A reduced model for Darcy's problem in networks of fractures

DOI:10.1051/m2an/2013132 www.esaim-m2an.org

A REDUCED MODEL FOR DARCY’S PROBLEM IN NETWORKS OF FRACTURES^∗

Luca Formaggia

, Alessio Fumagalli

, Anna Scotti

and Paolo Ruffo

Abstract. Subsurface flows are influenced by the presence of faults and large fractures which act as preferential paths or barriers for the flow. In literature models were proposed to handle fractures in a porous medium as objects of codimension 1. In this work we consider the case of a network of intersecting fractures, with the aim of deriving physically consistent and effective interface conditions to impose at the intersection between fractures. This new model accounts for the angle between fractures at the intersections and allows for jumps of pressure across intersections. This fact permits to describe the flow when fractures are characterized by different properties more accurately with respect to other models that impose pressure continuity. The main mathematical properties of the model, derived in the two-dimensional setting, are analyzed. As concerns the numerical discretization we allow the grids of the fractures to be independent, thus in general non-matching at the intersection, by means of the extended finite element method (XFEM). This increases the flexibility of the method in the case of complex geometries characterized by a high number of fractures.

Mathematics Subject Classification. 65N30, 76S05, 86A60.

Received February 12, 2013. Revised September 18, 2013.

Published online July 2, 2014.

1. Introduction

The presence of fractures can largely influence the flow in porous media in geophysical applications. The literature on flow in fractured media is rather extensive, we give here just some main references [2,3,8,10] and we mention the applications to reservoir simulations [21,26,28].

Fractured rocks are characterized by fractures covering a large range of space scales, from small joints to large faults. In particular, large fractures and faults can act, according to their different permeability, as barriers or

Keywords and phrases.Reduced models, fractured porous media, XFEM.

∗This work has been supported by ENI s.p.a. The first and third authors also acknowledge the support of MIUR through the PRIN09 project n. 2009Y4RC3B 001.

1 MOX - Dipartimento di Matematica “F. Brioschi” – Politecnico di Milano - via Bonardi 9, 20133 Milan, Italy.

[email protected]; [email protected]

2 IFP Energies nouvelles – 1 and 4, avenue de Bois-Prau, 92852 Rueil-Malmaison Cedex, France.

[email protected]

3 ENI Spa – Exploration and Production Division – 5^◦Palazzo Uffici, Room 4046 E, GEBA Dept. - via Emilia 1, San Donato Milanese, 20097 (MI), Italy.[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2014

preferential paths to the flow. This is due to the fact that fractures may be produced at different geological times and may undergo infilling and other geochemical processes that may modify their permeability considerably [17].

At a different space scale, micro fractures can alter, according to their density and orientation, the overall permeability of the porous medium. Numerical simulations of problems related to groundwater flow such as CO₂ storage, oil migration and recovery or groundwater contamination should be able to account for the presence of fractures to yield accurate results.

In the applications we are considering, the porous medium is usually characterized by the presence of several fractures that can intersect each other. Moreover the characteristic thickness, or aperture, of the fractures is very small compared to their length and, in particular, compared to the typical size of the domain of interest.

This geometric complexity makes the simulations particularly challenging for standard methods.

In [5,24,27] the authors propose a model reduction strategy to overcome part of the aforementioned prob- lems by using a domain decomposition approach where fractures are represented as one-codimension interfaces inside the porous domains. The proposed model can successfully reduce the number of unknowns in the sim- ulation since, instead of refining the grid to capture a thin n-dimensional region we are replacing it with a n−1-dimensional interface. This approach, originally developed for single-phase Darcy problems, has been suc- cessfully extended to passive transport in porous media [20] and to two-phase flow [19,25], with suitable reduced models to describe the flow in the fracture.

However, the aforementioned works consider just the restricted case of non-intersecting fractures that cut the domain into two separated sub-domains. In [7] this assumption is relaxed to include fractures that do not cut entirely the domain, i.e. fractures with tips immersed in the enclosing porous medium, with the constraint of mesh conformity between fractures and porous medium.

