hand, are especially built to get a high order of approxi-mation of the velocity, as they approximate simultaneously the two variables withoutderiving one from the other.

As can be seen from Figure 6.6 the total velocity field qualitatively reproduces thepermeability pattern. This behavior is further enhanced by the saturation contoursof Figure 6.7. The flow lines, in fact, accurately avoid the low permeability zones andtwo main channels are formed. It can also be clearly seen from the figure that the twoflow channels avoid the whole right side of the domain.

This kind of study can be very important even in the case of an enhanced oilrecovery process when a fully miscible component is injected into the reservoir. If the


0

0.1

0.2

0.3

0.4

0.5

0.6

0 100 200 300 400 500 600 700 800 900 1000

200 DAYS

750 DAYS

Oil component density

100

120

140

160

180

200

0 100 200 300 400 500 600 700 800 900 1000

200 DAYS

750 DAYS

Gas component density

0.1

0.15

0.2

0.25

0.3

0.35

0 100 200 300 400 500 600 700 800 900 1000

200 DAYS

750 DAYS

Water component density

Fig. 6.4. Test case 1. Component densities at 250 (solid line) and 750 days (dotted line).

permeability field is not as simple (from a physical point of view) as in our test case,only accurate numerical simulations can predict the location of the undepleted regionswhere one might want to drill other injector or producer wells.

6.3. Influence of gravity effects. In all the test cases presented above, it canbe easily seen that all the eigenvalues are nonnegative; therefore the Engquist–Osherflux reduces to a simple upwind evaluation. For the last test case we tried our codeagainst a cross-sectional problem [7], which takes place in the x–z plane with gravityacting in the negative z direction. The cross section considered is 400 feet in lengthand 50 feet in height, discretized with a rectangular 80×40 mesh. We inject fromthe left face of the cross section a fluid with composition specified by the followingsaturations: sinj = [0, 0, 1]. The injection pressure of 2000 psi is imposed at the top ofthe face and a pressure determined by a condition of hydrostatic equilibrium down onthe rest of the face. The initial reservoir saturations have been set to sres = [1, 0, 0],with a pressure along the top of 1800 psi and the pressures below being distributedwith a similar procedure as the injection face. A production pressure of 1600 psi wasassigned along the whole production face at the right end of the reservoir. In thisexample, counter-current flow occurs in the vertical direction due to the effect ofgravity on the fluid phases of different densities. Hence, for the z-directional fluxes,


0

0.5

1

0

0.5

1

Liquid phase

0

0.5

1

0

0.5

1

Vapor phase

0

0.5

1

0

0.5

1

Aqueous phase

Fig. 6.5. Test case 2. Liquid, vapor, and aqueous saturation contours at 250 and 750 days ofsimulation (from top to bottom).

0 100 200 3000

100

200

300

1

301

601

901

1.2e+03

1.5e+03

1.8e+03

0 100 200 3000

100

200

300

0

100

200

300

Fig. 6.6. Sand-shale permeability and total velocity fields.

the phase construction of the Riemann problem is needed to calculate the numericalflux.

From the results of the simulations, we found out that the two solvers behave ina similar way, however, from a theoretical point of view, the Engquist–Osher schemeis more reliable, since it can detect and resolve coalescent eigenvector directions. Theextra amount of work introduced by the Engquist–Osher flux calculation is not morethan about 15 % of the total CPU time, as can be seen from Table 6.1, and can justifythe use of such a sophisticated technique also in view of a production code.


Liquid phase Vapor phase Aqueous phase

Fig. 6.7. Test case 3. Liquid, vapor, and aqueous saturations at 3, 28.5, and 45 days of simu-lation (from top to bottom).

Table 6.1

Times (in percentage over the total CPU time) for the most expensive procedures in the simu-lation of test case 4 when using different schemes for flux calculation.

