Two-Phase Incompressible Flow with Dynamic Capillary Pressure in a Heterogeneous Porous Media

Abstract

We prove the existence of weak solutions of a two-incompressible immiscible wetting and non-wetting fluids phase flow model in porous media with dynamic capillary pressure. This model is a coupled system which includes a nonlinear parabolic saturation equation and an elliptic pressure–velocity equation. In the regularized case, the existence and uniqueness of the weak solution are obtained. We let the regularization parameter be $\eta \to 0$ to prove the existence of weak solutions.

1. Introduction

In this paper, we are interested in the study of a system modeling two-phase flow, immiscible and incompressible, wetting and non-wetting fluids, with dynamic capillary pressure. Firstly, we describe a nonlinear problem modeling the displacement of two incompressible and immiscible fluids in a porous media.

The functional capillary pressure models that define the $p_c-s_w$ relation are determined by well-defined laboratory experiments using samples of the porous medium, see [1,2,3]. In these experiments, $p_c$ is calculated by measuring the difference between the average pressures of non-wetting $p_o$ and wetting $p_w$ liquids. By expressing this difference as a function of saturation of the wetting phase $s_w$,

$$p_c=p_o-p_w=p_c\left(s_w\right),$$

these models define the capillary pressure under equilibrium conditions ${\partial }_t{s}_w=0$. However, fluids do not necessarily flow under equilibrium conditions, especially for short durations when the variation of $s_w$ with respect to time is fast, which implies that the derivative of the saturation in time ${\partial }_t{s}_w$ can be large. Under these circumstances, the relationship $p_c-s_w$ strongly depends on the two quantities $s_w$ and ${\partial }_t{s}_w$. The goal of this work, considering the dynamic capillary pressure is a function of variables $s_w$ and ${\partial }_t{s}_w$,

$$p_c=p_c(s_w,{\partial }_t{s}_w),$$

where the function $p_c\left(s_w\right)$ is increasing and stands for classical static capillary pressure, and we define the damping coefficient $\kappa =\kappa \left(s_w\right)$ as a function of variable $s_w$. In this work, we generalize the problem for $\kappa$ as a function of s, where we previously observed that the problem for the damping coefficient $\kappa >0$ is a positive real constant, see [4]. Next, we mathematically analyze the existence of weak solutions in the model.

In this dimension, to better adapt to the conditions of equilibrium (interface), it is necessary to develop equilibrium laws at the macroscopic level, and the appropriate constitutive relations for the correct calculation of the interfacial equilibrium, as well as the properties of the phases, flow in the porous medium.

Accordingly, a thermodynamic theory for two-phase flow in a porous medium has been worked out in detail. The main constitutive assumption in this theory is the dependence of Helmholtz free energy functions for phases and interfaces on state variables, such as the density, mass, temperature, saturation, porosity, interfacial surface density, and solid-phase stress tensor. Certainly, the explicit inclusion of “interface effects” and “surface properties” at the two-phase flow level is an essential feature of this new approach. It is well recognized in the literature [5,6] that interfaces play an important role in determining the thermodynamic state of the whole system. In this way, in the case of two-phase flow in a porous medium, the three phases and three interfaces are also involved in the context of this new theory. The need for studies on the dynamic effects of the $p_c-s_w$ relationship has been demonstrated through several experimental studies. The articles [7,8] present a review of this experimental work. Water and air are combined into one system in most previous studies and research. In summary, it can be seen that for a given saturation, a transient drainage process (${\partial }_t{s}_w<0$) exhibits higher capillary pressure than under equilibrium conditions, and a transient imbibition process (${\partial }_t{s}_w>0$) exhibits lower capillary pressure. In [9], a review of some experimental studies and also of existing approaches in the literature for models that seek to give dynamic effects a theoretical basis is presented.

We insist on the model of Hassanizadeh and Gray [10]. Likewise, Kalaydjian [11] provides a good and concise description that combines fluids and interfaces, and this leads to the classical or microscopic extension of capillary pressure. The same can be said for Stover, for example [12], who gave a different expression for dynamic capillary pressure, somewhat similar to what Hassan Zabdeh, Gray, and Klaydjian proposed in their work. In the same context, this model is the first to reflect dynamic capillary pressure in a quantitative manner, as it is based solely on experimental results. On the other hand, the Barenblatt model [13] is cited, which looks at another form of modeling dynamic effects on capillary pressure, which simultaneously concerns relative dynamic permeability. As part of the experimental studies, in [14], two liquids of the same viscosity were used, unlike the most commonly used water–air system, to study the pressure dependence of the interface discontinuity of an imbibition process.

A similar work by Kalaydjian [11] will be briefly described since his model constitutes the interfaces between fluids. Both approaches lead to an extension of classical (macroscopic) capillary pressure by a non-equilibrium term consisting of the rate of change in saturation. Stauffer [15] found another expression for dynamic capillary pressure similar to those of Hassanizadeh, Gray, and Kalaydjian. This model, based solely on experimental results, was the first model to attempt to produce dynamic capillary pressure quantitatively. Finally, the Barenblatt model [13] is cited, which looks at another form of modeling dynamic effects on capillary pressure, which simultaneously concerns relative dynamic permeability. As part of the experimental studies, in [14], two liquids of the same viscosity were used, in contrast to the most commonly used water–air system, to study the pressure dependence of the interface discontinuity of an imbibition process. A capillary pressure threshold of $\Delta p_c$ is required for the movement of the interface, and from this threshold, a nonlinear dependence of $\Delta p_c$ and the velocity of l interface are revealed. At very high speeds, even an inversion of the interface leading to a concave curvature with respect to the wetting phase is observed. Models that take dynamic capillary pressure into account are also called [9] extended models.

We symbolize with ($p_c^e$) the classical or so-called macroscopic capillary pressure, also called the equilibrium capillary pressure, and it represents the difference between the phase pressures $p_c=p_c^e$ in thermodynamic equilibrium. One of the results that can be obtained is to find $p_c^e$ using classical patterns $p_c-s_w$.

Among the works that examine static drainage experiments in different vertical columns, we mention what was done by the researcher Stauffer in 1978 [15], where he plotted the relevant capillary differences against the rates of change, then Stauffer decided to take the linear dependence into account and, in turn, determined the dynamic capillary pressure $p_c^d$ as follows:

$$p_c^d=p_c^e-{\alpha }_s{\rho }_{\omega }g\varphi {\displaystyle \frac{\partial s_e}{\partial t}},$$

(1)

where ${\alpha }_s$ represents the dynamic effect and is also given by

$${\alpha }_s={\displaystyle \frac{{\beta }_s{p}_d^2}{{\rho }_w^2{g}^2{K}_f}}[\mathrm{m}\mathrm{s}].$$

(2)

In most of Stauffer’s work, the value of ${\alpha }_s$ belongs to the field $[7\mathrm{ms},24\mathrm{ms}]$. We recall that (${\beta }_s,p_d$) are coefficients of the dimensionless scale and capillary pressure, respectively, while $p_c^e$ represents the equilibrium capillary pressure. As directed by Brooks-Corey $p_c\left(s_e\right)=p_d{s}_e^{-{\displaystyle \frac{1}{\lambda }}}$, Stauffer implemented $p_c-s$.

Also, supposing the density of water is constant, we define new ${\tau }_s$ as follows:

$${\tau }_s={\alpha }_s{\rho }_{w}g\varphi [\mathrm{kgm}/\mathrm{s}].$$

Using (1), the model $p_c-s$ is described as follows:

$$p_c^d=p_c^e-{\kappa }_s{\displaystyle \frac{\partial s_e}{\partial t}}.$$

(3)

We supposing that Newtonian fluids exist immiscible and incompressible in the displacement process with a rigid porous medium and a spherical fluid/fluid interface. In [11,16], the following model for dynamic capillary pressure is suggested:

$$p_c^d={\displaystyle \frac{2{\sigma }_{wn}}{r}}-\kappa Ka{\displaystyle \frac{\partial s_w}{\partial t}}.$$

(4)

Here, $p_c^e, {\sigma }_{wn}, r$ and $\kappa Ka$ represent the equilibrium capillary pressure defined by the equation of Laplace, the surface tension between the two phases, wetting and non-wetting, the spherical interface radius, and the non-equilibrium effects. As in [15], the capillary pressure differential at equilibrium is given by the rate of change in saturation of the wetting phase.

In a series of articles, Hassanizadeh and Gray [10,12,17] developed a theory of multiphase flow in porous media based on thermodynamics. They constructed balance equations for mass, motion, and energy. These equilibrium equations were first formulated at the microscopic scale and then extended to the macroscopic scale. In sequence, the equilibrium equations are superseded in the second law of thermodynamics. Thus, new expressions can be obtained for effective parameters such as relative permeability and capillary pressure, which are based on deep physical considerations. Thus, by means of the general conservation equations at the microscopic scale and the second law of thermodynamics, and taking into account the continuity of the interface, the following inequality was obtained:

$$-{\displaystyle \frac{1}{\varphi }}{\displaystyle \frac{\partial s_w}{\partial t}}(p_c^d-p_c^e)\ge 0.$$

(5)

Based on this inequality, we can deduce that if (${\displaystyle \frac{\partial s_w}{\partial t}}>0$), then ($(p_c^d-p_c^e)<0$). On the other hand, if (${\displaystyle \frac{\partial s_w}{\partial t}}<0$), then ($(p_c^d-p_c^e)>$0). Considering a linear relation between ${\displaystyle \frac{\partial s_w}{\partial t}}$ and $(p_c^d-p_c^e)$, which by hypothesis remains close to thermodynamic equilibrium states, we obtain the following:

$$p_c^d=p_c^e-\kappa {\displaystyle \frac{\partial s_w}{\partial t}},$$

(6)

In this case, the non-negative coefficient of delay $\kappa [Pa.s]$, defined by Hassanizadeh and Gray, is given by the linear expression (6).

Another model that takes into account dynamic effects on capillary pressure has been proposed by Barenblatt in [13]. Specifically, this alternative approach refers to dynamic models for both capillary pressure and relative permeabilities, considering non-equilibrium effects related to the phase distribution in the porous medium. In this model, it is suggested that, in a process with rapid changes in the phase saturations, the time required to modify the phase arrangement, defined as the redistribution time $t_r$, must be taken into account. In [13], it is postulated that the relative permeabilities, $k_w$ and $k_o$, and the capillary pressure, $p_c$, can be obtained by the usual parametrizations obtained at a steady state, but calculating these quantities from an “apparent saturation” $s_{aw}$, which is the saturation imbalance, can be seen as the actual saturation in a future state. Estimating $s_{aw}$ is proposed as a simplified empirical model:

$$s_{aw}=s_w+t_r{\displaystyle \frac{\partial s_w}{\partial t}},$$

(7)

where the capillary dynamic pressure becomes,

$$p_c^d=p_c^e\left(s_{aw}\right)=p_c^e(s_w+t_r{\displaystyle \frac{\partial s_w}{\partial t}}),$$

(8)

recalling that $p_c^e$ is a monotone decreasing function. Thus, the pair of imbibition processes ($s_{aw}>s_w$) and capillary pressure during fast procedures are less than for slow processes, which agrees with the model mentioned above. Considering the dynamic effects of capillary pressure, we can make a relation, from a formal point of view, between the Barenblatt model (7) and (8) and the models mentioned above, in particular, the Hassanizadeh–Gray model (6). Here, we make an extension of the Taylor series of (8),

$$p_c^e(s_w+t_r{\displaystyle \frac{\partial s_w}{\partial t}})=p_c^e\left(s_w\right)+t_r{\displaystyle \frac{d{p}_c^e}{d{s}_w}}{\displaystyle \frac{\partial s_w}{\partial t}}+o\left(t_r^2\right).$$

(9)

Here, it is considered that $t_r$ is much smaller than the characteristic timescale of the [13] process, while we make the following approximation,

$$p_c^d=p_c^e\left(s_w\right)+t_r{\displaystyle \frac{d{p}_c^e}{d{s}_w}}{\displaystyle \frac{\partial s_w}{\partial t}}.$$

(10)

where we have identified the dynamic effect coefficient,

$$\kappa =t_r{\displaystyle \frac{d{p}_c^e}{d{s}_w}}.$$

(11)

Thus, a relationship can be established between the dynamic effect coefficient and the phase redistribution time, and a link between the different empirical and theoretical models can also be drawn.

