Uniform-in-Time Convergence of Numerical Schemes for a Two-Phase Discrete Fracture Model

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Jerome Droniou, Julian Hennicker, Roland Masson

To cite this version:

Jerome Droniou, Julian Hennicker, Roland Masson. Uniform-in-Time Convergence of Numerical

Schemes for a Two-Phase Discrete Fracture Model. FVCA 2017 - International Conference on Finite

Volumes for Complex Applications VIII, Jun 2017, Lille, France. �10.1007/978-3-319-57397-7_20�.

�hal-01541150�

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Uniform-in-time convergence of numerical

schemes for a two-phase discrete fracture model

J. Droniou, J. Hennicker, R. Masson

Abstract Flow and transport in fractured porous media are of paramount impor-tance for many applications such as petroleum exploration and production, geo-logical storage of carbon dioxide, hydrogeology, or geothermal energy. We con-sider here the two-phase discrete fracture model introduced in [3] which represents explicitly the fractures as codimension one surfaces immersed in the surrounding matrix domain. Then, the two-phase Darcy flow in the matrix is coupled with the two-phase Darcy flow in the fractures using transmission conditions accounting for fractures acting either as drains or barriers. The model takes into account complex networks of fractures, discontinuous capillary pressure curves at the matrix frac-ture interfaces and can be easily extended to account for gravity including in the width of the fractures. It also includes a layer of damaged rock at the matrix fracture interface with its own mobility and capillary pressure functions. In this work, the convergence analysis carried out in [3] in the framework of gradient discretizations [2] is extended to obtain the uniform-in-time convergence of the discrete solutions to a weak solution of the model.

Key words: Discrete fracture model, two-phase Darcy flow, uniform-in-time con-vergence, gradient discretization method

J´erˆome Droniou

School of Mathematical Sciences, Monash University, Victoria 3800, Australia, e-mail: jerome. [email protected]

Julian Hennicker

Université Côte d’Azur, Inria, CNRS, Laboratoire J.A. Dieudonné, team Coffee, France; and CSTJF, TOTAL S.A. - Avenue Larribau, 64018 Pau, France, e-mail: julian.hennicker@ unice.fr

Roland Masson

Université Côte d’Azur, Inria, CNRS, Laboratoire J.A. Dieudonné, team Coffee, France e-mail: [email protected].

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2 J. Droniou, J. Hennicker, R. Masson

1 Continuous model

We give here a brief overview of the notations, and refer to [3] for more details.Ω is a bounded polytopal domain of Rd_{(d = 2,3), partitioned into a fracture domain}

Γ and a matrix domain Ω\Γ . The network of fractures is Γ =Si∈IΓi, where eachΓi

is planar and has therefore two faces a+_{(i) and a}−_{(i). Set}_{χ = {a}+_(i),a−₍_{i) | i ∈ I}}

the set all faces and write, for simplicity,Γa+_(i)=Γ_a−(i)=Γi. For a ∈ χ, γais the

one-sided trace operator onΓaandnadenotes the unit normal vector directed from

the face a to the matrix domain. The following notations, in which uαµ is the phase

pressure in the mediumµ and phase α, are used throughout the paper. Mm=Ω , Mf=Γ and Ma=Γa; s+=max(0,s), s−= (−s)+;

(pm,pf) = (u1m− u2m,u1f− u2f)(capillary pressures); JuαKa=γauαm− uαf.

The assumptions in the rest of this paper are:

• The matrix-valued functions ΛmandΛf, permeability tensors in the matrix and

fracture domains, respectively, are uniformly coercive tensors.

• The functions Tf (half-normal transmissibility in the fracture network),φmand

φf (porosities of the matrix and fracture, respectively), and df (fracture width)

are bounded measurable and uniformly positive.

• The phase mobilities kαµ: Mµ× [0,1] → R are bounded uniformly positive

Caratheodory functions, hαµ ∈ L2((0,T ) × Mµ)andη > 0.