Realistic simulations in a three dimensional domain are presented in [6], where suitable coupling conditions are imposed at the intersections between fractures. In particular, the continuity of pressure and mass conserva- tion are enforced. These conditions however, also used in [5], may lead to inaccurate results if two intersecting fractures have different permeability. In this case one may expect strong variations of pressure near the inter- section, thus pressure continuity does not seems an appropriate condition to represent this behavior in a model reduction approach. We mention also the recent works [11,12] on three dimensional discrete fracture network flows.

In this work, we focus on the development of a reduced model that generalizes the coupling conditions of [5,6]

to account for different properties of the fractures such as different permeability and thickness and to include the effect of the intersection angle. The new coupling conditions allow for pressure and velocity jumps at the intersection, similarly to the conditions derived in [27] for the matrix-fracture system. Hence, we account for the fact that in a fracture system one fracture can act as a barrier or a preferential path with respect to the other.

We analise the resulting coupled system of equations to derive its well posedness, and assess its conservation and positivity properties. The analysis is focused on the two dimensional case, where fractures are modeled as one dimensional manifolds. Although fractures are clearly 3D features, the study of 2D networks is still significant, particularly when the normal to the fracture mid-surface lays on a plane [23,30]. Moreover they are less computationally expensive than a three dimensional counterpart and thus useful for preliminary studies.

We propose a discretization method that allows for non matching grids at the intersection points with the intent of providing the maximal flexibility when dealing with complex networks. More precisely, we employ an extended finite element (XFEM) strategy to treat intersecting fractures.

We focus just on the fracture network neglecting flow in the surrounding medium. This choice can be regarded as an intermediate step for the development of a fully coupled model with intersecting fractures immersed in a permeable medium, but also as a reasonable approximation of realistic situations where the rock has low permeability and flow occurs mainly through the fracture network.

The paper is structured as follows. In Section 2 we introduce the governing equations and provide the setting for the derivation of the reduced model. In Section 3 the reduced model for the intersecting fractures is derived. The corresponding weak formulation in mixed form and these analyzes are presented in Section 4,

d₂ D

I Γ^p

Γ^u

γ1

∂D

∂Ω\∂D Ω22

γ₂ Ω₂₁ d₁

Ω11

Ω12

Figure 1.Example of a network of fractures and its subdivision (2.2).

while in Section 5 we address the numerical discretization and its main properties. In Section 6 we present some numerical test cases to assess the theoretical results. Finally, Section 7 is devoted to conclusions.

2. The governing equations

In this work we consider the case of fractures in a domain of R². For the sake of simplicity, let us consider two crossing fracturesΩ₁, Ω₂∈R², included in a regionD⊂R²intersecting the fracture boundary. The results illustrated in this section may be extended rather easily to the case of several fractures, as the examples in Section6 show.

Following [27] we suppose that, for eachΩ_i, there exists a non auto-intersecting one dimensional manifoldγ_i of classC² such thatΩ_i may be defined as

Ω_i =

x∈Rⁿ :x=s+rn_i,s∈γ_i, r∈

−d_i(s) 2 ,d_i(s)

2

, (2.1)

where d_i∈C²(γ_i) is the thickness of Ω_i andn_i the unit normal of γ_i. We assume that |γ_i| d_i, for i= 1,2.

Furthermore, we assume that there existc₁, c₂∈R⁺, withc₂ “small”, such that d_i(s)> c₁, |d_i(s)|< c₂ ∀s∈γ_i fori= 1,2.

In other words, we assume that the thickness of the fracture varies slowly and is small compared to the other dimensions of the fracture.

Remark 2.1. The requirement thatγ_i be of classC² may be partially dispensed with. Indeed, it is sufficient thatγ_i be a piecewiseC²curve.

