SIAM J. SCI. COMPUT. Vol. 20, No. 3, pp. 970–997
c 1998 Society for Industrial and Applied Mathematics
A MIXED FINITE ELEMENT–FINITE VOLUME FORMULATION OF THE BLACK-OIL MODEL∗ LUCA BERGAMASCHI† , STEFANO MANTICA‡ , AND GIANMARCO MANZINI§ Abstract. In this paper a sequential coupling of mixed finite element and shock-capturing finite volume techniques is proposed, in order to numerically solve the system of partial differential equations 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 system of mass conservation laws is solved by a higher order Godunov-type scheme, here extended to trianglebased unstructured grids. Numerical results on 1-D and 2-D test cases prove the effectiveness and the robustness of the coupling, which seems particularly suited to handling high heterogeneities and at 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 reservoir under a particular recovery strategy is a crucial factor for its economical exploitation. In the last decade several mathematical models describing physical phenomena occurring in reservoirs have been proposed, and a variety of numerical methods for their approximate solution has been investigated rather extensively. The mathematical models usually considered describe the fluid flowing in a reservoir as composed by many chemical components which combine in different phases. In this paper we mainly address the compressible model where three independent components (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 phases allowing 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 the vapor phase. Within the reservoir, pressure and temperature are different than STC: oil and gas combine in a different mass density ratio according to a thermodynamic equilibrium constraint. The Black-Oil model consists of a set of partial differential equations describing the conservation of mass for each component and the time evolution of the phase pressures 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 EC contract 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
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with a very different nature. The conservation of mass equations form a nonlinear hyperbolic system if dispersion effects are neglected, or a strongly advection-dominated parabolic system if these latter ones are included. Instead, in the volume discrepancy formulation adopted by TB, phase pressures obey a parabolic equation which tends to be elliptic in the limit of null total (fluid plus rock) compressibility. A reliable numerical model must be able to take into account the complexity of a real reservoir. The irregularity in the geological and geometrical morphology affects the computational domain and empirical data such as the permeability and the porosity fields involved in the calculations. In the present approach, unstructured triangle-based grids have been preferred because of their inherent flexibility and of the possibility of introducing in a straightforward manner local refinements around complex geometrical patterns. The reservoir permeability field is a full tensor quantity that presents very large local variations, up to 8 or 10 orders of magnitude [23]. This results in highly discontinuous terms in the discretized form of the equation that may lead to inaccurate solutions if not appropriately treated. Moreover, the strong nonlinear hyperbolic nature of the mass conservation equations is responsible for the formation of sharp fronts moving through all the reservoir. Therefore, an effective numerical resolution of the Black-Oil model deserves special attention, and appropriate methods should be devised to ensure both space and time accuracy. The experience acquired so far in numerically treating flow models in porous media demonstrates the effectiveness of blending mixed finite elements for the parabolic equations and shock-capturing finite volume methods for the hyperbolic equations [21, 15]. These numerical strategies directly exploit the mathematical properties of the model, taking proper care of the completely different nature of the equations in the system. In particular, this approach is capable of accurate approximation of pressure and velocity flow fields in the presence of strong reservoir heterogeneities as well as of capturing unsteady discontinuous fronts. Mixed finite element methods [14] for the simultaneous solution of fluid pressure and velocity yield a very accurate numerical discretization and have been extensively studied; see [17, 26]. In our framework, velocities are approximated using the lowest order Brezzi–Douglas–Marini elements (BDM1 ) with piecewise-constant pressure. Their accuracy in treating heterogeneous permeability situations has been evaluated in a previous work [11]. Shock-capturing finite volume methods exhibit the property of accurately resolving steep gradients without spurious numerical oscillations while taking numerical dissipation effects at a very low level. A conservative formulation is usually considered, allowing an accurate numerical prediction of the moving discontinuities appearing in the physical solution. The discretization is based on an evaluation of the fluxes by generalizing the Engquist–Osher numerical flux for systems [24, 25] and on a second-order triangle-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 that mass concentrations are known. Then, an explicit predictor–corrector scheme with characteristic tracing updates the hyperbolic mass conservation equations using the velocity field computed in the previous stage. As yet, the idea of splitting the equations into two subsystems to be solved separately in an alternate way has been exploited by researchers in porous media flows and is documented in literature (see, e.g., [9]). It is the purpose of this work to de-
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velop a new sequential approach for the Black-Oil formulation, considering in this framework both mixed finite element and unstructured shock-capturing finite volume discretizations. The major issue of this paper, from the numerical point of view, relies on illustrating by direct numerical experiences the effectiveness of this methodology when applied to the complex physical phenomena predicted by the Black-Oil model. Our discretization model generalizes the Durlofsky approach proposed for slightly compressible 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 the velocity 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 using finite volume methods in oil modeling problems is relatively new. Bell, Colella, and Trangenstein [7] and Bell, Dawson, and Shubin [8] did some pioneering work in this direction though their discretizations rely upon a finite difference formulation of the model equations. It is worth pointing out that the mixed finite element formulation for pressure and velocity is particularly well suited as it directly generates up to an optimal order of accuracy the interelement fluxes which in turn can be easily incorporated into the finite volume solver. More importantly, we stress again that this blended approach allows one to exploit the different nature of the two governing equations by two different methods, each of them being the best suited for the equation at hand. The paper proceeds as follows. In section 2 we describe the mathematical formulation of the Black-Oil model as proposed by TB. Mass density conservation equations and Darcy’s law for phase velocities and pressures are introduced and a total velocity is defined. In section 3 we present the sequential approach and derive the parabolic pressure equation via the discrepancy error formulation. In section 4 we report the numerical discretization and time evolution scheme of this equation using mixed finite elements. In section 5 we illustrate the discretization of the mass conservation equations using an explicit predictor–corrector time marching scheme with characteristic tracing 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 approach by a set of test cases describing realistic situations. A final discussion on the performances of the present method and its future development closes the paper in section 7. 2. The Black-Oil formulation. In this section we formulate the equations describing the Black-Oil model for reservoir simulation. A reservoir fluid is considered composed 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 and with water to form three different phases. These phases, called “liquid,” “vapor,” and “aqueous,” and indicated by the subscripts l, v, and a, simultaneously flow through the reservoir. At STC, the liquid and vapor phases are, respectively, composed of oil and gas hydrocarbon, so we can identify each phase with its principal components. At reservoir pressure and temperature, oil can also be present in the vapor phase, gas in the liquid and aqueous phase. This mass redistribution is governed by thermodynamics constraints, and situations of undersaturation, in which some of the phases are not formed, must be appropriately treated by the simulator.