2. Mathematical Model

This is a non-linear partial differential equation describing the motion of a fluid i. It is written as follows:

$$\varphi {\partial }_t\left({\rho }_i{s}_i\right)+\mathrm{div}\left({\rho }_i{\mathbf{u}}_i\right)=0,$$

(12)

where $\varphi$, ${\rho }_{\alpha }$, $s_{\alpha }$, and ${\mathbf{u}}_{\alpha }$ represent, respectively, the porosity, density, saturation, and velocity of the $\alpha$ phase. We apply Equation (12) to the wetting $i=w$ and non-wetting phases $i=o$; moreover, the two fluids are assumed to be incompressible, so the density ${\rho }_i$ remains constant in space and time. We divide each equation of the system (12) by ${\rho }_i$ in order to obtain the following system:

$$\varphi {\partial }_t{s}_i+\mathrm{div}\left({\mathbf{u}}_i\right)=0,i=o,w.$$

(13)

For Darcy’s law in the two-phase case, see [18], it is always expressed as a function of a pressure gradient $\nabla p$ and the permeability term K. However, this time, the permeability coefficient is weighted by the absolute permeability and the relative permeability:

$${\mathbf{u}}_i=-\mathbf{K}{\displaystyle \frac{k_{r_{\alpha }}\left(s_i\right)}{{\mu }_i}}\left(\nabla p_i-{\rho }_i\mathbf{g}\right),i=o,w,$$

(14)

where $\mathbf{K}$ is the permeability tensor of the porous medium, $k_{ri}$ is the relative permeability of phase i, ${\mu }_i$ is the viscosity, $p_i$ is the pressure, ${\rho }_i$ is the density of phase i, and $\mathbf{g}$ is the gravity term. We define the mobility of phase i by $M_i\left(s_i\right)=k_{r_i}\left(s_i\right)/{\mu }_i$ with $i=o,w$. Suppose that the two phases occupy all the pores of the medium; then, the saturations verify the following:

$$s_o+s_w=1.$$

(15)

The overall pressure design for static capillary pressure is given in [19,20,21,22]. We define the same concept for dynamic capillary pressure, considering that

$$p_o-p_w=p_c\left(s_w\right)+\kappa \left(s_w\right){\partial }_t{s}_w,$$

(16)

where the function $p_c\left(s_w\right)$ is increasing and stands for classical static capillary pressure, and $\kappa \left(s_w\right)$ is the damping coefficient. This global pressure p can be written as follows:

$$\begin{array}{c}\hfill p=p_o+\tilde{p}\left(s_w\right)=p_w-\overline{p}\left(s_w\right)-\kappa \left(s_w\right){\partial }_t{s}_w,\end{array}$$

(17)

where the artificial pressures are denoted by $\overline{p}$ and $\tilde{p}$ and are defined by the following:

$$\begin{array}{ccc}\hfill \tilde{p}\left(s_w\right)& =& {\int }_0^{s_w}{\displaystyle \frac{M_w\left(z\right)}{M\left(z\right)}}{p}_c^{\prime }\left(z\right)dz={\int }_0^{s_w}{\nu }_w\left(z\right)p_c^{\prime }\left(z\right)dz,\hfill \\ \hfill \overline{p}\left(s_w\right)& =& -{\int }_0^{s_w}{\displaystyle \frac{M_o\left(z\right)}{M\left(z\right)}}{p}_c^{\prime }\left(z\right)dz=-{\int }_0^{s_w}{\nu }_o\left(z\right)p_c^{\prime }\left(z\right)dz,\hfill \end{array}$$

(18)

where $M\left(s_w\right)=M_w\left(s_w\right)+M_o\left(s_o\right)$ is the total mobility, and ${\nu }_i=M_i/M$ is the mobility of phase $i=o,w$. The term capillary is also defined by

$$\alpha \left(s_o\right)={\displaystyle \frac{M_w\left(s_w\right)M_o\left(s_o\right)}{M\left(s_w\right)}}{\displaystyle \frac{d{p}_c}{d{s}_w}}\left(s_w\right)\ge 0.$$

(19)

Now, when we write the expression of the velocity of each phase respecting the global pressure and the capillary term, we find that

$${\mathbf{u}}_w=\mathbf{K}{M}_w(\nabla p-{\rho }_w\mathbf{g})+\alpha \left(s_w\right)\mathbf{K}\nabla s_w+\kappa \left(s_w\right)M_w\nabla {\partial }_t{s}_w,$$

(20)

$${\mathbf{u}}_o=\mathbf{K}{M}_o(\nabla p-{\rho }_o\mathbf{g})-\alpha \left(s_w\right)\mathbf{K}\nabla s_w.$$

(21)

We denote $s_o=s$, and using the velocity expressions (20), we obtain the following:

$$\varphi {\partial }_{t}s-\mathrm{div}\left(M_w\mathbf{K}(\nabla p-{\rho }_w\mathbf{g})\right)+\mathrm{div}(\mathbf{K}\alpha \left(s\right)\nabla s)+\kappa \left(s_w\right)\mathrm{div}(M_w\nabla {\partial }_{t}s)=0,$$

(22)

$$\varphi {\partial }_{t}s+\mathrm{div}\left(M_o\mathbf{K}(\nabla p-{\rho }_o\mathbf{g})\right)+\mathrm{div}(\mathbf{K}\alpha \left(s\right)\nabla s)=0.$$

(23)

To obtain the pressure equation, we take the difference between the two previous equations, and we find

$$\mathrm{div}\left(M\left(s\right)\mathbf{K}(\nabla p-\rho (s\left)\mathbf{g}\right)\right)+\kappa \left(s_w\right)\mathrm{div}(M_w\left(s\right)\nabla {\partial }_{t}s)=0,$$

(24)

with $\rho \left(s\right)={\rho }_o{\nu }_o+{\rho }_w{\nu }_w$. We assume that the flow takes place in the time interval $\left]0;T\right[$. We can write the system obtained above in the following form:

$$\varphi {\partial }_{t}s+\mathrm{div}\left(f\left(s\right)\mathbf{u}\right)-\mathrm{div}\left(\mathbf{K}(\alpha \left(s\right)\nabla s-k\left(s\right)\mathbf{g})\right)=0in\Omega \times \left]0,T\right[={\Omega }_T,$$

(25)

$$\mathrm{div}\left(\mathbf{u}\right)-\kappa \left(s_w\right)\mathrm{div}(M_w\nabla {\partial }_{t}s)=0,\mathbf{u}=-M\mathbf{K}\left[\nabla p-\rho \left(s\right)\mathbf{g}\right]\mathrm{in}{\Omega }_T.$$

(26)

The reservoir is denoted as $\Omega$, a bounded open set of ${\mathbb{R}}^d$. We set ${\Omega }_T=(0,T)\times \Omega$, ${\Sigma }_T=(0,T)\times \partial \Omega$, where $\partial \Omega =\Gamma$ denotes the boundary of $\Omega$. The first equation of $\left(S\right)$ is called the saturation equation, and the second is the pressure equation. We define the functions $f,\alpha ,k$ and $\rho$ as following $f\left(s\right)=M_o\left(s\right)/M\left(s\right)$, with $f\left(0\right)=0, f\left(1\right)=1$ and $0\le f\left(s\right)\le 1$. Next, $\alpha \left(s\right)=M\left(s\right){\nu }_o\left(s\right){\nu }_w\left(s\right)p_c^{\prime }\left(s\right)$, where $\alpha \left(0\right)=\alpha \left(1\right)=0$ and $\alpha \left(s\right)\ge 0$ for all $s\in [0,1]$. After that, $k\left(s\right)=M{\nu }_0\left(s\right){\nu }_w\left(s\right)\left({\rho }_o-{\rho }_w\right)$; hence, $k\left(0\right)=k\left(1\right)=0.$ Finally, $\rho \left(s\right)={\rho }_o{\nu }_o+{\rho }_w{\nu }_w$; therefore, $\rho \left(s\right)>0.$ If the total mobility is a convex function and

$$M\left(s\right)\ge \underset{s\in \left[0;1\right]}{min}M\left(s\right)={\displaystyle \frac{1}{{\rho }_0+{\rho }_w}}>0,$$

we use the previous notations. As a result, we can write

$${\mathbf{u}}_o=f\left(s\right)\mathbf{u}-\mathbf{K}\alpha \left(s\right)\nabla s+\mathbf{K}k\left(s\right)\mathbf{g}.$$

The boundary conditions $\left(BC\right)$ are written as folows:

$$\left.\begin{array}{c}\mathbf{u}·\mathbf{n}=-q_e,\left({\mathbf{u}}_o·\mathbf{n}=0\right),s=0\mathrm{on}{\Gamma }_e.\\ \mathbf{u}·\mathbf{n}={\mathbf{u}}_o·\mathbf{n}=0,\mathrm{on}{\Gamma }_i.\\ \mathbf{u}·\mathbf{n}=q_s;\mathbf{K}\left[\alpha \left(s\right)\nabla s-k\left(s\right)\mathbf{g}\right]·\mathbf{n}=0\mathrm{on}{\Gamma }_s\end{array}\right\}\left(BC\right),$$

where $\mathbf{n}$ is the outward unit normal to $\Omega ={\Gamma }_e\dot{\cup }{\Gamma }_i\dot{\cup }{\Gamma }_s$, and the positive measure of ${\Gamma }_e$ is not equal to zero. The initial condition $\left(IC\right)$ is written as follows:

$$s(x,t=0)=s_0\left(x\right)\mathrm{in}\Omega .$$

(27)

The preceding considerations lead us to adapt the following strategy for proving the existence of a weak solution for the problem $\left(S\right)$ with conditions $\left(BC\right)$ and $\left(IC\right)$:

First step: Replace the non-regular problem with a regularized problem, replacing $\alpha \left(s\right)$ by $\alpha \left(s\right)+\eta$ with $\eta >0$, a constant. Here, for $\eta$ to be fixed, we will study the regular nonlinear problem. Here, we will prove the existence of a solution.

Second step: Study the non-regular problem. By letting the $\eta$ veer toward zero, we will try to obtain a solution to this problem.

3. Hypotheses and Main Result

Let us begin by providing the main assumptions on the coefficient functions of the system $\left(S\right)$. In our hypotheses, the following are assumed:

Hypothesis 1 (H1).

$f\in C^1\left(\left[0;1\right];{\mathbb{R}}_{+}\right)withf\left(0\right)=0,0\le f\left(s\right)\le 1andf\left(1\right)=1.$

Hypothesis 2 (H2).

$k,\rho \in C^1\left(\left[0;1\right];{\mathbb{R}}_{+}\right)withk\left(0\right)=k\left(1\right)=0$, and there is a constant $\underline{M}>0$, such that

$$M_{\alpha }\left(s\right)\ge \underline{M},\forall s\in \left[0;1\right].$$

Hypothesis 3 (H3).

When $\alpha \in C^1\left(\left[0;1\right];{\mathbb{R}}_{+}\right)with\alpha \left(0\right)=\alpha \left(1\right)=0,let$

$$\beta \left(s\right)=\underset{0}{\overset{s}{\int }}\alpha \left(\xi \right)d\xi .$$

We assume that β is strictly increasing on $[0,1]$ and that its inverse function ${\beta }^{-1}$ is Holderian (there exists a constant $C>0$, such that for all $s_1,s_2\in [0,\beta \left(1\right)]$, one has $|{\beta }^{-1}\left(s_1\right)-{\beta }^{-1}\left(s_2\right)|\le C|s_1-s_2{|}^{\theta }$) on $[0,\beta (1\left)\right]$ of exponent $\theta \in ]0,1]$. This means there is a constant $C>0$, such that

$$\left|{\beta }^{-1}\left({\xi }_1\right)-{\beta }^{-1}\left({\xi }_2\right)\right|\le C{\left|{\xi }_1-{\xi }_2\right|}^{\theta },\forall {\xi }_1,{\xi }_2\in \left[0;\beta \left(1\right)\right].$$

Hypothesis 4 (H4).

$q_e\in L^{\infty }\left(0,T;L^2\left({\Gamma }_e\right)\right)et{q}_s\in L^{\infty }\left(0,T;L^2\left({\Gamma }_s\right)\right)with$

$$\underset{{\Gamma }_e}{\int }{q}_{e}d\gamma \left(x\right)=\underset{{\Gamma }_s}{\int }{q}_{s}d\gamma \left(x\right),a.e.t\in \left]0;T\right[.$$

Hypothesis 5 (H5).

The function ϕ is in $L^{\infty }(\Omega )$, and there exist two constants, ${\varphi }_{*}>0$ and ${\varphi }^{*}>0$, such that ${\varphi }_{*}\le \varphi \left(x\right)\le {\varphi }^{*}$ a.e. $x\in \Omega$.