We setI:=Ω₁∩Ω₂ and we assume that eachΩ_i can be subdivided into three disjoint and non-empty sub- regionsΩ_i1,Ω_i2andI,i.e. a T shaped intersection is not allowed (see Fig.1). For convenience let us introduce the following sets, fori, j= 1,2

Ω:=Ω₁∪Ω₂, γ:=γ₁∪γ₂, i_p:=γ₁∩γ₂ and ∂I_ij :=∂I∩∂Ω_ij. (2.2) It is implicit in these definitions that we assume thatγ₁ andγ₂ intersect each other at a single point, indicated withi_p. The extension to multiple intersection is straightforward.

We assume a Lipschitz-continuous boundary for bothDandΩ. We indicate withn_ij,n_Ωandn_Dthe outward unit normals to∂I_ij,∂Ω and∂D, respectively. Here and in the sequel we indicate with the lower case subscripts iandij the restriction of data and unknowns toΩ_i orΩ_ij, respectively, and with the subscriptIthe restriction toI. For instance, foru_iin Ω_i,u_ij indicates the function inΩ_ij such thatu_ij= u_i|_Ωij and so on.

We are interested in computing the steady pressure field pand the velocity fielduin the whole network Ω, governed by the following Darcy problems formulated inΩ_i andI,

∇·u_i=f_i

K⁻¹_i u_i+∇pi=0 inΩ_i\Ifori= 1,2 and

∇·u_I =f_I

K⁻¹_I u_I+∇pI =0 inI. (2.3a) Here K_i ∈ [L^∞(Ω)]^2×2 and K_I ∈ [L^∞(Ω)]^2×2 denote the permeability tensors, which are symmetric and positive definite, andf ∈L²(Ω) is a source term which represents a possible volume source.

We consider the following physical coupling conditions betweenI andΩ\I

p_ij=p_I

u_ij·n_ij =u_I·n_ij on∂I_ij fori, j= 1,2, (2.3b) and boundary conditions

⎧⎪

⎨

⎪⎩

u·n_Ω = 0 on∂Ω\∂D, p=g onΓ^p, u·n_D=b onΓ^u.

(2.3c)

The first condition of (2.3c) means that we are considering the fractures as immersed in an impermeable medium D\Ω. In the remaining part of∂Ω we impose boundary condition for the pressure onΓ^p withg∈H^1/2(Γ^p), or for the flux on Γ^u with b ∈ H^−1/2(Γ^u). We have Γ^p∪Γ^u :=∂D∩∂Ω with Γ^p∩Γ^u =∅, moreover we require thatΓ^p=∅ and that∂Ω_ij∩∂D belongs toΓ^p or toΓ^ufori, j= 1,2. We consider a subdivision ofΓ^p andΓ^u for each fracture, indicated asΓ_i^pandΓ_i^u, respectively. Finally, according to the latter sub-division, we set∂γ_i^p:=∂γ_i∩Γ_i^p and∂γ_i^u:=∂γ_i∩Γ_i^u. Introducing the vector functional space

Hdiv(Ω) :=

v∈H_div(Ω) : v·n_D−b, w= 0,∀w∈H_0,Γ¹ u(Ω) ,

withH_0,Γ¹ u(Ω) :=

w∈H¹(Ω) : w= 0 inΓ^u

, we have the following standard result for the Darcy problem, see [14,16,29].

Theorem 2.2. Under the given hypothesis on the data, problem (2.3) is well posed. In particular, we have (u, p)∈Hdiv(Ω)×L²(Ω).

3. Derivation of a reduced model

The derivation of the reduced model follows the approach presented, in a different framework, in [27].

We indicate the projection operators in the normal and tangential direction of γ_i as N_i := n_i⊗n_i and T_i := I−N_i respectively, I being the identity tensor. Given two regular functions g and q, we define the tangential gradient and divergence for eachγ_i as

∇_τig:=T_i∇g and ∇_τi·q:=T_i:∇q, (3.1) respectively. We require that the permeability tensorK_i inΩ_i\I,i= 1,2, can be written asK_i =K_i,nN_i+ K_i,τT_i, with K_i,n, K_i,τ ∈L^∞(Ω_ij) and strictly positive. This is a reasonable request since we are assuming, in the two-dimensional case, that the permeability tensor is diagonal in a frame of reference that is aligned with the fracture. In the three dimensional case this assumption also implies that the permeability should be isotropic in the tangent plane of the fracture.