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One of the major features of the Black-Oil model is its capability of predicting mass transfer effects between phases when simulating primary—pressure depletion— and secondary—water injection—recovery. These capabilities have contributed to imposing the Black-Oil method as the mathematical model mainly used in petroleum reservoir engineering. The Black-Oil mathematical formulation introduced by TB is based on the following assumptions: • isothermal thermodynamic phase-equilibrium: how components combine to form phases; • Darcy’s law: how the phase volumetric flow rates are related to the corresponding 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 model is taken into account by a set of thermodynamic relations which express how any chemical component redistributes over the phases and how the phase fluid occupies the porous volume. These relations are usually given as a function of the liquid pressure in the reservoir. Each phase is described by a velocity and a pressure field, related by Darcy’s law. Moreover, the phase pressures can be related to a total velocity field—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. The equation of state is usually formulated in terms of the phase saturations—the ratio between the volume occupied by the phase and the total pore volume. This equation states 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 at each step of the computation and one of the constitutive relations of the model has to be relaxed. Following TB, we linearize in time the equation of state so that it is only satisfied approximately, thus introducing a volume discrepancy error, and impose exactly the thermodynamic equilibrium, Darcy’s law, and the component conservation equations. 2.1. Primary and secondary variables. When mathematically formulating the Black-Oil model, it is useful to distinguish between a set of “primary” and “secondary” variables. Time evolution of the previous ones is directly governed by the equations of the model, while the secondary variables depend on the first ones by some functional relations. Primary and secondary variables can again be subdivided for sake of presentation in “component unknowns” and “phase unknowns,” due to the physical nature of the quantities they are related to. Component primary variables are the mass concentration densities (mass per unit pore volume) indicated by z = [zo , zg , zw ]T . These unknowns express the density of each chemical component integrated over the different phases in which it is distributed. Let us also introduce as “secondary variables” the generic element zij giving the concentration density of the component i in phase j, and the component density matrix Z:
(2.1)
zol Z = (zij ) = zgl 0
zov zgv 0
0 zga . zwa
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L. BERGAMASCHI, S. MANTICA, AND G. MANZINI
The matrix Z and the vector z satisfy the following condition: X (2.2) zij = zi , i = o, g, w, or Ze = z, j=l,v,a
T
with e = [1, 1, 1] , which simply casts the component conservation integrated over the phases. In our mixed formulation, phase primary variables are the pressure p and the total velocity ~vt given as the sum of the phase velocities: ~vt = ~vl + ~vv + ~va . These latter T ones, collected into a phase velocity vector ~v = (~vl , ~vv , ~va ) , are “secondary” variables and can be obtained at any time step once the total velocity has been updated. Phase secondary variables are also the saturations u, defined as the ratio between the pore volume occupied by the phase and the total pore volume. As a consequence of the sequential formulation, real saturations no longer satisfy the equation of state and are substituted by a vector of “effective saturations” s—which will be referred to simply as “saturations” in the following. The saturations s depend on the saturations u by uj (2.3) j = l, v, a . sj = ul + uv + ua The equation of state is finally replaced by an obvious normalization condition, with no 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 gas hydrocarbon, and the aqueous phase by water. Hence, it is possible to identify for a phase a principal component, i.e., the component present at STC. Let us now introduce a dimensionless matrix R whose generic term rij represents the ratio between the 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 0 1 Rv 0 zol 0 0 , Z = RDz = Rl 1 Ra 0 zgv (2.4) 0 0 1 0 0 zwa where (2.5)
Rl =
zgl , zol
Rv =
zov , zgv
Ra =
zga , zwa
and Dz = diag[zol , zgv , zwa ]. In this way we write the density component matrix Z as a function of the primary variable z and the entry of matrix R, which are given as functions of the pressure p, either explicitly or (more usually) tabulated in an empirical way. In what follows it will be useful to define the matrix T as the left inverse of the matrix R (i.e., I = T R). Let us also introduce three formation volume factors, defined as the ratio between phase saturations and densities of the principal components. These terms are also known as empirical functions of the pressure uv ua ul (2.6) , Bv = , Ba = . Bl = zol zgv zwa Or, in matrix form: (2.7)
B = diag(Bl , Bv , Ba ) = Du Dz−1 .
The definition of the matrices R and B allows us to express an explicit dependence of the two variables z and u as (2.8)
u = BT z.
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2.3. Undersaturation. Although the Black-Oil model involves three components, this does not necessarily mean that three phases are always formed in the whole reservoir. When the three components are present, two cases of undersaturation are possible: • 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 in either liquid or vapor. Furthermore, only one undersaturated situation is allowed at a time, because oil must be present in liquid or in vapor phase and it cannot dissolve into the aqueous phase. We say that a phase is missing if the corresponding component of the diagonal matrix Dz (or equivalently, by (2.4), of the vector T z) 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 component of T z 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 for p the relation (2.9)
zgv ≡ (T (p)z)v ≡ −Rl (p)zo + zg − Ra (p)zw = 0.
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 1 0 ¯ = R¯l R¯a , (2.10) R 0 1 ¯ is no longer nonsingular, so we where R¯l ≡ Rl (pb) and R¯a ≡ Ra (pb). Now matrix R ¯ need to define a new matrix T which plays the role of T in the saturated case yielding ¯ = I: T¯R 1 0 0 ¯ (2.11) . T = 0 0 1
Since there is no vapor phase, all of the oil must be in liquid and all of the water must be in aqua, giving (2.12)
u = [ul , ua ]T ,
Dz = diag(zo zw ),
B = diag(Bl , Ba ),
and (2.13)
zo ¯ z = R¯l zo Z = RD 0
0 R¯a zw . zw
2.3.2. Liquid phase missing. When the liquid component of T z is negative we have that the liquid phase is missing. To force this component to be zero we need to solve (2.14)
zgl ≡ (T z)l ≡ Rv zg − Rv Ra zw + zo = 0,
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L. BERGAMASCHI, S. MANTICA, AND G. MANZINI
which gives R¯v (p) =
zo . zg − Ra (p)zw
As in the vapor missing case we can define R¯v 0 ¯ = 1 Ra , (2.15) R 0 1
¯ and T¯ as follows: R T¯ =
0 0
1 −Ra 0 1
.
