Energy stable discretization for two-phase porous media flows

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Energy stable discretization for two-phase porous media flows

Clément Cancès, Flore Nabet

To cite this version:

Clément Cancès, Flore Nabet. Energy stable discretization for two-phase porous media flows. Finite

Volumes for Complex Applications IX, Jun 2020, Bergen, Norway. �hal-02442233v2�

Energy stable discretization for two-phase porous media flows

Cl´ement Canc`es and Flore Nabet

Abstract We propose aP1finite-element scheme with mass-lumping for a model of two incompressible and immiscible phases in a porous media flow. We prove the dissipation of the free energy and the existence of a solution to the nonlinear scheme.

We also present numerical simulations to illustrate the behavior of the scheme.

Key words: Two-phase porous media flows, energy stable finite-elements MSC(2010): 65M60, 65M12, 35K65

1 Immiscible two-phase flows in porous media

We are interested in the numerical approximations of the equations governing an immiscible incompressible two-phase flow in a porous medium. LetΩ ⊂R^d(d= 2,3) be an open bounded polyhedral subset with Lipschitz boundary condition and lett_f>0 be an arbitrary finite time horizon. Then the conservation of the wetting (subscript w) and non-wetting phases (subscript n) are given by

φ ∂_ts_α−∇·(η_α(s_α)Λ∇p_α) =q_α(s_α), α∈ {w,n}, (1) where the unknowns are the phase saturationssα, which satisfy

sn+sw=1, (2)

Cl´ement Canc`es

Inria, Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlev´e, F-59000 Lille, France e-mail: clement.cances@inria.fr

Flore Nabet

CMAP, Ecole polytechnique, CNRS, I.P. Paris, 91128 Palaiseau, France e-mail: flore.nabet@polytechnique.edu

1

2 Cl´ement Canc`es and Flore Nabet

and the phase pressures pα. The porosityφ ∈(0,1)is given, as well as the intrin- sic permeabilityΛ, which is assumed to be symmetric and uniformly elliptic. The mobilityηα:[0,1]→Ris assumed to be continuous and strictly increasing, with ηα(0) =0 andηα(s)>0 ifs>0. They are extended to the wholeRbyηα(s) =0 ifs<0 andηα(s) =ηα(1)ifs>1. The sourcesqα are such that

q_α(x,s_α) =q_inj(x) η_α(c_α)

ηw(c_w) +ηn(c_n)−q_sink(x) η_α(s_α)

ηw(s_w) +ηn(s_n), (3) wherecw∈(0,1] and cn=1−cw is the prescribed composition of the injected mixture, and where q_inj,q_sink ∈L^∞(Ω) are nonnegative, bounded, and such that R

Ωqinj=^R_Ωqsink. The phase pressures are linked by the capillary pressure relation

p_n−p_w=γ(s_n), (4)

whereγ ∈L¹(0,1)is strictly increasing, nonnegative, and blows up ass_ntends to 1. This function is extended fors<0 byγ(s) =γ(0) +2s. We further assume that s7→ηw(1−s)γ(s)∈L^∞(0,1)ands7→ηw(1−s)γ⁰(s)∈L¹(0,1). These assump- tions are satisfied by the usual models of the literature (see for instance [1]). The system is complemented with no-flux boundary conditions and initial conditions sⁱⁿⁱ_α ∈L^∞(Ω;[0,1])that are compatible with (2). Note that sinceγ∈L¹(0,1), then Γ :s7→^R₀^sγ(a)dais bounded on[0,1]. The phase pressures being only defined up to a constant, we enforce additionally that^R_Ω p_n=0.