I d1

∂I1,2

∂I1,1

τ1

n1

γ1

θ d1

d^∗1

γ2

i_p

Figure 2. Example of an intersection.

We indicate with the symbol ˆ·the reduced variables defined onγ_i. In particular, for anys_i∈γ_i, we introduce the reduced pressure ˆpand flux ˆuas

uˆ_i(s_i) :=

^di

2

−^di₂

T_iu_i(s_i+rn_i) dr and pˆ_i(s_i) := 1 d_i

^di

2

−^di₂ p_i(s_i+rn_i) dr. (3.2) Note that we have preferred to use the flux and not the average velocity in the reduced model since it simplifies the model in the case of fractures with variable thickness.

Moreover, a reduced source term and the inverse of the scaled permeabilities are defined as fˆ_i(s_i) :=

^di

2

−^di₂ f_i(s_i+rn_i) dr, η_γi := d_i

K_i,n and ηˆ_i:= 1 d_iK_i,τ,

respectively.

Using (3.2) and approximating Γ_i^u with ∂γ_i^u and Γ_i^p with ∂γ_i^p, so that ∂γ_i^u∩∂γ_i^p = ∅, we obtain the corresponding reduced boundary data from the last two expressions of (2.3c),

ˆb_ij:=

∂Ωij∩∂Db_ij(σ) dσ and gˆ_ij := 1

|∂Ω_ij∩∂D|

∂Ωij∩∂Dg_ij(σ) dσ.

Indeed, by the definition in (3.2) we have that ˆu_i·n_i= 0 onγ_i,i.e. uˆ_i is aligned to the tangent ofγ_i.

The reduced model on eachΩ_ij is obtained by integrating (2.3a) along the fracture thickness and can then be written as

∇_τi·uˆ_i= ˆf_i ˆ

η_iuˆ_i+∇_τipˆ_i=0 in γ_i\i_p and

uˆ_i·n_D = ˆb_i on∂γ_i^u ˆ

p_i = ˆg_i on∂γ_i^p fori= 1,2.

We derive now a reduced model for the flow in the intersecting region I in order to find proper coupling conditions. To this aim, we assume that I can be modeled as a quadrilateral with parallel sides, i.e. the thicknessesd_i can be considered constant inI. Furthermore, we have assumed that the permeability tensorK_I can be taken constant in I. Let us indicate with τ_i the tangential unit vector to γ_i, and with τ_i,ip its value at i_p, and define d^∗_i :=d_i/sinθ, with sinθ =

1−

τ_1,ip·τ_2,ip2

. Then |I|=d^∗₁d^∗₂|sinθ|. Note thatθ is the angle between fractures at the intersection, as shown in Figure2. We can write the intersecting region as

I=

x∈Ω: x=i_p+x₁τ_1,ip+x₂τ_2,ip, x_i∈

−d^∗_j 2 , d^∗_j

2

fori=j= 1,2

.

The reduction process approximates I with i_p and assumes that the fluxes ˆu_i and the pressures ˆp_i can be discontinuous ati_p. Thus, we denote with ˆp_I ∈Rthe reduced value of the pressure at the intersection, defined as

ˆ p_I := 1

|I|

I

p_I(x) dx. (3.3)

The intersection point divides each line γ_i into two parts, γ_i1 andγ_i2 respectively, where the indexes 1 and 2 refer to the orientation induced by the tangent vectors. Again, a double subscriptij, withi, j= 1,2, will be used to indicate quantities onγ_ij. Furthermore, since ˆu_i is by definition aligned alongγ_i we may write ˆu_i=u_iτ_i.

We can then define the jump and mean operator acrossi_p as aiip:=a_i1−a_i2 and {{ai}}_ip:= a_i1+a_i2

2 , for i= 1,2.

It is reasonable to make the following approximation, (2.3b)

∂Iij

u_I·n

τ_i≈uˆ_ij(i_p) and 1 d^∗_i

∂Iij

p_I ≈pˆ_ij(i_p), fori, j= 1,2.