Since there is no liquid phase, the amount of gas in vapor zgv must be equal to zg −zga and all the water must be in aqua; that is, (2.16)
u = [uv , ua ]T ,
Dz = diag(zg − Ra zw , zw ),
B = diag(Bv , Ba ),
and R¯v (zg − Ra zw ) ¯ z= zg − R a zw Z = RD 0
(2.17)
0 R a zw . zw
2.4. The equation of state. The basic physical principle defined by the equation of state is that the fluid which flows in the reservoir completely fills the rock pore volume. This fact is formally expressed by the constraint that the total sum of the phase volume is equal to the total porous volume and given by (2.18)
ul + uv + ua = 1.
The saturation u depends on the pressure and the component mass densities. Since these latter fields are time dependent, the equation of state can be directly exploited to 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 the model. We adopted here the approach proposed by TB, where a linearized form of the equation of state is considered. This approach yields an equation for the time variation of the pressure at a price of introducing a “volume discrepancy error” because the equation of state is not exactly satisfied. The linearization of the equation of state also allows to separate the system of equations of the Black-Oil model in two subsets which can be treated sequentially, using ad hoc numerical methodologies. 2.5. Darcy’s law for the total velocity. The three phase velocities ~vj are related to phase pressures by Darcy’s law: i h ~ , j = l, v, a, ~ + pjl ) − ρj g ∇h (2.19) ~vj = −Mj ∇(p
where p is the pressure for the liquid phase, pjl is the difference between the pressure in the phase j and the liquid phase, g is the gravity, h is the depth, and ρj is the density in phase j. In what follows, in order to simplify the notation, we will neglect the 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 (2.20)
Mj = K
krj , µj
krj /µj Mj fj = P = , k /µ Mt s s rs
Mt =
X j
Mj ,
NUMERICAL FORMULATION OF THE BLACK-OIL MODEL
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where K is the absolute permeability tensor, krj the relative permeability of the phase j, and µj the phase viscosity. We can introduce a total velocity ~vt as a function of the liquid phase pressure: i h X X ~ . ~ − ρj g ∇h (2.21) ~vj = − Mj ∇p ~vt = j
j
For subsequent convenience we define the phase vector fˆ fˆj = fj ρj ,
j = l, v, a,
and
fˆ = [fˆl , fˆv , fˆa ]T ,
and we rewrite (2.21) in the compact format: ~ . ~ + eT fˆ g ∇h Mt−1~vt = −∇p
(2.22)
Inversely, the velocity of phase j can be written as a function of the total velocity (2.21): i h X ~ (2.23) , j, s = l, v, a. ~vj = fj ~vt − Ms (ρs − ρj )g ∇h s6=j
Moreover, we set
tj = fj
X
Ms (ρj − ρs ) g,
j, s = l, v, a,
s6=j
and f = [fl , fv , fa ]T ,
t = [tl , tv , ta ]T
to obtain a compact expression for the phase velocity vector (2.24)
~ ~v = f~vt + t∇h.
The quantities µj and krj are functions of the pressure p and of the saturations s, respectively. Usually, the relative permeabilities are given as tabulated values in conjunction with special models for three-phase flow (see the Appendix). 2.6. Mass conservation equations. The conservation of the mass for the three different components is given by the balance of the contribution of each phase to the mass fluxes. In the saturated case they read zol~vl zov ~vv ∂(zo φ) ~ + = 0, +∇· ul uv ∂t zgl~vl zgv ~vv zga~va ∂(zg φ) ~ +∇· + + (2.25) = 0, ∂t ul uv ua zwa~va ∂(zw φ) ~ +∇· =0 . ∂t ua Using the definitions of Z and Du previously introduced, these relations can be expressed in a more compact vector form, which is valid also in the case of undersaturation, provided that ~v is the vector of existing phase velocities such as (2.26)
∂(zφ) ~ + ∇ · (ZDu−1~v ) = 0. ∂t
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L. BERGAMASCHI, S. MANTICA, AND G. MANZINI
It is convenient to reformulate the mass conservation laws given by (2.26), in terms of the matrices R and B. These matrices are known functions of the pressure p, which is the solution at each time step, together with ~vt , of the mixed pressure-velocity system. We have (2.27)
∂(zφ) ~ + ∇ · (RB −1~v ) = 0, ∂t
which holds even in the general case of missing phases, once the proper definition of undersaturated B and R is employed. 3. The sequential approach. The sequential method is one of the approaches used in literature to solve the equations of the Black-Oil model and relies upon a linearization of the equation of state. Such a method assumes that the state equation is 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 the total velocity, taking at the previous time step the values for the saturations and the density z. Then, the hyperbolic system is solved as a function of z. The nonlinear character of this equation depends on the phase velocities, which are functions of total velocity and saturations. The mass component densities are evolved in time by freezing pressure and total velocity as solution of the mixed equation at the current time step. 3.1. The pressure equation. Following TB, we develop a differential equation for the pressure assuming that the state equation is not exactly satisfied, and introducing in this way a volume discrepancy error. We start by writing the state equation as a function of the pressure at the previous time step and of the component density vector z. By the chain rule, we can write the time derivative of u as ∂u ∂u ∂p ∂u ∂z = + . ∂t ∂p ∂t ∂z ∂t Introducing a linear approximation of ∂∂tu in the time interval ∆t (3.1)