• The saturation S1µ : Mµ× R → [0,1] of the non wetting phase is a Caratheodory

function; for a.e. x ∈ Mµ, S1µ(x,·) is a non-decreasing Lipschitz continuous

function on R; S1

µ(·,q) is piecewise constant on a finite partition (Mµj)j∈Jµ of

polytopal subsets of Mµ, for all q ∈ R. Not indicating the phase in the saturation

means thatα = 1, that is, S_µ=S1

µ. Of course, S2µ=1 − S1µThe initial capillary

pressures (p_m,0,p_{f ,0})belong to H1(Ω \Γ )×L2(Γ ). Forϕ_µ _{∈ L}2₍₍_{0,T ) × M}_µ₎_{and a.e. (t,}_{x) ∈ (0,T) × M}_µ_{, we let}

Sαµ(ϕµ)(t,x) = Sαµ(x,ϕµ(t,x)) and [kS]µα(ϕµ)(t,x) = kαµ(x,Sαµ(x,ϕµ(t,x))).

The PDEs model writes: find phase pressures (uαm,uαf)and velocities (qαm,qαf)

(α = 1,2), such that                φ_m∂tSαm(pm) +div(qαm) =hαm on (0,T ) × Ω \Γ qαm=−[kS]αm(pm)Λm∇uαm on (0,T ) × Ω \Γ φ_fdf∂tSαf(pf) +divτ(qαf)−

∑

a∈χ Qα_{f ,a}=dfhαf on (0,T ) ×Γ qαf =−df[kS]αf(pf)Λf∇τuf on (0,T ) ×Γ (pm,pf)|t=0= (pm,0,pf ,0) on (Ω \Γ )×Γ, (1a)

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qα m· na+Qαf ,a=η∂tSαa(γapm) Qα f ,a= [kS]αf(pf)TfJuαK−a− [kS]αa(γapm)TfJuαK+a. (1b) Sf(pf) Qα f,a Sα a(γapm) qα m qα m·na qα f df

Fig. 1 Illustration of the coupling con-dition. It can be seen as an upwind two point approximation of Qα_{f ,a}. The up-winding takes into account the damaged rocktype of porous thicknessη at the matrix-fracture interfaces.

To give the weak formulation of this model, set V0₌_V0

m×Vf0with

V0

m={v ∈ H1(Ω\Γ ) | γ∂Ωv = 0 on∂Ω},

V0

f ={v ∈ H1(Γ ) | γ∂Γiv = 0 on∂Γi∩ ∂ Ω for all i ∈ I}.

The space H1_{(Γ ) is made of functions whose restriction to each Γ}

ibelong to H1(Γi),

and whose traces are continuous at fracture intersections∂Γi∩ ∂Γj. Here,∂Γiis the

boundary ofΓirespective to the hyperplane containingΓi, andγ is the trace operator.

We abridge∑_{µ∈{m, f }},∑a∈χand∑α=12 into, respectively,∑µ,∑aand∑α.

Definition 1 (Weak solution of the model). A weak solution of the model is (uα

m,uαf)α=1,2∈ [L2(0,T ;Vm0)× L2(0,T ;Vf0)]2 such that, for anyα = 1,2 and any

(ϕα m,ϕαf)∈ C0∞([0,T ) × Ω) ×C0∞([0,T ) ×Γ ),

∑

µ − Z T 0 Z Mµ φ_µSαµ(pµ)∂tϕαµdτµdt + Z T 0 Z Mµ[kS] α µ(pµ)Λµ∇uαµ· ∇ϕαµdτµdt − Z Mµ φ_µSαµ(pµ,0)ϕαµ(0,·)dτµ +

∑

a Z T 0 Z Γa F (γapm,pf,JuαKa)JϕαKadτdt −

∑

a Z T 0 Z Γa ηSα a(γapm)∂tγaϕαmdτdt + Z Γa ηSα a(γapm,0)γaϕαm(0,·)dτ =

∑

µ Z T 0 Z Mµh α µϕαµdτµ, (2) where F (s1,s2,s3) =Tf([kS]αa(s1)s+3 − [kS]αf(s2)s−3), dτm(x) = dx and dτf(x) =

d_f(x)dτ(x) (dτ being the (d − 1)-dimensional measure on the fractures).