Mass conservation implies that 2 k=1

ˆu_k·τ_kip= ˆf_I with fˆ_I:=

I

f_I(x) dx.

We integrate the first of (2.3a) on I, approximating the integral involving the velocityu_I by the trapezoidal rule on each fracture, to find

I

K⁻¹_I u_I ≈K⁻¹_I |I|

2 2 k=1

1

d^∗_k ( ˆu_k1+ ˆu_k2) =K⁻¹_I |I|²

k=1

1

d^∗_k {{ˆu_k}}_ip. Furthermore, the integral of the gradient of the pressurep_I in the intersection can be written as

I

∇pI = 2 i,j=1

n_ij

∂Iij

p_I ≈(ˆp₁₂−pˆ₁₁)n₂d^∗₁+ (ˆp₂₂−pˆ₂₁)n₁d^∗₂=−pˆ_1ipn₂d^∗₁−ˆp_2ipn₁d^∗₂.

Then, we obtain

K⁻¹_I |I|

2 k=1

1

d^∗_k {{uˆ_k}}_ip=ˆp₁ipn₂d^∗₁+ˆp₂ipn₁d^∗₂.

Multiplying the above relation by τ₁, or similarly by τ₂, using the identity d₁ =d^∗₁n₂·τ₁ and the fact that uˆ_i= ( ˆu_i·τ_i)τ_i we obtain,

|I| d_i

2 k=1

η_ik^I

d^∗_k {{uˆ_k·τ_k}}_ip =pˆ_iip ini_p, i= 1,2, (3.4) where

η^I_ij:=τ_i,i

p·K⁻¹_I τ_j,ip. (3.5)

τ₁ n₁

τ₂ n₂

∂I_2,2

γ2

∂I1,1 ∂I_1,2

γ₁ i_p

∂I_2,1 x₁

x₂

Figure 3.Example of a bi-dimensional intersection between two fractures.

Ifγ₁and γ₂ are orthogonal and the permeability tensorK_I is such that

K⁻¹_I =η_γ1τ_1,ip⊗τ_1,ip+η_γ2τ_2,ip⊗τ_2,ip, (3.6) coupling conditions (3.4) can be simplified into

η_γid_j

d_i {{uˆ_i·τ_i}}_ip=ˆp_iip in i_p, fori, j= 1,2, j=i.

To close the system we derive a model for the pressure at the intersection. For each fracture in the first half of the transverse section we approximate the value of the pressure ini_p by the following truncated Taylor expansion,

p_j(i_p) =p_I(x₁) +d^∗_j

2∇pI(θ₁)·τ_i i, j= 1,2 withi=j, (3.7) where θ₁=i_p−τ_iξ₁d^∗_j/2 withξ₁ ∈[0,1], see Figure 3. In the second transverse section we use an analogous approximation, namely

p_j(i_p) =p_I(x₂)−d^∗_j

2∇p_I(θ₂)·τ_i i, j= 1,2 withi=j, (3.8) whereθ₂=i_p+τ_iξ₂d^∗_j/2 with ξ₂∈[0,1]. Using (2.3a) and (2.3b) we find

p_j(i_p) =p_i,k+ (−1)^kd^∗_j

2 τ_i ·K⁻¹_I u_I(θ_k) fork= 1,2. (3.9) The values ofu_I in bothθ₁andθ₂are unknown, therefore we express them by the following convex combination for each fracture

u_I(θ₁) =ξ₁u_1,1+ (1−ξ₁)u_1,2+1

2(u_2,1+u_2,2) forξ₁∈[0,1], u_I(θ₂) =ξ₂u_1,2+ (1−ξ₂)u_1,1+1

2(u_2,2+u_2,1) forξ₂∈[0,1].