Flux scheme Matrix System Fluxassembly solution TVD recon. Predict estimation

Engquist–Osher 15.95 37.89 2.67 4.89 25.40Exact Riemann solver 20.66 48.87 3.51 6.34 10.21

7. Conclusions and future developments. In this work, we have reportedthe development of a sequential coupled mixed finite element/shock-capturing finite-volume method and its application to the numerical approximation of the solution ofthe time-dependent 2-D equations of the Black-Oil model.

The present methodology appears to be a very powerful approach to time-depend-ent problems in Oil Recovery simulation when moving discontinuities are presentin the solution and the reservoir is characterized by strong heterogeneities in thepetrophysical data (permeability and porosity fields).

In fact, high-order mixed finite elements are one of the most accurate numericalschemes to simultaneously treat the phase pressure and velocity. On the other hand,


shock-capturing finite-volume methods exhibit the property of accurately resolvingsteep gradients without spurious numerical oscillations while taking numerical dissi-pation effects at a very low level. A conservative formulation is usually consideredin this case, allowing an accurate numerical prediction of the moving discontinuitiesappearing in the physical solution. It is worth stressing again that the inherently con-servative form of the mixed finite element methods has been shown to be particularlywell adapted to the integral conservative formulation on which shock-capturing finitevolume schemes are based.

The use of such sophisticated techniques can appear quite expensive when com-pared with more traditional methods (such as Galerkin finite elements or structuredfinite differences). However, the high quality of the results obtained completely jus-tifies this approach. In order to reduce the computational effort demanded by thecomputation of the mixed finite element method, a more detailed study is being car-ried on to develop efficient algebraic solvers for the solution of the linear systems whicharise from this discretization. A more sophisticated coupling scheme should also bedeveloped to better solve the full system of equations forming the Black-Oil model,beyond the sequential approximation.

The present method has also been designed in order to exploit the inherent flex-ibility of unstructured grids. It allows in a straightforward manner local refinementsaround complex geometrical patterns. In fact, a crucial point of this numerical schemerests in its triangle-based formulation, which allows an easy treatment of complexreservoir geometries. The geometrical complexity of a reservoir and the presence offault surfaces and large local variation of soil properties actually require the develop-ment of suitable mesh-adaptive algorithms with high flexibility, capable of automati-cally producing localized refined solutions. A more detailed study is required in orderto include these adaptivity features. Research work is underway to develop this newcapability.

Finally, we remark that our approach can be largely applied in a number ofsituations of geophysical interest, such as, e.g., more complex oil recovery modeling(polymer flooding, compositional models), groundwater flow, and passive contaminanttransport in porous media.

Appendix. Pressure-volume functional forms. The spatial coordinates haveunits of feet, and time t is measured in days. Pressure p is measured in psi, viscosity incentipoise, and total permeability k is measured in ft2 cp/psi days, or .006328 timesthe value in millidarcies. In this way the gravitational acceleration g used in Darcy’slaw must be removed because it is incorporated in this constant. A factor of 1/144has been included in the gravitational term to ensure consistency of units. Phasedensities are measured in lb/ft−3. For the computations of this paper, the reservoirhad porosity φ = 0.2(1 + 10−5p).

For our examples, we have used the standard square saturation relative perme-ability functions with Stone’s first model for three phase flows:

krv = s2v, kra = s2a, krl =(1 − sv − sa)klvkla(1 − sv)(1 − sa)

, kla = (1 − sa)2 klv = (1 − sv)

2,

where kla and klv are the liquid relative permeability of a liquid–aqueous and liquid–vapor system, respectively. For simplicity, we have assumed that the connate aqueoussaturation and the residual liquid saturation are both zero. Otherwise, the normalizedform of Stone’s first model proposed by Aziz and Settari [3] should be preferred.


The component densities z are measured in cubic feet at some reference condition(temperature and pressure) per reservoir cubic foot. Following [18] we have chosen thefollowing functional forms:

• Entries of matrix R

Rl(p) = .05p,Rv(p) = 9 × 10−5 − 6 × 10−8p+ 1.6 × 10−11p2,Ra(p) = .005p.