un+1 − un ∂u = , ∂t ∆t
(3.2)
we impose that at time tn+1 the state equation is exactly satisfied, or eT un+1 = 1. Equation (3.1) can be rewritten, after multiplication by eT , as (3.3)
∂u ∂p ∂u ∂z 1 − eT u + eT , = eT ∂p ∂t ∂z ∂t ∆t
where, for simplicity, we have dropped the superscript n. Actually, the pressure p is computed in order to correct the errors introduced in the state equation. Substituting ∂z ∂t from (2.27) in (3.3), we get ∂u ∂u ∂φ ∂p ∂u ~ eT u − 1 φ = −φ eT + eT z + eT ∇ · RB −1~v . (3.4) ∆t ∂p ∂z ∂p ∂t ∂z Observe now that, by (2.8), the saturation u is a homogeneous function of the first degree in z. In fact,
(3.5)
∂u = BT, ∂z
which implies
∂u z = u. ∂z
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NUMERICAL FORMULATION OF THE BLACK-OIL MODEL
Equation (3.4) can then be written as eT u − 1 φ= ∆t
(3.6)
−φ eT
∂u ∂φ + eT u ∂p ∂p
∂p ~ · RB −1~v , + eT BT ∇ ∂t
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 introduces an O(∆t) error in the numerical method which is usually referred to as “volume discrepancy error.” This essentially indicates the accuracy by which the equation of state eT u = 1 is solved. Hence, the pressure computed from the parabolic equation solves only approximately the volume balance equation. However, this O(∆t) error can be removed by correcting the pressure in order to exactly satisfy the equation of state. This technique is based on an algebraic iterative solution (by, e.g., Newton’s method) of the equation of state, regarded as a nonlinear function of pressure only f (p) ≡ eT u(p) − 1 = 0. The application of this technique results in a relevant reduction of the volume discrepancy error, as investigated by Dicks in [18]. 4. Mixed finite element formulation. The mixed finite element approach starts 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 domain Ω into triangular elements whose maximum diameter is h, and let {Wh }, {Ph } be two families 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 {Ph } should satisfy the discrete inf-sup condition [12, 4]. A proper choice for Ph and Wh is represented by the space of piecewise constant functions P0 (T ), and the BDM1 space defined as Wh = {~q ∈ H(div; Ω) | ∀T ∈ Th , ~q|T ∈ (P1 (T ))2 }, respectively. This space coupling ensures first-order spatial accuracy on the approximation of pressures and second-order accuracy for velocities, while second-order accuracy for pressure can be found at the element centroids [20]. A proper choice of degrees of freedom yields a flux conservative scheme. The discrete weak formulation of the mixed problem is easily obtainable by the projection of (3.6) and (2.21) onto the spaces {Ph } and {Wh } under their usual scalar products, while the discretization in time is performed by a simple forward difference: (4.1a)
eT u − 1 φ ψ dA = ∆t
∂u ∂φ pn+1 − pn T −φ e +e u n ψ dA ∂pn ∂p ∆t ZΩ ~ · RB −1~v ψ dA eT BT ∇ ∀ψ ∈ Ph , +
Z
Ω
Z
T
Ω
(4.1b)
Z
Ω
Mt−1~vt · ω ~ dA = −
Z
Ω
~ ·ω ∇p ~ dA +
Z
~ ·ω ~ dA eT fˆ g ∇h
∀ω ∈ Wh .
Ω
We designate, now, the total number of elements in the discretized domain as Ne and the total number of edges as Nf . The total velocity and pressure fields can be
980
L. BERGAMASCHI, S. MANTICA, AND G. MANZINI
represented in terms of degrees of freedom pk and qm and basis functions ψk and ω ~m as (4.2)
~vt =
2Nf X
qm ω ~m ;
p=
m=1
Ne X
pk ψ k .
k=1
Here, we choose the basis functions in order to satisfy ψk = 1 over the element Tk and zero elsewhere and, for j = 0, 1, Z 1 if j Nf < m ≤ (j + 1) Nf and mod (m, Nf ) = i, j ω ~ m · ~ni l dl = 0 otherwise, ei where ei is the edge of index i and ~ni is the respective normal. ¯ T¯ in the undersaturated case) and B appearing in (4.1a) The matrices R, T (R, are the ratio matrices defined in section 2.2, and their elements, known functions of the pressure at the previous time level, can be considered constant over each element Tj of the discretization mesh. Choosing the test function ψ in (4.1a) to be the pressure basis function ψj characteristic of element Tj , it is possible to simplify the last integral therein as Z Z Z ~ · ~vt dA . ~ · ~v dA = ~ · RB −1~v ψj dA = (4.3) ∇ eT ∇ eT BT ∇ Tj
Tj
Ω
We can thus rewrite (4.1a) and (4.1b) expanding the two unknowns ~vt and pn+1 in terms of the basis functions ω ~ m and ψj :
(4.4a)
2Nf X
qm
m=1
(4.4b)
2Nf X
m=1
qm
Z
Tj
Z
Ω
~ ·ω ∇ ~ m dA + pj hnj =
Mt−1 ω ~m
·ω ~ q dA −
Ne X
k=1
Z
Tj
pk
Z
eT u − 1 φ dA + hnj pnj , ∆t
~ ·ω ∇ ~ q ψk dA =
Z
j = 1, . . . , Ne,
~ ·ω ~ q dA, eT fˆ g ∇h
Ω
Ω
q = 1, . . . , 2Nf,
where we have, for simplicity, considered homogeneous Dirichlet boundary conditions in the application of the divergence theorem to (4.1b), and set (in view of (2.8)) Z 1 ∂φ n T ∂BT T hj = −φe z + e u n dA . (4.5) ∆t Tj ∂pn ∂p The mixed system, built with (4.4a) and (4.4b), can be written in compact matrix form as follows: (4.6)
Mx = b ,
with (4.7)
M=
A CT
C −D
,
x=
q p
,
b=
r1 r2
.
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981
The system matrix M is symmetric but not positive definite. In fact, the pressure stiffness matrix D has positive entries due to the parabolic character of the pressure equation (see [35]). Nevertheless, the simple block structure of M suggests its Schur decomposition with respect to D, which leads to a single equation in the variable q: (4.8)
Sq = c ,
with
S = A + CD−1 C T ,
c = r 1 + CD−1 r 2 .
