Two-phase geothermal model with fracture network and multi-branch wells

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Two-phase geothermal model with fracture network and multi-branch wells

Antoine Armandine Les Landes, Daniel Castanon Quiroz, Laurent Jeannin, Simon Lopez, Roland Masson

To cite this version:

Antoine Armandine Les Landes, Daniel Castanon Quiroz, Laurent Jeannin, Simon Lopez, Roland

Masson. Two-phase geothermal model with fracture network and multi-branch wells. 2021. �hal-

03273589�

Two-phase geothermal model with fracture network and multi-branch wells

A. Armandine Les Landes ^∗ , D. Castanon Quiroz ^† , L. Jeannin ^‡ , S. Lopez ^§ , R. Masson ^¶ June 29, 2021

Abstract

This paper focuses on the numerical simulation of geothermal systems in complex geological settings. The physical model is based on two-phase Darcy flows coupling the mass conservation of the water component with the energy conservation and the liquid vapor thermodynamical equilibrium. The discretization exploits the flexibility of unstructured meshes to model complex geology including conductive faults as well as complex wells. The polytopal and essentially nodal Vertex Approximate Gradient scheme is used for the approximation of the Darcy and Fourier fluxes combined with a Control Volume approach for the transport of mass and energy.

Particular attention is paid to the faults which are modelled as two-dimensional interfaces defined as collection of faces of the mesh and to the flow inside deviated or multi-branch wells defined as collection of edges of the mesh with rooted tree data structure. By using an explicit pressure drop calculation, the well model reduces to a single equation based on complementarity constraints with only one well implicit unknown. The coupled systems are solved fully implicitely at each time step using efficient nonlinear and linear solvers on parallel distributed architectures.

The convergence of the discrete model is investigated numerically on a simple test case with a Cartesian geometry and a single vertical producer well. Then, the ability of our approach to deal efficiently with realistic test cases is assessed on a high energy faulted geothermal reservoir operated using a doublet of two deviated wells.

1 Introduction

Deep geothermal systems are often located in complex geological settings, including faults or fractures. These geological discontinuities not only control fluid flow and heat transfer, but also provide feed zones for production wells. Modeling the operation of geothermal fields and the exchange of fluids and heat in the rock mass during production requires explicitly taking into account objects of different characteristic sizes such as the reservoir itself, faults and fractures, which have a small thickness compared to the characteristic size of geological formations and wells (whose radius is of the order of a few tens of centimeters).

A common way to account for these highly constrated spatial scales is based on a reduction of dimension both for the fault/fracture and the well models. Following [23, 4, 10, 21, 31, 37,

∗

BRGM, 3 avenue Claude-Guillemin, BP 36009, 45060 Orl´ eans Cedex 2, France, A.ArmandineLesLandes@brgm.fr

†

Universit´ e Cˆ ote d’Azur, Inria, CNRS, LJAD, UMR 7351 CNRS, team Coffee, Parc Valrose 06108 Nice Cedex 02, France, danielcq.mathematics@gmail.com

‡

STORENGY, 12 rue Raoul Nordling - Djinn - CS 70001 92274 Bois Colombes Cedex, France, lau- rent.jeannin@storengy.com

§

BRGM, 3 avenue Claude-Guillemin, BP 36009, 45060 Orl´ eans Cedex 2, France, s.lopez@brgm.fr

¶

Universit´ e Cˆ ote d’Azur, Inria, CNRS, LJAD, UMR 7351 CNRS, team Coffee, Parc Valrose 06108 Nice Cedex 02,

France, roland.masson@unice.fr

26, 5, 11, 15, 34] faults/fractures will be represented as co-dimension one manifolds coupled with the surrounding matrix domain leading to the so-called hybrid-dimensional or Discrete Fracture Matrix (DFM) models. This reduction of dimension is obtained by averaging both the equations and unknowns in the fracture width and using appropriate transmission conditions at matrix fracture interfaces. In our case, the faults/fractures will be assumed to be conductive both in terms of permeability and thermal conductivity in such a way that pressure and temperature continuity can be assumed as matrix fracture transmission conditions [4, 10, 37]. This setting has been extended to two-phase Darcy flows in [12, 13] and to multi-phase compositional non-isothermal Darcy flows in [45].

The well will be modelled as a line source defined by a 1D graph with tree structure. It will be coupled to the 3D matrix domain and to the 2D faults/fractures possibly intersecting the well using Peaceman’s approach. It is a widely used approach in reservoir simulation for which the Darcy or Fourier fluxes between the reservoir and the well are discretized by a two-point flux approximation with a transmissivity accounting for the unresolved pressure or temperature singularity. This leads to the concept of well or Peaceman’s index defined at the discrete level and depending on the type of cell, on the well radius and geometry and on the scheme used for the discretization. Let us refer to [35] for its introduction in the framework of a two-point cell-centered finite volume scheme on square cells, to [36] for its extension to non square cells and anisotropic permeability field and to [43, 1, 17]

for extensions to more general well geometries and different discretizations. The coupling with the faults/fractures is considered in [9]. Let us also refer to [22] for a related approach also based on a removal of the singularity induced by the well line source but at the continuous level.

This paper focuses on the liquid vapor single water component non-isothermal Darcy flow model based on mass and energy conservation equations coupled with thermodynamical equilibrium and volume balance. The extension to hybrid-dimensional models follows [45] with pressure and temper- ature continuity at matrix fracture interfaces. The thermal well model is a simplified version of the drift flux model [30, 41] neglecting transient terms, thermal losses and cross flow in the sense that all along the well, the well behaves either as a production or an injection well. It results that using an explicit approximation of the mixture density along the well, the well model can be reduced to a single unknown, the so-called bottom hole pressure, implicitely coupled to the reservoir.

The discretization of hybrid-dimensional Darcy flow models has been the object of many works using cell-centered Finite Volume schemes with either Two Point or Multi Point Flux Approximations [27, 5, 24, 42, 38, 2, 3], Mixed or Mixed Hybrid Finite Element methods [4, 31, 26], Hybrid Mimetic Mixed Methods [20, 6, 11, 15], and Control Volume Finite Element Methods (CVFE) [10, 37, 33, 24, 32]. This article focus on the Vertex Approximate Gradient (VAG) scheme accounting for polyhedral meshes. It has been introduced for the discretization of multiphase Darcy flows in [19] and extended to hybrid-dimensional models in [12, 11, 44, 15, 45, 16, 14].