Using the previous expression foru_I and integrating in I, equation (3.9) becomes ˆ

p_I = ˆp_i,1−τ_i ·K⁻¹_I 1

2 d_j

d_i {{ˆu_i}}_ip+{{uˆ_j}}_i

p

+ ˆξ_0,1d_j d_iuˆ_iip

,

ˆ

p_I = ˆp_i,2+τ_i ·K⁻¹_I 1

2 d_j

d_i {{ˆu_i}}_ip+{{uˆ_j}}_ip

−ξˆ_0,2d_j d_iuˆ_iip

,

where ˆξ_0,k := (2ξ_k −1)/4 for k= 1,2. Finally using (3.4) and the fact that pressure ˆp_I is single valued, thus ξˆ_0,k = ˆξ₀ fork= 1,2, we obtain the last coupling condition of our reduced model

ξˆ₀d_j

d_iη_ii^Iuˆ_i·τ_iip={{pˆ_i}}_ip−pˆ_I ini_p. (3.10) To sum up, the complete reduced model that describes the evolution of ˆu_i, ˆp_i and ˆp_I consists of the following system of partial differential equations

∇τi·uˆ_i= ˆf_i ˆ

η_iuˆ_i+∇τipˆ_i=0 in γ_i\i_p and

uˆ_i·n_D = ˆb_i on∂γ_i^u ˆ

p_i = ˆg_i on∂γ_i^p fori= 1,2. (3.11a) and the coupling conditions for the fracture-fracture system

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩ 2 k=1

uˆ_k·τ_kip= ˆf_I

|I|

d_i 2 k=1

η^I_ik

d^∗_k {{uˆ_k·τ_k}}_ip=ˆp_iip

ξˆ₀d_j

d_iη^I_iiuˆ_i·τ_iip ={{pˆ_i}}_ip−pˆ_I

ini_p, i, j= 1,2, i=j. (3.11b)

If the intersection region has a high permeability thenη^I_ij ≈0 and conditions (3.11b) reduce to those in [4,6], i.e. continuity of pressure and mass conservation. However, our model is more general as it allows for different choices ofK_I, and it is useful in practical situations where fractures have rather different permeability and may even act as barrier to the flow. This fact will be illustrated in the section dedicated to numerical experimentation.

4. Weak formulation and functional setting

We describe here the functional setting for homogeneous essential boundary conditions, i.e. all possible ˆb_i are set to zero, since the non-homogeneous case may be recovered by standard lifting techniques. For a given regular curveγ: (0, L)→R²with tangentτ defined almost everywhere onγwe define the vector space

H_div(γ) :=

w∈ L²(γ)₂

: ∇_τ ·w∈L²(γ),w·n= 0 and w·n_D= 0 on∂γ

,

with norm

w²_Hdiv(γ):=w²_L2(γ)+∇_τ·w²_L2(γ).

Furthermore, we assume that elements of w ∈ H_div(γ) are aligned with γ, i.e. for a w ∈ H_div(γ) we have w=wτ withw∈H¹(γ). We setW_ij:=H_div(γ_ij) andW_i :=W_i1×W_i2 with norm

w_i²_Wi :=w_i1²_Wi1+w_i2²_Wi2+w_i·τ_i²_ip where w_i= (w_i1,w_i2)∈W_i.

Let us define Q_ij :=L²(γ_ij) andQ_i:=Q_i1×Q_i2.Q_i may be identified withL²(γ_i). We set W :=W₁×W₂ with norm

w²_W :=w₁²_W1+w₂²_W2, where w= (w₁,w₂)∈W, andQ:=Q₁×Q₂×Rwith norm

q²_Q:=q₁²_Q1+q₂²_Q2+q²₃, where q= (q₁, q₂, q₃)∈Q.

All those spaces are in fact Hilbert spaces equipped with scalar products associated with the chosen norms.

To obtain the weak formulation of (3.11) we take a test functionq_i ∈Q_i and integrate on each branch γ_ij the first equation in (3.11a) to obtain, summing overj

(∇τi·uˆ_i, q_i)_γ

i= fˆ_i, q_i

γi

∀qi∈Q_i, i= 1,2.