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 the sparsity pattern of matrix S is the same as that of A—with at most ten nonzero entries per row. The condition number of S is strictly related to the time step, which is at the denominator of the elements of matrix D. If the time step is very small, then the condition number of S is very close to that of A. By contrast, if ∆t has a large value (of the order of 10 days in the units used in the Black-Oil model), then the ill-conditioning of the divergence matrix CC T can influence the condition number of S. 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) with respect to the matrix A and consists of the iterative solution of the, once again, SPD system C T A−1 C + D p = r 2 + C T A−1 r 1
for the special preconditioning techniques, which are required in the solution of this system; see, e.g., [27, 31, 10] and the references therein. 4.1. Second order in time pressure correction. In section 4 we developed a parabolic equation which is second-order accurate in space and only first-order accurate in time. A globally second-order algorithm would require a more accurate time discretization of the pressure equation resulting in a nonlinear iterative scheme and adding considerable computational complexity to the algorithm. For sake of completeness we mention that second-order methods, which do not require the solution of a nonlinear equation, are reported in literature. An example of such procedures (based on 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 developed to describe primary and secondary oil recovery processes—characterized by a slow variation of the pressure field—and hence the time truncation errors are expected to be small. 5. Shock-capturing finite volume formulation. The sequential approach introduced in section 3 separates the pressure and velocity equations from the mass balance equations. Assuming pressure and velocity fields given by the solution at the previous time step, mass balance equations form a hyperbolic system which is discretized on 2-D unstructured meshes in a finite volume framework via an explicit higher order Godunov-type method. This method allows an accurate approximation of the moving discontinuities which appear in the physical solution. This is possible because, in the conservation form of our shock-capturing finite volume approach, discontinuities are computed as part of the numerical solution [29]. The main drawback of most of these schemes is that steep gradients are approximated by continuous transitions which usually spread the discontinuity over many cell points. A higher
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order shock-capturing scheme yields a solution which is at least second-order accurate in smooth regions and provides nonoscillating sharp shock profiles. The multidimensional scheme developed in the present work takes origin from the class of multidimensional methods historically proposed by Colella in [16]. These methods were originally developed to numerically approximate the solution of strictly hyperbolic systems of conservation laws. They were subsequently extended to general nonstrictly hyperbolic systems with local linear degeneracies in [7] and applied to oil recovery situations. In the 1-D version, a Taylor series extrapolation in space and time along characteristic curves gives two approximations of the solution at each cell-interface, one from the cell on the left (z L ) and the other from the cell on the right (z R ). Then, a local Riemann problem is solved and the Godunov flux, required to update the cellaveraged solution, is defined as the physical flux calculated at the Riemann solution state. If z L and z R are simply taken to be the cell-averaged values, considering in this way a constant representation of the spatial dependence of the solution within any cell, a first-order accurate scheme is obtained. Second-order accuracy is instead given 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 of the cell i are defined as approximations of ∆t ∂z ∆x ∂z n (5.1) + , zi ± 2 ∂x 2 ∂t
z ni being the i th cell-averaged state at time tn . This approach requires a monotonized slope computation with some form of limiting to avoid numerical oscillations at discontinuities. The resulting piecewise linear solution is traced along the characteristic families approximated by the eigenstructure computed at some mean state. In the present work, a cell-averaged approximation of the solution logically associated to the centroid of each cell is advanced in time via an explicit predictor–corrector time stepping scheme using a conservative flux balance estimation. The higher order Godunov method 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 procedures; 3. update the cell-averaged solution within any cell. Two main variants have been introduced, with respect to the original scheme of Colella to make possible the calculation on a triangle-based unstructured mesh. First, the central difference monotonized slope considered in [7] for the piecewise linear reconstruction has been substituted by the triangle-based TVD estimation of gradients of Barth and Jespersen [6]. Then, tracing forward along characteristics is done in the direction normal to each cell-interface. This approach differs qualitatively from the unsplit approach on structured grids of [7], whose multidimensional method considers contributions to fluxes both in the direction 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 the robustness of our approach. 5.1. The finite volume conservative framework. Integrating (2.26) over the element Tk yields Z Z ∂φ ∂z ~ · RB −1~v dA . z ∇ (5.2) + φ dA = ∂t ∂t Tk Tk
NUMERICAL FORMULATION OF THE BLACK-OIL MODEL
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The application of the divergence theorem allows a finite volume formulation of the mass conservation laws: (5.3)
Z
Tk
z
3 Z X ∂z ∂φ + φ dA = − RB −1~v · ~ndl, ∂t ∂t i=1 ei
where the summation is extended over all the edges of the element Tk . The vector ~v is the phase velocity vector at time step tn+1 from the mixed system. Let us call z k,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 conservation laws is given by Z ∂φk ∆pk 1 z k,n+1 − z k,n ~ (5.4) φk · ~n dl, RB −1 f~vt + t∇h + z k,n+1 =− ∆t ∂p ∆t Ak ∂Tk where f is the vector of fractional flows expressed as a function of mass densities through the saturations by (2.20), known at time step n. The pressure field p and the total velocity field ~vt are given by the initial “parabolic” step of the sequential method and considered constant in the time interval (tn , tn+1 ]. The porosity φk (p) as a function of p can also be assumed constant within each cell. If, in addition, φk is considered constant in time, we can view (5.4) as the finite volume formulation corresponding to the following hyperbolic system of conservation laws written in differential form: (5.5)
φ
∂z ~ ~ · ~g(z) = − ∂~g · ∇z, = −∇ ∂z ∂t
~ 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 focus 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) around the centroid position ~xc , which produces a linear correction in space and time of the cell-averaged solution z at time tn . We use this linear representation of the solution in Tk × (tn , tn+1 ] to compute the initial states of the local Riemann problems at the midpoint of each cell-interface ~xf , as required by the algorithm. The first-order Taylor expansion yields ∆t ~ + ∆t ∂z = z(~xc , tn ) + (~xf − ~xc ) · ∇z z ~xf , tn + (5.6) 2 2 ∂t and, after substitution of (5.5) into (5.6), ∆t ~ − ∆t 1 ∂~g · ∇z ~ . z ~xf , tn + (5.7) = z(~xc , tn ) + (~xf − ~xc ) · ∇z 2 2 φ ∂z x xf − ~xc , where the factor 1/2 is introduced only for convenience Let us define ∆~ 2 = ~ reasons, in order to have the same final form of the analogous relations in [7]. We g ∂~ decompose ∆~x and ∂ z onto the directions ~η and ~τ , normal and tangent to the face f at ~xf , respectively, as shown in Figure 5.1:
(5.8)
∆~x = (∆~x · ~η )~η + (∆~x · ~τ )~τ ,
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L. BERGAMASCHI, S. MANTICA, AND G. MANZINI
C τ xc
A
η
xf
B
Fig. 5.1. The geometry of triangle-based decomposition for the local characteristic tracing.