Hypothesis 6 (H6).

The permeability tensor $\mathbf{K}$ is in ${\left(L^{\infty }(\Omega )\right)}^{d\times d}$, and there exist two constants $k_0>0,k_{\infty }>0$, such that ${\parallel \mathbf{K}\parallel }_{{\left(L^{\infty }(\Omega )\right)}^{d\times d}}\le k_{\infty }$ and

$$\langle \mathbf{K}\left(\mathbf{x}\right)\xi ,\xi \rangle \ge k_0{\left|\xi \right|}^{2}a.e.x\in \Omega ,\forall \xi \in {\mathbb{R}}^d.$$

Hypothesis 7 (H7).

The damping coefficient $\kappa \left(s_w\right)$ is in $C^1([0,1];{\mathbb{R}}_{+})$ and $\kappa \left(s_w\right)>0$ for all $s_w\in [0,1]$.

Note that

$$\begin{array}{c}{\Omega }_T=\Omega \times ]0,T[ \mathrm{and} H_{{\Gamma }_e}^1\left(\Omega \right)=\left\{s\in H^1\left(\Omega \right)|s=0 \mathrm{on} {\Gamma }_e\right\},\hfill \end{array}$$

and

$$\begin{array}{c}\hfill H_{moy}^1(\Omega )=\{v\in H^1\left(\Omega \right)|\underset{\Omega }{\int }vdx=0\}.\end{array}$$

Theorem 1.

Suppose the hypotheses $(H1-H7)$ are satisfied and that the initial condition $s_0$ satisfies $0\le s_0\left(x\right)\le 1$ almost everywhere (a.e.) in Ω. Thefore, there are a couple of $(s,p)$ functions such as the following:

$$\begin{array}{c}0\le s\left(x,t\right)\le 1,a.e.in{\Omega }_T\hfill \\ s\in L^{{\displaystyle \frac{\theta }{2}}}(0,T;W^{\theta ,{\displaystyle \frac{2}{\theta }}}\left(\Omega \right)), \beta \left(s\right)\in L^2\left(0,T;H_{{\Gamma }_e}^1\left(\Omega \right)\right), s_t\in L^2\left(0,T;H^1(\Omega )\right),\hfill \\ p\in L^{\infty }\left(0,T;H_{moy}^1\left(\Omega \right)\right),\hfill \\ \mathbf{u}\in L^{\infty }\left(0,T;L^2\left(\Omega \right)\right),\hfill \\ s\left(x,0\right)=s_0\left(x\right)a.e.in\Omega ,\hfill \end{array}$$

and verify the integral identities as follows:

$$\begin{array}{ccc}& & \langle \varphi {\partial }_{t}s,\phi \rangle -\underset{\Omega }{\int }f\left(s\right)\mathbf{u}·\nabla \phi dx+\underset{\Omega }{\int }\nabla \beta \left(s\right)·\nabla \phi dx\hfill \\ & & -\underset{\Omega }{\int }\mathbf{K}k\left(s\right)\nabla \phi ·\mathbf{g}dx+\underset{{\Gamma }_s}{\int }f\left(s\right)q_s\phi d\gamma =0,\forall \phi \in H_{{\Gamma }_e}^1\left(\Omega \right), a.e.int\in \left]0;T\right[;\hfill \\ & & \underset{\Omega }{\int }M\mathbf{K}\left[\nabla p-\rho \left(s\right)\mathbf{g}\right]·\nabla \psi dx-\underset{\Omega }{\int }\kappa \left(s\right)M_w\left(s\right)\nabla {\partial }_{t}s·\nabla \psi dx\hfill \\ & & =\underset{{\Gamma }_e}{\int }{q}_e\psi d\gamma -\underset{{\Gamma }_s}{\int }{q}_s\psi d\gamma ,\forall \psi \in H^1\left(\Omega \right),a.e.int\in \left]0;T\right[.\hfill \end{array}$$

(28)

The bracket $\langle ·,·\rangle$ denotes the duality between $H_{{\Gamma }_e}^1\left(\Omega \right)$ and its dual.

The proof of Theorem 1 is carried out in two steps. The first consists of regularizing the non-regular problem and proving the existence of a solution to this regular problem. The second consists of passing the limit to obtain a solution for the initial system.

4. Existence of Weak Solutions Regular Case

The dissipation coefficient $\alpha \left(s\right)$ is replaced by

$${\alpha }_{\eta }\left(s\right)=\alpha \left(s\right)+\eta \mathrm{when}\eta >0\mathrm{is} \mathrm{constant}.$$

(29)

Then, consider the regularized problem: find two functions $s_{\eta }$ and $p_{\eta }$ solutions of the regularized system:

$$\left(S_{\eta }\right)\left\{\begin{array}{c}\varphi {\partial }_t{s}_{\eta }+\mathrm{div}\left(f\left(s_{\eta }\right){\mathbf{u}}_{\eta }\right)-\mathrm{div}\left[\mathbf{K}\left({\alpha }_{\eta }\left(s_{\eta }\right)\nabla s_{\eta }-k\left(s_{\eta }\right)\mathbf{g}\right)\right]=0,\\ \mathrm{div}{\mathbf{u}}_{\eta }-\kappa \left(s_{\eta }\right)\mathrm{div}(M_w\left(s_{\eta }\right)\nabla {\partial }_t{s}_{\eta })=0,{\mathbf{u}}_{\eta }=-M\left(s_{\eta }\right)\mathbf{K}\left[\nabla p_{\eta }-\rho \left(s_{\eta }\right)\mathbf{g}\right],\\ \mathrm{with} \mathrm{the} \mathrm{conditions}\left(BC\right)+\left(IC\right)\mathrm{corresponding}.\end{array}\right.$$

Definition 1.

We call the weak solution of the $\left(S_{\eta }\right)$ problem a pair $\left(s_{\eta },p_{\eta }\right)$ of functions satisfying all the conditions and identities of Theorem 1, where $\left(s,p\right)$ is replaced by $\left(s_{\eta },p_{\eta }\right)$.

Theorem 2.

Under the hypotheses of Theorem 1, the problem $\left(S_{\eta }\right)$ admits at least one weak solution. Although the possible solutions depend on η, for simplicity, we denote $\left(s,p\right)$ instead of $\left(s_{\eta },p_{\eta }\right)$.

To obtain the proof of Theorem 2, and to prove the existence of a solution of the system $\left(S_{\eta }\right)$, we will use the Galerkin approximation method. As we do not know a priori that the solutions sought are bounded, we are obliged to extend by continuity the coefficient functions of the system $\left(S_{\eta }\right)$ out of their domains “natural” by definition.

$$\begin{array}{c}\overline{\alpha }\left(s\right)=\alpha \left(s\right)\mathrm{if}s\in \left[0;1\right],\alpha \left(s\right)=0\mathrm{if}s\le 0\mathrm{or}s\ge 1;\hfill \\ \overline{f}\left(s\right)=f\left(s\right)\mathrm{if}s\in \left[0;1\right],f\left(s\right)=0\mathrm{if}s\le 0,f\left(s\right)=f\left(1\right)\mathrm{if}s\ge 1;\hfill \\ \overline{k}\left(s\right)=k\left(s\right)\mathrm{if}s\in \left[0;1\right],k\left(s\right)=0\mathrm{if}s\le 0\mathrm{or}s\ge 1;\hfill \\ \overline{\kappa }\left(s\right)=\kappa \left(s\right)\mathrm{if}s\in \left[0;1\right],\kappa \left(s\right)=\kappa \left(0\right)\mathrm{if}s\le 0,\kappa \left(s\right)=\kappa \left(1\right)\mathrm{if}s\ge 1.\hfill \end{array}$$

Remark 1.

The preceding extension is not necessarily of class $C^1$ over $\mathbb{R}$. To have an extension of class $C^1$ on $\mathbb{R}$ with a bounded derivative on all $\mathbb{R}$, we can proceed as follows. Let h be a real function of class $C^1$ on the compact interval $[a;b]$. Let us first extend by continuity the derivative $h^{\prime }$ to all $\mathbb{R}$ and that by setting

$$\overline{h^{\prime }}\left(r\right)=\left\{\begin{array}{c}{h}^{\prime }\left(a\right)ifr\le a,\\ h^{\prime }\left(r\right)ifa<r<b,\\ h^{\prime }\left(b\right)ifr\ge b.\end{array}\right.$$

We can then verify that the function

$$\overline{h}\left(r\right)=\underset{a}{\overset{r}{\int }}\overline{h^{\prime }}\left(\xi \right)d\xi$$

is an extension of h to all $\mathbb{R}$ and it is of class $C^1$ on $\mathbb{R}$ with $\underset{\mathbb{R}}{sup}\left|{\left(\overline{h}\right)}^{\prime }\right|=\underset{\left[a;b\right]}{max}\left|h^{\prime }\right|$, so that $\overline{h}$ is Lipschitz on $\mathbb{R}$.

4.1. Galerkin’s Method for the Saturation Equation

We assume here that $\mathbf{u}\in L^{\infty }\left(0,T;L^2{(\Omega )}^3\right)$. Let ${\left\{v_n\right\}}_{n\ge 1}$ be a Hilbertian basis of $H_{{\Gamma }_e}^1\left(\Omega \right)$, orthogonal for the dot product of $L^2(\Omega )$. Let $H_N$ be the subspace generated by the first N functions of this basis, $H_N=\left[v_1,v_2,\cdots ,v_N\right]$. We will look for an approximate solution of the saturation equation in the following form:

$$s_N\left(x,t\right)=\sum _{i=1}^N{\xi }_i\left(t\right)v_i\left(x\right),(x,t)\in {\Omega }_T,$$

with functions $\xi ,i=1,\dots ,N$, of class $C^1$ on $\left[0,T\right]$. So, $s_N\in C^1\left(\left[0,T\right];H_N\right)$. The function $s_N$ must verify the same integral identity as s but on $H_N$ and the “condition initial”:

$$\left(G_N\right)\left\{\begin{array}{c}\underset{\Omega }{\int }\varphi {\displaystyle \frac{d{s}_N}{dt}}{\phi }_{N}dx-\underset{\Omega }{\int }f\left(s_N\right)\mathbf{u}·\nabla {\phi }_{N}dx\\ +\underset{\Omega }{\int }\mathbf{K}{\alpha }_{\eta }\left(s_N\right)\nabla s_N·\nabla {\phi }_{N}dx\\ -\underset{\Omega }{\int }\mathbf{K}k\left(s_N\right)\mathbf{g}·\nabla {\phi }_{N}dx\\ +\underset{{\Gamma }_s}{\int }f\left(s_N\right){\phi }_N{q}_{s}d\gamma =0,\forall {\phi }_N\in H_N\\ \begin{array}{c}{s}_N\left(0\right)={\mathbb{P}}_N\left(s_N\right),{\mathbb{P}}_N=\mathrm{projection} \mathrm{orthogonal} \mathrm{on}{H}_N\hfill \end{array},\end{array}\right.$$

where projection ${\mathbb{P}}_N$ on $H_N$ is taken with respect to the scalar product of $L^2$.

To verify the integral identity in $\left(G_N\right)$, it suffices to verify it on the basis functions $H_N$, i.e., for ${\phi }_N=v_i,i=i,\dots ,N$. With this choice of functions ${\phi }_N$, the problem $\left(G_N\right)$ is equivalent to the Cauchy problems for the system of first order differential equations

$${\displaystyle \frac{d\xi }{dt}}=F\left(\xi \right),\xi \left(t\right)={\left({\xi }_i\left(t\right)\right)}_{i=1}^N,$$

where F is a continuous function with respect to $\xi \in {\mathbb{R}}^N$. The Cauchy–Peano theorem guarantees the local existence of a solution of class $C^1$ in time defined on an interval $\left[0,T_N\right]$ with $0<T_N\le T$. To show that $T_N=T$, we use the theorem on the finite time explosion of ordinary differential equations. To show that there is no explosion in finite time, it is necessary to establish energy estimates on $s_N$, independent of N. This finite time explosion theorem then guarantees that we have $T_N=T$.

4.2. Energy Estimates

Let us take $\left(G_N\right)$ as test function ${\phi }_N=s_N$.

