Numerical analysis of a two-phase flow discrete fracture matrix model

Abstract

We present a new model for two phase Darcy flows in fractured media, in which fractures are modelled as submanifolds of codimension one with respect to the surrounding domain (matrix). Fractures can act as drains or as barriers, since pressure discontinuities at the matrix-fracture interfaces are permitted. Additionally, a layer of damaged rock at the matrix-fracture interfaces is accounted for. The numerical analysis is carried out in the general framework of the Gradient Discretisation Method. Compactness techniques are used to establish convergence results for a wide range of possible numerical schemes; the existence of a solution for the two phase flow model is obtained as a byproduct of the convergence analysis. A series of numerical experiments conclude the paper, with a study of the influence of the damaged layer on the numerical solution.

1 Introduction

Flow and transport in fractured porous media are of paramount importance for many applications such as petroleum exploration and production, geological storage of carbon dioxide, hydrogeology, or geothermal energy. Two classes of models, dual continuum and discrete fracture matrix models, are typically employed and possibly coupled to simulate flow and transport in fractured porous media. Dual continuum models assume that the fracture network is well connected and can be homogenised as a continuum coupled to the matrix continuum using transfer functions. On the other hand, discrete fracture matrix models (DFM), on which this paper focuses, represent explicitly the fractures as co-dimension one surfaces immersed in the surrounding matrix domain. The use of lower dimensional rather than equi-dimensional entities to represent the fractures has been introduced in [4, 8, 31, 36, 37] to facilitate the grid generation and to reduce the number of degrees of freedom of the discretised model. The reduction of dimension in the fracture network is obtained from the equi-dimensional model by integration and averaging along the width of each fracture. The resulting so called hybrid-dimensional model couple the 3D model in the matrix with a 2D model in the fracture network taking into account the jump of the normal fluxes as well as additional transmission conditions at the matrix-fracture interfaces. These transmission conditions depend on the mathematical nature of the equi-dimensional model and on additional physical assumptions. They are typically derived for a single phase Darcy flow for which they specify either the continuity of the pressure in the case of fractures acting as drains [4, 9] or Robin type conditions in order to take into account the discontinuity of the pressure for fractures acting either as drains or barriers [5, 11, 31, 37].

Fewer works deal with the extension of hybrid-dimensional models to two-phase Darcy flows. Most of them build directly the model at the discrete level as in [8, 34, 40] or are limited to the case of continuous pressures at the matrix-fracture interfaces as in [8, 10, 40]. In [35], an hybrid-dimensional two-phase flow model with discontinuous pressures at the matrix-fracture interfaces is proposed using a global pressure formulation. However, the transmission conditions at the interface do not take into account correctly the transport from the matrix to the fracture.

In this paper, a new hybrid-dimensional two-phase Darcy flow model is proposed accounting for complex networks of fractures acting either as drains or barriers. The model takes into account discontinuous capillary pressure curves at the matrix-fracture interfaces. It also includes a layer of damaged rock at the matrix-fracture interface with its own mobility and capillary pressure functions. This additional layer is not only a modelling tool. It also plays a major role in the convergence analysis of the model by giving time estimates on the approximate interfacial saturations, which yield their compactness (see Remark 4.7) and enables the identification of their limit. Moreover, when solving the discrete equations with a Newton-Raphson method, a non-zero distribution of volume at the interfacial unknowns is in general required for the Jacobian not to be degenerate. The sensitivity of the discrete solution as well as of the computational performance on interfacial parameters is studied in the test case section. The results suggest that the model converges with vanishing interfacial volume. However, this is still an open question.

The discretisation of hybrid-dimensional Darcy flow models has been the object of many works using cell-centred Finite Volume schemes with either Two Point or Multi Point Flux Approximations (TPFA and MPFA) [2, 3, 5, 33, 36, 41, 43], Mixed or Mixed Hybrid Finite Element methods (MFE and MHFE) [4, 34, 37], Hybrid Mimetic Mixed Methods (HMM, which contains Mixed/Hybrid Finite Volume and Mimetic Finite Difference schemes [22]) [6, 9, 11, 30], Control Volume Finite Element Methods (CVFE) [8, 33, 38,39,40], and the Vertex Approximate Gradient (VAG) scheme [9,10,11, 44, 45]. Let us also mention that non-matching discretisations of the fracture and matrix meshes are studied for single phase Darcy flows in [7, 14, 32, 42]. The convergence analysis for single-phase flow models with a single fracture is established in [4, 37] for MFE methods, in [14] for non matching MFE discretisations, and in [5] for TPFA discretisations. The case of single-phase flows with complex fracture networks is studied in the general framework of the gradient discretisation method in [9] for continuous pressure models and in [11] for discontinuous pressure models. For hybrid-dimensional two-phase flow models, the only convergence analysis is to our knowledge done in [10] for the VAG discretisation of the continuous pressure model with fractures acting only as drains. Let us recall that the gradient discretisation method (GDM) enables convergence analysis of both conforming and non conforming discretisations for linear and non-linear second order elliptic and parabolic problems. It accounts for various discretisations such as conforming Finite Element methods, MFE and MHFE methods, some TPFA and symmetric MPFA schemes, and the VAG and HHM schemes [24]. The main advantage of this framework is to provide, for a given model, a convergence proof for all schemes satisfying some abstract conditions, at the reduced cost of a single convergence analysis; see e.g. [19, 20, 23, 28, 29]. We refer to the monograph [21] for a detailed presentation of the GDM.

The main purpose of this paper is to propose an extension of the gradient discretisation method to our hybrid-dimensional two-phase Darcy flow model. This provides, in an abstract framework, the convergence of the approximate solution to a weak solution of the model; as a by-product, this proves the existence of a solution to this continuous model. The numerical analysis is partially based on the previous work [29] dealing with the gradient discretisation method for single medium two-phase Darcy flows. The main new difficulty addressed in this work compared with the analysis of [29] and [10] comes from the transmission conditions at the matrix-fracture interfaces; these conditions involve an upwinding between the fracture phase pressures and the traces of the matrix phase pressures. Note that, as in [29] and [10], the convergence analysis assumes that the phase mobilities do not vanish.

The outline of this paper is as follows. Section 2 introduces the geometry of the fracture network, the function spaces, the strong and weak formulations of the model as well as the assumptions on the data. Section 3 details the gradient discretisation method, including the definition of the abstract reconstruction operators, of the discrete variational formulation (gradient scheme), and of the coercivity, consistency, limit conformity and compactness properties. Section 4 proves the main result of this paper which is the convergence of the gradient scheme solution to a weak solution of the model. This convergence is established using compactness arguments, and requires us to establish various compactness results on the approximation solutions: averaged in time and space, uniform-in-time and weak-in-space, etc. The Minty monotonicity trick is used to identify the limit of the non-linear term resulting from the the upwinding between the fracture and matrix phase pressures. Section 5 studies on a 2D numerical example the influence of the additional layer of damaged rock at the matrix-fracture interface on the solution of the model. The discretisation used in this test case is based on the VAG scheme which can be shown from [11] to satisfy the assumptions of our gradient discretisation method. Note that numerical comparisons of our model with the equi-dimensional model as well as with the continuous pressure model of [10] can be found in [1, 12] without the accumulation term in the interfacial layer, which plays a minor role in the numerical tests when this layer is thin with respect to the fracture (see Sect. 5). It is shown that the discontinuous pressure model analysed in this paper is more accurate than the continuous pressure model of [10] even in the case of fractures acting only as drains; this improved accuracy is due to more accurate transmission conditions at the matrix-fracture interfaces.

2 Notation and model

2.1 Geometry

Let $\Omega$ denote a bounded domain of ${\mathbb {R}}^d$ ($d=2,3$), polyhedral for $d=3$ and polygonal for $d=2$. To fix ideas the dimension will be fixed to $d=3$ when it needs to be specified, for instance in the naming of the geometrical objects or for the space discretisation in the next section. The adaptations to the case $d=2$ are straightforward.