(5.9)
∂~g = ∂z
∂~g ∂~g · ~η ~η + · ~τ ~τ . ∂z ∂z
Substituting them in (5.7), we have 1 1 ∂~g ∆t n n ~ (5.10) = z(~xc , t ) + ∆~x · ~η − ∆t · ~η ~η · ∇z z ~xf , t + 2 2 φ ∂z 1 1 ∂~g ~ · ~τ ~τ · ∇z. ∆~x · ~τ − ∆t + φ ∂z 2 For nonlinear problems, as suggested in [7], the estimate provided by (5.10) must be modified in order to disregard the wave components which do not correspond to waves traveling towards the cell-interface. This procedure is coherent with the upwind nature of the scheme and has been reported in literature to yield a more robust scheme even in strongly nonlinear cases. Let Hη and Hτ be the normal and the tangent component of the Jacobian, respectively; i.e., Hη = (∂~g/∂z) · ~η , and Hτ = (∂~g/∂z) · ~τ . The subtraction of waves traveling in wrong directions is accomplished via a local projection operator, defined as P = XΛ+ X −1 , where X is the matrix of right eigenvectors of the Jacobian in the η direction; i.e., X −1 Hη X = 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: ∆t 1 1 n n ~ z ~xf , t + (5.11) = z(~xc , t ) + P ∆~x · ~η − ∆t Hη ~η · ∇z 2 2 φ 1 1 ~ ∆~x · ~τ − ∆t Hτ ~τ · ∇z. + φ 2 ~ which appears in (5.11) must satisfy a TVD stability condition to The gradient ∇z ensure that numerical oscillations do not appear whenever a discontinuity forms in the solution. This is directly imposed as a monotonicity constraint in the triangle-based reconstruction algorithm which we present in the next section. 5.3. The TVD triangle-based linear reconstruction. A linear reconstruction of the flow variables within a generic triangular cell Tk can be written in the following way: (5.12)
~ k, z(~x, t) = z(~xk , t) + (~xk − ~x) · ∇z|
NUMERICAL FORMULATION OF THE BLACK-OIL MODEL
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(j)
k
ni
p
(i)
Fig. 5.2. The triangle-based stencil for the linear TVD reconstruction.
where ~xk is the centroid of the triangle Tk and z k = z(~xk , t) is the value logically associated to it at time t. All the relations involving quantities like z k are to be ~ k inside Tk must interpreted componentwisely. An estimate of a “mean” gradient ∇z| z | and ∂ z | . The be somehow provided, which implies an estimate of the two terms ∂∂x k ∂y k application of the divergence theorem can easily yield these derivatives which can be taken with second-order accuracy in space as the values at centroids. Let us consider ~: the divergence theorem applied to a generic vector field V I Z ~ · ~nds, ~ ·V ~ dΩ = (5.13) V ∇ Ω
∂Ω
where ∂Ω is the close boundary which defines a generic domain of integration Ω and ~n is the outward normal to this boundary. ~ = (z, 0) Let us take for Ω the triangular cell Tk and consider two vector fields V ~ = (0, z). Hence, applying the relation (5.13), we obtain the required estimates and V z and ∂ z . for ∂∂x ∂y The satisfaction of a TVD condition is obtained by introducing a limiting function θ on the slopes given by the gradient in the piecewise linear reconstruction. Hence, the reconstruction takes the form (5.14)
~ k . z(~x, t) = z(~xk , t) + θ(~xk − ~x) · ∇z|
The limiter θ takes values in the range between 0 and 1. Any limiter introduced in the literature can be used; see [34] for a review. However, these limiters are essentially monodimensional. A really 2-D limiter has been introduced by Barth and Jespersen (see, e.g., [6]) and is reported here. R Let us indicate with z L i and z i , respectively, the left and right reconstructed values on the i th edge ei of the mesh. Referring to Figure 5.2, the piecewise linear reconstruction inside Tk yields the left state z L i while the piecewise linear reconstruc. “Left” and “right” are defined according to tion inside Tp yields the right state z R i the orientation of the normal vector at the edge ei which points outward to Tk and inward to Tp . The limiter proposed by Barth and Jespersen is designed in order to ensure that the following two properties are satisfied by the reconstructed values: (A) the reconstruction must not exceed the minimum and maximum of the neighboring cell averages; L (B) the difference in the interpolated values z R i − z i at the i th edge and the difference in the corresponding cell averages z(~xk , t) − z(~xp , t) should have the same sign.
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L. BERGAMASCHI, S. MANTICA, AND G. MANZINI
These two properties can be summarized as (5.15)
1≥
L zR i − zi ≥0. z(~xk , t) − z(~xp , t)
Barth and Jespersen showed that to satisfy property (A) it is sufficient to design the limiter as follows: • Compute z max and z min as the maximum and minimum of the cell averages k k in the neighbors of cell Tk . • Then a local limiting value θj is introduced for each vertex (j) of the cell Tk : z max −z min(1, zk j −z k k ), if z j − z k > 0, z min −z (5.16) θj = min(1, zk j −z k k ), if z j − z k < 0, 1, if z j − z k = 0,
where z j 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 reconstructed values. If these values do not satisfy condition (B), they will be substituted by their mean. 5.4. Flux estimation. A standard way to define a numerical flux in shockcapturing 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 the left fluxes and then in correcting that by a dissipative term. The dissipative correction is introduced to stabilize the computation and to ensure convergence to the correct entropy weak solution. The procedure used in Godunov-like methods to define the numerical flux is generally referred to as the Riemann solver, since it considers as a model of the wave propagation of the numerical information an exact or an approximate solution of the Riemann problem locally defined by the left and the right interface states. The solution of a Riemann problem for a hyperbolic system of N equations with initial conditions z L , z R consists of a composition of a finite number of simple waves progressing from z L to z R . A path connecting the two initial states must be built via a sequence of intermediate states which correspond to the elementary waves—shocks and rarefactions—forming the solution. This procedure is a common approach to provide constructive proofs of existence and uniqueness theorems. A general theory is described by Smoller in [33] for systems of conservation laws satisfying a strict hyperbolicity and a genuine nonlinearity constraint. However, this condition is not satisfied in the Black-Oil model considered here and more generally in 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 complex situations where local linear degeneracies and loss of strict hyperbolicity occur. Let us call r l the l th right eigenvector of the eigenvalue matrix Λ—i.e., the l th column of the matrix X. In the case of small amplitude jumps z l − z l−1 = αl r l + O(|z R − z L |2 ); that is, the right eigenvectors approximate the wave curves in phase space to the second order in the jump between the left and the right state. This jump can thus be expressed by (5.17)
zR − zL =
N X l=1
αl r l + O(|z R − z L |2 ) .