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}\varphi {\displaystyle \frac{d}{dt}}{∥s_N∥}_{L^2(\Omega )}^2& +\underset{\Omega }{\int }\mathbf{K}{\alpha }_{\eta }\left(s_N\right){∥\nabla s_N∥}^{2}dx\hfill \\ \hfill & \le \underset{\Omega }{\int }\left|f\left(s_N\right)\mathbf{u}·\nabla s_N\right|dx+k_{\infty }\underset{\Omega }{\int }\left|k\left(s_N\right)\mathbf{g}·\nabla s_N\right|dx+\underset{{\Gamma }_s}{\int }\left|f\left(s_N\right)s_N{q}_s\right|dx.\hfill \end{array}$$

Then,

$$\left|f\left(s\right)\right|\le 1;\left|k\left(s\right)\right|\le \overline{k}=\underset{[0,1]}{max}\left|k\right|;{\alpha }_{\eta }\left(s\right)\ge \eta ,\forall s\in \mathbb{R}.$$

Now, we write

• $\left|\underset{\Omega }{\int }f\left(s_N\right)\mathbf{u}·\nabla s_{N}dx\right|\le {∥\mathbf{u}∥}_{L^2(\Omega )}{∥\nabla s_N∥}_{L^2(\Omega )},$
• $\left|\underset{\Omega }{\int }f\left(s_N\right)\mathbf{u}·\nabla s_{N}dx\right|\le \overline{k}\underset{\Omega }{\int }\left|\nabla s_N\right|dx\le \overline{k}{\left|\Omega \right|}^{1/\phantom{12}2}{∥\nabla s_N∥}_{L^2(\Omega )},$
• $\left|\underset{{\Gamma }_s}{\int }f\left(u_N\right)q_s{s}_{N}d\gamma \right|\le {∥q_s∥}_{L^2\left({\Gamma }_s\right)}{∥s_N∥}_{L^2\left({\Gamma }_s\right)}\le C{∥q_s∥}_{L^2\left({\Gamma }_s\right)}{∥\nabla s_N∥}_{L^2(\Omega )}.$

The last inequality was obtained using the continuity of the map trace. Therefore,

$${\displaystyle \frac{1}{2}}{\varphi }_{*}{\displaystyle \frac{d}{dt}}{∥s_N∥}_{L^2(\Omega )}^2+\eta {∥\nabla s_N∥}_{L^2(\Omega )}^2\le \gamma \left(t\right){∥\nabla s_N∥}_{L^2(\Omega )},$$

(30)

with

$$\gamma \left(t\right)={∥\mathbf{u}∥}_{L^2(\Omega )}+\overline{k}{\left|\Omega \right|}^{1/\phantom{12}2}+C{∥q_s∥}_{L^2\left({\Gamma }_s\right)},$$

and we have by hypothesis $\underset{t\in \left[0,T\right]}{sup}\gamma \left(t\right)\le \overline{\gamma }$. Let us now recall the Cauchy inequality in “$\epsilon$”: for all positive real numbers a and b, we have

$$ab\le {\displaystyle \frac{\epsilon }{2}}{a}^2+{\displaystyle \frac{1}{2\epsilon }}{b}^2,\forall \epsilon >0.$$

For $a=\overline{\gamma },b={∥\nabla s_N∥}_{L^2(\Omega )}\mathrm{and}\epsilon ={\displaystyle \frac{\eta }{2}}$, we obtain the following inequality:

$$\overline{\gamma }\Vert \nabla s_N{\Vert }_{L^2}\le {\displaystyle \frac{{\overline{\gamma }}^2}{2\eta }}+{\displaystyle \frac{\eta }{2}}{\Vert \nabla s_N\Vert }_{L^2(\Omega )}^2.$$

Using this estimate in the (30) inequality, we obtain

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{∥s_N∥}_{L^2(\Omega )}^2+{\displaystyle \frac{\eta }{2}}{∥\nabla s_N∥}_{L^2(\Omega )}^2\le {\displaystyle \frac{{\overline{\gamma }}^2}{2\eta }}.$$

By integrating over the time interval $\left[0,T\right]$ with $t\le T_n\le T$, we get the following:

$${\displaystyle \frac{1}{2}}{∥s_N\left(t\right)∥}_{L^2(\Omega )}^2+{\displaystyle \frac{\eta }{2}}\underset{0}{\overset{t}{\int }}{∥\nabla s_N\left(t\right)∥}_{L^2(\Omega )}^{2}dt\le {\displaystyle \frac{{\overline{\gamma }}^2}{2\eta }}T+{\displaystyle \frac{1}{2}}{∥s_N\left(0\right)∥}_{L^2(\Omega )}^2.$$

(31)

But, $s_N\left(0\right)={\mathbb{P}}_N{s}_0$ with ${\mathbb{P}}_N$, and the projection operator on $H_N$ in the Hilbert space $L^2(\Omega )$. So, ${∥{\mathbb{P}}_N∥}_{{\left(L^2\right)}^{\prime }}=1$, and this implies that

$${∥s_N\left(0\right)∥}_{L^2(\Omega )}^2\le {∥{\mathbb{P}}_{\mathbb{N}}∥}_{L^2}{∥s_0∥}_{L^2}={∥s_0∥}_{L^2(\Omega )}.$$

Therefore, we proved that

$${∥s_N\left(t\right)∥}_{L^2(\Omega )}^2+\eta \underset{0}{\overset{t}{\int }}{∥\nabla s_N\left(t\right)∥}_{L^2(\Omega )}^{2}dt\le {\displaystyle \frac{{\overline{\gamma }}^2}{\eta }}T+{∥s_0∥}_{L^2(\Omega )}^2,\forall t\in \left[0,T_N\right].$$

(32)

This estimate shows that the norms of approximate solutions $s_N$ in $L^2(\Omega )$ are bound independently of N. So, the solutions are global, i.e., $T_N=T$. The estimate (32) also shows that, for all $N\in \mathbb{N}$, we have

$$\underset{t\in \left[0,T\right]}{sup}{∥s_N∥}_{L^2(\Omega )}\le C,$$

(33)

$$\eta {∥s_N∥}_{L^2\left(0,T;H_{{\Gamma }_e}^1(\Omega )\right)}^2\le C,$$

(34)

with C as a positive constant independent of N.

For the compactness of the sequence of approximate solutions in an adequate space, we need to establish a uniform estimate (independent of N) of ${\partial }_t{s}_N$.

Let $\phi \in L^2\left(0,T;H_{{\Gamma }_e}^1(\Omega )\right)$. As $s_N\in C^1\left(0,T;H_N\right)$, then we have $s_N\left(t\right)\in H_N$ and ${\displaystyle \frac{d{s}_N}{dt}}\in H_N$.

For $t\in \left]0,T\right[$, decompose $\phi \left(t\right)$ using ${\mathbf{P}}_N$. Then, the orthogonal projection on $H_N$ in the Hilbert space $H=H_{{\Gamma }_e}^1(\Omega )$ is as follows:

$$\phi \left(t\right)={\mathbf{P}}_N\phi \left(t\right)+{\mathbf{Q}}_N\phi \left(t\right),{\mathbf{Q}}_{N}i{d}_{H_{{\Gamma }_e}^1}-{\mathbf{P}}_N,$$

where $i{d}_{H_{{\Gamma }_e}^1}$ is the identity map of $H_{{\Gamma }_e}^1(\Omega )$. The assistant operator ${}_{ }{}^{t}P_N^{ }$ is a continuous linear operator on $H^{\prime }$ of norm $\le 1$, giving us the following:

$${〈\varphi {\displaystyle \frac{d{s}_N}{dt}}\left(t\right),\phi \left(t\right)〉}_{H^{\prime },H}={〈\varphi {\displaystyle \frac{d{s}_N}{dt}}\left(t\right),{\mathbf{P}}_N\phi \left(t\right)+{\mathbf{Q}}_N\phi \left(t\right)〉}_{H^{\prime },H}={〈\varphi {\displaystyle \frac{d{s}_N}{dt}}\left(t\right),{\mathbf{P}}_N\phi \left(t\right)〉}_{H^{\prime },H}.$$

We can then use the equation $\left(G_N\right)$ integrated in time. Using hypotheses $\left(H5\right)$ and $\left(H6\right)$ and the Cauchy–Schwartz inequality, we obtain the following:

$$\begin{array}{cc}\hfill {\varphi }_{*}\left|\underset{0}{\overset{T}{\int }}\left({\displaystyle \frac{d{s}_N}{dt}},{\mathbf{P}}_N\phi \right)dt\right|& \le {∥\mathbf{u}∥}_{L_t^2\left(L_x^2\right)}{∥\nabla \left({\mathbf{P}}_N\phi \right)∥}_{L_t^2\left(L_x^2\right)}\hfill \\ \hfill & +k_{\infty }(\overline{\alpha }+\eta ){∥\nabla s_N∥}_{L^2\left(L^2\right)}{∥\nabla \left({\mathbf{P}}_N\phi \right)∥}_{L_t^2\left(L_x^2\right)}\hfill \\ \hfill & +k_{\infty }\overline{k}{∥\nabla \left({\mathbf{P}}_N\phi \right)∥}_{L_t^1\left(L_x^1\right)}+{∥q_s∥}_{L_t^{\infty }\left(L^2\left({\Gamma }_s\right)\right)}{∥{\mathbf{P}}_N\phi ∥}_{L_t^2\left(L^2\left({\Gamma }_s\right)\right)}.\hfill \end{array}$$

We have ${∥{\mathbf{P}}_N∥}_{H_{{\Gamma }_e}^1}\le 1$. Then,

$$\left|\underset{0}{\overset{T}{\int }}〈\varphi {\displaystyle \frac{d{s}_N}{dt}},\phi 〉dt\right|\le C{∥\phi ∥}_{L^2\left(H_{{\Gamma }_e}^1\right)},\forall \phi \in L^2\left(0,T;H_{{\Gamma }_e}^1(\Omega )\right),$$

where C is a constant independent of N, which can depend on $\eta$. In conclusion, the following is true:

$${\left\{{\displaystyle \frac{d{s}_N}{dt}}\right\}}_{N=1}^{\infty }\begin{array}{c}\mathrm{remains} \mathrm{bounded} \mathrm{to}\hfill \end{array}{L}^2\left(0,T;{\left(H_{{\Gamma }_e}^1(\Omega )\right)}^{\prime }\right).$$

(35)

4.3. Passing the Limit with Respect to N at Fixed $\eta >0$

We will have to use a compactness result from Jacques Simon [23] (Page 85, Corollary 4) of which here is a translation:

Proposition 1.

(Simon) Let X, B, and Y be three Banach spaces with X injecting compactly into B and the injection of this space into Y continues. We note

$$X\underset{compact}{\hookrightarrow }B\underset{continuous}{\hookrightarrow }Y.$$

• Let F be a bounded subset in $L^q\left(0,T;X\right)$ with $1\le q<\infty$, such that

  $${\displaystyle \frac{\partial F}{\partial t}}=\left\{{\displaystyle \frac{\partial f}{\partial t}}\mid f\in F\right\}$$

  can be bounded in $L^1\left(0,T;Y\right)$. Then, F is relatively compact in $L^q\left(0,T;B\right)$.
• Let F be a bounded subset in $L^{\infty }\left(0,T;X\right)$, such that ${\displaystyle \frac{\partial F}{\partial t}}$ can be bounded in $L^r\left(0,T;Y\right)$ with $r>1$. Then, F is relatively compact in $C\left(0,T;B\right)$.
  Let us recall the following classic injections:

  $$H_{{\Gamma }_e}^1\left(\Omega \right)\underset{compact}{\hookrightarrow }{L}^2\left(\Omega \right)\underset{continuous}{\hookrightarrow }{\left(H_{{\Gamma }_e}^1\left(\Omega \right)\right)}^{\prime }.$$

It is then possible to use the previous 1 proposition with $q=2, X=H_{{\Gamma }_e}^1\left(\Omega \right), B=L^2\left(\Omega \right)$ and $Y={\left(H_{{\Gamma }_e}^1\left(\Omega \right)\right)}^{\prime }$. So, the injection

$$v\in L^2\left(0,T;H_{{\Gamma }_e}^1\left(\Omega \right)\right)|v^{\prime }={\displaystyle \frac{dv}{dt}}\in L^2\left(0,T;{\left(H_{{\Gamma }_e}^1\left(\Omega \right)\right)}^{\prime }\right)$$

in $L^2\left(0,T;L^2\left(\Omega \right)\right)$ is compact, see [24,25]. For fixed $\eta >0$, using the estimates (34) and (35), we see that the sequence of approximate solutions ${\left\{s_N\right\}}_{N=1}^{\infty }$ is relatively compact in $L^2\left(0,T;L^2\left(\Omega \right)\right)$. So, to extract a subsequence, we assume that

$$s_N\to s\mathrm{strongly}in{L}^2\left(0,T;L^2\left(\Omega \right)\right)anda.e.\mathrm{in}{\Omega }_T$$

Then,

$$\begin{array}{c}{s}_N\to s\mathrm{in}{L}^2\left(0,T;H_{{\Gamma }_e}^1\left(\Omega \right)\right)\mathrm{weakly},s_N\to sin{L}^2\left(0,T;{\left(H_{{\Gamma }_e}^1\left(\Omega \right)\right)}^{\prime }\right)\mathrm{weakly}.\hfill \end{array}$$

We get the following:

$$s_N\to sin{L}^2\left(0,T;L^2\left(\partial \Omega \right)\right).$$

We can then pass to the limit in the weak formulation $\left(G_N\right)$. Indeed, let ${\phi }_k\in H_N$, k fixed, $k\le N$, and let $\psi =\psi \left(t\right)\in \mathcal{D}\left(\left]0,T\right[\right)$. Multiply $\left(G_N\right)$ and integrate from 0 to T.