Let $\overline{\Gamma }= \bigcup _{i\in I} \overline{\Gamma }_i$ and its interior $\Gamma = \overline{\Gamma }{\setminus } \partial \overline{\Gamma }$ denote the network of fractures $\Gamma _i\subset \Omega$, $i\in I$. Each $\Gamma _i$ is a planar polygonal simply connected open domain included in a plane ${\mathcal {P}}_i$ of ${\mathbb {R}}^d$. It is assumed that the angles of $\Gamma _i$ are strictly smaller than $2\pi$, and that $\Gamma _i\cap \overline{\Gamma }_j=\emptyset$ for all $i\ne j$. For all $i\in I$, let us set $\Sigma _i = \partial \Gamma _i$, with $\mathbf n _{\Sigma _i}$ as unit vector in ${\mathcal {P}}_i$, normal to $\Sigma _i$ and outward to ${\Gamma _{}}_i$. Further $\Sigma _{i,j}= \Sigma _i\cap \Sigma _j$ for $i\not =j$, $\Sigma _{i,0} = \Sigma _i\cap \partial \Omega$, $\Sigma _{i,N} = \Sigma _i{\setminus }(\bigcup _{j\in I\setminus \{i\}}\Sigma _{i,j}\cup \Sigma _{i,0})$, $\Sigma = \bigcup _{(i,j)\in I\times I, i\ne j} ( \Sigma _{i,j}{\setminus }\Sigma _{i,0} )$ and $\Sigma _0 = \bigcup _{i\in I} \Sigma _{i,0}$. It is assumed that $\Sigma _{i,0} = \overline{\Gamma }_i\cap \partial \Omega$.

Fig. 1

Example of a 2D domain $\Omega$ and 3 intersecting fractures $\Gamma _i,i=1,2,3$. We define the fracture plane orientations by ${{\mathfrak {a}}^\pm (i)\in \chi }$ for $\Gamma _i$, $i\in I$

We define the two unit normal vectors $\mathbf n _{{\mathfrak {a}}^{\pm (i)}}$ at each planar fracture ${\Gamma _{}}_i$, such that $\mathbf n _{{\mathfrak {a}}^+(i)} + \mathbf n _{{\mathfrak {a}}^-(i)} = 0$ and oriented outward to the matrix side ${\mathfrak {a}}^\pm (i)$ (cf. Fig. 1). We define the set of indices $\chi = \{{\mathfrak {a}}^+(i), {\mathfrak {a}}^-(i)\mid i\in I\}$, such that $\#\chi = 2\# I$. For ease of notation, we use the convention ${\Gamma _{}}_{{\mathfrak {a}}^+(i)} = {\Gamma _{}}_{{\mathfrak {a}}^-(i)} = {\Gamma _{}}_i$.

For $\AA = {\mathfrak {a}}^\pm (i)\in \chi$, we denote by $\gamma _{{\mathfrak {a}}}$ the one-sided trace operator on ${\Gamma _{}}_{\mathfrak {a}}$. It satisfies the condition , where .

On the fracture network ${\Gamma _{}}$, the tangential gradient is denoted by $\nabla _\tau$, and is such that

$$\begin{aligned} \nabla _\tau v = (\nabla _{\tau _i} v_i )_{i\in I}, \end{aligned}$$

where, for each $i\in I$, the tangential gradient $\nabla _{\tau _i}$ is defined by fixing a reference Cartesian coordinate system of the plane ${\mathcal {P}}_i$ containing ${\Gamma _{}}_i$. In the same manner, we denote by $\mathrm{div}_{\tau } \mathbf q = (\mathrm{div}_{\tau _i} \mathbf q _i )_{i\in I}$ the tangential divergence operator.

2.2 Continuous model and hypotheses

We describe here the continuous model and assumptions that are implicitly made throughout the paper. In the matrix domain $\Omega {\setminus }\overline{{\Gamma _{}}}$, let us denote by $\Lambda _m\in L^{\infty }(\Omega )^{d\times d}$ the symmetric permeability tensor, chosen such that there exist $\overline{\lambda }_m\ge \underline{\lambda }_m > 0$ with

$$\begin{aligned} \underline{\lambda }_m|\zeta |^2 \le \Lambda _m(\mathbf x )\zeta \cdot \zeta \le \overline{\lambda }_m|\zeta |^2 \text{ for } \text{ all } \zeta \in {\mathbb {R}}^d, \mathbf x \in \Omega . \end{aligned}$$

Analogously, in the fracture network ${\Gamma _{}}$, we denote by $\Lambda _f\in L^{\infty }(\Gamma )^{(d-1)\times (d-1)}$ the symmetric tangential permeability tensor, and assume that there exist $\overline{\lambda }_f\ge \underline{\lambda }_f > 0$, such that

$$\begin{aligned} {\underline{\lambda }_f|\zeta |^2 \le \Lambda _f(\mathbf x )\zeta \cdot \zeta \le \overline{\lambda }_f|\zeta |^2 \text{ for } \text{ all } \zeta \in {\mathbb {R}}^{d-1}, \mathbf x \in \Gamma .} \end{aligned}$$

On the fracture network ${\Gamma _{}}$, we introduce an orthonormal system $(\varvec{\tau }_1(\mathbf x ),\varvec{\tau }_2(\mathbf x ),\mathbf n (\mathbf x ))$, defined a.e. on ${\Gamma _{}}$. Inside the fractures, the normal direction is assumed to be a permeability principal direction. The normal permeability $\lambda _{f,\mathbf n } \in L^{\infty }(\Gamma )$ is such that $\underline{\lambda }_{f,\mathbf n } \le \lambda _{f,\mathbf n }(\mathbf x ) \le \overline{\lambda }_{f,\mathbf n }$ for a.e. $\mathbf x \in \Gamma$ with $0 < \underline{\lambda }_{f,\mathbf n } \le \overline{\lambda }_{f,\mathbf n }$. We also denote by $d_f \in L^\infty ({\Gamma _{}})$ the width of the fractures, assumed to be such that there exist ${\overline{d}}_f\ge {\underline{d}}_f > 0$ with ${\underline{d}}_f \le d_f(\mathbf x ) \le {\overline{d}}_f$ for a.e. $\mathbf x \in \Gamma$. The half normal transmissibility in the fracture network is denoted by

$$\begin{aligned} {T_f = \frac{2\lambda _{f,\mathbf n }}{d_f}.} \end{aligned}$$

Furthermore, $\phi _m$ and $\phi _f$ are the matrix and fracture porosities, respectively, $\rho ^\alpha \in {\mathbb {R}}^+$ denotes the density of phase $\alpha$ (with $\alpha =1$ the non-wetting and $\alpha =2$ the wetting phase) and $\mathbf g \in {\mathbb {R}}^d$ is the gravitational vector field. We assume that $\underline{\phi }_{m,f}\le \phi _{m,f}\le \overline{\phi }_{m,f}$, for some $\underline{\phi }_{m,f},\overline{\phi }_{m,f}> 0$. $(k_m^\alpha ,k_f^\alpha )$ and $(S_m^\alpha ,S_f^\alpha )$ are the matrix and fracture phase mobilities and saturations, respectively. Hypothesis on these functions are stated below.

The PDEs model writes: find phase pressures and velocities ($\alpha = 1,2$), such that

(1a)

The matrix-fracture coupling condition on $(0,T)\times {\Gamma _{}}_{\mathfrak {a}}$ (for all ${\mathfrak {a}}\in \chi$) are

(1b)

where , with representing the interfacial width and the interfacial porosity (cf. Fig. 2). We assume that each of these parameters is uniformly bounded below. In these equations, we have

(1c)

Fig. 2

Illustration of the coupling condition. It can be seen as an upwind two point approximation of . The upwinding takes into account the damaged rock type at the matrix-fracture interfaces. The arrows show the positive orientation of the normal fluxes and

In the above, we used the shorthand notations

as well as, for ${{\mu \in \{m,f\}}\cup \chi }$, and a.e. $(t,\mathbf x )\in (0,T)\times {M_\mu }$,

Here and in the following, ${M_\mu }$ is defined by

$$\begin{aligned} {M_\mu }= \left\{ \begin{array}{ll} {\Omega }&{} \text { if }\mu = m \\ {\Gamma _{}}&{} \text { if }\mu = f \\ {\Gamma _{{\mathfrak {a}}}} &{} \text { if }\mu = {\mathfrak {a}}\in \chi . \end{array}\right. \end{aligned}$$

The various boundary conditions imposed on the domain are: homogeneous Dirichlet conditions at the boundary of the domain, pressure continuity and flux conservation at the fracture-fracture intersections, and zero normal flux at the immersed fracture tips. In other words,

Let us define $L^2(\Gamma ) = \{v = (v_i)_{i\in I}, v_i\in L^2(\Gamma _i), i\in I\}$. The assumptions under which the model is considered are:

• and ,

• For ${\mu \in \{m,f\}}$ and $\alpha =1,2$, ,

• For ${{\mu \in \{m,f\}}\cup \chi }$: is a Caratheodory function; for a.e. $\mathbf x \in {M_\mu }$, is a non-decreasing Lipschitz continuous function on ${\mathbb {R}}$; for all $q\in {\mathbb {R}}$, is piecewise constant on a finite partition of polytopal subsets of ${M_\mu }$.