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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 to be of unit length (5.18)
zR − zL =
N X
αl ̺l .
l=1
The generalized eigenvectors ̺l can be numerically approximated by computing them ¯ l = r l (¯ ¯ = (z L + z R )/2 as ̺ z ). In the same way, we can introduce a in a mean state z ¯ generalized eigenvalue λl which is an approximation of the wave speed along the line ¯ l . An approximation of the segments corresponding to the generalized eigenvectors ̺ exact solution of the Riemann problem [1] can now be estimated at the interface, by a linear combination of the contributions associated with the waves traveling from the left (positive eigenvalues) and those coming from the right (negative eigenvalues): X z Riemann = z z L , z R = z L + (5.19) αl ̺l . λl 0.2, j − sj | (6.2) ∆s = 0 otherwise. 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-undersaturated flow 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 length with constant permeability of 100 millidarcies and no dip. Two wells, an injector, and a 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) as well as the relative permeability data adopted in the simulation can be found in the Appendix. The referred test case is one dimensional in the original work by TB. Nevertheless, the code implementing the full 2-D algorithm has been used to produce the result reported here. The 1-D nature of the reservoir was simulated using the 100 × 1 regular grid shown in Figure 6.1. The number of degrees of freedom necessary for the discretization of the parabolic equation is over-dimensioned in this case. All the edges parallel to the flow direction are actually treated as “no flow” boundaries (the physical flux is set to zero across them). Moreover, as this example does not include any gravitational effect, the flow direction is unique from left to right. The Engquist–Osher
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1
1
AQUA
AQUA 0.8
0.8
LIQUID
LIQUID
0.6
0.6
0.4
0.4
VAPOR
VAPOR
0.2
0.2
0
0 0
100
200
300
400
500
600
700
800
900
1000
0
100
200
300
400
500
600
700
800
900
1000
Fig. 6.2. Test case 1. Saturations at 250 (left) and 750 days (right). The lower curve represents the 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 algebraic effort of higher order mixed finite elements (BDM1 ) is not justified here, due to the constant permeability and porosity fields. However, we stress that the purpose of such computations is just to validate the algorithm in some interesting test problems from the physical and mathematical viewpoint by a comparison with results previously reported in literature. We do expect the effectiveness of the coupled MFE-FV approach proposed in this work to be evident on truly 2-D problems with highly heterogeneous permeability and porosity fields, and in presence of counter-current flows. Test case 1 (TB 5) is representative of flows with sharp interfaces between phases due to relevant composition changes. The injector and producer wells operate at 4300 and 4100 psi, respectively, and the reservoir initial pressure is 4200 psi. The injected and initial reservoir component densities are given by .668 .0179 z inj = 178.68 , z res = 100.19 . .0668 .357
With these definitions, at the beginning of the computation, the reservoir conditions are undersaturated with the vapor phase missing, where the gas component is completely dissolved into the liquid and aqueous phases. The composition of the injected fluid, a two-phase mixture with the liquid phase missing, is such that a threephase region suddenly develops and a sharp front of gas, followed by a rarefaction wave, 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 problem 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 a Buckley–Leverett rarefaction-shock compound pattern. As the simulation proceeds, the liquid is displaced from the region near the injection well and thermodynamic equilibrium between the aqueous and vapor phases is there settled. At 250 days, as shown in Figure 6.2, three regions separated by sharp fronts are present in the reservoir: liquid and aqua coexist near the production well, aqua and vapor 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, as can be seen from the oil rate and gas-oil ratio (GOR) plots of Figure 6.3. In fact,
NUMERICAL FORMULATION OF THE BLACK-OIL MODEL
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5.5 5 4.5 4
OIL RATE
3.5 3 2.5
GOR
2 1.5 1
WOR 0.5 0 0
500
1000
1500
2000
2500
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 sudden increase in the GOR curve slope clearly gives evidence of breakthrough time. From a numerical viewpoint we remark that, on the linear 200 element mesh considered here, the proposed method is capable of resolving, with high-order spatial accuracy, the sharp gradients which form in the solution without being affected by spurious numerical oscillations. Moreover, these results are definitely comparable, both qualitatively and quantitatively, to that of the TB computations. 6.2. 2-D Simulations without gravity. In this section we present results from 2-D test cases with constant and variable permeability fields. We analyze the standard case of a quarter of five spot: a square domain with an injector and a producer well located 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 regular mesh of 3200 triangles. The simulations for the test cases of this section are performed extending to 2-D the 1-D test case of the previous section. From the saturation plots of Figure 6.5 it can be seen how steep gradients separating different zones of the reservoir are well captured by our scheme without spurious oscillations even in 2-D test cases. The next test case (3) is somewhat more representative of the typical situations occurring in reservoir simulation. We consider a sand-shale permeability field, depicted in 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 rendering the velocity fields when highly discontinuous geological properties are present. Mixed finite elements, on the other hand, are especially built to get a high order of approximation of the velocity, as they approximate simultaneously the two variables without deriving one from the other. As can be seen from Figure 6.6 the total velocity field qualitatively reproduces the permeability pattern. This behavior is further enhanced by the saturation contours of Figure 6.7. The flow lines, in fact, accurately avoid the low permeability zones and two main channels are formed. It can also be clearly seen from the figure that the two flow channels avoid the whole right side of the domain. This kind of study can be very important even in the case of an enhanced oil recovery process when a fully miscible component is injected into the reservoir. If the
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200 0.6
200 DAYS 200 DAYS 750 DAYS 750 DAYS
180
0.5
0.4 160
0.3 140 0.2
120 0.1
0
100 0
100
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300
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600
700
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Oil component density
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Gas component density
0.35 200 DAYS 750 DAYS
0.3
0.25
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500
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800
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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 regions where one might want to drill other injector or producer wells. 6.3. Influence of gravity effects. In all the test cases presented above, it can be easily seen that all the eigenvalues are nonnegative; therefore the Engquist–Osher flux reduces to a simple upwind evaluation. For the last test case we tried our code against a cross-sectional problem [7], which takes place in the x–z plane with gravity acting in the negative z direction. The cross section considered is 400 feet in length and 50 feet in height, discretized with a rectangular 80×40 mesh. We inject from the left face of the cross section a fluid with composition specified by the following saturations: sinj = [0, 0, 1]. The injection pressure of 2000 psi is imposed at the top of the face and a pressure determined by a condition of hydrostatic equilibrium down on the 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 distributed with a similar procedure as the injection face. A production pressure of 1600 psi was assigned along the whole production face at the right end of the reservoir. In this example, counter-current flow occurs in the vertical direction due to the effect of gravity on the fluid phases of different densities. Hence, for the z-directional fluxes,
993
NUMERICAL FORMULATION OF THE BLACK-OIL MODEL 1
1
0.5
0.5
0
0
1
1
0.5
0.5
0
0
1
0.5
0
1
0.5
0
Liquid phase
Vapor phase
Aqueous phase
Fig. 6.5. Test case 2. Liquid, vapor, and aqueous saturation contours at 250 and 750 days of simulation (from top to bottom). 300
300
1.8e+03
1.5e+03
200
200 1.2e+03
901
100
601
100
301
1
0
0 0
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 numerical flux. From the results of the simulations, we found out that the two solvers behave in a similar way, however, from a theoretical point of view, the Engquist–Osher scheme is more reliable, since it can detect and resolve coalescent eigenvector directions. The extra amount of work introduced by the Engquist–Osher flux calculation is not more than about 15 % of the total CPU time, as can be seen from Table 6.1, and can justify the use of such a sophisticated technique also in view of a production code.