We obtain

$$\begin{array}{cc}\hfill \underset{0}{\overset{T}{\int }}〈\varphi {\displaystyle \frac{d{s}_N}{dt}},{\phi }_k〉\psi \left(t\right)dt& -\underset{{\Omega }_T}{\int }f\left(s_N\right)\mathbf{u}·\nabla {\phi }_k\psi \left(t\right)dxdt\hfill \\ \hfill & +\underset{{\Omega }_T}{\int }\mathbf{K}{\alpha }_{\eta }\left(s_N\right)\nabla s_N·\nabla {\phi }_k\psi \left(t\right)dxdt\hfill \\ \hfill & -\underset{{\Omega }_T}{\int }\mathbf{K}k\left(s_N\right)\mathbf{g}·\nabla {\phi }_k\psi \left(t\right)dxdt\hfill \\ \hfill & +\underset{0}{\overset{T}{\int }}\underset{{\Gamma }_s}{\int }f\left(s_N\right){\phi }_k{q}_s\psi \left(t\right)d\gamma dt.\hfill \end{array}$$

Keeping k fixed, let N tend to veer toward infinity. Then, we have (a), since ${\partial }_t{s}_N\rightharpoonup {\partial }_{t}s$ in $L^2\left(0,T;V^{\prime }\right)$ (with $V=H_{{\Gamma }_e}^2\left(\Omega \right)$); hence,

$$\underset{0}{\overset{T}{\int }}{〈\varphi {\displaystyle \frac{d{s}_N}{dt}},{\phi }_k\psi 〉}_{V^{\prime },V}dt⟶\underset{0}{\overset{T}{\int }}{〈\varphi {\displaystyle \frac{ds}{dt}},{\phi }_k\psi 〉}_{V^{\prime },V}dt.$$

(b) As $s_N\to s$ a.e. in ${\Omega }_T$ and f continue, we have $f\left(s_N\right)\to f\left(s\right)$ a.e. in ${\Omega }_T$ and $\left|f(·)\right|\le 1$. After using the convergence theorem is dominated, we can pass to the limit, and we obtain

$$\underset{N\to \infty }{lim}\underset{{\Omega }_T}{\int }f\left(s_N\right)\mathbf{u}·\nabla {\phi }_k\psi \left(t\right)dxdt=\underset{{\Omega }_T}{\int }f\left(s\right)\mathbf{u}·\nabla {\phi }_k\psi \left(t\right)dxdt.$$

(c) In the same way as in the previous point, we see that

$$\underset{n\to \infty }{lim}\underset{{\Omega }_T}{\int }\mathbf{K}k\left(s_N\right)\mathbf{g}·\nabla {\phi }_k\psi \left(t\right)dxdt=\underset{{\Omega }_T}{\int }\mathbf{K}k\left(s\right)\mathbf{g}·\nabla {\phi }_k\psi \left(t\right)dxdt.$$

(d) On the one hand, we have $\nabla s_N\rightharpoonup \nabla s$ in $L^2\left(0,T;L^2\left(\Omega \right)\right)$ low. On the other hand, we have ${\alpha }_{\eta }\left(s_N\right)\to {\alpha }_{\eta }\left(s\right)$ a.e. in ${\Omega }_T$ and ${\alpha }_{\eta }$ bounded, so (in denoting by d the dimension of the ambient space of $\Omega$)

$$\mathbf{K}{\alpha }_{\eta }\left(s_N\right)\nabla {\phi }_k\psi \left(t\right)\to \mathbf{K}{\alpha }_{\eta }\left(s\right)\nabla {\phi }_k\psi \left(t\right)\mathrm{in}{L}^2{\left(0,T;L^2\left(\Omega \right)\right)}^d\mathrm{strongly},$$

since

$$\underset{N\to \infty }{lim}\underset{{\Omega }_T}{\int }\mathbf{K}{\alpha }_{\eta }\left(s_N\right)\nabla s_N·\nabla {\phi }_k\psi \left(t\right)dxdt+\underset{{\Omega }_T}{\int }\mathbf{K}{\alpha }_{\eta }\left(s\right)\nabla s_N·\nabla {\phi }_k\psi \left(t\right)dxdt.$$

(e) To pass to the limit in the “border” term, $\underset{0}{\overset{T}{\int }}\underset{{\Gamma }_s}{\int }f\left(s_N\right){\phi }_k{q}_s\psi \left(t\right)d\gamma dt$, we use the fact that (for example) $H^1\left(\Omega \right)$ injects compactly into $H^{3/\phantom{34}4}\left(\Omega \right)$ and the continuity of the trace application of this space in $H^{1/\phantom{14}4}\left(\Gamma \right)$. For regular $\Gamma$, these results are given, for example, in Lions and Magenes [26], pages 47 and 110. We deduce from this that we have

$$s_N\to s\mathrm{in}{L}^2\left(0,T;L^2\left(\partial \Omega \right)\right)\mathrm{strongly}\mathrm{and}\mathrm{a}.\mathrm{e}.\mathrm{on}\left(0,T\right)\times \Gamma .$$

This result allows us to pass the limit to see that we have

$$\underset{N\to \infty }{lim}\underset{0}{\overset{T}{\int }}\underset{{\Gamma }_s}{\int }f\left(s_N\right){\phi }_k{q}_s\psi \left(t\right)d\gamma dt=\underset{0}{\overset{T}{\int }}\underset{{\Gamma }_s}{\int }f\left(s\right){\phi }_k{q}_s\psi \left(t\right)d\gamma dt.$$

Finally, for $\phi \in H_{{\Gamma }_e}^1\left(\Omega \right),$ by setting ${\phi }_k={\mathbb{P}}_k\phi$, orthogonal projection from $\phi$ on the space generated by the functions $v_1,\cdots ,v_k$, we see that $\underset{k\to \infty }{lim}{\phi }_k=\phi$ in $H_{{\Gamma }_e}^1\left(\Omega \right)$ because the family ${\left\{v_k\right\}}_k$ is total in this space. This makes it possible to replace passages in the limit preceding ${\phi }_k$ by any function $\phi$ of $H_{{\Gamma }_e}^1\left(\Omega \right)$. Finally, a conclusion is achieved using the fact that the functions $\left\{\varphi \left(t\right)\phi \left(x\right)\right\}$ with $\psi \left(t\right)\in \mathcal{D}\left(\left]0,T\right[\right)$ and $\phi \left(x\right)\in H_{{\Gamma }_e}^1\left(\Omega \right)$ are dense in $L^2\left(0,T;H_{{\Gamma }_e}^1\left(\Omega \right)\right)$.

In summary, we have shown the existence of a function $s_{\eta }$, such that

$$\begin{array}{c}\hfill s_{\eta }\in C^0\left(0,T;L^2\left(\Omega \right)\right)\cap L^2\left(0,T;H_{{\Gamma }_e}^1\left(\Omega \right)\right) \mathrm{with} {\partial }_{t}s\in L^2\left(0,T;H^1(\Omega )\right),\end{array}$$

verifying

$$\begin{array}{c}\hfill {〈\varphi {\partial }_t{s}_{\eta },\phi 〉}_{L^2\left(0,T;{\left(H_{{\Gamma }_e}^1\right)}^{\prime }\right),L^2\left(0,T;H_{{\Gamma }_e}^1\right)}-\underset{{\Omega }_T}{\int }f\left(s_{\eta }\right)\mathbf{u}·\nabla \phi dxdt\\ \hfill +\underset{{\Omega }_T}{\int }\mathbf{K}{\alpha }_{\eta }\left(s_{\eta }\right)\nabla s_{\eta }·\nabla \phi dxdt-\underset{{\Omega }_T}{\int }\mathbf{K}k\left(s_{\eta }\right)\mathbf{g}·\nabla \phi dxdt\\ \hfill +\underset{0}{\overset{T}{\int }}\underset{{\Gamma }_s}{\int }f\left(s_{\eta }\right)\phi q_{s}d\gamma dt=0\forall \phi \in L^2\left(0,T;H_{{\Gamma }_e}^1\left(\Omega \right)\right).\end{array}$$

4.4. Pressure Equation

We prove the existence of a weak solution for following problem:

$$\left\{\begin{array}{c}\mathrm{div}{\mathbf{u}}_{\eta }-\kappa \left(s\right)\mathrm{div}(M_w\left(s_{\eta }\right)\nabla {\partial }_t{s}_{\eta })=0,\\ {\mathbf{u}}_{\eta }=-M\mathbf{K}\left(s_{\eta }\right)\left[\nabla p_{\eta }-\rho \left(s_{\eta }\right)\mathbf{g}\right] \mathrm{in} \Omega ,\mathrm{a}.\mathrm{e}.t\in \left]0,T\right[,\\ {\mathbf{u}}_{\eta }·\mathbf{n}=-q_e \mathrm{on} {\Gamma }_e,{\mathbf{u}}_{\eta }·\mathbf{n}=q_s \mathrm{on} {\Gamma }_s \mathrm{and} {\mathbf{u}}_{\eta }·\mathbf{n}=0 \mathrm{on} {\Gamma }_i.\end{array}\right.$$

Proposition 2.

Let $s:\Omega \times ]0;\left]0,T\right[\to \mathbb{R}$ be a given bounded function. Then, if the compatibility condition

$$\underset{{\Gamma }_e}{\int }{q}_e\left(x,t\right)d\gamma \left(x\right)-\underset{{\Gamma }_s}{\int }{q}_s\left(x,t\right)d\gamma \left(x\right)=0,a.e.t\in \left]0,T\right[$$

is verified, there is only one function $p\in L^{\infty }\left(0,T;H_{moy}^1\left(\Omega \right)\right)$ solution of

$$\begin{array}{c}\underset{\Omega }{\int }M\mathbf{K}\left(s\left(x,t\right)\right)\left[\nabla p-\rho \left(s\right)\mathbf{g}\right]·\nabla \psi dx=\underset{\Omega }{\int }\kappa \left(s\right)M_w\left(s\right)\nabla {\partial }_{t}s·\nabla \psi dx\hfill \\ +\underset{{\Gamma }_e}{\int }{q}_e\left(x,t\right)\psi d\gamma \left(x\right)-\underset{{\Gamma }_s}{\int }{q}_s\left(x,t\right)\psi d\gamma \left(x\right),\forall \psi \in H^1\left(\Omega \right)a.e.t\in \left]0,T\right[\hfill \end{array}.$$

(36)

Proof. 

For fixed $t\in \left]0,T\right[$, we apply the Lax–Milgram theorem on the Hilbert space $H_{moy}^1\left(\Omega \right)$.

The Poincaré inequality is true on this space, which allows us to obtain the following:

$$H_{moy}^1\left(\Omega \right)\times H_{moy}^1\left(\Omega \right)\ni \left(\xi ,\psi \right)\mapsto a\left(\xi ,\psi \right)=\underset{\Omega }{\int }\mathbf{K}M\left(s\left(x,t\right)\right)\nabla \xi ·\nabla \psi dx\in \mathbb{R},$$

where the bilinear form $a(·,·)$ is continuous, and

$$\begin{array}{cc}\hfill \left|a\left(\xi ,\psi \right)\right|& \le C{∥\nabla \xi ∥}_{H^1}{∥\nabla \psi ∥}_{H^1},\hfill \end{array}$$

where C is a positive constant depending on $k_{\infty }$ and $\underset{t\in \left[0,T\right]}{sup}M\left(s\left(x,t\right)\right)$.