Multiplying (1) bypα, summing overα∈ {n,w}, integrating overΩ, and using (2) and (4) yields

d dt

Z

Ω

φ Γ(s_n) + Z

Ω

∑

α∈{n,w}

ηα(s_α)Λ∇pα·∇pα= Z

Ω

∑

α∈{n,w}

qα(s_α)p_α. (5) Following [6], we define the global pressure P byP=p_n−r(s_n) withr:s_n7→

Rsn

0

ηw(1−a)

ηn(a)+η_w(1−a)γ⁰(a)da. The definition ofPyields

∑

α

ηα(s_α)|∇p_α|²= (η_n(s_n) +ηw(s_w))|∇P|²+ ηn(s_n)η_w(s_w)

η_n(s_n) +η_w(s_w)|∇γ(s_n)|². (6) In view of the particular form (3) of the source terms,

α∈{n,w}

∑

q_α(s_α)p_α≤ q_inj−q_sink

(P+r(s_n)) +q_sinkk(s_n), (7)

withk(s_n) = ^ηw(1−sn)

ηw(1−s_n)+ηn(sn)γ(s_n). Since η_w(1− ·)γ⁰∈L¹(0,1) andη_w(1− ·)γ ∈ L^∞(0,1), bothrandkare bounded on(0,1). Moreover, the extensions outside(0,1) ofηα andγensure that for allε>0, there existsCεsuch that

|s|+|k(s)|+|r(s)| ≤εΓ(s) +Cε, ∀s∈(−∞,1). (8)

Combining (7) with (6) in (5) together with the uniform ellipticity ofΛ,η_n(s) + ηw(1−s)≥δ>0 for alls, and (8) we get that

d dt

Z

Ω

φ Γ(s_n) + Z

Ω

|∇P|²+

∑

α∈{n,w}

ηα(s_α)|∇p_α|²

!

≤C. (9) This estimate is enough to establish the existence of a weak solution. In this pa- per, our goal is to show that this stability is still encoded in very natural numerical schemes. For the sake of simplicity, we present our analysis in the framework ofP1

finite-elements with mass-lumping, but our approach can be extended to a wide fam- ily of schemes having the structure highlighted in [3, Section 3]. We show here how to transpose estimate (9) to the discrete setting and to infer the existence of discrete solutions therefrom. A full convergence study will be carried out in a forthcoming contribution. While deeply inspired from [7], the goal of this paper is to exploit more finely the energy estimate which allows to relax some stringent conditions on the anisotropy, on the mesh and the non-linearities presented in [7].

2 An energy stable finite-element scheme

We study the problem (1)–(4) using a P1conforming finite-element scheme with mass-lumping for the space discretization. LetT be a conforming simplicial dis- cretization of Ω. We denote by T ∈T a simplex, VT is the set of all the ver- tices a andVT ⊂VT the set of the(d+1)vertices a₀,· · ·,a_d of the simplexT. We also denote byV_h={u_h∈C(Ω):u_h|_T is affine for allT ∈T}the usual con- forming P1 finite-element space corresponding to the meshT and by(ϕ_a)_a∈VT the basis ofV_h. In order to deal with the mass-lumping procedure, for any ver- tex a∈V_T, we define the set s_a, the boundary ∂s_a of which being defined by the hyperplanes joining the centers of mass of the simplices, edges (and faces if d=3) sharinga as a vertex. We can now define the functional spaceX_h:={u∈ L^∞(Ω):u|_sais constant for alla∈V_T},and the linear mappingsπX:C(Ω)→X_h andπV:C(Ω)→V_hbyπ`u(a) =u(a), for anya∈VT, for anyu∈C(Ω),`=X,V. In order to lighten the notations, for anyu_h∈V_hwe writeπXu_h=u_h. We will use the following Poincar´e inequality that can be established as in [2]: there existC₁, C₂>0 depending only on the mesh regularity such that for anyu_h∈V_h,

u_h− 1

|Ω| Z

Ω

u_h L²(Ω)

≤C1

u_h− 1

|Ω| Z

Ω

u_h L²(Ω)