2 The gradient scheme

Definition 2 (Gradient Discretization (GD)). A spatial gradient discretisation for a DFN is DS= (X0, (Π_Dµ_S,∇_Dµ_S)_{µ∈{m, f }}, (J·Ka,DS)a∈χ, (TaDS)a∈χ), where

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• X0is a finite dimensional space of degrees of freedom, • ΠDµS: X

0_{→ L}2_(M_µ₎_{reconstructs a function on M}_µ_{from the DOFs,}

• ∇µDS : X

0

→ L2(Mµ)dimMµ reconstructs a gradient on Mµ from the DOFs,

• J·Ka,DS : X0→ L2(Γa)reconstructs, from the DOFs, a jump onΓabetween the

matrix and fracture, • Ta

DS: X

0_{→ L}2_(Γ

a)reconstructs, from the DOFs, a trace onΓafrom the matrix.

Here,Π_Dµ_Sand Ta

DSare piecewise constant reconstructions in the sense of [2], which

implies that if g : R → R then ΠDµSg(w) = g(Π

µ DSw) and T a DSg(w) = g(T a DSw). DSis

extended into a space-time GD D = (DS, (Iµ_D)_{µ∈{m, f }}, (tn)n=0,...,N)with

• 0 = t0<t1<··· < tN=T a discretisation of the time interval [0,T ],

• ImD: H1(Ω \Γ ) → X0and I f

D: L2(Γ ) → X0are operators designed to interpolate

initial conditions.

The spatial operators are extended into space-time operators the following way. If w = (w_n)n=0,...,N+1∈ (X0)N+1, andΨDS=Π µ DS,∇ µ DS, J·Ka,DSor TaDS, thenΨDw is defined on [0,T ] × Mµ or [0,T ] ×Γaby ΨDw(0,·) = ΨDSu0and, ∀n ∈ {0,...,N − 1}, ∀t ∈ (tn,tn+1]ΨDw(t,·) = ΨDSwn+1.

We also define the discrete time derivativeδtw : (0,T ] → X0by, for the same n and

t as above,δtw(t) =wtn+1_n+1−w−tnn.

The gradient scheme for (2) is: find (uα)α=1,2∈ [(X0)N+1]2 such that, setting p = u1_{− u}2_{, we have p}

0= (ImDpm,0,IDf pf ,0)and, forα = 1,2 and vα∈ (X0)N+1,

∑

µ Z T 0 Z Mµ φ_µΠ_Dµδ_tSαµ(p) Π_Dµvα+ [kS]α_µ(Π_Dµp)Λ_µ∇µ_Duα· ∇µDvα dτµdt +

∑

a Z T 0 Z Γa

F (TaDp,ΠDfp,JuαKa,D)JvαKa,Ddτdt

+ Z T 0 Z Γa ηTa D δtSαa(p) Ta Dvαdτdt =

∑

µ Z T 0 Z Mµh α µΠDµvαdτµdt. (3)

3 Main result

Theorem 1. Under the assumptions of Section 1, let (Dl₎

l∈N be a coercive,

con-sistent, limit-conforming and compact sequence of space-time GD (see [3]), and let (uα,l)_l∈N be such that uα,l ∈ (Xl0)Nl+1 is a sequence of solutions of (3) with

D = Dl. Then, there exists a weak solution (uαm,uαf)α=1,2of the model such that, for

allµ ∈ {m, f } and a ∈ χ, Sµ(pµ): [0,T ] → L2(Mµ)and Sa(γapm): [0,T ] → L2(Γa)

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Π_DµlSµ(pl)−→ Sµ(pµ)in L∞(0,T ;L2(Mµ)),

Ta_DlSa(pl)−→ Sa(γapm)in L∞(0,T ;L2(Γa)).