The VAG scheme uses nodal and fracture face unknowns in addition to the cell unknowns which can be eliminated without any fill-in. Thanks to its essentially nodal nature, it leads to a sparse discretization on tetrahedral meshes which are particulary adapted to discretize complex geological features like faults defined as a collection of faces and slanted or multi-branch wells defined as a col- lection of edges with tree structure. Compared with other nodal approaches such as CVFE methods, the VAG scheme avoid the mixing of the control volumes at the matrix fracture interfaces, which is a key feature for its coupling with a transport model. As shown in [12] for two-phase flow problems, this allows to use a coarser mesh size at the matrix fracture interface.

The remainder of this paper is organized as follows. Section 2 presents the physical model

describing the flow and transport in the matrix domain coupled to the fracture/fault network in

the hybrid-dimensional setting. Section 3 presents the VAG discretization of this liquid vapor non- isothermal hybrid-dimensional model. It is based on the discrete mass and energy conservations on each control volume coupled with thermodynamical equilibrium and the sum to one of the saturations.

Then, the well modelling is addressed starting with the description of the well geometry as a collection of edges defining a rooted tree data structure. The source terms connecting the well to the reservoir at each well node are based on two-point fluxes with transmissivities defined by Peaceman’s indexes.

The derivation of the simplified well model is detailed both for production and injection wells starting from the drift flux model. We discuss at the end of Section 3 the algorithms used to solve the nonlinear and linear systems on distributed parallel architectures at each time step of the simulation. Finally, to demonstrate the efficiency of our approach, we present in Section 4 two numerical tests. The first test case checks the numerical convergence of the model for a vertical production well connected to an homogeneous reservoir on a family of refined Cartesian meshes. The second test case simulates the development plan of a high enthalpy faulted geothermal reservoir with slanted production and injection wells.

2 Hybrid-dimensional non-isothermal two-phase Discrete Frac- ture Model

This section recalls, in the particular case of a non-isothermal single-component two-phase Darcy flow model, the hybrid-dimensional model introduced in [45].

2.1 Discrete Fracture Network

Let Ω denote a bounded domain of R

³

assumed to be polyhedral. Following [4, 21, 31, 11, 15] the fractures are represented as interfaces of codimension 1. Let J be a finite set and let Γ = S

j∈J

Γ

_j

and its interior Γ = Γ \ ∂ Γ denote the network of fractures Γ

j

⊂ Ω, j ∈ J , such that each Γ

j

is a planar polygonal simply connected open domain included in a plane of R

³

. The fracture width is

Figure 1: Example of a 2D domain with 3 intersecting fractures Γ

₁

, Γ

₂

, Γ

₃

.

denoted by d

_f

and is such that 0 < d

_f

≤ d

_f

(x) ≤ d

_f

for all x ∈ Γ. We can define, for each fracture

j ∈ J , its two sides + and −. For scalar functions on Ω, possibly discontinuous at the interface Γ

(typically in H

¹

(Ω \ Γ)), we denote by γ

^±

the trace operators on the side ± of Γ. Continuous scalar

functions u at the interface Γ (typically in H

¹

(Ω)) are such that γ

⁺

u = γ

⁻

u and we denote by γ the

trace operator on Γ for such functions. At almost every point of the fracture network, we denote by

n

^±

the unit normal vector oriented outward to the side ± of Γ such that n

⁺

+ n

⁻

= 0. For vector

fields on Ω, possibly discontinuous at the interface Γ (typically in H div(Ω \ Γ), we denote by γ

_n^±

the normal trace operator on the side ± of Γ oriented w.r.t. n

^±

.

The gradient operator in the matrix domain Ω \ Γ is denoted by ∇ and the tangential gradient operator on the fracture network is denoted by ∇

τ

such that

∇

_τ

u = ∇u − (∇u · n

⁺

)n

⁺

.

We also denote by div

_τ

the tangential divergence operator on the fracture network, and by dτ (x) the Lebesgue measure on Γ.

We denote by Σ the dimension 1 open set defined by the intersection of the fractures excluding the boundary of the domain Ω, i.e. the interior of S

{(j,j⁰)∈J×J|j6=j⁰}

∂Γ

_j

∩ ∂Γ

_j⁰

\ ∂ Ω.

For the matrix domain, Dirichlet (subscript D) and Neumann (subscript N ) boundary conditions are imposed on the two dimensional open sets ∂Ω

_D

and ∂Ω

_N

respectively where ∂ Ω

_D

∩ ∂Ω

_N

= ∅,

∂ Ω = ∂Ω

_D

∪∂Ω

_N

. Similarly for the fracture network, the Dirichlet and Neumann boundary conditions are imposed on the one dimensional open sets ∂Γ

_D

and ∂Γ

_N

respectively where ∂Γ

_D

∩ ∂Γ

_N

= ∅,

∂ Γ ∩ ∂Ω = ∂Γ

_D

∪ ∂Γ

_N

.

2.2 Non-isothermal two-phase flow model

We consider in this work a two-phase liquid gas, single water component, and non-isothermal Darcy flow model. The liquid (`) and gas (g) phases are described by their pressure p (neglecting capillary effects), temperature T and pore volume fractions or saturations s

^α

, α ∈ {`, g}. Let us also introduce the mass fraction c

^α

of the water component in phase α, equal to 1 for a present phase α but lower than 1 for an absent phase. It will be used below to express the thermodynamical equilibrium as complementary constraints.

For each phase α, we denote by ρ

^α

(p, T ) its mass density, by µ

^α

(p, T ) its dynamic viscosity, by e

^α

(p, T ) its specific internal energy, and by h

^α

(p, T ) its specific enthalpy. The rock energy density is denoted by E

_r

(p, T ).

The reduction of dimension in the fractures leading to the hybrid-dimensional model is obtained by integration of the conservation equations along the width of the fractures complemented by transmission conditions at both sides of the matrix fracture interfaces (see [45]). In the following, p

m

, T

m

, s

^α_m

, c

^α_m

denote the pressure, temperature, saturations, and mass fractions in the matrix do- main Ω \ Γ, and p

_f

, T

_f

, s

^α_f

, c

^α_f

are the pressure, temperature, saturations and mass fractions in the fractures averaged along the width of the fractures. The permeability tensor is denoted by K

_m

in the matrix domain and we denote by K

f

the tangential permeability tensor in the fractures (average value along the fracture width assuming that the permeability tensor in the fracture has the normal as principal direction). The porosity (resp. thermal conductivity of the rock and fluid mixture) is denoted by φ

m

(resp. λ

m

) in the matrix domain and by φ

f

(resp. λ

f

) along the fracture network (average values along the fracture width). The relative permeability of phase α as a function of the phase saturation is denoted by k

_r,m^α

in the matrix and by k

^α_r,f

in the fracture network. The gravity acceleration vector is denoted by g.