• For $\alpha =1,2$ and ${{\mu \in \{m,f\}}\cup \chi }$: there exist constants , such that is a Caratheodory function.

Recall that a Caratheodory function is measurable w.r.t. its first argument and continuous w.r.t. its second argument.

2.3 Weak formulation

The subspace $H^1(\Gamma )$ of $L^2(\Gamma )$ consists in functions $v = (v_i)_{i\in I}$ such that $v_i\in H^1(\Gamma _i)$ for all $i\in I$, with continuous traces at the fracture intersections $\Sigma _{i,j}$ for all $i\not =j$. Its subspace of functions with vanishing traces on $\Sigma _0$ is denoted by $H^1_{\Sigma _0}(\Gamma )$.

Let us now define the hybrid-dimensional function spaces that are used as variational spaces for the Darcy flow model. Starting from

$$\begin{aligned} V = H^1(\Omega {\setminus } \overline{{\Gamma _{}}})\times H^1({\Gamma _{}}), \end{aligned}$$

consider the subspace

$$\begin{aligned} V^0 = V_m^0\times V_f^0 \end{aligned}$$

where (with the trace operator on $\partial {\Omega }$)

The weak formulation of (1) amounts to finding satisfying the following variational equalities, for any $\alpha = 1,2$ and any :

(2)

Here,

with $\mathrm{d}\tau ({x})$ the $d-1$ dimensional Lebesgue measure on $\Gamma$.

3 The gradient discretisation method

The gradient discretisation method consists in selecting a set (called a gradient discretisation) of a finite-dimensional space and reconstruction operators on this space, and in substituting them for their continuous counterpart in the weak formulation of the model. The scheme thus obtained is called a gradient scheme. Let us first define the set of discrete elements that make up a gradient discretisation.

Definition 3.1

(Gradient discretisation (GD)) A spatial gradient discretisation for a discrete fracture matrix model is , where

• is a finite-dimensional space of degrees of freedom (DOFs),

• For , reconstructs a function on ${{M_\mu }}$ from the DOFs,

• For , reconstructs a gradient on ${{M_\mu }}$ from the DOFs,

• For ${\mathfrak {a}}\in \chi$, reconstructs, from the DOFs, a jump on ${{\Gamma _{{\mathfrak {a}}}}}$ between the matrix and fracture,

• For ${{\mathfrak {a}}\in \chi }$, reconstructs, from the DOFs, a trace on ${{\Gamma _{{\mathfrak {a}}}}}$ from the matrix.

These operators must be chosen such that the following expression defines a norm on :

The spatial gradient discretisation ${\mathcal {D}_S}$ is extended to a space-time gradient discretisation by setting with

• a discretisation of the time interval [0, T],

• an operator designed to interpolate the initial condition.

The space-time operators act on a family the following way: for all ${n = 0,\dots ,N-1}$ and all ,

(3)

We extend these functions at ${t=0}$ by considering the corresponding spatial operators on .

If is a family in , the discrete time derivatives are defined such that, for all ${n = 0,\dots ,N-1}$ and all , with ,

Let be a basis of . If , we write . Then, for ${g\in C({\mathbb {R}})}$, we define by . In other words, g(w) is defined by applying g to each degree of freedom of w. Although this definition depends on the choice of basis , we do not explicitly indicate this dependency. This definition of g(w) is particularly meaningful in the context of piecewise constant reconstructions, see Remark 3.3 below.

The gradient scheme for (1) consists in writing the weak formulation (2) with continuous spaces and operators replaced by their discrete counterparts, after a formal integration-by-parts in time. In other words, the gradient scheme is: find such that, with ,

(4)

and, for any ${\alpha = 1,2}$ and ,

(5)

3.1 Properties of gradient discretisations

The convergence analysis of the GDM is based on a few properties that sequences of GDs must satisfy.

Definition 3.2

(Piecewise constant reconstruction operator) Let be the basis of chosen in Sect. 3. For , an operator is called piecewise constant if it has the representation

where is a partition of ${{M_\mu }}$ up to a set of zero measure, and ${{\mathbb {1}}_{\omega _\nu ^\mu }}$ is the characteristic function of ${\omega _\nu ^\mu }$.

In the following, all considered function reconstruction operators and are assumed to be piecewise constant.

Remark 3.3

Recall that, if ${g\in C^0({\mathbb {R}})}$ and , then is defined by the degrees of freedom . Then, any piecewise constant reconstruction operator ${\Pi }$ commutes with g in the sense that .

The coercivity property enables us to control the functions and trace reconstruction by the norm on . This is a combination of a discrete Poincaré inequality and a discrete trace inequality.

Definition 3.4

(Coercivity of spatial GD) Let

A sequence ${(\mathcal {D}_S^l)_{l\in {\mathbb {N}}}}$ of gradient discretisations is coercive if there exists ${\mathcal {C}_P > 0}$ such that

(6)

This consists in choosing a family

The consistency ensures that a certain interpolation error goes to zero along sequences of GDs.

Definition 3.5

(Consistency of spatial GD) For and , define

and . A sequence of gradient discretisations is GD-consistent (or consistent for short) if, for all ,

(7)

To define the notion of limit-conformity, we need the following two spaces:

where ${C_b^\infty ({\Omega }\setminus \overline{{\Gamma _{}}})\subset C^\infty ({\Omega }\setminus \overline{{\Gamma _{}}})}$ is the set of functions ${\varphi }$, such that for all $\mathbf{x \in {\Omega }}$ there exists ${r > 0}$, such that for all connected components ${\omega }$ of one has ${\varphi \in C^\infty (\overline{\omega })}$, and such that all derivatives of ${\varphi }$ are bounded. The limit-conformity imposes that, in the limit, the discrete gradient and function reconstructions satisfy a natural integration-by-part formula (Stokes’ theorem).

Definition 3.6

(Limit-conformity of spatial GD) For all , and , define

and . A sequence ${(\mathcal {D}_S^l)_{l\in {\mathbb {N}}}}$ of gradient discretisations is limit-conforming if, for all and all ,

(8)

Remark 3.7

(Domain of ) Usually, the measure of limit-conformity is defined on spaces in which the Darcy velocities of solutions to the model are expected to be, not smooth spaces as $\mathbf{C ^\infty _{\Omega }\times \mathbf C ^\infty _{\Gamma }}$ [21, Definition 2.6]. However, if we do not aim at obtaining error estimates (which is the case here, given that such estimates would require unrealistic regularity assumptions on the data and the solution), only needs to be defined and to converge to 0 on spaces of smooth functions—see Lemma A.2.

For any space-dependent function f, define . Likewise, for any time-dependent function g, let . The compactness property ensures a sort of discrete Rellich theorem (compact embedding of ${H^1_0}$ into ${L^2}$). By the Kolmogorov theorem, this compactness is equivalent to a uniform control of the translates of the functions.

Definition 3.8

(Compactness of spatial GD) For all and , with and , where ${\tau ({\mathcal {P}}_i)}$ is the (constant) tangent space of ${{\mathcal {P}}_i}$, define

where all the functions on ${{\Omega }}$ (resp. ${{\Gamma _{}}_i}$) have been extended to ${{\mathbb {R}}^d}$ (resp. ${{\mathcal {P}}_i}$) by 0 outside their initial domain. Let . A sequence ${(\mathcal {D}_S^l)_{l\in {\mathbb {N}}}}$ of gradient discretisations is compact if

(9)

All these properties for spatial GDs naturally extend to space–time GDs with, for the consistency, additional requirements on the time steps and on the interpolants of the initial conditions.

Definition 3.9

(Properties of space-time gradient discretisations) A sequence of space-time gradient discretisations ${(\mathcal {D}^l)_{l\in {\mathbb {N}}}}$ is

• 1. Coercive if ${(\mathcal {D}_S^l)_{l\in {\mathbb {N}}}}$ is coercive.

• 2. Consistent if

  • (i) ${(\mathcal {D}_S^l)_{l\in {\mathbb {N}}}}$ is consistent,

  • (ii) as ${l\rightarrow \infty }$, and

  • (iii) For all , letting we have, as ${l\rightarrow \infty }$,

    

• 3. Limit-conforming if ${(\mathcal {D}_S^l)_{l\in {\mathbb {N}}}}$ is limit-conforming.

• 4. Compact if ${(\mathcal {D}_S^l)_{l\in {\mathbb {N}}}}$ is compact.