Notations and preliminary results. Before proving this theorem, we recall some convergence results established in [3], under the assumptions of Theorem 1. Here, if (wl₎

l∈Nis a sequence of functions in L2((0,T ) ×M) for some measured space M,

“wl_{→ w in L}2_{” means that the convergence holds in L}2₍₍_{0,T ) × M).}

There exists a weak solution u = (um,uf)such that, up to a subsequence as l → ∞,

for allµ ∈ {m, f } and a ∈ χ, with p = u1_{− u}2_{and p}

µ=u1µ− u2µ,

Π_Dµluα,l⇀uαµ,∇µ_Dluα,l⇀∇uαµ and Juα,lKa,Dl ⇀JuαKaweakly in L2, (4)

Π_Dµ_lSµ(pl)→ Sµ(pµ) and TaDlSa(pl)→ Sa(γapm)strongly in L2. (5)

The functions S_µ(p_µ): [0,T ] → L2(Mµ)and S_a(γapm): [0,T ] → L2(Γa)are

contin-uous for the weak topologies of L2_(M

µ)and L2(Γa), respectively. Moreover, for any

(Tl₎

l∈N⊂ [0,T ] that converges to some T∞,

Π_DµlSµ(pl)(Tl)→ Sµ(pµ)(T∞)weakly in L2(Mµ), and Ta_DlSa(pl)(Tl)→ Sa(γapm)(T∞)weakly in L2(Γa).

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There existsρa∈ L2((0,T ) ×Γa)such that

F (Ta Dlp l_,_Πf Dlp l_,_Juα,l_K a,Dl)→ ρaweakly in L2, (7)

and, for allϕ ∈ [L2_{(0,T ;V}0

m)× L2(0,T ;Vf0)]2,

∑

α,a Z T 0 Z Γa ρaJϕαKadτdt =

∑

α,a Z T 0 Z Γa F (γapm,pf,JuαKa)JϕαKadτdt. (8)

For ρ = µ ∈ {m, f } or ρ = a ∈ χ, let RSρ(x,·) be the range of Sρ(x,·) and

[Sρ(x,·)]i: RSρ(x,·)→ R be its pseudo-inverse defined by

[Sρ(x,·)]i(q) =    inf{z ∈ R|Sρ(x,z) = q} if q > Sρ(x,0), 0 if q = Sρ(x,0), sup{z ∈ R|Sρ(x,z) = q} if q < Sρ(x,0). Let Bρ(x,·) : R → [0,∞] be given by Bρ(x,q) =R_Sq_ρ_(x,0)[Sρ(x,·)]i(τ)dτ if q ∈ RSρ(x,·),

Bρ(x,q) = +∞ otherwise. Bρ(x,·) is convex l.s.c. and Bρ(x,Sρ(x,·)) has a

sub-quadratic growth: Bρ(x,Sρ(x,r)) ≤ Kr2for some K not depending onx or r.

The following continuous (based on [1, Lemma 3.6]) and discrete energy rela-tions hold. For all T0∈ [0,T ],

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∑

µ Z Mµφµ h Bµ(Sµ(pµ)(T0))dτµ− Z MµφµBµ(Sµ(pµ)(0)) i dτµ +

∑

a Z Γa h ηB_a(S_a(γapm)(T0))dτµ− Z Γa ηB_a(S_a(γapm)(0)) i dτ +

∑

α,µ Z T0 0 Z Mµ[kS] α µ(pµ)Λµ∇uαµ· ∇uαµdτµdt +

∑

α,a Z T0 0 Z Γa F (γapm,pf,JuαKa)JuαKadτdt =

∑

α,µ Z T0 0 Z Mµh α µuαµdτµdt (9)

and, if k is chosen such that T0∈ (tk,tk+1],

∑

µ Z Mµ φ_µBµ(Sµ(Π_Dµl Sp l_)(T 0))− Bµ(Sµ(Π_Dµl Sp0)) dτµ +

∑

a Z Γa ηhBa(Sa(Ta_Dl Sp l_)(T 0))− Ba(Sa(Ta_Dl Sp0)) i dτ +

∑

α,µ Z T0 0 Z Mµ [kS]α_µ(Π_Dµlpl)Λµ∇_Dµluα,l· ∇_Dµluα,ldτµdt +

∑

α,a Z T0 0 Z Γa F (Ta_Dlpl,Π_Dflpl,Juα,lKa,Dl)Juα,lKa,Dldτdt ≤

∑

α,µ Z tk+1 0 Z Mµh α µΠ_Dµluα,ldτµdt. (10)

Proof of Theorem 1. The proof follows the ideas initially introduced in [1]. By the characterisation [2, Lemma 4.8] of uniform-in-time convergence, it suffices to prove that, for any sequence (Tl₎

l∈N⊂ [0,T ] converging to some T∞,

Π_DµlSµ(pl)(Tl)→ Sµ(pµ)(T∞)in L2(Mµ), Ta_DlSa(pl)(Tl)→ Sa(γapm)(T∞)in L2(Γa).