The set of equations couples the mass, energy and volume balance equations in the matrix



 

 

 

 

 

 

 

  φ

m

∂

t

X

α∈{`,g}

ρ

^α

(p

m

, T

m

)s

^α_m

c

^α_m

+ div(q

^h_m^2o

) = 0, φ

_m

∂

_t

X

α∈{`,g}

ρ

^α

(p

_m

, T

_m

)e

^α

(p

_m

, T

_m

)s

^α_m

c

^α_m

+ (1 − φ

_m

)∂

_t

E

_r

(p

_m

, T

_m

) + div(q

^e_m

) = 0, X

α∈{`,g}

s

^α_m

= 1,

(1)

in the fracture network



 

 

 

 

 

 

 

 

 

 

d

f

φ

f

∂

t

X

α∈{`,g}

ρ

^α

(p

f

, T

f

)s

^α_f

c

^α_f

+ div

τ

(q

^h_f^2o

) − γ

_n⁺

q

^h_m^2o

− γ

_n⁻

q

^h_m^2o

= 0, d

_f

φ

_f

∂

_t

X

α∈{`,g}

ρ

^α

(p

_f

, T

_f

)e

^α

(p

_f

, T

_f

)s

^α_f

c

^α_f

+ d

_f

(1 − φ

_f

)∂

_t

E

_r

(p

_f

, T

_f

) + div

τ

(q

^e_f

) − γ

_n⁺

q

^e_m

− γ

_n⁻

q

^e_m

= 0, X

α∈{`,g}

s

^α_f

= 1,

(2)

with the thermodynamical equilibrium for i = m, f



 

 

 

 

c

^g_i

p

_i

− p

_sat

(T

_i

)c

^`_i

= 0, min

s

^`_i

, 1 − c

^`_i

= 0, min

s

^g_i

, 1 − c

^g_i

= 0,

(3)

where p

_sat

(T ) is the vapor saturated pressure as a function of the temperature T . The Darcy and Fourier laws provide the mass and energy fluxes in the matrix

q

^h_m^2o

= X

α∈{`,g}

q

^α_m

, q

^α_m

= c

^α_m

ρ

^α

(p

_m

, T

_m

)

µ

^α

(p

_m

, T

_m

) k

_r,m^α

(s

^α_m

)V

_m^α

, q

^e_m

= X

α∈{`,g}

h

^α

(p

m

, T

m

)q

^α_m

− λ

m

∇T

m

,

(4)

and in the fracture network

q

^h_f^2o

= X

α∈{`,g}

q

^α_f

, q

^α_f

= c

^α_f

ρ

^α

(p

_f

, T

_f

)

µ

^α

(p

_f

, T

_f

) k

_r,f^α

(s

^α_f

)V

_f^α

, q

^e_f

= X

α∈{`,g}

h

^α

(p

_f

, T

_f

)q

^α_f

− d

_f

λ

_f

∇

_τ

T

_f

,

(5)

where

V

^α_m

= −K

_m

∇p

_m

− ρ

^α

(p

_m

, T

_m

)g

, V

^α_f

= −d

_f

K

_f

∇

_τ

p

_f

− ρ

^α

(p

_f

, T

_f

)g

_τ

, and g

_τ

= g − (g · n

⁺

)n

⁺

.

The system (1)-(2)-(4)-(5) is closed with transmission conditions at the matrix fracture interface Γ. These conditions state the continuity of the pressure and temperature at the matrix fracture interface assuming that the fractures do not act as barrier neither for the Darcy flow nor for the thermal conductivity (see [4, 21, 31, 45]).

γ

⁺

p

_m

= γ

⁻

p

_m

= γp

_m

= p

_f

,

γ

⁺

T

_m

= γ

⁻

T

_m

= γT

_m

= T

_f

. (6)

At fracture intersections Σ, note that we assume mass and energy flux conservation as well as the

continuity of the pressure p

_f

and temperature T

_f

. Homogeneous Neumann boundary conditions are

applied for the mass q

^h_f^2o

and energy q

^e_f

fluxes at the fracture tips ∂Γ \ ∂Ω.

3 VAG Finite Volume Discretization

3.1 Space and time discretizations

The VAG discretization of hybrid-dimensional two-phase Darcy flows introduced in [12] considers generalized polyhedral meshes of Ω in the spirit of [18]. Let M be the set of cells that are disjoint open polyhedral subsets of Ω such that S

K∈M

K = Ω, for all K ∈ M, x

_K

denotes the so-called

“center” of the cell K under the assumption that K is star-shaped with respect to x

_K

. The set of faces of the mesh is denoted by F and F

_K

is the set of faces of the cell K ∈ M. The set of edges of the mesh is denoted by E and E

_σ

is the set of edges of the face σ ∈ F . The set of vertices of the mesh is denoted by V and V

_σ

is the set of vertices of the face σ. For each K ∈ M we define V

_K

= S

σ∈F_K

V

_σ

.

The faces are not necessarily planar. It is just assumed that for each face σ ∈ F , there exists a so-called “center” of the face x

_σ

∈ σ \ S

a∈E_σ

a such that x

_σ

= P

s∈V_σ

β

_σ,s

x

_s

, with P

s∈V_σ

β

_σ,s

= 1, and β

_σ,s

≥ 0 for all s ∈ V

_σ

; moreover the face σ is assumed to be defined by the union of the triangles T

_σ,a

defined by the face center x

_σ

and each edge a ∈ E

_σ

. The mesh is also supposed to be conforming w.r.t. the fracture network Γ in the sense that for each j ∈ J there exists a subset F

_Γj

of F such that

Γ

_j

= [

σ∈FΓj

σ.

We will denote by F

_Γ

the set of fracture faces F

_Γ

= [

j∈J

F

_Γj

,

and by

V

_Γ

= [

σ∈F_Γ

V

_σ

,

the set of fracture nodes. This geometrical discretization of Ω and Γ is denoted in the following by D.

In addition, the following notations will be used

M

_s

= {K ∈ M | s ∈ V

_K

}, M

_σ

= {K ∈ M | σ ∈ F

_K

}, and

F

_Γ,s

= {σ ∈ F

_Γ

| s ∈ V

_σ

}.

For N

_tf

∈ N

^∗

, let us consider the time discretization t

⁰

= 0 < t

¹

< · · · < t

ⁿ⁻¹

< t

ⁿ

· · · < t

^Ntf

= t

_f

of the time interval [0, t

_f

]. We denote the time steps by ∆t

ⁿ

= t

ⁿ

− t

ⁿ⁻¹

for all n = 1, · · · , N

_tf

.