Elements of are identified with functions by setting, for with ,

(10)

This definition is compatible with the choices of space-time operators made in Definition 3.1, in the sense that, for any ${t\in (0,T]}$, (and similarly for the other reconstruction operators). With the identification (10), the norm on is

4 Convergence analysis

In the rest of this paper, when the phase parameter ${\alpha }$ is absent this implicitly means that it is equal to 1. For example, we write for . The main convergence result is the following.

Theorem 4.1

(Convergence Theorem) Let ${(\mathcal {D}^l)_{l\in {\mathbb {N}}}}$ be a coercive, consistent, limit-conforming and compact sequence of space-time gradient discretisations, with piecewise constant reconstructions. Then for any ${l\in {\mathbb {N}}}$ there is a solution of (5) with ${\mathcal {D}=\mathcal {D}^l}$.

Moreover, there exists solution of (2) such that, up to a subsequence as ${l\rightarrow \infty }$,

• 1. The following weak convergences hold, for ,

  

  (11)

• 2. The following strong convergences hold, with and :

  

  (12)

Remark 4.2

(Uniform-in-time strong-in-space convergence) It is additionally proved in [26] that the saturations converge uniformly-in-time strongly in ${L^2}$ (that is, in ${L^\infty (0,T;L^2({\Omega }))}$).

Remark 4.3

(Discretisation spaces varying with the time step) As mentioned in [13, Remark 3.5] for a different model, it is also possible to consider gradient schemes in which the gradient discretisation changes at each time step. This consists in choosing a family${\widetilde{\mathcal {D}}_{S}=(\mathcal {D}_{S,n})_{n=0,\ldots ,N_l}}$ of spatial gradient discretisations ${\mathcal {D}_{S,n}}$ (as in Definition 3.1), in considering unknowns and in defining the space-time operators (3) with , , and respectively replaced by , , and . The gradient scheme is then written as in (5). For a sequence ${(\widetilde{\mathcal {D}}^l_{S})_{l\in {\mathbb {N}}}}$ of such families of spatial GDs, the notions coercivity, consistency, limit-conformity and compactness are defined by writing the bound and convergences in (6), (7), (8) and (9) with , , and replaced by

With these notions, Theorem 4.1 still holds.

By using spatial GDs that change at each time step, one can represent in the GDM framework numerical methods with moving or dynamically refined meshes, or whose gradient reconstruction involves time-dependent parameters (as in ${\mathbb {RT}_k}$ Mixed Finite Elements with a diffusion tensor that depends on some unknown of the system; see [13, Section 4.1]).

Before delving into the proof of the theorem, let us give an overview of the strategy. The convergence of the solutions to the gradient schemes (5) is established by a compactness technique, as briefly described in [18, Section 1.2]: (i) prove a priori estimates on the solutions to the scheme, (ii) using discrete compactness theorems, deduce from these estimates that the (reconstructions of the) approximate solutions are compact in appropriate spaces, (iii) prove that any limit, in these spaces, of the approximate solutions is a solution to the continuous model (2).

• (i) A priori estimates. The first a priori estimates are classically obtained by using the approximate solution itself as a test function in the scheme (5). After summing the two equations corresponding to each phase, the diffusion terms then directly yield an estimate on . The time derivative term form the discrete counterpart of which, after integration in time, would yield an estimate on with . To make explicit that this estimate is actually an estimate on the saturation, we re-write as for a well-chosen ${B_\mu }$. These a priori estimates are stated in Lemma 4.4. These initial estimates only concern spatial derivatives of the approximate solution (they are a discrete equivalent of ${L ^2(0,T;H^1_0)}$ estimates). Since this solution depends on both time and space, estimates are also required on its (discrete) time derivative to establish the compactness in an appropriate space. These time derivative estimates are the purpose of Lemma 4.6 and, classically for parabolic PDEs, they are obtained in a weak spatial norm (a sort of discrete ${H^{-1}}$ norm). They are obtained on and, thanks to the modelling of the damaged rock type at the matrix-fracture interface (term in (1b)), also on . These estimates are instrumental to obtain the compactness in time and space of all the saturations in the model.

• (ii) Compactness. The estimates on the discrete spatial and temporal derivatives, together with the compactness property of the gradient discretisations, yield estimates on the spatial and temporal translates of the saturations (Lemmas 4.8 and 4.9). A use of the Kolmogorov theorem and of the consistency of the gradient discretisations (to identify, through Lemma A.2, weak limits of reconstructed gradients and traces as the gradient and trace of the limit of the approximate solutions) then give the convergences (11) and (12); this is stated in Theorem 4.11. A discontinuous Ascoli-Arzela theorem (Theorem A.1) is then applied in Theorem 4.13 to obtain the convergence of the saturations uniformly-in-time and weakly in ${L^2(\Omega )}$. This uniform-in-time convergence is essential to pass to the limit, in (iii) below, in the energy estimate (16) (which involves pointwise-in-time values of the saturations).

• (iii) The limit is a solution of the model. The conclusion, presented in Sect. 4.3, consists in proving that the limit of the approximate solutions is a solution to the continuous model. As we do not have strong convergence of the phase pressures , the main challenge in analysing this limit arises from the non-linear upwinding terms . The limit of this term is obtained by using the monotony properties of this upwinding, a Minty trick, and the discrete energy estimate (16).

4.1 Preliminary estimates

Let us introduce some useful auxiliary functions. These functions are the same as in [19, 25], with adjustments to account for the fact that the saturations depends on $\mathbf{x }$ and might not vanish at ${p=0}$. For ${{{\mu \in \{m,f\}}\cup \chi }}$, let ${R_{S_\mu (\mathbf x ,\cdot )}}$ be the range of ${S_\mu (\mathbf x ,\cdot )}$. The pseudo-inverse of ${S_\mu (\mathbf x ,\cdot )}$ is the mapping ${[S_\mu (\mathbf x ,\cdot )]^i:R_{S_\mu (\mathbf x ,\cdot )}\rightarrow {\mathbb {R}}}$ defined by

That is, ${[S_\mu (\mathbf x ,\cdot )]^i(q)}$ is the point z in ${R_{S_\mu (\mathbf x ,\cdot )}}$ that is the closest to ${S_\mu (\mathbf x ,0)}$ and such that ${S_\mu (\mathbf x ,z)=q}$. The function ${B_\mu (\mathbf x ,\cdot ):{\mathbb {R}}\rightarrow [0,\infty ]}$ is given by

${B_\mu (\mathbf x ,\cdot )}$ is convex lower semi-continuous (l.s.c.) and satisfies the following properties [25]

$$\begin{aligned}&B_\mu (\mathbf x ,S_\mu (\mathbf x ,r))=\int _0^r \tau \frac{\partial S_\mu }{\partial q}(\mathbf x ,\tau )\mathrm{d}\tau , \end{aligned}$$

(13)

$$\begin{aligned}&\forall a,b\in {\mathbb {R}}\,,\; a(S_\mu (\mathbf x ,b)-S_\mu (\mathbf x ,a))\le B_\mu (\mathbf x ,S_\mu (\mathbf x ,b))-B_\mu (\mathbf x ,S_\mu (\mathbf x ,a)) \qquad \end{aligned}$$

(14)

and, for some ${K_0}$, ${K_1}$ and ${K_2}$ not depending on $\mathbf{x }$ or r,

$$\begin{aligned} K_0S_\mu (\mathbf x ,r)^2-K_1\le B_\mu (\mathbf x ,S_\mu (\mathbf x ,r))\le K_2 r^2. \end{aligned}$$

(15)

In the following, we write ${A\lesssim B}$ for “${A\le MB}$ for a constant M depending only on an upper bound of and on the data in the assumptions of Sect. 2.2”.

Lemma 4.4

(Energy estimates) Under the assumptions of Sect. 2.2, let ${\mathcal {D}}$ be a gradient discretisation with piecewise constant reconstructions , . Let be a solution of the gradient scheme of (5). Take ${T_0\in (0,T]}$ and ${k\in \{0,\ldots ,N-1\}}$ such that ${T_0\in (t_k,t_{k+1}]}$. Then

(16)

As a consequence,

(17)

Proof

We remove the spatial coordinate $\mathbf{x }$ in the arguments, when not needed. Reasoning as in [19, Lemma 4.1], Property (14) gives

(18)

where we have used, by definition, . Similarly,

(19)

Equation (16) is then obtained by taking (for ${\alpha =1,2}$) in the gradient scheme (5), by summing the resulting equations over ${\alpha =1,2}$, by using (18) and (19), and by reducing the time integrals in the left-hand side from ${[0,t_{k+1}]}$ to ${[0,T_0]}$, due to the non-negativity of the integrands.