The bilinear form $a(·,·)$ is coercive:

$$\begin{array}{cc}\hfill a\left(\xi ,\xi \right)& \ge C_1{∥\xi ∥}_{H^1}^2,\hfill \end{array}$$

where $C_1$ is a positive constant depending on $k_0$, ${\rho }_o$, ${\rho }_w$, and the measure of $\Omega$.

Hence, $a(·,·)$ is coercive, and the linear form ℓ is continuous:

$$\begin{array}{ccc}\hfill H_{moy}^1\left(\Omega \right)\ni \psi \mapsto \ell \left(\psi \right)& =& \underset{\Omega }{\int }\mathbf{K}M\left(s\left(x,t\right)\right)\rho \left(s\right)\mathbf{g}·\nabla \psi dx+\kappa \left(s_w\right)\underset{\Omega }{\int }{M}_w\left(s\right)\nabla {\partial }_{t}s·\nabla \psi dx\hfill \\ & & +\underset{{\Gamma }_e}{\int }{q}_e\left(x,t\right)\psi d\gamma \left(x\right)-\underset{{\Gamma }_s}{\int }{q}_s\left(x,t\right)\psi d\gamma \left(x\right)\in \mathbb{R}.\hfill \end{array}$$

There is then a single function $p=p\left(.,t\right)\in H_{moy}^1\left(\Omega \right)$, verifying

$$a\left(p,\psi \right)=\ell \left(\psi \right),\forall \psi \in H_{moy}^1\left(\Omega \right).$$

In fact, we have

$$a\left(p,\psi \right)=\ell \left(\psi \right),\forall \psi \in H^1\left(\Omega \right).$$

To see this, for $\psi$ belonging to $H^1\left(\Omega \right)$, we notice that $\psi -{\displaystyle \frac{1}{\left|\Omega \right|}}\underset{\Omega }{\int }\psi dx$ is in $H_{moy}^1\left(\Omega \right)$; taking this as a test function, we obtain $a\left(p,\psi -{\displaystyle \frac{1}{\left|\Omega \right|}}\underset{\Omega }{\int }\psi dx\right)=a\left(p,\psi \right)$ and

$$\begin{array}{ccc}\hfill \ell \left(\psi -{\displaystyle \frac{1}{\left|\Omega \right|}}\underset{\Omega }{\int }\psi dx\right)& =& \underset{\Omega }{\int }\mathbf{K}M\left(s\right)\rho \left(s\right)\mathbf{g}·\nabla \psi dx-\underset{\Omega }{\int }\kappa \left(s\right)M_w\left(s\right)\nabla {\partial }_{t}s·\nabla \psi dx\hfill \\ & & +\underset{{\Gamma }_e}{\int }{q}_e\left(\psi -{\displaystyle \frac{1}{\left|\Omega \right|}}\underset{\Omega }{\int }\psi dx\right)d\gamma -\underset{{\Gamma }_s}{\int }{q}_s\left(\psi -{\displaystyle \frac{1}{\left|\Omega \right|}}\underset{\Omega }{\int }\psi dx\right)d\gamma \hfill \\ & =& \underset{\Omega }{\int }\mathbf{K}M\left(s\right)\rho \left(s\right)\mathbf{g}·\nabla \psi dx-\kappa \left(s_w\right)\underset{\Omega }{\int }{M}_w\left(s\right)\nabla {\partial }_{t}s·\nabla \psi dx\hfill \\ & & +\underset{{\Gamma }_e}{\int }{q}_e\psi d\gamma -\underset{{\Gamma }_s}{\int }{q}_s\psi d\gamma \hfill \\ & =& \ell \left(\psi \right).\hfill \end{array}$$

The penultimate equality results from the compatibility condition. To complete the proof of the proposition, it remains to prove that $p\in L^{\infty }\left(0,T;H_{moy}^1\left(\Omega \right)\right)$. Take in the integral identity (36) $\psi =P\left(.,t\right)$. Therefore (for simplicity we do not write the variable $\left(x,t\right)$,

$$\underset{\Omega }{\int }M\left(s\right){\left|\nabla p\right|}^{2}dx=\underset{\Omega }{\int }\mathbf{K}\rho \left(s\right)\mathbf{g}·\nabla pdx+\underset{\Omega }{\int }\kappa \left(s\right)M_w\left(s\right)\nabla {\partial }_{t}s·\nabla pdx+\underset{{\Gamma }_e}{\int }{q}_{e}pd\gamma -\underset{{\Gamma }_s}{\int }{q}_{s}pd\gamma .$$

□

4.5. End of the Proof of Theorem 2

We will build an iterative method to show the existence of a solution for the complete “saturation–pressure” problem. Let $s_0\in L_2(\Omega )$. We build the sequence ${\left\{\left(p_n,s_n\right)\right\}}_n$ solution of the system as follows:

$$\left\{\begin{array}{c}\mathrm{div}{\mathbf{u}}_n-\kappa \left(s_w\right)\mathrm{div}(M_w\left(s_n\right)\nabla {\partial }_t{s}_n)=0,{\mathbf{u}}_n=-M\left(s_{n-1}\right)\mathbf{K}\left[\nabla p_n-\rho \left(s_{n-1}\right)\mathbf{g}\right]\\ {\partial }_t{s}_n+\mathrm{div}\left(f\left(s_n\right){\mathbf{u}}_n\right)-\mathrm{div}\left[\mathbf{K}\left({\alpha }_n\left(s_n\right)\nabla s_n-k\left(s_n\right)\mathbf{g}\right)\right]=0\\ \mathrm{with}\mathrm{the}\mathrm{conditions}\left(BC\right)+\left(IC\right)\mathrm{corresponding}.\end{array}\right.$$

We prove, as before, that the following is true:

$${\left\{p_n\right\}}_{n\ge 1} \mathrm{is} \mathrm{bounded} \mathrm{in} L^{\infty }\left(0,T;H_{moy}^1\left(\Omega \right)\right),$$

$${\left\{{\mathbf{u}}_n\right\}}_{n\ge 1} \mathrm{is} \mathrm{bounded} \mathrm{in} L^{\infty }\left(0,T;L^2\left(\Omega \right)\right),$$

$${\left\{s_n\right\}}_{n\ge 1} \mathrm{is} \mathrm{bounded} \mathrm{in} L^2\left(0,T;H_{{\Gamma }_e}^1\left(\Omega \right)\right),$$

$${\left\{{\partial }_t{s}_n\right\}}_{n\ge 1} \mathrm{is} \mathrm{bounded} \mathrm{in} L^2\left(0,T;{\left(H_{moy}^1\left(\Omega \right)\right)}^{\prime }\right),$$

$${\left\{\nabla {\partial }_t{s}_n\right\}}_{n\ge 1} \mathrm{is} \mathrm{bounded} \mathrm{in} L^2\left(0,T;\left(L^2(\Omega )\right)\right).$$

Up to extracted subsequences, we have the following:

$$\nabla p_n\rightharpoonup \nabla p \mathrm{in} L^{\infty }\left(0,T;L^2\left(\Omega \right)\right) \mathrm{weak},$$

$$s_n\rightharpoonup s \mathrm{in} L^2\left(0,T;H_{{\Gamma }_e}^1\left(\Omega \right)\right) \mathrm{weak},$$

$${\partial }_t{s}_n\rightharpoonup {\partial }_{t}s \mathrm{in} L^2\left(0,T;{\left(H_{{\Gamma }_e}^1\left(\Omega \right)\right)}^{\prime }\right) \mathrm{weak},$$

$$\nabla \left({\partial }_t{s}_n\right)\rightharpoonup \nabla \left({\partial }_{t}s\right) \mathrm{dans} L^2\left(0,T;L^2(\Omega )\right) \mathrm{weak},$$

$$s_n⟶s \mathrm{in} C^0\left(\left[0,T\right];L^2\left(\Omega \right)\right) \mathrm{strong},$$

$${\mathbf{u}}_n\rightharpoonup \mathbf{w} \mathrm{in} L^{\infty }\left(0,T;L^2\left(\Omega \right)\right) \mathrm{weak}-*.$$

Let us start by identifying $\mathbf{w}$. We obtain the following:

• BY using Lebesgue, $M\left(s_{n-1}\right)\to M\left(s\right)\mathrm{and}\rho \left(s_{n-1}\right)\to \rho \left(s\right)$ in $L^q\left(0,T;L^q\left(\Omega \right)\right)$ for all finite $1\le q$.
• $\nabla p_n\rightharpoonup \nabla p\mathrm{in}{L}^{\infty }\left(0,T;L^2\left(\Omega \right)\right)\mathrm{weak},-*$

According to a previous result, this implies that ${\mathbf{u}}_n\rightharpoonup M\left(s\right)\mathbf{K}\left[\nabla p-\rho \left(s\right)\mathbf{g}\right]=\mathbf{u}$ weakly in $L^2\left(0,T;L^2\left(\Omega \right)\right)$, so $\mathbf{w}=\mathbf{u}$. For convergence for pressure p, let $\psi \in H^1\left(\Omega \right)$. There fore, we have the following:

$$\underset{\Omega }{\int }\mathbf{K}M\left(s_{n-1}\right)\left[\nabla p_n-\rho \left(s_{n-1}\right)\mathbf{g}\right]·\nabla \psi dx-\underset{\Omega }{\int }\kappa \left(s\right)M_w\left(s_{n-1}\right)\nabla {\partial }_t{s}_{n-1}·\nabla \psi dx=\underset{{\Gamma }_e}{\int }{q}_e\psi d\gamma -\underset{{\Gamma }_s}{\int }{q}_s\psi d\gamma .$$

Using Lebesgue and the previous convergences, we see that by letting n tend to infinity we get the following:

$$\underset{\Omega }{\int }M\left(s\right)\mathbf{K}\left[\nabla p-\rho \left(s\right)\mathbf{g}\right]·\nabla \psi dx-\underset{\Omega }{\int }\kappa \left(s\right)M_w\left(s\right)\nabla {\partial }_{t}s·\nabla \psi dx=\underset{{\Gamma }_e}{\int }{q}_e\psi d\gamma -\underset{{\Gamma }_s}{\int }{q}_s\psi d\gamma ,\forall \psi \in H^1\left(\Omega \right).$$

For convergence for saturation s, let $\phi \in L^2\left(0,T;H_{{\Gamma }_e}^1\left(\Omega \right)\right)$. Therefore, we have the following:

$$\begin{array}{cc}\hfill {〈\varphi {\partial }_t{s}_n,\phi 〉}_{L^2\left(0,T;\left({H_{{\Gamma }_e}^1}^{\prime }\right)\right),L^2\left(0,T;H_{{\Gamma }_e}^1\right)}& -\underset{{\Omega }_T}{\int }f\left(s_n\right){\mathbf{u}}_n·\nabla \phi dxdt\hfill \\ \hfill & +\underset{{\Omega }_T}{\int }\mathbf{K}{\alpha }_{\eta }\left(s_n\right)\nabla s_n·\nabla \phi dxdt\hfill \\ \hfill & -\underset{{\Omega }_T}{\int }\mathbf{K}k\left(s_n\right)\mathbf{g}·\nabla \phi dxdt\hfill \\ \hfill & +\underset{0}{\overset{T}{\int }}\underset{{\Gamma }_s}{\int }f\left(s_n\right)\phi q_{s}d\gamma dt=0.\hfill \end{array}$$