3.2 VAG fluxes and control volumes

The VAG discretization is introduced in [18] for diffusive problems on heterogeneous anisotropic media. Its extension to the hybrid-dimensional Darcy flow model is proposed in [12] based upon the following vector space of degrees of freedom:

V

D

= {v

_K

, v

_s

, v

_σ

∈ R , K ∈ M, s ∈ V , σ ∈ F

_Γ

}.

The degrees of freedom are illustrated in Figure 2 for a given cell K with one fracture face σ in bold.

The matrix degrees of freedom are defined by the set of cells M and by the set of nodes V \ V

_Γ

excluding the nodes at the matrix fracture interface Γ. The fracture faces F

_Γ

and the fracture nodes V

Γ

are shared between the matrix and the fractures but the control volumes associated with these degrees of freedom will belong to the fracture network (see Figure 3). The degrees of freedom at the fracture intersection Σ are defined by the set of nodes V

_Σ

⊂ V

_Γ

located on Σ. The set of nodes at the Dirichlet boundaries ∂Ω

D

and ∂Γ

D

is denoted by V

D

.

The VAG scheme is a control volume scheme in the sense that it results, for each non Dirichlet degree of freedom in a mass or energy balance equation. The matrix diffusion tensor is assumed to be cellwise constant and the tangential diffusion tensor in the fracture network is assumed to be facewise constant. The two main ingredients are therefore the conservative fluxes and the control volumes. The VAG matrix and fracture fluxes are illustrated in Figure 2. For u

D

∈ V

D

, the matrix fluxes F

K,ν

(u

D

) connect the cell K ∈ M to the degrees of freedom located at the boundary of K, namely ν ∈ Ξ

_K

= V

_K

∪ (F

_K

∩ F

_Γ

). The fracture fluxes F

_σ,s

(u

D

) connect each fracture face σ ∈ F

_Γ

to its nodes s ∈ V

_σ

. The expression of the matrix (resp. the fracture) fluxes is linear and local to the cell (resp. fracture face). More precisely, the matrix fluxes are given by

F

_K,ν

(u

D

) = X

ν⁰∈Ξ_K

T

_K^ν,ν0

(u

_K

− u

_ν⁰

),

with a symmetric positive definite transmissibility matrix T

_K

= (T

_K^ν,ν0

)

_(ν,ν⁰_)∈ΞK×ΞK

depending only on the cell K geometry (including the choices of x

_K

and of x

_σ

, σ ∈ F

_K

) and on the cell matrix diffusion tensor. The fracture fluxes are given by

F

_σ,s

(u

D

) = X

s∈Vσ

T

_σ^s,s0

(u

_σ

− u

_s⁰

),

with a symmetric positive definite transmissibility matrix T

σ

= (T

_σ^s,s0

)

_(s,s⁰)∈Vσ×Vσ

depending only on the fracture face σ geometry (including the choice of x

_σ

) and on the fracture face width and tangential diffusion tensor. Let us refer to [12] for a more detailed presentation and for the definition of T

K

and T

σ

.

Figure 2: For a cell K and a fracture face σ (in bold), examples of VAG degrees of freedom u

_K

, u

_s

, u

_σ

, u

_s⁰

and VAG fluxes F

_K,σ

, F

_K,s

, F

_K,s⁰

, F

_σ,s

.

The construction of the control volumes at each degree of freedom is based on partitions of the cells and of the fracture faces. These partitions are respectively denoted, for all K ∈ M, by

K = ω

_K

[



 [

s∈V_K\V_D

ω

_K,s



 ,

and, for all σ ∈ F

_Γ

, by

σ = Σ

_σ

[



 [

s∈V_σ\V_D

Σ

_σ,s



 .

The practical implementation of the scheme does not require to build explicitly the geometry of these partitions but only need to define the matrix volume fractions

α

_K,s

= R

ωK,s

dx R

K

dx , s ∈ V

_K

\ (V

_D

∪ V

_Γ

), K ∈ M, constrained to satisfy α

_K,ν

≥ 0, and P

s∈V_K\(V_D∪V_Γ)

α

_K,s

≤ 1, as well as the fracture volume fractions α

_σ,s

=

R

Σσ,s

d

_f

(x)dτ (x) R

σ

d

_f

(x)dτ(x) , s ∈ V

_σ

\ V

_D

, σ ∈ F

_Γ

, constrained to satisfy α

_σ,s

≥ 0, and P

s∈V_σ\V_D

α

_σ,s

≤ 1, where we denote by dτ (x) the 2 dimensional Lebesgue measure on Γ. Let us also set

φ

K

= (1 − X

s∈V_K\(V_D∪V_Γ)

α

K,s

) Z

K

φ

m

(x)dx for K ∈ M, and

φ

_σ

= (1 − X

s∈Vσ\V_D

α

_σ,s

) Z

σ

φ

_f

(x)d

_f

(x)dτ (x) for σ ∈ F

_Γ

, as well as

φ

_s

= X

K∈Ms

α

_K,s

Z

K

φ

_m

(x)dx for s ∈ V \ (V

_D

∪ V

_Γ

), and

φ

_s

= X

σ∈F_Γ,s

α

_σ,s

Z

σ

φ

_f

(x)d

_f

(x)dτ (x) for s ∈ V

_Γ

\ V

_D

,

which correspond to the porous volumes distributed to the degrees of freedom excluding the Dirichlet nodes. The rock complementary volume in each control volume ν ∈ M ∪ F

_Γ

∪ (V \ V

_D

) is denoted by ¯ φ

_ν

.

As shown in [12], the flexibility in the choice of the control volumes is a crucial asset, compared with usual CVFE approaches and allows to significantly improve the accuracy of the scheme when the permeability field is highly heterogeneous. As exhibited in Figure 3, as opposed to usual CVFE approaches, this flexibility allows to define the control volumes in the fractures with no contribution from the matrix in order to avoid to artificially enlarge the flow path in the fractures.

Figure 3: Example of control volumes at cells, fracture face, and nodes, in the case of two cells K

and L separated by one fracture face σ (the width of the fracture is enlarged in this figure). The

control volumes are chosen to avoid mixing fracture and matrix rocktypes.

A rocktype is assigned to each cell, node and fracture face. In our case, for cells and for nodes not located along the fractures, the matrix rocktype is assigned. For fracture nodes and faces at the inter- face between the matrix and the fracture rocktypes, the fracture rocktype is assigned corresponding to the most pervious rock type consistently with the choice of the control volumes (see [12]). For convenience’s sake, in the following, we will denote by k

_r,ν^α

the corresponding relative permeability function for ν ∈ M ∪ V ∪ F

Γ

.

In the following, we will keep the notation F

_K,s

, F

_K,σ

, F

_σ,s

for the VAG Darcy fluxes defined with the cellwise constant matrix permeability K

m

and the facewise constant fracture width d

f

and tangential permeability K

_f

. Since the rock properties are fixed, the VAG Darcy fluxes transmissibility matrices T

_K

and T

_σ

are computed only once.