The inequality (17) is the consequence of a few simple estimates on the terms of (16) with ${T_0=T}$. For the symmetric diffusion terms (for ${\alpha =1,2}$ and ), we write

(20)

where if ${\mu =m}$. The matrix–fracture coupling terms are handled by noticing that, for any , and , so that for and ,

(21)

Here, we have used , and ${|s|^2=(s^+)^2+(s^-)^2}$. Plugging estimates (15), (20) and (21) in (16) (with ${T_0=T}$) and invoking Cauchy–Schwarz inequalities leads to

The proof of (17) is complete by noticing that the left-hand side is equal to , and by using Young’s inequality and the definition of in the right-hand side. $\square$

The existence of a solution to the gradient scheme follows by a standard fixed point argument based on the Leray–Schauder topological degree, see e.g. [10, proof of Lemma 3.2] or [23, Step 1 in the proof of Theorem 3.1].

Corollary 4.5

Under the assumptions of Lemma 4.4, there exists a solution to the gradient scheme (5).

We now want to obtain estimates on the discrete time derivatives. Let the dual norm of be defined by

(22)

Lemma 4.6

(Weak estimate on time derivatives) Under the assumptions of Sect. 2.2, let ${\mathcal {D}}$ be a gradient discretisation with piecewise constant reconstructions , . Let be a solution of the gradient scheme of (5). Then,

Remark 4.7

(Damaged rock modelling) The modelling of the damaged rock type (term in (1b)) is essential to obtain the estimate on above. These estimates are required to obtain the compactness of this discrete saturation (see Theorems 4.11 and 4.13).

Proof

Take and apply (5) with ${\alpha =1}$ to the test function , where is at an arbitrary position n. This shows that, for all and

where we have used the definition of in the last step. Taking the supremum over all such that shows that

(23)

Take the square of this relation, use ${(a+b)^2\le 2a^2+2b^2}$, and apply Jensen’s inequality to introduce the square inside the time integral. Multiply then by and sum over n to conclude. $\square$

Lemma 4.8

(Estimate on time translates) Under the assumptions of Sect. 2.2, let ${\mathcal {D}}$ be a gradient discretisation with piecewise constant reconstructions , . For any ${h>0}$ and any solution of (5),

(24)

where we recall that and , and where all functions of time have been extended by 0 outside (0, T).

Proof

Let us start by assuming that ${h\in (0,T)}$, and let us consider integrals over ${(0,T-h)}$ (we therefore do not use extensions outside (0, T) yet). By the Lipschitz continuity and monotonicity of the saturations we have . Thus, setting for all ${s\in {\mathbb {R}}}$,

(25)

In the last line, we simply wrote as the sum of the jumps if between s and ${s+h}$ (likewise for ).

For a fixed s, define by

With this choice,

(26)

We keep s fixed and concentrate on the integrand of the outer integral in the right-hand side of (25). Estimate (23), the definition (22) of , and Young’s inequality yield

Returning to (25), integrate the previous estimate over ${s\in (0,T-h)}$. In this step, it is crucial to realise that

Hence, recalling the definition of v,

Since , this proves (24) with ${L^2(0,T-h)}$ norms in the left-hand side, instead of ${L^2(0,T)}$ norms. The complete form of (24) follows by recalling that , so that (and similarly for other saturation terms). $\square$

Lemma 4.9

(Estimate on space translates) Under the assumptions of Sect. 2.2, let ${\mathcal {D}}$ be a gradient discretisation with piecewise constant reconstructions , . Let be a solution of (5), and let , with and , where ${\tau ({\mathcal {P}}_i)}$ is the (const.) tangent space of ${{\mathcal {P}}_i}$. Then, extending the functions and by 0 outside ${{M_\mu }}$,

where we recall that , and is given in Definition 3.8.

Proof

Let us focus on the matrix ${{\Omega }}$, and remember that, as a function of $\mathbf{x }$, is piecewise constant on a polytopal partition . Write

(27)

Let be the set of points $\mathbf{x }$ that do not belong to the same element ${\Omega _j}$ as their translate . By assumption on ,

Moreover, since each ${{\Omega }_j}$ is polytopal, . Hence,

(28)

On the other hand, by definition of ${{\mathcal {T}}_{D_S}}$ and the Lipschitz continuity of ,

(29)

Plugging (28) and (29) into (27) and reasoning similarly for $\iff S$ and concludes the proof. $\square$

Remark 4.10

This proof is the only place where the assumption that each is polytopal is used; this is to ensure that (and likewise for fracture and interfacial terms). Obviously, this asssumption on the sets could be relaxed (e.g., into “each has a Lipschitz-continuous boundary”), but assuming that these sets are polytopal is not restrictive for practical applications.

4.2 Initial convergences

We can now state our initial convergence theorem for sequences of solutions to gradient schemes. This theorem does not yet identify the weak limits of such sequences.

Theorem 4.11

(Averaged-in-time convergence of approximate solutions) Let ${(\mathcal {D}^l)_{l\in {\mathbb {N}}}}$ be a coercive, consistent, limit-conforming and compact sequence of space-time gradient discretisations, with piecewise constant reconstructions. Let be such that is a solution of (5) with ${\mathcal {D}=\mathcal {D}^l}$. Then, there exists such that, up to a subsequence as ${l\rightarrow \infty }$, the convergences (11) and (12) hold.

Proof

Combining Lemmata 4.4 and A.2 immediately gives (11) up to a subsequence. By assumption, and therefore, by Lemmata 4.8 and 4.9 and the Kolmogorov compactness theorem, there exists a subsequence of that strongly converges in and a subsequence of that strongly converges in . Also, by assumption, are non-decreasing functions, which allows us to identify the limits in (12) by applying Corollary A.3. $\square$

Let be the subspace of functions in vanishing on a neighbourhood of the boundary $\partial \Omega$. Define also as the image of ${C_0^\infty ({\Omega })}$ through the trace operator .

The following lemma and theorem add a uniform-in-time weak $L^2$ convergence property to the convergences established in Theorem 4.11.

Lemma 4.12

(Uniform-in-time, weak-in-space translate estimates) Under the assumptions of Sect. 2.2, let ${\mathcal {D}}$ be a gradient discretisation with piecewise constant reconstructions , . Let be a solution of the gradient scheme (5), and . Then, for all and all ${s,t\in [0,T]}$,

(30)

where ${C_\varphi }$ only depends on ${\varphi }$, ${\iff d}$ is the width of the fractures, and .

Proof

Let us introduce an interpolant such that, for all , . As in the proof of Lemma 4.8, let for all ${r\in [0,T]}$. Denote by L the left-hand side of (30) and introduce in the first sum and in the second sum to write

(31)

(32)

(33)

Here, the terms (31) and (32) have been estimated by using and the definition of . Let ${L_1}$ be the second addend in (33). Assuming that ${t<s}$, and hence ${n(t)\le n(s)}$, write and as the sum of their jumps, and recall the definition (22) of to obtain

Use now Lemmata 4.6 and the Cauchy–Schwarz inequality to infer

(34)

By the triangle inequality,

Plugging this into (34) and the resulting inequality into (33) concludes the proof. $\square$

Theorem 4.13

(Uniform-in-time, weak-in-space convergence) Under the assumptions of Theorem 4.11, for all and , and are continuous for the weak topologies of ${L^2({M_\mu })}$ and ${L^2({\Gamma _{}}_{{\mathfrak {a}}})}$, respectively, and

(35)

where the definition of the uniform-in-time weak $L^2$ convergence is recalled in “Appendix A.1”.

Proof

The proof hinges on the discontinuous Arzelà-Ascoli theorem (Theorem A.1 in the appendix). Consider first the matrix saturation. The space is dense in $L^2({\Omega })$. Apply (30) to . Since , only the term involving remains in the left-hand side. The resulting estimate and the property show that the sequence of functions satisfies the assumptions of Theorem A.1 with . Hence, has a subsequence that converges uniformly on [0, T] weakly in $L^2({\Omega })$. Given (12), the weak limit of this sequence must be .

A similar reasoning, based on the space —which is dense in $L^2(\Gamma )$—and using $\varphi =(0,\iff \varphi )$ in (30), gives the uniform-in-time weak $L^2(\Gamma )$ convergence of towards .

Let us now turn to the convergence of the trace saturations. Take such that the support of is non empty for exactly one ${\mathfrak {a}}\in \chi$. Considering in (30) leads to

(36)

Since it was established that converges uniformly-in-time weakly in $L^2({\Omega })$, the sequence is equi-continuous and the last term in (36) therefore tends to 0 uniformly in l as ${s-t\rightarrow 0}$. Hence, (36) enables the usage of Theorem A.1, by noticing that is dense in ${{L^2({\Gamma _{}}_{{\mathfrak {a}}})}}$, and gives the uniform-in-time weak ${{L^2({\Gamma _{}}_{{\mathfrak {a}}})}}$ convergence of . $\square$

4.3 Proof of Theorem 4.1

The proof of the main convergence theorem can now be given.