Taking then a test functionw_i∈W_i and integrating on eachγ_ij the second equation in (3.11a), we obtain (ˆη_iuˆ_i,w_i)_γ

i−( ˆp_i,∇τi·w_i)_γ

i+ˆp_iw_i·τ_iip+

ij:∂γ^p_ij=∅

ˆ

g_i(w_i·n_D)|_∂γ^p

ij = 0 ∀w_i∈W_i, i= 1,2.

Note that we have integrated by parts the pressure term and used the natural boundary conditions. Thanks to the identityab=a{{b}}+{{a}}bwe can include the coupling conditions (3.11b) substituting the expression of the pressure jump and average as

pˆ_iw_i·τ_iip=ˆp_iip{{w_i·τ_i}}_ip+{{ˆp_i}}_ipw_i·τ_iip.

Introducing ˆu:= ( ˆu₁,uˆ₂)∈W and ˆp:= (ˆp₁,pˆ₂, pˆ_I)∈Qand summing overi, the weak formulation of the coupled problem (3.11) can be now written as: find (ˆu,p)ˆ ∈W ×Qsuch that

A( ˆu,w) +B(ˆp,w) =F(w) ∀w∈W

B(q,u) =ˆ Q(q) ∀q∈Q. (4.1)

The functionals and bilinear forms in (4.1) are defined as A(u,w) :=

2 i=1

a_i(u_i,w_i) + 2 i,j=1 i=j

η^I_ij{{u_j·τ_j}}_ip{{w_j·τ_j}}_ip, (4.2a)

B(q,w) :=

2 i=1

−(q_i,∇τi·w_i)_L2(γi)+q₃w_i·τ_iip, (4.2b)

F(w) :=

ij:∂γ_ij^p=∅

−ˆg_i(w_i·n_D)|_∂γij^p (4.2c)

Q(q) :=

2 i=1

− fˆ_i, q_i

L²(γi)+ ˆf_Iq₃. (4.2d)

The bilinear formsa_i inAare given, fori, j= 1,2 and i=j, by a_i(u,w) := (ˆη_iu,w)_L2(γ

i)+η^I_iid_j d_i

ξˆ₀u·τ_iipw·τ_iip+{{u·τ_i}}_ip{{w·τ_i}}_ip

. (4.3)

We have the following

Lemma 4.1 (Boundedness ofA). There exist a constantC∈R⁺ such that

|A(u,w)| ≤Cu_Ww_W ∀u,w∈W.

Proof. A is clearly a bilinear form on W. Since each u_ij and w_ij are aligned along γ_ij, that is u_ij =u_ijτ_i, the request thatu_ij ∈H_div(γ_ij) implies thatu_ij ∈H¹(γ_ij). Then the boundedness ofAcan be obtained from

the application Cauchy–Schwarz inequality together with Sobolev embeddings and trace inequalities [1,13], by which we can finally state that ∃C∈R⁺ such that

|A(u, w)| ≤Cu_Ww_W where C=C

ˆη_L∞(γ1∪γ2), λ_max, d, c_γ withd:= max

i=j=1,2d_i/d_j andc_γ depends on the trace inequality constants for eachγ_ij.

Lemma 4.2 (Coercivity ofA). There exist a constantα∈R⁺ such that A(u,u)≥αu²_W ∀u∈W₀ whereW₀:={w∈W :B(q,w) = 0, ∀q∈Q}.

Proof. By definition ofBwe have that a w∈W₀ is characterized by∇τi·w_ij = 0 almost everywhere on γ_ij and w_i·τ_iip = 0, for i, j = 1,2. Therefore, for aw ∈W₀ we have w²_W =₂

i=1w_i²_L2(γi). Moreover, if w∈W₀ we have

A(w,w) = 2

i=1

η_i^1/2w_i²

L²(γi)+ 2 i,j=1 i=j

η_ii^Id_j

d_i {{w_i·τ_i}}²_ip+η_ij^I {{w_i·τ_i}}_ip{{w_j·τ_j}}_ip

.