The VAG Fourier fluxes are denoted in the following by G

K,s

, G

K,σ

, G

σ,s

. They are obtained with the isotropic matrix and fracture thermal conductivities averaged in each cell and in each fracture face using the previous time step fluid properties. Hence VAG Fourier fluxes transmissibility matrices need to be recomputed at each time step.

3.3 Multi-branch non-isothermal well model

Let W denote the set of wells. Each multi-branch well ω ∈ W is defined by a set of oriented edges of the mesh assumed to define a rooted tree oriented away from the root. This orientation corresponds to the drilling direction of the well. The set of nodes of a well ω ∈ W is denoted by V

_ω

⊂ V and its root node is denoted by s

_ω

. A partial ordering is defined on the set of vertices V

_ω

with s <

ω

s

⁰

if and only if the unique path from the root s

_ω

to s

⁰

passes through s. The set of edges of the well ω is denoted by E

_ω

and for each edge a ∈ E

_ω

we set a = ss

⁰

with s <

ω

s

⁰

(i.e. s is the parent node of s

⁰

, see Figure 4). It is assumed that V

_ω1

∩ V

_ω2

= ∅ for any ω

₁

, ω

₂

∈ W such that ω

₁

6= ω

₂

.

We focus on the part of the well that is connected to the reservoir through open hole, production liners or perforations. In this section, exchanges with the reservoir are dominated by convection and we decided to neglect heat losses as a first step. The latest shall be taken into account when modeling the wellbore flow up to the surface. It is assumed that the radius r

_ω

of each well ω ∈ W is small compared to the cell sizes in the neighborhood of the well. It results that the Darcy flux between the reservoir and the well at a given well node s ∈ V

_ω

is obtained using the Two Point Flux Approximation

V

_s^ω

= WI

_s

(p

_s

− p

^ω_s

),

where p

_s

is the reservoir pressure at node s and p

^ω_s

is the well pressure at node s. The Well Index WI

_s

is typically computed using Peaceman’s approach (see [35, 36, 17]) and takes into account the unresolved singularity of the pressure solution in the neighborhood of the well. Fourier fluxes between the reservoir and the well could also be discretized using such Two Point Flux Approximation but they are assumed to be small compared with thermal convective fluxes and will be neglected in the following well model. At each well node s ∈ V

_ω

the temperature inside the well is denoted by T

_s^ω

and the volume fractions by s

^α_s,ω

, α ∈ {`, g}. The temperature in the reservoir at node s is denoted by T

_s

, the saturations by s

^α_s

, and the phase mass fractions by c

^α_s

for α ∈ {`, g}.

For any a ∈ R , let us define a

⁺

= max(a, 0) and a

⁻

= min(a, 0). The mass flow rates between the reservoir and the well ω at a given node s ∈ V

_ω

are defined by the following phase based upwind approximation of the mobilities:

q

^r→ω_s,α

= β

_ω^inj

ρ

^α

(p

^ω_s

, T

_s^ω

)

µ

^α

(p

^ω_s

, T

_s^ω

) k

^α_r,s

(s

^α_s,ω

)(V

_s^ω

)

⁻

+ β

_ω^prod

c

^α_s

ρ

^α

(p

_s

, T

_s

)

µ

^α

(p

_s

, T

_s

) k

^α_r,s

(s

^α_s

)(V

_s^ω

)

⁺

, q

^r→ω_s,h

2o

= X

α∈{`,g}

q

_s,α^r→ω

, (7)

and the energy flow rate is defined similarly by q

_s,e^r→ω

= X

α∈{`,g}

h

^α

(p

^ω_s

, T

_s^ω

)(q

_s,α^r→ω

)

⁻

+ h

^α

(p

_s

, T

_s

)(q

_s,α^r→ω

)

⁺

. (8) The well coefficients β

_ω^inj

and β

_ω^prod

are used to impose specific well behavior. The general case corresponds to β

_ω^inj

= β

_ω^prod

= 1. Yet, for an injection well, it will be convenient as explained in subsection 3.3.2, to impose that the mass flow rates q

_s,h^r→ω

2o

are non positive for all nodes s ∈ V

_ω

corresponding to set β

_ω^inj

= 1 and β

_ω^prod

= 0. Likewise, for a production well, it will be convenient as explained in subsection 3.3.3, to set β

_ω^inj

= 0 and β

_ω^prod

= 1 which corresponds to assume that the mass flow rates q

^r→ω_s,h2o

are non negative for all nodes s ∈ V

_ω

. These simplifying options currently prevent the modeling of cross flows where injection and production occur in different places of the same well, as it sometimes happen in geothermal wells, typically in closed wells.

3.3.1 Well physical model

Our conceptual model inside the well assumes that the flow is stationary at the reservoir time scale along with perfect mixing and thermal equilibrium. The Fourier fluxes and the wall friction are neglected and the pressure distribution is assumed hydrostatic along the well.

For the sake of simplicity, the flow rate between the reservoir and the well is considered concen- trated at each node s of the well. For each edge a ∈ E

_ω

, let us denote by q

^α_a

the mass flow rate of phase α along the edge a oriented positively from s

⁰

to s with a = ss

⁰

(let us recall that s is the parent node of s

⁰

).

Let α ∈ {`, g}, the set of well unknowns is defined at each node s ∈ V

_ω

by the well pressure p

^ω_s

, the well temperature T

_s^ω

, the well saturations s

^α_s,ω

, and at each edge a ∈ E

_ω

by the mass flow rates q

_a^α

. These well unknowns are complemented by the well mass flow rates q

_ω^α

which are non negative for production wells and non positive for injection wells (see Figure 4).