First step: passing to the limit in the gradient scheme.

Let us introduce the family of functions :

and their continuous counterparts :

The following properties are easy to check. Firstly, since $\iff T$, and are positive and $s\mapsto s^+$ and $s\mapsto -s^-$ are non-decreasing,

(37)

Secondly, by the convergences (12), for ${(\beta _l)_{l\in {\mathbb {N}}}\subset {L^2({\Gamma _{}}_{{\mathfrak {a}}})}}$ and ${\beta \in {L^2({\Gamma _{}}_{{\mathfrak {a}}})}}$,

(38)

Thirdly, by Lemma 4.4, the sequences (, $\alpha =1,2$) are bounded in and there exists thus such that, up to a subsequence,

(39)

Consider , where and . Take as “test function” in (5). Here, is defined as in the proof of Lemma 4.12. Apply the discrete integration-by-parts of [21, Section D.1.7] on the accumulation terms in (5), let and use standard convergence arguments [19, 21] based on Theorem 4.11 to see that

(40)

Note that Equation (40) also holds for any smooth , by density of tensorial functions in smooth functions [17, Appendix D]. Recalling the weak formulation (2), proving Theorem 4.1 is now all about showing that

(41)

This is achieved by using Minty’s trick.

Second step: proof that

(42)

Having in mind to employ the energy inequality (16) with $T_0=T$, we first establish, for and , the following convergences as ${l\rightarrow \infty }$:

(43)

(44)

(45)

The convergence (43) is obvious by Theorem 4.11. From the choice (4) of the scheme’s initial conditions, together with the consistency of the interpolation operator , in ${{L^2({M_\mu })}}$ and in ${{L^2({\Gamma _{}}_{{\mathfrak {a}}})}}$, as ${l\rightarrow \infty }$. Then, (15) and [27, Lemma A.1] yield (44) and (45).

We further show that

(46)

(47)

(48)

By the uniform-in-time weak $L^2$ convergences of Theorem 4.13, in ${{L^2({M_\mu })}}$ and in ${{L^2({\Gamma _{}}_{{\mathfrak {a}}})}}$, as ${l\rightarrow \infty }$. Note also that, since (by assumption) and are not explicitly space-dependent on each open set of the formerly introduced partitions of ${{M_\mu }}$ and ${{\Gamma _{{\mathfrak {a}}}}}$, respectively, so are and . On these partitions, and are convex l.s.c. and an easy adaptation of [25, Lemma 4.6] (which essentially states the $L^2$-weak l.s.c. of strongly l.s.c. convex functions on $L^2$), to account for the terms and $\eta$, thus shows that (46) and (47) hold. To prove (48), apply the Cauchy-Schwarz inequality to write

and take the inferior limit as ${l\rightarrow \infty }$, using the strong convergence of and weak convergence of to pass to the limit in the left-hand side and the first term in the right-hand side.

Let us now come back to the proof of (42). Plugging the convergences (43)–(48) into (16) with $T_0=T$ yields

(49)

Recall that ${C_0^\infty ([0, T ))\otimes [C_{\Omega }^\infty \times C_{{\Gamma _{}}}^\infty ]}$ is dense in . Owing to “Appendix A.3”, we infer from (40) that , that (where is the adjoint of ), and that, for any ,

Note that the duality product between and is taken respective to the measure , and remember the abuse of notation (57). Apply this to . Recalling that , we have and thus

(50)

[19, Lemma 3.6] establishes a temporal integration-by-parts property by using arguments purely based on the time variable, and that can easily be adapted to our context, even considering the “combined” time derivatives and the heterogeneities of the media treated here—i.e. the presence of , see assumptions in Sect. 2.2. This adaptation yields

and

Plugging these relations into (50) and using (49) concludes the proof of (42).

Third step: conclusion.

As in the first step, take and set . Developing the monotonicity property (37) of , integrating over ${(0,T)\times {\Gamma _{{\mathfrak {a}}}}}$ and summing over yields

Use (38), (39) and (11) to pass to the limit in the second and third integral terms:

Use (42) and the density of the tensorial function spaces ${C_0^\infty ([0, T ))\otimes [C_{\Omega }^\infty \times C_{{\Gamma _{}}}^\infty ]}$ in (cf. [11, proposition 2.3]) to obtain

for all . The conclusion is now standard in the Minty trick (see e.g. [21, Proof of Theorem 3.34]): for any smooth , choose and let ${\epsilon \rightarrow 0}$ to derive (41) and conclude the proof. $\square$

5 Two-phase flow test cases

We present in this section a series of test cases for two-phase flow through a fractured 2 dimensional reservoir of geometry as shown in Fig. 3. The domain ${\Omega }$ is of extension $(0,10)\mathrm {m}\times (0,20)\mathrm {m}$ and the fracture width $d_f$ is assumed constant equal to $1\,{\mathrm {cm}}$. We consider isotropic permeability in the matrix and in the fracture. The following geological configuration is considered: the matrix and fracture permeabilities are Darcy and $\iff \lambda = 100$ Darcy, respectively; the matrix and fracture porosities are and $\iff \phi = 0.4$, respectively.

Initially, the reservoir is saturated with water (density $\rho ^2 = 1000\ {\mathrm {kg}}/\mathrm {m}^3$, viscosity $\kappa ^2 = 0.001\,{\mathrm {Pa.s}}$) and oil (density $\rho ^1 = 700\ {\mathrm {kg}}/\mathrm {m}^3$, viscosity $\kappa ^1 = 0.005\,{\mathrm {Pa.s}}$) is injected from below. Also, hydrostatic distribution of pressure is assumed. The oil then rises by gravity, thanks to its lower density compared to water. At the lower boundary of the domain, we impose constant capillary pressure of 0.1 bar and water pressure of 3 bar; at the upper boundary, the capillary pressure is constant equal to 0 bar and the water pressure is 1 bar. Elsewhere, homogeneous Neumann conditions are imposed.

Fig. 3

Geometry of the reservoir under consideration. Fracture in red and matrix domain in blue. ${\Omega }= (0,10)\hbox {m}\times (0,20)\hbox {m}$ and $d_f = 0.01\hbox {m}$

We use the VAG scheme to obtain solutions for the DFM. We refer to [11] for a presentation of the scheme as a gradient scheme, and for proofs that, under standard regularity assumptions on the meshes, the corresponding sequences of gradient discretisations are coercive, GD-consistent, limit-conforming and compact. The tests are driven on a triangular mesh extended to a 3D mesh with one layer of prisms (we use a 3D implementation of the VAG scheme). The resulting numbers of cells and degrees of freedom are exhibited in Table 1. The mesh size is of order $10 d_f$.

The non-linear system of equations occurring at each time step is solved via a Newton algorithm with relaxation. To solve the linear system obtained at each step of the Newton iteration, we use the sequential version of the SuperLU direct sparse solver [15, 16]. The stopping criterion on the $L^1$ relative residual is ${\mathrm{crit}}_{\mathrm{Newton}}^{\mathrm{rel}}$. To ensure well defined values for the capillary pressure, after each Newton iteration, we project the (oil) saturation on the interval $[0,1-10^{-14}]$. The time stepping is progressive, i.e. after each iteration, the upcoming time step is deduced by multiplying the previous one by 2, while imposing a maximal time step $\Delta t_{max}$. If at a given time iteration the Newton algorithm does not converge after 35 iterations, then the actual time step is divided by 4 and the time iteration is repeated. The number of time step failures at the end of a simulation is indicated by $\mathbf N _{\mathrm{Chop}}$.

Table 1 Nb Cells is the number of cells of the mesh; Nb DOF is the number of discrete unknowns; Nb DOF el. is the number of discrete unknowns after elimination of cell unknowns without fill-in. Time steps used in the simulations in days (d)

Inside the matrix domain the capillary pressure function is given by Corey’s law $p_m = -a_m\log (1-S_m)$ with $a_m = 1$ bar. Inside the fracture network, we suppose $p_f = -a_f\log (1-S_f)$ with $a_f = 0.02$ bar. The matrix and fracture relative permeabilities of each phase $\alpha$ are given by Corey’s laws and , and the phase mobilities are defined by , (see Fig. 4). The phase saturations at the interfacial layers are defined by the interpolation

(51)

with parameter ${\theta \in [0,1]}$. The mapping is a diffeomorphism so the choice

is valid, since this function can be written as with . Finally, the interfacial porosity is set to 0.2 and

with parameter $\varepsilon >0$. The parameter $\eta$ is then defined by

Let us start with some remarks. From the capillary pressure functions (cf. Fig. 4), it is obvious that for given p, the one-sided jump of the oil saturation is negative, i.e.