Let us introduce the vectors a_i =

d_j/d_i{{w_i·τ_i}}_ipτ_i, for i, j = 1,2 and j = i and the scalar product (a₁,a₂)_K=a^T₁K⁻¹_I a₂. We recall the definition ofη^I_ij in (3.5) to note that the last sum in the previous equality may be written as

(a₁,a₁)_K+ (a₂,a₂)_K+ 2(a₁,a₂)_K = (a₁+a₂,a₁+a₂)_K ≥0.

Therefore, the wanted inequality is proved withα= infess

x∈γ1∪γ2

ˆ

η(x).

We indicate with M the set of indexes ij corresponding to portions of fracture where we impose pressure boundary conditions, that is

M :=

(i, j) : i, j= 1,2 and∂γ_ij∩∂γ_ij^p =∅

and n_d:=M.

Theorem 4.1(Inf-sup condition). If n_d>0, then for all p∈Q there exist aw∈W withw=0such that B(p,w)≥βp_Qw_W,

for β∈R⁺ independent onpandw.

Proof. Given p= ( ˆp₁,pˆ₂,pˆ_I)∈Qwe construct the following auxiliary problems. For (i, j)∈M we look for a functionφ_ij∈H²(γ_ij) such that

⎧⎪

⎪⎨

⎪⎪

⎩

−∇τi·(∇τiφ_ij) = ˆp_ij in γ_ij,

∂φ_ij

∂τ_i = pˆ_I

n_d(−1)^j+1|γ| oni_p,

φ_ij = 0 on∂γ_ij∩∂γ_ij^p,

(4.4)

with|γ|=

i|γi|. While, for all other values of the indexesiandj we look for theφ_ij solution of

⎧⎪

⎪⎨

⎪⎪

⎩

−∇_τi·(∇_τiφ_ij) = ˆp_ij in γ_ij,

∂φ_ij

∂τ_i = 0 on∂γ_ij∩∂γ_ij^u, φ_ij = 0 oni_p.

(4.5)

Both problems are well posed and enjoy elliptic regularity.

We considerw_ij =∇τiφ_ij. We have, by construction, that the solution of (4.4) provides at the intersection pointi_p

w_ij·τ_i= pˆ_I

n_d(−1)^j+1|γ|. (4.6)

As for the solution of (4.5), by simple computations we derive that ati_p w_ij·τ_i=−

γij

(−1)^j+1pˆ_ijdγ. (4.7)

Furthermore,

B(p,w) = ²

i,j=1

pˆ_ij²_L2(γij)+|γ|pˆ²_I +

(i, j)∈M

γij

ˆ p_ijpˆ_I.

Thanks to Young’s inequality applied to the third term, we have that B(p,w)≥ 1

2

⎛

⎝²

i,j=1

pˆ_ij²_L2(γij)+|γ|pˆ²_I

⎞

⎠≥cp²_Q,

withc=¹₂min{1,|γ|}. Exploiting standard stability results for the solution of (4.4) and (4.5), we infer that 2

i=1

w_ij²_L2(γij)= 2 i=1

∇_τiφ_ij²_L2(γij)≤C

ˆ p²_I+

2 i=1

pˆ_ij²_L2(γij)

.

Moreover, we have 2 i=1

∇τi·w_ij²_L2(γij)= 2

i=1

∇τi·(∇τiφ_ij)²_L2(γij)=

= 2

i=1

φij²_H2(γij)≤C

ˆ p²_I+

2 i=1

pˆ_ij²_L2(γij)

.

Thus,

ijw_ij²_Wij p²_Q. Furthermore,w_i²_ippi²_Qi+ ˆp²_I because of (4.7) and (4.6). In conclusion there exist a constantC∈R⁺ such thatw_W ≤Cp_Q. This result allows us to complete the proof.

Remark 4.2. The conditionn_d≥1 in the previous proof is needed, otherwise we are not able to control the pressure ˆp_I. However, if all boundary conditions are imposed on the velocity we are still able to find a solution provided that the boundary velocity satisfies a global mass conservation. In this case, however, ˆp_ij ∈L²(γ)\R and ˆp_I may take any arbitrary value.