For each edge a = ss

⁰

∈ E

_ω

, and each phase α, let us define the following phase based upwind approximations of the specific enthalpy, mass density and saturation

h

^α_a

=

( h

^α

(p

^ω_s0

, T

_s^ω0

) if q

_a^α

≥ 0, h

^α

(p

^ω_s

, T

_s^ω

) if q

^α_a

< 0. ρ

^α_a

=

( ρ

^α

(p

^ω_s0

, T

_s^ω0

) if q

_a^α

≥ 0, ρ

^α

(p

^ω_s

, T

_s^ω

) if q

_a^α

< 0. s

^α_a

=

( s

^α_s0,ω

if q

_a^α

≥ 0,

s

^α_s,ω

if q

_a^α

< 0. (9) For all ss

⁰

= a ∈ E

_ω

, let us set κ

_a,s⁰

= −1 and κ

_a,s

= 1. The well equations account for the mass and energy conservations at each node of the well combined with the sum to one of the saturations and the thermodynamical equilibrium. Let E

_s^ω

⊂ E

_ω

denote the set of well edges sharing the node s ∈ V

_ω

, then for all s ∈ V

_ω

we obtain the equations



 

 

 

 

 

 

 

 

 

 

 

 

q

_s,h^r→ω_2o

+ X

a∈E_s^ω

X

α∈{`,g}

κ

_a,s

q

_a^α

= δ

_s^sω

X

α∈{`,g}

q

_ω^α

, q

_s,e^r→ω

+ X

a∈E_s^ω

X

α∈{`,g}

h

^α_a

κ

a,s

q

_a^α

= δ

_s^sω

X

α∈{`,g}

¯ h

^α_ω

(q

_ω^α

)

⁻

+ h

^α

(p

^ω_s

, T

_s^ω

)(q

_ω^α

)

⁺

, s

^`_s,ω

+ s

^g_s,ω

= 1,

p

^ω_s

= p

_sat

(T

_s^ω

) if s

^g_s,ω

> 0 and s

^`_s,ω

> 0,

p

^ω_s

≥ p

sat

(T

_s^ω

) if s

^g_s,ω

= 0, p

^ω_s

≤ p

sat

(T

_s^ω

) if s

^`_s,ω

= 1,

(10)

where δ stands for the Kronecker symbol, and ¯ h

^α_ω

for prescribed specific enthalpies in the case of injection wells. Inside the well, the hypothesis of hydrostatic pressure distribution implies that

p

^ω_s

− p

^ω_s0

+ ρ

_a

g(z

_s

− z

_s⁰

) = 0, (11)

Figure 4: Example of multi-branch well ω with its root node s

_ω

, one edge a = ss

⁰

and the main physical quantities: the well mass flow rates q

_ω^α

, the mass and energy flow rates between the reservoir and the well q

^r→ω_s,h

2o

, q

_s,e^r→ω

, the well node pressure, temperature and saturations p

^ω_s

, T

_s^ω

, s

^α_s,ω

, and the edge mass flow rates q

_a^α

.

for each edge ss

⁰

= a ∈ E

_ω

, where ρ

_a

is the mass density of the liquid gas mixture. The system is completed by a slip closure law expressing the slip between the liquid velocity u

^`_a

and the gas velocity u

^g_a

at each edge a ∈ E

_ω

with

q

_a^α

= πr

²_ω

ρ

^α_a

s

^α_a

u

^α_a

.

In the following simplified well models developed in subsections 3.3.2 and 3.3.3, a zero slip law will be assumed for simplicity in such a way that u

^`_a

= u

^g_a

. Note that these simplified well models could be easily extended to account for non-zero slip laws as well as for an explicit approximation of the wall friction along the wells. The two fundamental assumptions to obtain these simplified well models are (i) prescribed sign of the mass flow rates q

_s,α^r→ω

, s ∈ V

_ω

, forced to be all non-negative for production

wells and all non-positive for injection wells,

(ii) neglected Fourier fluxes compared with thermal convection fluxes.

The well boundary conditions prescribe a limit total mass flow rate ¯ q

_ω

and a limit bottom hole pressure ¯ p

_ω

. Then, complementary constraints accounting for usual well monitoring conditions, are imposed between q

ω

− q ¯

ω

and p

ω

− p ¯

ω

using the notations

p

_ω

= p

^ω_sω

and q

_ω

= X

α∈{`,g}

q

_ω^α

.

In the following subsections, we consider the particular case of injection wells assuming a pure liquid

phase, and the case of production wells. The flow rates are enforced to be non positive (resp. non

negative) at all well nodes for injection wells (resp. production wells). It corresponds to set β

_ω^inj

= 1,

β

_ω^prod

= 0 for an injection well and β

_ω^inj

= 0, β

_ω^prod

= 1 for a production well. The limit bottom

hole pressure ¯ p

_ω

is a maximum (resp. minimum) pressure and the limit total mass flow rate ¯ q

_ω

is

a minimum non positive (resp. maximum non negative) flow rate for injection (resp. production)

wells.

In both cases, using an explicit computation of the hydrostatic pressure drop, the well model will be reduced to a single equation and a single implicit unknown corresponding to the well reference pressure p

ω

(see e.g. [7]).

3.3.2 Liquid injection wells

The injection well model sets β

_ω^inj

= 1, β

_ω^prod

= 0 and prescribes the minimum well total mass flow rate ¯ q

_ω

≤ 0, the well maximum bottom hole pressure ¯ p

_ω

and the well specific liquid enthalpy ¯ h

^`_ω

. It is assumed that the injection is in liquid phase and that no gas will appear in the well during the simulation as it is usually the case in geothermal systems.

Since β

_ω^inj

= 1 and β

_ω^prod

= 0, the mass flow rates q

^α_a

are enforced to be non negative and it results from (10), and the assumption that the gas phase does not appear in the well that h

^`_a

= ¯ h

^`_ω

for all a ∈ E

_ω

and that s

^`_s,ω

= 1 − s

^g_s,ω

= 1 for all s ∈ V

_ω

.

Given the previous time step well reference pressure p

ⁿ⁻¹_ω

= p

^ω,n−1_sω

, we first compute the pressures along the well solving the equations

p

^ω_s

− p

^ω_s0

+ ρ

_a

g(z

_s

− z

_s⁰

) = 0 for all a = ss

⁰

∈ E

_ω

, p

^ω_sω

= p

^ω,n−1_sω

,

ρ

_a

= ρ

^`

(p

^ω_s

, T

_s^ω

) for all a = ss

⁰

∈ E

_ω

, h

^`

(p

^ω_s

, T

_s^ω

) = ¯ h

^`_ω

for all s ∈ V

_ω

. We deduce the explicit pressure drops

∆p

^ω,n−1_s

= p

^ω_s

− p

ⁿ⁻¹_ω

,

which provide for all s ∈ V

_ω

the pressures p

^ω,n_s

and temperatures T

_s^ω,n

along the well at the current time step n such that

p

^ω,n_s

= p

ⁿ_ω

+ ∆p

^ω,n−1_s

, h

^`

(p

^ω,n_s

, T

_s^ω,n

) = ¯ h

^`_ω

.

The mass and energy flow rates at each node s ∈ V

ω

between the reservoir and the well are defined by (7)-(8) with β

_ω^inj

= 1 and β

_ω^prod

= 0 and depend only on the implicit unknowns p

ⁿ_ω

and p

ⁿ_s

. They are respectively denoted by q

_s,h^r→ω

2o

(p

ⁿ_s

, p

ⁿ_ω

) and q

_s,e^r→ω

(p

ⁿ_s

, p

ⁿ_ω

).