(52)

To account for the interfacial zone properly, the mobilities have to be adjusted by choosing the model parameter $\theta$ depending on the rock type characteristics of the layer. Obviously, $\theta = 0$ refers to a fracture rock type and $\theta = 1$ to a matrix rock type.

On the other hand, with larger $\eta$, the volume of the interfacial layers gets augmented and the interfacial accumulation terms play a more important role. The availability of the supplementary volume has a direct impact on the phase front speed inside the fracture during its filling: (51)–(52) show that the volume of oil in the interfacial layers is strictly decreasing as a function of $\theta$, given a distribution of capillary pressures. This indicates that, from the accumulation point of view, the fracture front speed should grow with growing $\theta$, and this effect should be enhanced by a larger $\eta$.

Fig. 4

Curves for capillary pressures and relative permeabilities

Fig. 5

Fracture oil saturation for time $t = 6\text {h}$

Figure 5a indicates that, for a fixed $\theta = 0, 0.5, 1$, the solutions are not sensitive to small variations of $\varepsilon$. Quantitatively, we see that the solution for $\varepsilon =0.1$ is close to the solution for $\varepsilon = 10^{-6}$. With respect to the computational performance exposed in Table 2, we thus see that choosing $\varepsilon =0.1$ is a good compromise between accuracy and cost. This point is presented in more detail for the intermediate rock type, i.e. $\theta =0.5$, in Fig. 6. Figure 5b confirms the aforementioned feature of extended (large $\varepsilon$) interfacial layers to delay the propagation of the oil in the drain. As suggested, this effect is even more important, with decreasing $\theta$. In Fig. 5c, we study the impact of the choice of the interfacial mobility for parameters $\theta = 0, 0.5, 1$ on the solution. Here, the interfacial accumulation is negligible due to an $\varepsilon$ close to zero. Let us remark that in the limit of a vanishing interfacial layer, i.e. $\eta =0$, we aim at recovering the fracture mobilities for the mass exchange fluxes between the matrix-fracture interface and the fracture. Hence, in this case, the right choice of $\theta$ would be 0. We observe that changing the mobilities does not much influence the solution, due to the fact that fluxes are mostly oriented from the fracture towards the interfacial layers. The regions where a difference is observed in the fracture oil front for the different models are those with a small positive oil saturation. There, the relative permeabilities for $\theta = 0$ and $\theta = 0.5$ are very close and the difference to $\theta = 1$ is at its peak; this explains the behaviour of the fracture front for the three models.

Table 2 Computational cost

Fig. 6

Volume occupied by oil in the matrix, fracture and oil volume normalised by $\varepsilon$ in the interfacial layers, for $\theta = 0.5$, as a function of time

Table 2 shows that the computational cost increases with decreasing $\varepsilon$ and that, in the case of $\varepsilon =0$, the Jacobian becomes singular. Furthermore, the efficiency severely deteriorates for $\theta = 1$. In this case, $S_{\mathfrak {a}}'(p)$ is (significantly) smaller during the filling of the fracture (for capillary pressures p below a characteristic ), since $S_m'(p)\ll S_f'(p)$. When oil fluxes oriented from the fracture to the interface are present, the Jacobian is thus ill-conditioned.

6 Conclusion

We introduced a new discrete fracture matrix model for two phase Darcy flow, permitting pressure discontinuity at the matrix-fracture interfaces. It respects the heterogeneities of the media and between the matrix and the fractures, since it takes into account saturation jumps due to different capillary pressure curves in the respective domains. It also considers damaged layers located at the matrix-fracture interfaces. Another feature of the model are upwind fluxes between these interfacial layers and the fractures. The upwinding is needed for transport dominated flow in normal direction to the fractures. The extension to gravity is straightforward (cf. [12]).

We developed the numerical analysis of the model in the framework of the gradient discretisation method, which contains for example the VAG and HMM schemes. Based on compactness arguments, we showed in Theorem 4.1 the strong $L^2$ convergence of the saturations and the weak $L^2$ and $H^1$ convergences for the pressures to a solution of Model (1). In Theorem 4.13, we established uniform-in-time, weak $L^2$ in space convergence for the saturations, a result that is extended to uniform-in-time, strong $L^2$ in space convergence in [26].

Finally, we presented a series of test cases, with the objective to study the impact of the interfacial layer on the solution. The observed behaviour of the solutions for the different situations corresponds to the expectations. It exhibits significant differences, during the filling of the fracture, for large interfacial layers and small differences for small layers. In terms of computational cost, we saw that the presence of a damaged zone at the matrix-fracture interface is needed in order to solve the linear system of the discrete problem, occurring at each time step. We also observed that for a large contrast between the drain’s and the interfacial layer’s capillary pressures, the simulation becomes expensive. Therefore, we see that, in order to cope with both, fractures acting as drains or as barriers, the possibility to deal with mixed rock types for the damaged zone is essential.

A Appendix

1.1 A.1 Uniform-in-time weak $L^2$ convergence

Let A be a subset of ${{\mathbb {R}}^n}$, endowed with the standard Lebesgue measure, and $\{\varphi _\ell \,:\,\ell \in {\mathbb {N}}\}$ be a dense countable set in $L^2(A)$. On any bounded ball of $L^2(A)$, the weak topology can be defined by the following distance:

A sequence ${(v_m)_{m\in {\mathbb {N}}}}$ of bounded functions $[0,T]\rightarrow L^2(A)$ converges uniformly on [0, T] weakly in ${L^2(A)}$ to some v if it converges uniformly for the weak topology of ${L^2(A)}$, meaning that, for all ${\phi \in L^2(A)}$, ${\langle v_m(\cdot ),\phi \rangle _{L^2(A)} \rightarrow \langle v(\cdot ),\phi \rangle _{L^2(A)}}$ uniformly on [0, T] as ${m\rightarrow \infty }$.

With this introductory material, the following result is a consequence of [21, Theorem 4.26] or [19, Theorem 6.2] (see also the reasoning at the end of [19, Proof of Theorem 3.1]).

Theorem A.1

(Discontinuous weak ${L^2}$ Ascoli–Arzela theorem) Let ${\mathcal {R}}$ be a dense subset of ${L^2(A)}$ and $(v_m)_{m\in {\mathbb {N}}}$ be a sequence of functions ${[0,T]\rightarrow L^2(A)}$ such that

• $\sup _{m\in {\mathbb {N}}}\sup _{t\in [0,T]}\Vert v_m(t)\Vert _{L^2(A)}<+\infty$,

• for all $\varphi \in \mathcal {R}$, there exist $\omega _\varphi :[0,T]^2\rightarrow [0,\infty )$ and $(\delta _m(\varphi ))_{m\in {\mathbb {N}}}\subset [0,\infty )$ satisfying

  $$\begin{aligned}&\omega _\varphi (s,t)\rightarrow 0 \text{ as } s-t\rightarrow 0\,,\;\delta _m(\varphi )\rightarrow 0 \text{ as } m\rightarrow \infty \,, \text{ and } \\&\forall (s,t)\in [0,T]^2\,,\;\forall m\in {\mathbb {N}}\,,\; |\langle v_m(s)-v_m(t),\varphi \rangle _{L^2(A)}|\le \delta _m(\varphi )+\omega _\varphi (s,t). \end{aligned}$$

Then, there exists a function $v:[0,T]\rightarrow L^2(A)$ such that, up to a subsequence as $m\rightarrow \infty$, $v_m\rightarrow v$ uniformly on [0, T] weakly in ${L^2(A)}$. Moreover, v is continuous on [0, T] for the weak topology of ${L^2(A)}$.

1.2 A.2 Generic results on gradient discretisations

The following lemma is a classical result in the context of the standard gradient discretisation method, see e.g. [21, Lemma 4.7]. We give a sketch of its proof for gradient discretisations adapted to discrete fracture matrix model.

Lemma A.2

(Regularity of the limit) Let ${(\mathcal {D}^l)_{l\in {\mathbb {N}}}}$ be a coercive and limit-conforming sequence of gradient discretisations, and let be such that , where $N_l$ is the number of time steps of $\mathcal {D}^l$. We assume that is bounded. Then, there exists such that, up to a subsequence, the following weak convergences hold:

(53)

Proof

By coercivity and since is bounded, all the sequences in (53) are bounded in their respective spaces. Up to a subsequence, we can therefore assume that there exists , , and such that , , and weakly in their respective $L^2$ spaces as $l\rightarrow \infty$.