≤C2k∇u_hk_L2(Ω). (10) Before detailing the numerical scheme, we have to define the discrete tensor field Λ_h:Ω→R^d×dalmost everywhere byΛ_h(x):=ΛT:=_|T¹_|^R_TΛifx∈T. From there, we define the matrixA_T := (α_i,^T_j)1≤i,j≤d∈R^d×dby

4 Cl´ement Canc`es and Flore Nabet

α_i,^T_j=α^T_j,i:=

Z

T

ΛT∇ϕa_i·∇ϕa_j

and for anyu_h,v_h∈V_hone has, Z

T

ΛT∇u_h·∇v_h=



 v_a1−v_a0

. . . v_ad−v_a0



·A_T





u_a1−u_a0 . . . u_ad−u_a0



. (11) Following [4], we can prove that there existsC₃>0 depending on the regularity of the mesh and on the anisotropy ratio ofΛ andC₄>0 depending, in addition, ond such that that for anyT∈T the matrixA_T satisfies

cond2(A_T)≤C3 and

d i=1

∑

d j=1

∑

|α_i,^T_j|

!

(u_ai)²≤C4u·A_Tu,∀u= (u_ai)∈R^d. (12) We are now in a position to give the numerical scheme using a backward Euler scheme for the time discretization. Let(tⁿ)n=0,···,Nbe a partition of the interval[0,t_f] and forn=1,· · ·,Nwe denote byτn=tⁿ−tⁿ⁻¹the time step. We define the discrete initial data bys⁰_α,h:=∑a∈V_Ts⁰_α,aϕa∈V_hwiths⁰_α,a=_|s¹

a| R

sasⁱⁿⁱ_α.

Letsⁿ⁻¹_α ∈V_hbe given, we search forsⁿ_α,pⁿ_α∈V_hsuch that for anyv_α,h∈V_hwith α= (n,w)one has,

φ Z

Ω

sⁿ_α,h−sⁿ⁻¹_α,h τn

v_α,h+ Z

Ω

η_α,hⁿ Λh∇pⁿ_α,h·∇v_α,h= Z

Ω

q_α(sⁿ_α,h)v_α,h, (13a) sⁿ_n,h+sⁿ_w,h=1, (13b) pⁿ_n,h−pⁿ_w,h=γ_n,hⁿ , (13c)

Z

Ω

pⁿ_n,h=0. (13d)

We have denoted byη_α,hⁿ =πVη(sⁿ_α,h),γ_n,hⁿ =πVγ(sⁿ_n,h)and, q_α(sⁿ_α,h) =q_inj ηα(c_α)

ηw(c_w) +ηn(c_n)−q_sink ηα(sⁿ_α,h) ηw(sⁿ_w,h) +ηn(sⁿ_n,h).

Mimicking the continuous case, we define the discrete global pressure and we can obtain the discrete counterpart of (6).

Proposition 1Let sⁿ_α,h,pⁿ_α,h∈V_hbe a solution to the scheme(13). Then there exists C₅>0depending on the regularity of the mesh, on the anisotropy ratio ofΛ, onδ and d such that

Z

Ω

Λh∇P_h·∇P_h≤C_{5 Z}

Ω

η_n,hⁿ Λh∇pⁿ_n,h·∇pⁿ_n,h+ Z

Ω

η_w,hⁿ Λh∇pⁿ_w,h·∇pⁿ_w,h

, where P_hⁿ=pⁿ_n,h−π_Vr(sⁿ_n,h)∈V_h.

Proof We define the functions fn(s) = ηn(s)

η_n(s) +η_w(1−s) and fw(s) = ηw(1−s) η_n(s) +η_w(1−s).

Then, noting that f_n+f_w=1 and using equation (13c), for anyT ∈T and for any verticesa₀,a_i∈VT, there existssⁿ_i ∈[min(a₀,a_i),max(a₀,a_i)]such that,

P_aⁿ

0−P_aⁿ

i=f_n(sⁿ_i) pⁿ_n,a

0−pⁿ_n,a

i

−f_w(sⁿ_i) pⁿ_w,a

0−pⁿ_w,a

i

.