The well equation at the current time step is defined by the following complementary constraints between the prescribed minimum well total mass flow rate and the prescribed maximum bottom hole pressure



 

 



 

 

 X

s∈Vω

q

^r→ω_s,h2o

(p

ⁿ_s

, p

ⁿ_ω

) − q ¯

ω

¯ p

ω

− p

ⁿ_ω

= 0, X

s∈Vω

q

^r→ω_s,h2o

(p

ⁿ_s

, p

ⁿ_ω

) − q ¯

_ω

≥ 0,

¯

p

_ω

− p

ⁿ_ω

≥ 0.

(12)

3.3.3 Production wells

The production well model sets β

_ω^inj

= 0, β

_ω^prod

= 1 and prescribes the maximum well total mass flow rate ¯ q

_ω

≥ 0 and the well minimum bottom hole pressure ¯ p

_ω

.

The solution at the previous time step n − 1 provides the pressure drop ∆p

^ω,n−1_s

at each node

s ∈ V

_ω

. This computation based on thermodynamical equilibrium is detailed below. As for the

injection well, we deduce the well pressures using the bottom well pressure at the current time step n

p

^ω,n_s

= p

ⁿ_ω

+ ∆p

^ω,n−1_s

.

The mass and energy flow rates at each node s ∈ V

_ω

between the reservoir and the well are defined by (7)-(8) with β

_ω^inj

= 0 and β

_ω^prod

= 1 and depend only on the implicit reservoir unknowns X

_sⁿ

setting

X

s

=

P

s

, T

s

, s

^`_s

, s

^g_s

, c

^`_s

, c

^g_s

,

and on the implicit well unknown p

ⁿ_ω

. They are respectively denoted by q

^r→ω_s,h

2o

(X

_sⁿ

, p

ⁿ_ω

) and q

_s,e^r→ω

(X

_sⁿ

, p

ⁿ_ω

).

The well equation at the current time step is defined by the following complementary constraints between the prescribed maximum well total mass flow rate and the prescribed minimum bottom hole pressure



 

 



 

 



¯

q

_ω

− X

s∈Vω

q

_s,h^r→ω_2o

(X

_sⁿ

, p

ⁿ_ω

)

p

ⁿ_ω

− p ¯

_ω

= 0,

¯

q

_ω

− X

s∈V_ω

q

^r→ω_s,h2o

(X

_sⁿ

, p

ⁿ_ω

) ≥ 0, p

ⁿ_ω

− p ¯

_ω

≥ 0.

(13)

Let us now detail the computation of the pressure drop at each node s ∈ V

_ω

using the previous time step solution n −1 consisting of the reservoir unknowns and the well pressures. We first compute the well temperature T

_s^ω,n−1

and saturations s

^α,n−1_s,ω

at each node s using equations (10). Summing the mass and energy equations of (10) over all nodes s

⁰⁰

≥

ω

s, we obtain for all a = s

⁰

s ∈ E

_ω

that X

α∈{`,g}

Q

^α,n−1_a

= X

s⁰⁰∈V_ω|s⁰⁰≥

ωs

q

_s^r→ω00,h2o

(X

_sⁿ⁻¹00

, p

ⁿ⁻¹_ω

) = Q

^ω_s,h2o

, X

α∈{`,g}

h

^α

(p

^ω,n−1_s

, T

_s^ω,n−1

)Q

^α,n−1_a

= X

s⁰∈Vω|s⁰⁰≥

ω

s

q

_s^r→ω0,e

(X

_sⁿ⁻¹00

, p

ⁿ⁻¹_ω

) = Q

^ω_s,e

,

with

Q

^α,n−1_a

= πr

²_ω

ρ

^α

(p

^ω,n−1_s

, T

_s^ω,n−1

)s

^α,n−1_s,ω

u

^α,n−1_a

, α ∈ {`, g}.

It results that the thermodynamical equilibrium at fixed well pressure p

^ω,n−1_s

, mass Q

^ω_s,h

2o

and energy Q

^ω_s,e

provides the well temperature T

_s^ω,n−1

and the well saturations s

^α,n−1_s,ω

at node s as follows.

Let us set p = p

^ω,n−1_s

. We first assume that both phases are present which implies that T

_sat

= (p

_sat

)

⁻¹

(p) and that the liquid mass fraction is given by

c

^`

=

h

^g

(p, T

_sat

) −

_Q^Qω^ωs,e s,h2o

h

^g

(p, T

_sat

) − h

^`

(p, T

_sat

) . The following alternatives are checked:

Two-phase state: if 0 < c

^`

< 1, the two-phase state is confirmed. Using the zero slip assumption, we obtain

T

_s^ω,n−1

= T

_sat

and s

^`,n−1_s,ω

= 1 − s

^g,n−1_s,ω

=

c^{`</sup> ρ<sup>`}(p,Tsat) c^`

ρ^`(p,Tsat)

+

_ρg(p,T^1−csat^` )

.

Liquid state: if c

^`

≥ 1, then only the liquid phase is present, we set s

^`,n−1_s,ω

= 1, s

^g,n−1_s,ω

= 0, and T

_s^ω,n−1

is the solution T of

h

^`

(p, T ) = Q

^ω_s,e

Q

^ω_s,h

2o

.

Gas state: if c

^`

≤ 0, then only the gas phase is present, we set s

^`,n−1_s,ω

= 0, s

^g,n−1_s,ω

= 1, and T

_s^ω,n−1

is the solution T of

h

^g

(p, T ) = Q

^ω_s,e

Q

^ω_s,h2o

. Then, the explicit pressure drop

∆p

^ω,n−1_s

= p

^ω_s

− p

ⁿ⁻¹_ω

, is obtained from

p

^ω_s

− p

^ω_s0

+ ρ

_a

g(z

_s

− z

_s⁰

) = 0 for all a = ss

⁰

∈ E

_ω

, p

^ω_sω

= p

^ω,n−1_sω

,

ρ

_a

= X

α∈{`,g}

s

^α,n−1_s,ω

ρ

^α

(p

^ω,n−1_s

, T

_s^ω,n−1

) for all a = ss

⁰

∈ E

_ω

.

3.4 Discretization of the hybrid-dimensional non-isothermal two-phase flow model

The time integration is based on a fully implicit Euler scheme to avoid severe restrictions on the time steps due to the small volumes and high velocities in the fractures. A phase based upwind scheme is used for the approximation of the mobilities in the mass and energy fluxes (see e.g. [8]). At the matrix fracture interfaces, we avoid mixing matrix and fracture rocktypes by choosing appropriate control volumes for σ ∈ F

_Γ

and s ∈ V

_Γ

(see Figure 3). In order to avoid tiny control volumes at the nodes s ∈ V

Σ

located at the fracture intersection, the volume is distributed to such a node s from all the fracture faces containing the node s.