• viscosities

µl =

.8 − 1 × 10−4p if liquid is saturated,(.8 − 1 × 10−4pb)(1 + cw(p− pb)) if liquid is undersaturated.

µv = .012 + 3 × 10−5,

µa =

.35 if aqua is saturated,

.35(1 + cw(p− pb)) if aqua is undersaturated.

• formation volume factors (entries of matrix B)

Bl =

1 + 1.5 × 10−4p if liquid is saturated,1+1.5×10−4pb1+co(p−pb) if liquid is undersaturated.

Bv =

16+.06p if vapor is saturated,

17+.06p + Rv

Rv

[

16+.06p − 1

7+.06p

]

if vapor is undersaturated.

Ba =

1 − 3 × 10−6p if aqua is saturated,1−3×10−6pb1+cw(p−pb) if aqua is undersaturated,

where cw = 6.78 × 10−5 (in psi−1) is the compressibility of water and co =2.31 × 10−5 psi−1 is the compressibility of oil.

• mass densities (at STC)

ρ = B−1RT [ρo, ρg, ρw]T, where ρo = 52.789 ρg = 0.07655 ρw = 62.382.

Acknowledgments. We wish to thank Alberto Cominelli for useful discussionsand Agip S.p.A. for the permission to publish this paper.

REFERENCES

[1] M. B. Allen, G. A. Behie, and J. A. Trangenstein, Multiphase Flow in Porous Media:Mechanics, Mathematics, and Numerics, Lecture Notes in Engrg. 34, Springer-Verlag, NewYork, 1988.

[2] K. Arrow, L. Hurwicz, and H. Uzawa, Studies in Nonlinear Programming, Stanford Univer-sity Press, Stanford, CA, 1958.

[3] K. Aziz and A. Settari, Petroleum Reservoir Simulation, Applied Science, London, 1979.[4] I. Babuska, Error bounds for the finite element method, Numer. Math., 16 (1971), pp. 322–333.[5] T. J. Barth, Aspects of unstructured grids and finite volume solvers for the Euler and Navier-

Stokes equations, in Proc. of the AGARD-VKI Lecture Series on Unstructured Grid Meth-ods for Advection Dominated Flows, AGARD Report 787, 1992.

[6] T. J. Barth and D. C. Jespersen, The Design and Application of Upwind Schemes on Un-structured Meshes, AIAA Ed. Ser. 89–0366, Washington, DC, 1989.

[7] J. B. Bell, P. Colella, and J. A. Trangenstein, High-order Godunov methods for generalsystems of hyperbolic conservation laws, J. Comput. Phys., 82 (1989), pp. 362–397.

[8] J. B. Bell, C. N. Dawson, and G. R. Shubin, An unsplit, higher order Godunov method forscalar conservation laws in multiple dimension, J. Comput. Phys., 74 (1988), pp. 1–24.


[9] L. Bergamaschi, C. Gallo, G. Manzini, C. Paniconi, and M. Putti, A mixed finite ele-ment/TVD finite volume scheme for saturated flow and transport in groundwater, in FiniteElements in Fluids, M. M. Cecchi et al., eds., Univ. of Padua, Italy, 1995, pp. 1223–1232.

[10] L. Bergamaschi and S. Mantica, Efficient algebraic solvers for mixed finite element modelsof porous media flows, in Proc. Int. Conf. Comp. Meth. Wat. Res. XI., A. A. Aldama, etal., eds., London, 1996, Computational Mechanics and Elsevier Applied Sciences, Compu-tational Mechanics MA, Billerica, MA, 1996, pp. 481–488.

[11] L. Bergamaschi, S. Mantica, and F. Saleri, Mixed Finite Element Approximation of Darcy’sLaw in Porous Media, Tech. rep. CRS4-AppMath-94-17, CRS4, Italy, 1994.

[12] F. Brezzi, On the existence, uniqueness and approximation of saddle point problems arisingfrom lagrangian multipliers, RAIRO Anal. Numer., 8 (1974), pp. 129–151.