Sinceηαis strictly increasing, for anyT∈T witha₀,· · ·,a_das vertices η_α,Tⁿ := 1

d+1

d

∑

i=0

η_α(sⁿ_α,a

i)≥ 1

d+1max

x∈T η_α(x)≥ 1

d+1η_α(sⁿ_i). (14) Thus using that fn,fw≤1 andηn(s) +ηw(1−s)≥δ >0 we obtain,

δ 2(d+1)

d

∑

i=0

P_aⁿ

0−P_aⁿ

i

2≤η_n,Tⁿ

d

∑

i=0

pⁿ_n,a

0−pⁿ_n,a

i

2+η_w,Tⁿ

d

∑

i=0

pⁿ_w,a

0−pⁿ_w,a

i

2.

Since for anyv₁,v₂,wsatisfying|v₁|²+|v₂|²≥cond₂(A_T)|w|²one has v₁·A_Tv₁+v₂·A_Tv₂≥w·A_Tw,

we use equality (11) associated with the fact that the condition number ofA_T is bounded, cf. (12). Then summing the resulting estimate overT ∈T and noting that the Lagrange vertex-quadrature formula is exact on P1 (see [5, Remark 2.2]) we

obtain the claim.

Proposition 2Let sⁿ⁻¹_α,h ∈V_hbe given and sⁿ_α,pⁿ_α∈V_hbe a solution to the scheme(13).

There exists C₆>0depending on the data of the continuous problem but neither on the meshT or nor the time stepτ_nsuch that,

φ Z

Ω

Γ(sⁿ_n,h) +τn

∑

α∈{n,w}

Z

Ω

η_α,hⁿ Λ_h∇pⁿ_α,h·∇pⁿ_α,h+τn Z

Ω

∇P_hⁿ·∇P_hⁿ

≤C₆

1+φ Z

Ω

Γ(sⁿ⁻¹_n,h )

. Proof Let us choosev_α,h=pⁿ_α,has test function in equation (13a) and then add the resulting equations. Then, sinceΓ is convex, thanks to relation (13c) we obtain

φ Z

Ω

Γ(sⁿ_n,h) +

∑

α∈{n,w}

τn Z

Ω

η_α,hⁿ Λ_h∇pⁿ_α,h·∇pⁿ_α,h

≤φ Z

Ω

Γ(sⁿ⁻¹_n,h ) +τ_n Z

Ω

∑

α∈{n,w}

q_α(sⁿ_α,h)pⁿ_α,h. (15)

6 Cl´ement Canc`es and Flore Nabet

As for the continuous case, one has

∑

α∈{n,w}

q_α(sⁿ_α,h)pⁿ_α,h≤ q_inj−q_sink

Pⁿ_h+r(sⁿ_n,h)

+q_sinkk(sⁿ_n,h). (16) Using the definition of the discrete global pressureP_hⁿand equation (13d), combined with the discrete Poincar´e inequality (10) and (8) give

kPⁿ_hk_L1(Ω)≤C₂|Ω|^1/2k∇P_hⁿk_L2(Ω)+ Z

Ω

r(sⁿ_n,h)

≤εk∇P_hⁿk²_L2(Ω)+ε

Γ(sⁿ_n,h)

L¹(Ω)+C_ε. (17) Sinceq_inj,q_sink∈L^∞(Ω), the use of the above inequality and of (8) in (16) leads to

Z

Ω

∑

α∈{n,w}

q_α(sⁿ_α,h)pⁿ_α,h≤εk∇P_hⁿk²_L2(Ω)+ε

Γ(sⁿ_n,h)

L¹(Ω)+C_ε (18) whateverε>0. Using (18) together with Proposition 1 in (15) provides the expected

bound.