For each ν ∈ M ∪ F

Γ

∪ V the set of reservoir pressure, temperature, saturations and mass fractions unknowns is denoted by X

_ν

=

P

_ν

, T

_ν

, s

^`_ν

, s

^g_ν

, c

^`_ν

, c

^g_ν

, where c

^α_ν

is the mass fraction of the water component in phase α used to express the thermodynamical equilibrium. We denote by X

D

, the set of reservoir unknowns

X

D

= {X

_ν

, ν ∈ M ∪ F

_Γ

∪ V},

and similarly by P

D

and T

D

the sets of reservoir pressures and temperatures. The set of well bottom hole pressures is denoted by P

W

= {p

ω

, ω ∈ W}.

The Darcy fluxes taking into account the gravity term are defined by



 

 

V

_K,ν^α

(X

D

) = F

_K,ν

(P

D

) − ρ

^α

(p

_K

, T

_K

) + ρ

^α

(p

_ν

, T

_ν

)

2 F

_K,ν

(G

D

), ν ∈ Ξ

_K

, K ∈ M, V

_σ,s^α

(X

D

) = F

_σ,s

(P

D

) − ρ

^α

(p

σ

, T

σ

) + ρ

^α

(p

s

, T

s

)

2 F

_σ,s

(G

D

), s ∈ V

_σ

, σ ∈ F

_Γ

,

(14)

where G

D

denotes the vector (g · x

_ν

)

ν∈M∪F_Γ∪V

.

For each Darcy flux, let us define the upwind control volume cv

^α_µ,ν

such that cv

^α_K,ν

=

( K if V

_K,ν^α

(X

_D

) > 0

ν if V

_K,ν^α

(X

D

) < 0 for ν ∈ Ξ

_K

, K ∈ M, for the matrix fluxes, and such that

cv

^α_σ,s

=

( σ if V

_σ,s^α

(X

_D

) > 0

s if V

_σ,s^α

(X

D

) < 0 for s ∈ V

_σ

, σ ∈ F

_Γ

,

for fracture fluxes. Using this upwinding, the mass and energy fluxes are given by q

_ν,ν^{α 0}

(X

D

) = c

^α_cv^α

ν,ν0

ρ

^α

(p

cv^α

ν,ν0

, T

cv^α

ν,ν0

) µ

^α

(p

_cv^α

ν,ν0

, T

_cv^α

ν,ν0

) k

_r,cv^{α α}

ν,ν0

(s

^α_cv^α

ν,ν0

)V

_ν,ν^α0

(X

D

), q

_ν,ν^h2o0

(X

D

) = X

α∈{`,g}

q

_ν,ν^α 0

(X

D

), q

_ν,ν^e 0

(X

D

) = X

α∈{`,g}

h

^α

(p

_cv^α

ν,ν0

, T

_cv^α

ν,ν0

)q

_ν,ν^α 0

(X

D

) + G

_ν,ν⁰

(T

D

).

In each control volume ν ∈ M ∪ F

Γ

∪ V , the mass and energy accumulations are denoted by A

_α,ν

(X

_ν

) = φ

_ν

ρ

^α

(p

_ν

, T

_ν

)s

^α_ν

c

^α_ν

,

A

h2o,ν

(X

ν

) = X

α∈{`,g}

A

α,ν

(X

ν

), A

_e,ν

(X

_ν

) = X

α∈{`,g}

e

^α

(p

_ν

, T

_ν

)A

_α,ν

(X

_ν

) + ¯ φ

_ν

E

_r

(p

_ν

, T

_ν

).

We can now state the system of discrete equations at each time step n = 1, · · · , N

_tf

which accounts for the mass (i = h

₂

o) and energy (i = e) conservation equations in each cell K ∈ M:

R

_K,i

(X

_Dⁿ

) := A

_i,K

(X

_Kⁿ

) − A

_i,K

(X

_Kⁿ⁻¹

)

∆t

ⁿ

+ X

s∈VK

q

_K,sⁱ

(X

_Dⁿ

) + X

σ∈FΓ∩FK

q

ⁱ_K,σ

(X

_Dⁿ

) = 0, (15) in each fracture face σ ∈ F

_Γ

:

R

_σ,i

(X

_Dⁿ

) := A

_i,σ

(X

_σⁿ

) − A

_i,σ

(X

_σⁿ⁻¹

)

∆t

ⁿ

+ X

s∈Vσ

q

ⁱ_σ,s

(X

_Dⁿ

) + X

K∈Mσ

−q

_K,σⁱ

(X

_Dⁿ

) = 0, (16) and at each node s ∈ V \ V

_D

:

R

_s,i

(X

_Dⁿ

, P

_Wⁿ

) := A

i,s

(X

_sⁿ

) − A

i,s

(X

_sⁿ⁻¹

)

∆t

ⁿ

+ X

σ∈F_Γ,s

−q

_σ,sⁱ

(X

_Dⁿ

) + X

K∈Ms

−q

ⁱ_K,s

(X

_Dⁿ

)

+ X

ω∈W|s∈V_ω

q

^r→ω_s,i

(X

_sⁿ

, p

^ω,n_s

) = 0.

(17)

It is coupled with the well equations for the injection wells ω ∈ W

inj

R

_ω

(X

_Dⁿ

, P

_Wⁿ

) := − min( X

s∈V_ω

q

_s,h^r→ω_2o

(X

_sⁿ

, p

ⁿ_ω

) − q ¯

_ω

, p ¯

_ω

− p

ⁿ_ω

) = 0, (18) and for the production wells ω ∈ W

_prod

R

_ω

(X

_Dⁿ

, P

_Wⁿ

) := min(¯ q

_ω

− X

s∈Vω

q

_s,h^r→ω

2o

(X

_sⁿ

, p

ⁿ_ω

), p

ⁿ_ω

− p ¯

_ω

) = 0, (19) reformulating respectively (12) and (13) using the min function.

The system is closed with thermodynamical equilibrium and the sum to one of the saturations R

₁

(X

_νⁿ

) := c

^g,n_ν

p

ⁿ_ν

− p

_sat

(T

_νⁿ

)c

^`,n_ν

= 0,

R

₂

(X

_νⁿ

) := min(s

^`,n_ν

, 1 − c

^`,n_ν

) = 0, R

₃

(X

_νⁿ

) := min(s

^g,n_ν

, 1 − c

^g,n_ν

) = 0, R

₄

(X

_νⁿ

) := s

^`,n_ν

+ s

^g,n_ν

− 1 = 0,

(20)