Take , and $\rho \in C^\infty _0(0,T)$. For F a function of $\mathbf x$, set $(\rho \otimes F)(t,\mathbf x )=\rho (t)F(\mathbf x )$. The definition of (see Definition 3.6) yields

The limit-conformity shows that the right-hand side of this inequality tends to 0. Hence,

Applying this to with , and using the density of tensorial functions in (see [17, Appendix D]) shows that . With where , we obtain . Considering now with and applying the divergence theorem gives . Finally, taking with a general yields . $\square$

With [29, Lemma 3.6], we can state the following.

Corollary A.3

Under the assumptions of Lemma A.2, if and are continuous, non-decreasing functions and if strongly converges in ${{L^2((0,T)\times {M_\mu })}}$ and strongly converges in , then

1.3 A.3 Identification of time derivatives

We discuss here how weak formulations, with derivatives on test functions, enable us to recover some regularity properties on time derivatives of quantities of interest.

Let us start with a classical situation, similar to [19, Remark 1.1]. Let ${(M,\nu )}$ be a measured space and E be a Banach space densely embedded in ${L^2(M)}$, so that ${E\hookrightarrow L^2(M)\hookrightarrow E'}$. Assume also that ${E'}$ is separable. Let ${{\mathcal {L}}:L^2(0,T;E)\rightarrow {\mathbb {R}}}$ be a continuous linear form and let ${{\mathcal {E}}\subset C^1_0([0,T);E)}$ be such that ${\mathcal {E}}_0=\{\Phi \in {\mathcal {E}}\,:\, \Phi (0,\cdot )=0\}$ is dense in ${L^2(0,T;E)}$. Suppose that ${U\in L^2(0,T;E)}$ and ${U_0\in L^2(M)}$ satisfy, for all ${\Phi \in {\mathcal {E}}}$,

$$\begin{aligned} -\int _0^T \int _M U(t,\mathbf x )\partial _t\Phi (t,\mathbf x )\mathrm{d}\nu (\mathbf x )\mathrm{d}t + \int _M U_0(\mathbf x )\Phi (0,\mathbf x )\mathrm{d}\nu (\mathbf x ) ={\mathcal {L}}(\Phi ). \end{aligned}$$

(54)

This relation shows that

$$\begin{aligned} \Xi :\Phi \mapsto -\int _0^T \int _M U(t,\mathbf x )\partial _t\Phi (t,\mathbf x )\mathrm{d}\nu (\mathbf x )\mathrm{d}t \end{aligned}$$

is linear (equal to ${\mathcal {L}}$) on ${\mathcal {E}}_0$, and continuous for the topology of ${L^2(0,T;E)}$. By density of ${{\mathcal {E}}_0}$ in this space, ${\Xi }$ can be extended into an element of ${(L^2(0,T;E))'=L^2(0,T;E')}$ (see [17, Theorem 1.4.1]). We denote this element by ${\partial _t U}$, as it clearly corresponds to the distributional derivative of U [17, Section 2.1.2]. By [17, Section 2.5.2] this shows that ${U:[0,T]\rightarrow L^2(M)}$ is continuous and, using [17, Proposition 2.5.2] to integrate by parts in (54), that ${U(0)=U_0}$ and

$$\begin{aligned} \forall \Phi \in {\mathcal {E}}\,,\; \langle \partial _t U,\Phi \rangle _{L^2(0,T;E'),L^2(0,T;E)}\mathrm{d}t= \int _0^T \langle \partial _t U(t),\Phi (t)\rangle _{E',E}\mathrm{d}t ={\mathcal {L}}(\Phi ). \end{aligned}$$

(55)

By density of ${{\mathcal {E}}}$ in ${L^2(0,T;E)}$, this relation actually holds for any ${\Phi \in L^2(0,T;E)}$.

We now consider the setting in the proof of Theorem 4.1 (see Section 4.3). Fixing ${M=M_f}$, ${\mathrm{d}\nu =\mathrm{d}\tau _f}$, ${E=V_f^0}$, ${{\mathcal {E}}=C^1([0,T];C^\infty _\Gamma )}$ and

and using (40) with and , for ${\alpha ,\beta =1,2}$ with ${\alpha \ne \beta }$, this identifies as an element of .

Let us now deal with a slightly more complicated case, in which the time derivatives of two functions need to be combined to exhibit a certain regularity. With the same M and E as above, take ${(N,\lambda )}$ a measured space and ${\gamma :E\rightarrow L^2(N)}$ a continuous linear mapping. Assume that ${U\in L^2(0,T;E)}$, ${V\in L^2(0,T;L^2(N))}$, ${U_0\in L^2(M)}$ and ${V_0\in L^2(N)}$, satisfy, for all ${\phi \in {\mathcal {E}}}$,

$$\begin{aligned} \begin{aligned} -\int _0^T \int _M U(t,\mathbf x )\partial _t{}&\Phi (t,\mathbf x )\mathrm{d}\nu (\mathbf x )\mathrm{d}t -\int _0^T \int _N V(t,\mathbf x )\partial _t\gamma (\Phi (t))(\mathbf x )\mathrm{d}\lambda (\mathbf x )\mathrm{d}t\\ {}&+ \int _M U_0(\mathbf x )\Phi (0,\mathbf x )\mathrm{d}\nu (\mathbf x )+ \int _N V_0(\mathbf x )\gamma (\Phi (0))(\mathbf x )\mathrm{d}\lambda (\mathbf x ) ={\mathcal {L}}(\Phi ). \end{aligned} \end{aligned}$$

(56)

The same reasoning as above shows that

$$\begin{aligned} \widetilde{\Xi }:\Phi \mapsto -\int _0^T \int _M U(t,\mathbf x )\partial _t\Phi (t,\mathbf x )\mathrm{d}\nu (\mathbf x )\mathrm{d}t -\int _0^T \int _N V(t,\mathbf x )\partial _t\gamma (\Phi (t))(\mathbf x )\mathrm{d}\lambda (\mathbf x )\mathrm{d}t \end{aligned}$$

can be extended into a linear continuous form on ${L^2(0,T;E)}$. Letting ${\gamma ^*: L^2(N)\rightarrow E'}$ be the adjoint of ${\gamma }$ (that is, ${\langle g,\gamma (\Phi )\rangle _{L^2(N)}=\langle \gamma ^*g,\Phi \rangle _{E',E}}$ for all ${g\in L^2(N)}$ and ${\Phi \in E}$), the form ${\widetilde{\Xi }}$ is naturally denoted by ${\partial _t U + \gamma ^* \partial _t V}$. Note that, in this sum, the two terms cannot be separated and it cannot, for example, be asserted that ${\partial _t U\in L^2(0,T;E')}$ and ${\gamma ^*\partial _t V\in L^2(0,T;E')}$. Then, a reasoning similar to the one in [17] shows that ${U+\gamma ^* V:[0,T]\rightarrow L^2(M)}$ is continuous with value ${U_0+\gamma ^*V_0}$ at ${t=0}$, and that, for all ${\Phi \in L^2(0,T;E)}$,

$$\begin{aligned} {\langle \partial _t U+\gamma ^*\partial _t V,\Phi \rangle _{L^2(0,T;E'),L^2(0,T;E)} ={\mathcal {L}}(\Phi ).} \end{aligned}$$

To write more natural equations, in the paper we sometimes make an abuse of notation and separate the two derivatives. We then write

$$\begin{aligned} \langle \partial _t U+\gamma ^*\partial _t V,\Phi \rangle _{L^2(0,T;E'),L^2(0,T;E)}= & {} \int _0^T\langle \partial _t U,\Phi \rangle \mathrm{d}t+ \int _0^T\langle \gamma ^*\partial _t V,\Phi \rangle \mathrm{d}t\nonumber \\= & {} \int _0^T\langle \partial _t U,\Phi \rangle \mathrm{d}t+ \int _0^T\langle \partial _t V,\gamma \Phi \rangle \mathrm{d}t,\nonumber \\ \end{aligned}$$

(57)

where, in the right-hand side, the duality brackets do not have indices, to avoid claiming that ${\partial _t U\in L^2(0,T;E')}$ or ${\gamma ^*\partial _t V\in L^2(0,T;E')}$, and to remember that these two terms must be understood together.

Used in (40) with ${\gamma =\gamma _{\mathfrak {a}}}$ for all ${{\mathfrak {a}}\in \chi }$, the above reasoning and notations enable us to identify the (combined) time derivatives of and as an element of .