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Applying the discrete energy relation (10) to T0=Tl yields

∑

µ Z MµφµBµ(Sµ(Π µ D_Slp l_)(Tl_))dτ µ+

∑

a Z Γa ηBa(Sa(Ta_Dl Sp l_)(Tl_))dτ ≤ Z MµφµBµ(Sµ(Π µ D_Slp0))dτµ+

∑

a Z Γa ηB_a(S_a(Ta_Dl Sp0))dτ −

∑

α,µ Z Tl 0 Z Mµ[kS] α µ(ΠDµlp l_)Λ µ∇_Dµluα,l· ∇µ_Dluα,ldτµdt −

∑

α,a Z Tl 0 Z Γa F (Ta_Dlpl,Π_Dflpl,Juα,lKa,Dl)Juα,lKa,Dldτdt

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+

∑

α,µ Z t_k(l)+1 0 Z Mµh α µΠ_Dµluα,ldτµdt = A1+ A2− A3− A4+ A5. (12)

where k(l) is such that Tl _{∈ (t}

k(l),tk(l)+1]. The consistency of (Dl)l∈N shows that

Π_Dµl Sp0=Π µ D_SlI µ Dlpµ,0 → pµ(0) in L 2_(M µ), Ta_Dl Sp0=T a D_SlI m Dlpm,0→ γapm(0) in L2_(Γ

a). Since Bρ◦ Sρ is sub-quadratic, we infer

A1+ A2→ Z MµφµBµ(Sµ(pµ(0)))dτµ+

∑

a Z Γa ηBa(Sa(γapm(0)))dτ. (13)

The convergence of A5is trivial from the weak convergence ofΠDµuα,land the fact

that tk(l)+1→ T∞: A5→

∑

α,µ Z T∞ 0 Z Mµh α µuαµdτµdt. (14)

Consider Lemma 1 applied to Fl_((t,_{x),ξ) = 1}

(0,Tl₎(t)[kS]α_µ(Π_Dµ_lpl)(t,x)Λ_µ(x)ξ and Wl₌_∇µ Dlu α,l_{. By (4) and (5), W}l_{→ W := ∇u}α µ weakly in L2((0,T ) × Mµ)and, up to a subsequence,1_(0,Tl₎Π_Dµ_lS_µ(pl)Λ_µ→ 1(0,T∞₎[kS]α_µ(p_µ)Λ_µa.e. on M_µ_{×(0,T )}

while remaining bounded. Since Fl _{is monotonic with respect to its second}

argu-ment, the assumptions of Lemma 1 are satisfied withρ = 1(0,T∞₎[kS]α_µ(p_µ)Λ_µ∇uα_µ,

and therefore liminf l→∞ A3≥_α,µ

∑

Z T∞ 0 Z Mµ[kS] α µ(pµ)Λµ∇uαµ· ∇uαµdτµdt. (15)

To study the limit of A4, we apply again Lemma 1, this time with Fl((t,x),ξ) =

F (Ta Dlp

l_(t,_x),Πf Dlp

l_(t,_{x),ξ) and W}l ₌_Juα,l_K

a,Dl. From the definition of F it

can be readily checked that Fl_{is monotonic with respect to its first argument.}

Us-ing therefore the strong convergences (5) of Sαa(Ta_Dlpl)and Sµ(Π_Dflpl), the weak

convergence (4) of Juα,lKa,Dl and the convergence property (7)–(8) of Fl(·,Wl) =

F (Ta Dlp

l_,_Πf Dlp

l_,_Juα,l_K

a,Dl), the assumptions of Lemma 1 are satisfied and

liminf l→∞ A4≥_α,a

∑

Z T 0 Z Γa F (γapm,pf,JuαKa)JuαKadτdt. (16)

Gathering (13), (14), (15) and (16) into (12) and using the energy equality (9) yields limsup l→∞