The passage to the limit in the term $\underset{{\Omega }_T}{\int }f\left(s_n\right){\mathbf{u}}_n·\nabla \phi dxdt$ is conducted by noting that, thanks to Lebesgue, $f\left(s_n\right)\nabla \phi$ strongly converges in $L^2\left(0,T;L^2\left(\Omega \right)\right)$ and ${\mathbf{u}}_n$ converge weakly to $\mathbf{u}$ in $L^2\left(0,T;L^2\left(\Omega \right)\right)$. The passages to the limit in the other terms are as before. In summary, we have proved the existence of a pair $\left(p,s\right)$ (weak) solution for the following regular system:

$$\left.\begin{array}{c}p\in H_{moy}^1\left(\Omega \right)\\ \underset{\Omega }{\int }\mathbf{K}M\left(s\right)\left[\nabla p-\rho \left(s\right)\mathbf{g}\right]·\nabla \psi dx-\underset{\Omega }{\int }\kappa \left(s\right)M_w\left(s\right)\nabla {\partial }_{t}s·\nabla \psi dx\\ =\underset{{\Gamma }_e}{\int }{q}_e\psi d\gamma -\underset{{\Gamma }_s}{\int }{q}_s\psi d\gamma ,\forall \psi \in H^1\left(\Omega \right)\\ \mathbf{u}=-M\left(s\right)\mathbf{K}\left[\nabla p-\rho \left(s\right)\mathbf{g}\right];\end{array}\right\}\left(pre\right),$$

$$\left.\begin{array}{c}{〈\varphi {\partial }_{t}s,\phi 〉}_{L^2\left(0,T;{\left(H_{{\Gamma }_e}^1\right)}^{\prime }\right),L^2\left(0,T;H_{{\Gamma }_e}^1\right)}-\underset{{\Omega }_T}{\int }f\left(s\right)\mathbf{u}·\nabla \phi dxdt\\ +\underset{{\Omega }_T}{\int }\mathbf{K}{\alpha }_{\eta }\left(s\right)\nabla s·\nabla \phi dxdt-\underset{{\Omega }_T}{\int }\mathbf{K}k\left(s\right)\mathbf{g}·\nabla \phi dxdt\\ +\underset{0}{\overset{T}{\int }}\underset{{\Gamma }_s}{\int }f\left(s\right)\phi q_{s}d\gamma dt=0,\forall \phi \in L^2\left(0,T;H_{{\Gamma }_e}^1\left(\Omega \right)\right)\end{array}\right\}\left(sat\right).$$

5. Proof of Theorem 1 (Irregular Problem)

We have proved the existence of functions depending on $\eta$,

$$p_{\eta }\in L^{\infty }\left(0,T;H_{moy}^1\left(\Omega \right)\right)et{s}_{\eta }\in C^0\left(0,T;L^2\left(\Omega \right)\right)\cap L^2\left(0,T;H_{{\Gamma }_e}^1\left(\Omega \right)\right)$$

with

$${\partial }_t{s}_{\eta }\in L^2\left(0,T;\left(H_{{\Gamma }_e}^1{\left(\Omega \right)}^{\prime }\right)\right)\mathrm{and}0\le s_{\eta }\left(x,t\right)\le 1\mathrm{a}.\mathrm{e}.\mathrm{in}{\Omega }_T$$

system solutions $\left(S_{\eta }\right)$. In what follows, we will establish on these functions independent estimates of $\eta$, which will allow us to make this parameter tend to veer toward zero to obtain a solution for the initial irregular system.

Lemma 1.

(Estimation 1) There is a constant $C>0$ (independent of η) such that

$$\eta {∥\nabla s_{\eta }∥}_{L^2\left({\Omega }_T\right)}\le Cet\eta {∥\sqrt{\alpha }\left(s_{\eta }\right)\nabla s_{\eta }∥}_{L^2\left({\Omega }_T\right)}^2\le C,\forall \eta \in \left]0,1\right[.$$

(37)

Proof. 

We take $\phi =s_{\eta }$ as a test function in the relation (Sat). We obtain the following:

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}\underset{0}{\overset{\tau }{\int }}\varphi {\displaystyle \frac{d}{dt}}{∥s_{\eta }∥}_{L^2\left(\Omega \right)}^{2}dt-\underset{{\Omega }_{\tau }}{\int }f\left(s_{\eta }\right)\mathbf{u}·\nabla s_{\eta }dxdt& +\underset{{\Omega }_{\tau }}{\int }\mathbf{K}\alpha \left(s_{\eta }\right){\left|\nabla s_{\eta }\right|}^{2}dxdt\hfill \\ \hfill +\eta \underset{{\Omega }_{\tau }}{\int }\mathbf{K}{\left|\nabla s_{\eta }\right|}^{2}dxdt& -\underset{{\Omega }_{\tau }}{\int }\mathbf{K}k\left(s_{\eta }\right)\mathbf{g}·\nabla s_{\eta }dxdt\hfill \\ \hfill & +\underset{0}{\overset{\tau }{\int }}\underset{{\Gamma }_s}{\int }f\left(s_{\eta }\right)s_{\eta }{q}_{s}d\gamma dt=0.\hfill \end{array}$$

Now, we use the Hypotheses (H5) and (H6), and using Poincaré and Cauchy–Schwartz inequalities, we get the following:

$$\begin{array}{cc}\hfill {\varphi }_{*}{∥s_{\eta }\left(\tau \right)∥}_{L^2\left(\Omega \right)}^2& +2k_0\underset{{\Omega }_{\tau }}{\int }\alpha \left(s_{\eta }\right){\left|\nabla s_{\eta }\right|}^{2}dxdt+2k_0\eta \underset{{\Omega }_{\tau }}{\int }{\left|\nabla s_{\eta }\right|}^{2}dxdt\hfill \\ \hfill & \le {∥s_0∥}_{L^2\left(\Omega \right)}^2+2{\left(\underset{{\Omega }_{\tau }}{\int }{\left|f\left(s_{\eta }\right)\mathbf{u}\right|}^{2}dxdt\right)}^{{\displaystyle \frac{1}{2}}}{∥\nabla s_{\eta }∥}_{L^2\left({\Omega }_{\tau }\right)}\hfill \\ \hfill & +2k_{\infty }{\left(\underset{{\Omega }_{\tau }}{\int }{\left|k\left(s_{\eta }\right)\mathbf{g}\right|}^{2}dxdt\right)}^{{\displaystyle \frac{1}{2}}}{∥\nabla s_{\eta }∥}_{L^2\left({\Omega }_{\tau }\right)}+2\underset{0}{\overset{\tau }{\int }}\underset{{\Gamma }_s}{\int }\left|f\left(s_{\eta }\right)\right|\left|s_{\eta }\right|q_{s}d\gamma dt\hfill \\ \hfill & \le {∥s_0∥}_{L^2\left(\Omega \right)}^2+2T^{{\displaystyle \frac{1}{2}}}{∥\mathbf{u}∥}_{L^{\infty }\left(0,T;L^2\left(\Omega \right)\right)}{∥\nabla s_{\eta }∥}_{L^2\left({\Omega }_{\tau }\right)}\hfill \\ \hfill & +2g{k}_{\infty }\overline{k}{\left(T\left|\Omega \right|\right)}^{{\displaystyle \frac{1}{2}}}{∥\nabla s_{\eta }∥}_{L^2\left({\Omega }_{\tau }\right)}+2T^{{\displaystyle \frac{1}{2}}}{∥q_s∥}_{L^{\infty }\left(0,T;L^2\left({\Gamma }_s\right)\right)}{∥s_{\eta }∥}_{L^2\left(0,T;L^2\left({\Gamma }_s\right)\right)}.\hfill \end{array}$$

The last term was estimated as follows:

$$\begin{array}{cc}\hfill \underset{0}{\overset{\tau }{\int }}\underset{{\Gamma }_s}{\int }\left|f\left(s_{\eta }\right)\right|\left|s_{\eta }\right|q_{s}d\gamma dt& \le \underset{0}{\overset{\tau }{\int }}\underset{{\Gamma }_s}{\int }\left|s_{\eta }\right|q_{s}d\gamma dt\hfill \\ \hfill & \le {\left(\underset{0}{\overset{\tau }{\int }}\underset{{\Gamma }_s}{\int }{\left|q_s\right|}^{2}d\gamma dt\right)}^{{\displaystyle \frac{1}{2}}}{\left(\underset{0}{\overset{\tau }{\int }}\underset{{\Gamma }_s}{\int }\left|s_{\gamma }\right|d\gamma dt\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le T^{{\displaystyle \frac{1}{2}}}{∥q_s∥}_{L^{\infty }\left(0,T;L^2\left({\Gamma }_s\right)\right)}{∥s_{\eta }∥}_{L^2\left(0,T;L^2\left({\Gamma }_s\right)\right)}.\hfill \end{array}$$

And, using the trace theorem and Poincaré’s inequality, we obtain the following:

$$\underset{0}{\overset{\tau }{\int }}\underset{{\Gamma }_s}{\int }\left|f\left(s_{\eta }\right)\right|\left|s_{\eta }\right|q_{s}d\gamma dt\le C{T}^{{\displaystyle \frac{1}{2}}}{∥q_s∥}_{L^{\infty }\left(0,T;L^2\left({\Gamma }_s\right)\right)}{∥\nabla s_{\eta }∥}_{L^2\left({\Omega }_{\tau }\right)}.$$

Therefore, we can see that there is a constant $C>0$, independent of $\eta$, such that the following is true:

$$\begin{array}{cc}\hfill {\varphi }_{*}{∥s_{\eta }\left(\tau \right)∥}_{L^2\left(\Omega \right)}^2& +2k_0\underset{{\Omega }_{\tau }}{\int }\alpha \left(s_{\eta }\right){\left|\nabla s_{\eta }\right|}^{2}dxdt+2k_0\eta {∥\nabla s_{\eta }∥}_{L^2\left({\Omega }_{\tau }\right)}^2\hfill \\ & \le {∥s_0∥}_{L^2\left(\Omega \right)}^2+C{∥\nabla s_{\eta }∥}_{L^2\left({\Omega }_{\tau }\right)}\le {∥s_0∥}_{L^2\left(\Omega \right)}^2+{\displaystyle \frac{C^2}{4\eta }}+\eta {∥\nabla s_{\eta }∥}_{L^2\left({\Omega }_{\tau }\right)}.\hfill \end{array}$$

As a result,

$${\varphi }_{*}{∥s_{\eta }\left(\tau \right)∥}_{L^2\left(\Omega \right)}^2+2k_0\underset{{\Omega }_{\tau }}{\int }\alpha \left(s_{\eta }\right){\left|\nabla s_{\eta }\right|}^{2}dxdt+\eta {∥\nabla s_{\eta }∥}_{L^2\left({\Omega }_{\tau }\right)}^2\le {∥s_0∥}_{L^2\left(\Omega \right)}^2+{\displaystyle \frac{C^2}{4\eta }}.$$

By multiplying the previous inequality by $\eta$, we get

$${\eta }^2{∥\nabla s_{\eta }∥}_{L^2\left({\Omega }_{\tau }\right)}^2\le \eta {∥s_0∥}_{L^2\left(\Omega \right)}^2+{\displaystyle \frac{C^2}{4}}\le {∥s_0∥}_{L^2\left(\Omega \right)}^2+{\displaystyle \frac{C^2}{4}},$$

and

$$2\eta \underset{{\Omega }_{\tau }}{\int }\alpha \left(s_{\eta }\right){\left|\nabla s_{\eta }\right|}^{2}dxdt\le {∥s_0∥}_{L^2\left(\Omega \right)}^2+{\displaystyle \frac{C^2}{4}}.$$

Hence, the following are estimates of this lemma. □

Lemma 2.

(Estimation 2) Let $\beta \left(s\right)=\underset{0}{\overset{s}{\int }}\alpha \left(z\right)dzandG\left(s\right)=\underset{0}{\overset{s}{\int }}\beta \left(z\right)dz.$ Then, there exists a constant $C>0$, such that the following is true:

$$\begin{array}{c}\underset{0\le t\le T}{sup}\underset{\Omega }{\int }G\left(s_{\eta }\left(t\right)\right)dx\le Cand\underset{{\Omega }_t}{\int }{\left|\nabla \beta \left(s_{\eta }\right)\right|}^{2}dxds\le C,\hfill \\ {\Omega }_t=\Omega \times \left]0,t\right[\left(t\in \left]0,T\right[\right),\forall \beta >0.\hfill \end{array}$$

(38)

Proof. 

We remark $\beta$ is bounded and is in $C^1\left(\left[0,1\right]\right)$. So, for $s_{\eta }\in L^2\left(0,T;H_{{\Gamma }_e}^1\left(\Omega \right)\right)$, we have $\beta \left(s_{\eta }\right)\in V=L^2\left(0,T;H_{{\Gamma }_e}^1\left(\Omega \right)\right)and\beta \left(0\right)=0$. Therefore, when we take this function as a test function in (Sat), we get the following:

$$\begin{array}{ccc}\hfill {〈\varphi {\partial }_t{s}_{\eta },\beta \left(s_{\eta }\right)〉}_{V^{\prime },V}& +& \underset{{\Omega }_T}{\int }f\left(s_{\eta }\right)\mathbf{u}·\nabla \beta \left(s_{\eta }\right)dxdt+\underset{{\Omega }_T}{\int }\mathbf{K}{\alpha }_{\eta }\left(s_{\eta }\right)\nabla s_{\eta }\nabla \beta \left(s_{\eta }\right)dxdt\hfill \\ & -& \underset{{\Omega }_T}{\int }\mathbf{K}k\left(s_{\eta }\right)\mathbf{g}·\nabla \beta \left(s_{\eta }\right)dxdt+{\int }_0^T{\int }_{{\Gamma }_s}f\left(s_{\eta }\right)\beta \left(s_{\eta }\right)q_{s}d\gamma dt=0,\hfill \end{array}$$

where the first term is written as follows:

$${〈\varphi {\partial }_t{s}_{\eta },\beta \left(s_{\eta }\right)〉}_{V^{\prime },V}={\displaystyle \frac{d}{dt}}\underset{\Omega }{\int }\varphi G\left(s_{\eta }\right)dx=\underset{\Omega }{\int }\varphi G^{\prime }\left(s_{\eta }\right){\partial }_t{s}_{\eta }dx=\underset{\Omega }{\int }\varphi \beta \left(s_{\eta }\right){\partial }_t{s}_{\eta }dx.$$

□

Lemma 3.