[13] F. Brezzi, J. Douglas, and L. D. Marini, Two families of mixed finite elements for secondorder elliptic problems, Numer. Math., 47 (1985), pp. 217–235.

[14] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, Berlin,1992.

[15] G. Chavent and J. Jaffre, Mathematical Models and Finite Elements for Reservoir Simu-lation, North–Holland, Amsterdam, 1986.

[16] P. Colella, Multidimensional upwind methods for hyperbolic conservation laws, J. Comput.Phys., 54 (1990), pp. 174–201.

[17] B. L. Darlow, R. E. Ewing, and M. F. Wheeler, Mixed finite element method for miscibledisplacement problems in porous media, Society of Petroleum Engineering Journal, (1984),pp. 391–398.

[18] M. E. Dicks, Higher Order Godunov Black-Oil Simulations for Compressible Flow in PorousMedia, Ph.D. thesis, University of Reading, Reading, UK March 1993.

[19] J. Douglas and T. Dupont, Galerkin methods for parabolic problems, SIAM J. Numer. Anal.,7 (1970), pp. 575–626.

[20] J. Douglas and J. E. Roberts, Global estimates for mixed methods for second order ellipticequations, Math. Comp., 44 (1985), pp. 39–52.

[21] L. J. Durlofsky, Development of a mixed finite-element-based compositional reservoir simu-lator, in SPE 1993 Reservoir Simulation Symposium, New Orleans, LA, 1993.

[22] L. J. Durlofsky, A triangle based mixed finite element-finite volume technique for modelingtwo phase flow through porous media, J. Comput. Phys., 105 (1993), pp. 252–266.

[23] L. J. Durlofsky, B. Engquist, and S. Osher, Triangle based adaptive stencils for the solutionof hyperbolic conservation laws, J. Comput. Phys., 98 (1992), pp. 64–75.

[24] B. Engquist and S. Osher, Stable and entropy satisfying approximations for transonic flowcalculations, Math. Comp., 34 (1980), pp. 45–75.

[25] B. Engquist and S. Osher, One sided difference approximations for nonlinear conservationlaws, Math. Comp., 36 (1981), pp. 321–351.

[26] M. Espedal, R. Hansen, P. Lanlgo, O. Soevareid, and R. Ewing, Heterogeneous porousmedia and domain decomposition methods, in 2nd European Conference on the Mathemat-ics of Oil Recovery, D. Guerrillot and O. Guillon, eds., Technip, Paris, 1990.

[27] R. Glowinski and P. L. Tallec, Augmented Lagrangian and Operator-Splitting Methods inNonlinear Mechanics, SIAM, Philadelphia, PA, 1989.

[28] C. Johnson, Numerical Solutions of Partial Differential Equations by the Finite ElementMethod, Cambridge University Press, Cambridge, 1987.

[29] P. Lax and B. Wendroff, Systems of conservation laws, Comm. Pure Appl. Math., 13 (1960),pp. 217–237.

[30] R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhauser-Verlag, Basel, Switzer-land, 1990.

[31] E. Ng, B. Nitrosso, and B. Peyton, On the solution of Stokes’s pressure system within N3Susing supernodal Cholesky factorization, in Finite Elements in Fluids: New Trends andApplications, K. Morgan et al., eds., Pineridge Press, Swansea, UK, 1993.

[32] P. A. Raviart and J. M. Thomas, A mixed finite element method for second order ellipticproblems, in Mathematical Aspects of the Finite Elements Method, Lecture Notes in Math.606, I. Galligani and E. Magenes eds., Springer-Verlag, New York, 1977.

[33] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, Berlin, 1982.[34] S. P. Spekreijse, Multigrid Solutions of the Steady Euler Equations, Ph.D. thesis, University

of Delft, Zuid–Holland, The Netherlands, 1987.[35] J. A. Trangenstein and J. B. Bell, Mathematical structure of the Black-Oil model for

petroleum reservoir simulation, SIAM J. Appl. Math., 49 (1989), pp. 749–783.

a mixed finite element--finite volume formulation of the black-oil model

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