Thanks to equations (13b) and (13c) we see that the saturations and the pressures of the wetting and non-wetting phases are linked. Thus we can choose the pressure of the wetting phase and the capillary pressure as main unknowns. Choosingv_α,h= ϕaas test functions in equations (13a) we can rewrite the scheme (13) as a nonlinear system of 2#VT algebraic equations Fⁿ((γ(sⁿ_n,a),pⁿ_w,a)a∈V_T) =0. Since γ(1) = +∞, the function Fⁿ is continuous but non uniformly continuous. However, we prove in the following lemma that this situation is avoided for a solution to the scheme (13).

Proposition 3Let sⁿ⁻¹_α,h ∈V_hbe such that^R_Ωsⁿ⁻¹_w,h ≥0and sⁿ_α,h,pⁿ_α,h∈V_hbe a so- lution the scheme(13). There existsστn,T,ετn,T >0depending on the data of the continuous problem,T,τ_nand sⁿ⁻¹_n,h such that,

−σ_τn,T ≤sⁿ_n,a≤1−ετn,T, ∀a∈VT.

Proof First of all, thanks to the extension ofγfors<0, the energy estimate given in Proposition 2 yields^R_Ω((sⁿ_n,h)⁻)²≤Cⁿ⁻¹,which provides the lower bound.

Then we prove a bound on the pressure of the non-wetting phasep_n,h. Thanks to inequality (17) and the definition ofP_hⁿone has

kpⁿ_n,hk_L1(Ω)≤Cⁿ⁻¹ ⇒ |pⁿ_n,a| ≤Cⁿ⁻¹

|s_a| , ∀a∈VT. (19) Now, let us note that proving the upper bound is equivalent to proving that there existsγ_τ^?

n,T such that for anya∈V_T,γ(s_n,a)≤γ_τ^?

n,T.

We choosevw,h=1 as test function in equation (13a), then sinceq_injis nonneg- ative,ηn(s) +ηw(1−s)≥δ >0 andcw>0 (and soηw(c_w)>0), one has

sⁿ_w,ai ≥ 1

|Ω| Z

Ω

sⁿ_w,h> 1

|Ω| Z

Ω

sⁿ⁻¹_w,h −τn

δ kq_sinkk_L∞(Ω) Z

Ω

ηw(sⁿ_n,h)

.

Note that we proved here by induction that^R_Ωsⁿ_w,h≥0. Sinces7→(s+η_w(s))⁻¹is Lipschitz, there existsa_i∈VT such thatsⁿ_w,a

i>0 that is there existsa_i∈VT such thatsⁿ_n,a

i <1.

Let a_f ∈VT be arbitrary and (a_q)<sub>q=0,···,`</sub> be a path from a<sub>i</sub> to a<sub>f</sub>. Let q ∈ {0,· · ·, `−1}. Using the property (12) of the matrixA_T and since the quadrature formula is exact onP1, Proposition 2 gives

∑

T∈T

η_w,Tⁿ

d

∑

∑

|α_i,j^T|

!

pⁿ_w,h(a_i)−pⁿ_w,h(a₀)2

≤C₄C₆ τ_n

1+φ

Z

Ω

Γ(sⁿ⁻¹_n,h )

.

We assume by induction that there existsετ_n,T >0 such thatsⁿ_n,a

q<1−ετ_n,T that is sⁿ_w,a

q>ετn,T. Thus, ifT is a simplex witha_q,a_q+1as vertices, the definition (14) of η_w,Tⁿ yieldsη_w,Tⁿ ≥^η(s

nw,aq) d+1 ≥ε_τ⁰

n,T. Thanks to equations (13c) and (19) it follows that,

γ(sⁿ_n,a

q)−γ(sⁿ_n,a

q+1) −

pⁿ_n,aq−pⁿ_n,aq+1

≤C_τn,T ⇒ γ(sⁿ_n,a

q+1)≤γ_τ^??_n,T.

We conclude the proof by induction along the path.