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L. BERGAMASCHI, S. MANTICA, AND G. MANZINI
Vapor phase
Liquid phase
Aqueous phase
Fig. 6.7. Test case 3. Liquid, vapor, and aqueous saturations at 3, 28.5, and 45 days of simulation (from top to bottom). Table 6.1 Times (in percentage over the total CPU time) for the most expensive procedures in the simulation of test case 4 when using different schemes for flux calculation. Flux scheme Engquist–Osher Exact Riemann solver
Matrix assembly 15.95 20.66
System solution 37.89 48.87
TVD recon. 2.67 3.51
Predict 4.89 6.34
Flux estimation 25.40 10.21
7. Conclusions and future developments. In this work, we have reported the development of a sequential coupled mixed finite element/shock-capturing finitevolume method and its application to the numerical approximation of the solution of the time-dependent 2-D equations of the Black-Oil model. The present methodology appears to be a very powerful approach to time-dependent problems in Oil Recovery simulation when moving discontinuities are present in the solution and the reservoir is characterized by strong heterogeneities in the petrophysical data (permeability and porosity fields). In fact, high-order mixed finite elements are one of the most accurate numerical schemes to simultaneously treat the phase pressure and velocity. On the other hand,
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shock-capturing finite-volume methods exhibit the property of accurately resolving steep gradients without spurious numerical oscillations while taking numerical dissipation effects at a very low level. A conservative formulation is usually considered in this case, allowing an accurate numerical prediction of the moving discontinuities appearing in the physical solution. It is worth stressing again that the inherently conservative form of the mixed finite element methods has been shown to be particularly well adapted to the integral conservative formulation on which shock-capturing finite volume schemes are based. The use of such sophisticated techniques can appear quite expensive when compared with more traditional methods (such as Galerkin finite elements or structured finite differences). However, the high quality of the results obtained completely justifies this approach. In order to reduce the computational effort demanded by the computation of the mixed finite element method, a more detailed study is being carried on to develop efficient algebraic solvers for the solution of the linear systems which arise from this discretization. A more sophisticated coupling scheme should also be developed 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 flexibility of unstructured grids. It allows in a straightforward manner local refinements around complex geometrical patterns. In fact, a crucial point of this numerical scheme rests in its triangle-based formulation, which allows an easy treatment of complex reservoir geometries. The geometrical complexity of a reservoir and the presence of fault surfaces and large local variation of soil properties actually require the development of suitable mesh-adaptive algorithms with high flexibility, capable of automatically producing localized refined solutions. A more detailed study is required in order to include these adaptivity features. Research work is underway to develop this new capability. Finally, we remark that our approach can be largely applied in a number of situations of geophysical interest, such as, e.g., more complex oil recovery modeling (polymer flooding, compositional models), groundwater flow, and passive contaminant transport in porous media. Appendix. Pressure-volume functional forms. The spatial coordinates have units of feet, and time t is measured in days. Pressure p is measured in psi, viscosity in centipoise, and total permeability k is measured in ft2 cp/psi days, or .006328 times the value in millidarcies. In this way the gravitational acceleration g used in Darcy’s law must be removed because it is incorporated in this constant. A factor of 1/144 has been included in the gravitational term to ensure consistency of units. Phase densities are measured in lb/ft−3 . For the computations of this paper, the reservoir had porosity φ = 0.2(1 + 10−5 p). For our examples, we have used the standard square saturation relative permeability functions with Stone’s first model for three phase flows: krv = s2v ,
kra = s2a ,
krl =
(1 − sv − sa )klv kla , (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 aqueous saturation and the residual liquid saturation are both zero. Otherwise, the normalized form of Stone’s first model proposed by Aziz and Settari [3] should be preferred.
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L. BERGAMASCHI, S. MANTICA, AND G. MANZINI
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 the following functional forms: • Entries of matrix R Rl (p) Rv (p) Ra (p)
= .05p, = 9 × 10−5 − 6 × 10−8 p + 1.6 × 10−11 p2 , = .005p.
• viscosities .8 − 1 × 10−4 p µl = (.8 − 1 × 10−4 pb)(1 + cw (p − pb)) µv = .012 + 3 × 10−5 , .35 µa = .35(1 + cw (p − pb))
if liquid is saturated, if liquid is undersaturated.
if aqua is saturated, if aqua is undersaturated.
• formation volume factors (entries of matrix B) ( 1 + 1.5 × 10−4 p if liquid is saturated, Bl = 1+1.5×10−4 pb if liquid is undersaturated. 1+co (p−pb) ( 1 if vapor is saturated, 6+.06p h i Bv = ¯v R 1 1 1 if vapor is undersaturated. 7+.06p + Rv 6+.06p − 7+.06p ( 1 − 3 × 10−6 p if aqua is saturated, Ba = 1−3×10−6 pb if aqua is undersaturated, 1+cw (p−pb) 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) T
ρ = B −1 RT [ρo , ρg , ρw ] ,
where
ρo = 52.789 ρg = 0.07655 ρw = 62.382.
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