∑

µ Z Mµ φ_µBµ(Sµ(Π_Dµl Sp l_)(Tl_))dτ µ+

∑

a Z Γa ηB_a(Sa(Ta_Dl Sp l_)(Tl_))dτ ≤

∑

µ Z Mµ φ_µBµ(Sµ(pµ)(T∞))dτµ+

∑

a Z Γa ηB_a(Sa(pf)(T∞))dτ. (17)

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8 J. Droniou, J. Hennicker, R. Masson On the other hand, the weak L2_{convergences (6) and the fact that the functions B}_ρ

are convex lower semi-continuous give, by [1, Lemma 3.4],

∑

µ Z Mµ φ_µBµ(Sµ(pµ)(T∞))dτµ≤ liminf l→∞

∑

_µ Z Mµ φ_µBµ(Sµ(Π_Dµl Sp l_)(Tl_))dτ µ (18)

∑

a Z Γa ηBa(Sa(pf)(T∞))dτ ≤ liminf l→∞

∑

a Z Γa ηBa(Sa(Ta_Dl Sp l_)(Tl_))dτ. ₍₁₉₎

Combining (17), (18) and (19) yields, by [2, Lemma 4.33],

∑

µ Z Mµ φ_µBµ(Sµ(pµ)(T∞))dτµ = lim l→∞

∑

_µ Z Mµ φ_µBµ(Sµ(Π_Dµl Sp l_)(Tl_))dτ µ

∑

a Z Γa ηBa(Sa(pf)(T∞))dτ = lim l→∞

∑

a Z Γa ηBa(Sa(Ta_Dl Sp l_)(Tl_))dτ.

The proof of (11), and thus of Theorem 1, is then completed using the exact same reasoning as in [1, Section 4.3]. ⊓⊔

Lemma 1 (Weak Fatou by monotonicity). Let k ≥ 1, M be a measured space, and let (Fl₎

l∈Nbe Caratheodory functions M ×Rk→ Rksuch that, for a.e.z ∈ M and all

ξ,η ∈ Rk_{, [F}l₍_z,ξ)−Fl₍_{z,η)]·[ξ −η] ≥ 0. Let (W}l₎

l∈Nsuch that, as l → ∞, Wl⇀

W weakly in L2_(M)k_{, (F}l₍_{·,W ))}

l∈Nconverges strongly in L2(M)k, and Fl(·,Wl) ⇀

ρ weakly in L2_(M)k_{. Then}R

Mρ(z) ·W(z)dz ≤ liminfl→∞RMFl(z,Wl(z)) ·Wl(z)dz.

Proof. We have [Fl_(z,Wl₎_{− F}l₍_{z,W)] · [W}l_{−W ] ≥ 0. Integrate and develop:}

0 ≤Z MF l_(z,Wl₎ ·Wldz − Z MF l_(z,Wl₎ ·W dz + Z MF l_{(z,W) ·}h_Wl −Widz. (20) The last term goes to 0 by strong convergence of Fl₍_{·,W ) and weak convergence of}

Wl_{. By weak convergence of F}l₍_·,Wl_{), the second term goes to}R

Mρ ·W. The proof

is concluded by taking the inferior limit of (20).

Acknowledgements We thank TOTAL S.A. and the Australian Research Council’s Discovery Projects funding scheme (project number DP170100605) for partially supporting this work.

References

1. Droniou, J., Eymard, R.: Uniform-in-time convergence of numerical methods for non-linear degenerate parabolic equations. Numer. Math. 132(4), 721–766 (2016). DOI 10.1007/ s00211-015-0733-6. URL http://dx.doi.org/10.1007/s00211-015-0733-6 2. Droniou, J., Eymard, R., Gallou¨et, T., Guichard, C., Herbin, R.: The gradient discretisation

method: A framework for the discretisation and numerical analysis of linear and nonlinear elliptic and parabolic problems (2016). URL https://hal.archives-ouvertes.fr/ hal-01382358

3. Droniou, J., Hennicker, J., Masson, R.: Numerical analysis of a two-phase flow discrete fracture model URL https://arxiv.org/abs/1612.07373. Submitted