The bound on the saturation associated with the definition (14) on η_w,Tⁿ yields η_w,Tⁿ ≥ηw(ε_τn,T). This, combined with the Poincar´e inequality (10) and since γ(sⁿ_n,a)≤γ(1−ε_τn,T) for any a∈V_T, allows us to obtain a discrete bound on the pressure.

Proposition 4There exists p^?_τ

n,T >0depending on the data of the continuous prob- lem,T,τ_nand sⁿ⁻¹_n,h such that^R_Ω|pⁿ_w,h|²≤p^?_τ

n,T.

Thanks to the material introduced above, it is possible to prove the existence of a solution to the discrete problem using the topological degree theory.

Theorem 1 (Existence of a solution)Let sⁿ⁻¹_n,h ∈V_h be given, there exists at least one solution to the scheme(13).

3 Numerical results

We present here numerical results obtained with the software FreeFem [8] in the two-dimension case by choosing as main unknowns the saturation of the non- wetting phase and the pressure of the wetting phase. To solve the nonlinear system we use a Newton method with a stopping criteria on the`^∞-norm between two suc- cessive iterations. The computational domain is the unit squareΩ =]0,1[²and the mesh is made up of triangles whose mesh size is approximately equal to 0.028. The

8 Cl´ement Canc`es and Flore Nabet

final time ist_f=0.015 and the time step is constantτ_n=10⁻³. We choose the poros- ityφ=0.3, the permeability tensor fieldΛ=

1 0 0 100

andc_w=0.2. Fors∈[0,1]

we define the mobility functions byηn(s) =s²andηw(s) =2s, the capillary pres- sure byγ(s) =^√1

1−s and the source functions are defined byq_inj=40.1[0,0.2]×[0.8,1]

andq_sink=40.1[0.8,1]×[0,0.2]. We plot in Figure 1 the approximate saturation of the non-wetting phase.

(a)t=0.002 (b)t=0.01 (c)t=0.015

Fig. 1: Approximate saturations_n,hinΩ for different timest.

One observes from the outset of the simulation the influence on the injection well q_injand of the anisotropy ratio in the longitudinal direction. Moreover we can see that the maximum does not exceedc_n=0.8.

Acknowledgements. This work was supported by the French National Research Agency (ANR) through grant ANR-13-JS01-0007-01 (GEOPOR project). Cl´ement Canc`es acknowledges the support of Labex CEMPI (ANR-11-LABX-0007).

References

1. Bear, J., Bachmat, Y.: Introduction to modeling of transport phenomena in porous media.

Kluwer Academic Publishers, Dordrecht, The Netherlands (1990)

2. Brenner, K., Masson, R.: Convergence of a vertex centered discretization of two-phase darcy flows on general meshes. Int. J. Finite Vol.10, 1–37 (2013)

3. Canc`es, C.: Energy stable numerical methods for porous media flow type problems. Oil & Gas Science and Technology-Rev. IFPEN73, 1–18 (2018)

4. Canc`es, C., Guichard, C.: Numerical analysis of a robust free energy diminishing finite volume scheme for parabolic equations with gradient structure. Found. Comput. Math.17(6), 1525–

1584 (2017)

5. Canc`es, C., Nabet, F., Vohral´ık, M.: Convergence and a posteriori error analysis for energy- stable finite element approximations of degenerate parabolic equations (2018). HAL: hal- 01894884

6. Chavent, G., Jaffr´e, J.: Mathematical Models and Finite Elements for Reservoir Simulation, vol. 17, stud. math. appl. edn. North-Holland, Amsterdam (1986)

7. Eymard, R., Herbin, R., Michel, A.: Mathematical study of a petroleum-engineering scheme.

M2AN Math. Model. Numer. Anal.37(6), 937–972 (2003)

8. Hecht, F.: New development in FreeFem++. J. Numer. Math.20(3-4), 251–265 (2012)