(Estimation 3) There is a constant $C>0$, such that

$$\underset{\eta >0}{sup}{∥{\partial }_t{s}_{\eta }∥}_{L^2\left(0,T;H^1(\Omega )\right)}\le C.$$

(39)

Proof. 

In order to pass the limit with respect to $\eta$, we will need the following result. □

Lemma 4.

(Holderian compactness) Let E be the set

$$E=\left\{s:{\Omega }_T\to \mathbb{R}|\beta \left(s\right)\in L^2\left(0,T;H^1\left(\Omega \right)\right),{\partial }_{t}s\in L^2\left(0,T;{\left(H^1\left(\Omega \right)\right)}^{\prime }\right)\right\}.$$

Suppose that ${\beta }^{-1}$ is a Holderian of order $0<\theta <1$. Then, the injection of any bounded E is relatively compact in $L^2\left(\Omega \times \left]0,T\right[\right)$. Moreover, if the functions s remain in a bounded E, their images $\gamma \left(s\right)$ form a relatively compact set in $L^2\left(\partial \Omega \times \left]0,T\right[\right)$; γ state the application traces to Γ.

Proof. 

Let $v=\beta \left(s\right)$ with $s\in L^2\left(0,T;H^1\left(\Omega \right)\right)ands={\beta }^{-1}\left(v\right)$ Holderian.

We have $H^1\left(\Omega \right)\hookrightarrow W^{r,2}\left(\Omega \right)$ for all $0<r<1$. So, $v\in L^2\left(0,T;W^{r,2}\left(\Omega \right)\right)$ and $v\left(t\right)\in W^{r,2}\left(\Omega \right)$ a.e. in t. This implies that $s={\beta }^{-1}\left(v\right)\in W^{\theta r,{\displaystyle \frac{2}{\theta }}}\left(\Omega \right)$ a.e. to t. Moreover,

$$\begin{array}{cc}\hfill {∥s\left(t\right)∥}_{W^{\theta r,{\displaystyle \frac{2}{\theta }}}\left(\Omega \right)}& ={∥{\beta }^{-1}\left(v\left(t\right)\right)∥}_{W^{\theta r,{\displaystyle \frac{2}{\theta }}}\left(\Omega \right)}\hfill \\ \hfill & ={\left[{∥{\beta }^{-1}\left(v\left(t\right)\right)∥}_{L^{2/\phantom{2\theta }\theta }}^{2/\phantom{2\theta }\theta }+\underset{\Omega }{\int }\underset{\Omega }{\int }{\displaystyle \frac{{\left|{\beta }^{-1}\left(v\left(t\right)\right)\left(x\right)-{\beta }^{-1}\left(v\left(t\right)\right)\left(y\right)\right|}^{\frac{2}{\theta }}}{{\left|x-y\right|}^{\frac{\theta }{2}\theta r+N}}}dxdy\right]}^{{\displaystyle \frac{\theta }{2}}}\hfill \\ \hfill & \le {\left[\underset{\Omega }{\int }{\left|{\beta }^{-1}\left(v\left(t\right)\right)\left(x\right)\right|}^{2/\phantom{2\theta }\theta }dx+C\underset{\Omega }{\int }\underset{\Omega }{\int }{\displaystyle \frac{{\left|v\left(t\right)\left(x\right)-v\left(t\right)\left(y\right)\right|}^2}{{\left|x-y\right|}^{2r+N}}}dxdy\right]}^{{\displaystyle \frac{\theta }{2}}}\hfill \\ \hfill & \le C{\left[\underset{\Omega }{\int }{\left|v\left(t\right)\left(x\right)\right|}^{2}dx+C\underset{\Omega }{\int }\underset{\Omega }{\int }{\displaystyle \frac{{\left|v\left(t\right)\left(x\right)-v\left(t\right)\left(y\right)\right|}^2}{{\left|x-y\right|}^{2r+N}}}dxdy\right]}^{{\displaystyle \frac{\theta }{2}}}\hfill \\ \hfill & =C{∥s\left(t\right)∥}_{W^{r,2}\left(\Omega \right)}^{\theta }.\hfill \end{array}$$

We obtain the following:

$${∥s\left(t\right)∥}_{W^{\theta r,{\displaystyle \frac{2}{\theta }}}\left(\Omega \right)}^{{\displaystyle \frac{1}{\theta }}}\le C^{{\displaystyle \frac{1}{\theta }}}{∥s\left(t\right)∥}_{W^{r,2}\left(\Omega \right)}\mathrm{and}\underset{0}{\overset{T}{\int }}{∥s\left(t\right)∥}_{W^{\theta r,{\displaystyle \frac{2}{\theta }}}\left(\Omega \right)}^{{\displaystyle \frac{2}{\theta }}}dt\le C^{{\displaystyle \frac{2}{\theta }}}\underset{0}{\overset{T}{\int }}{∥s\left(t\right)∥}_{W^{r,2}\left(\Omega \right)}^{2}dt.$$

This proves that

$${∥s\left(t\right)∥}_{L^{{\displaystyle \frac{2}{\theta }}}(0,T;W^{\theta r,{\displaystyle \frac{2}{\theta }}}\left(\Omega \right))}^{{\displaystyle \frac{2}{\theta }}}\le C{∥s∥}_{L^{{\displaystyle \frac{2}{\theta }}}\left(0,T;W^{r,2}\left(\Omega \right)\right)}^2,$$

where C is positive constant. Now, the s are bounded in the following set:

$$E_0=\left\{w:{\Omega }_T\to \mathbb{R}|w\in L^{{\displaystyle \frac{2}{\theta }}}(0,T;W^{\theta r,{\displaystyle \frac{2}{\theta }}}\left(\Omega \right)),{\partial }_{t}s\in L^2\left(0,T;{\left(H^1\left(\Omega \right)\right)}^{\prime }\right)\right\}.$$

Let $r^{\prime }=\theta r$. As $0<\theta <1$, we have $q={\displaystyle \frac{2}{\theta }}>2$, so taking $0<r^{\prime \prime }<r^{\prime }$, the following is true:

$$W^{r^{\prime },q}\left(\Omega \right)\underset{compact}{\hookrightarrow }{W}^{r^{\prime \prime },q}\left(\Omega \right)\underset{continuous}{\hookrightarrow }{L}^q\left(\Omega \right)\underset{continuous}{\hookrightarrow }{L}^2\left(\Omega \right)\underset{continuous}{\hookrightarrow }{\left(H^1\left(\Omega \right)\right)}^{\prime }.$$

Using the Aubin–Simon lemma, we see that for $0<r^{\prime \prime }<\theta$, the injection of E into $L^{{\displaystyle \frac{2}{\theta }}}(0,T;W^{r^{\prime \prime },{\displaystyle \frac{2}{\theta }}}\left(\Omega \right))$ is compact. Since the injection of $W^{r^{\prime \prime },q}\left(\Omega \right)$ into $L^2\left(\Omega \right)$ is continuous, the injection of E into $L^2\left(0,T;L^2\left(\Omega \right)\right)$ is compact. Concerning the traces, we use the fact that the injection of $W^{r^{\prime \prime },{\displaystyle \frac{2}{\theta }}}\left(\Omega \right)$ in $W^{r^{\prime \prime }-{\displaystyle \frac{\theta }{2}},{\displaystyle \frac{2}{\theta }}}\left(\Gamma \right)$ is continuous and the continuity of the injection of $W^{r^{\prime \prime }-{\displaystyle \frac{\theta }{2}},{\displaystyle \frac{2}{\theta }}}\left(\Gamma \right)$ into $L^2\left(\Gamma \right)$ for $r^{\prime \prime }-{\displaystyle \frac{\theta }{2}}>0$.

Thus, by choosing ${\displaystyle \frac{\theta }{2}}<r^{\prime \prime }<\theta$ for bounded E, the set $\gamma \left(E\right)$ is relatively compact in $L^2\left(0,T;L^2\left(\Gamma \right)\right)$.

□

Remark 2.

In the case of space

$$E=\left\{s:{\Omega }_T\to \mathbb{R}|s\in L^2\left(0,T;H^1\left(\Omega \right)\right),{\partial }_{t}s\in L^2\left(0,T;{\left(H^1\left(\Omega \right)\right)}^{\prime }\right)\right\},$$

we use the fact that

$$H^1\left(\Omega \right)\underset{compact}{\hookrightarrow }{H}^{1-\epsilon }\left(\Omega \right)\underset{continuous}{\hookrightarrow }{\left(H^1\left(\Omega \right)\right)}^{\prime }$$

to have the compactness of the injection of E in $L^2\left(0,T;H^{1-\epsilon }\left(\Omega \right)\right)$.

The continuity of the map $\gamma :H^{1-\epsilon }\left(\Omega \right)\to H^{{\displaystyle \frac{1}{2}}-\epsilon }\left(\Gamma \right)$ implies that the injection of $\gamma \left(E\right)$ in $L^2\left(0,T;L^2\left(\Gamma \right)\right)$ is compact.

Passing to the Limit with Respect to $\eta$

The estimates of (37)–(39) and the hypotheses on $\beta$ (${\beta }^{-1}$ are Holderian of order $\theta \in [0,1]$), which allows us to extract a subsequence $\left\{s_{\eta }\right\}$, such that

$$\begin{array}{c}{s}_{\eta }\to s\mathrm{in}{L}^2\left({\Omega }_T\right)\mathrm{strongly},\hfill \\ s_{\eta }\to s\mathrm{in}{L}^2\left(0,T;L^2\left(\partial \Omega \right)\right)\mathrm{strongly},\hfill \\ \nabla \beta \left(s_{\eta }\right)\mapsto \nabla \beta \left(s\right)\mathrm{in}{L}^2\left(0,T;L^2\left(\Omega \right)\right)\mathrm{weakly},\hfill \\ {\partial }_t{s}_{\eta }\mapsto {\partial }_{t}s\mathrm{in}{L}^2\left(0,T;H^1(\Omega )\right)\mathrm{weakly}\mathrm{for}\eta \to 0,\hfill \\ \nabla {\partial }_t{s}_{\eta }\mapsto \nabla {\partial }_{t}s\mathrm{in}{L}^2\left(0,T;\left(L^2(\Omega )\right)\right)\mathrm{weakly}\mathrm{for}\eta \to 0.\hfill \end{array}$$

These convergences make it possible to pass the limit in the weak formulation.

On the other hand, for $\phi \in L^2\left(0,T;H_{{\Gamma }_e}^1\left(\Omega \right)\right)$, we write the following:

$$\begin{array}{cc}\hfill {〈\varphi {\partial }_t{s}_{\eta },\phi 〉}_{L^2\left({\left(H_{{\Gamma }_e}^1\right)}^{\prime }\right),L^2\left(H_{{\Gamma }_e}^1\right)}& -\underset{{\Omega }_T}{\int }f\left(s_{\eta }\right){\mathbf{u}}_n·\nabla \phi dxdt\hfill \\ \hfill & +\underset{{\Omega }_T}{\int }\mathbf{K}\alpha \left(s_{\eta }\right)\nabla s_{\eta }·\nabla \phi dxdt+\eta \underset{{\Omega }_T}{\int }\mathbf{K}\nabla s_{\eta }·\nabla \phi dxdt\hfill \\ \hfill & -\underset{{\Omega }_T}{\int }\mathbf{K}k\left(s_{\eta }\right)\mathbf{g}·\nabla \phi dxdt+\underset{0}{\overset{T}{\int }}\underset{{\Gamma }_s}{\int }f\left(s_{\eta }\right)\phi q_{s}d\gamma dt=0.\hfill \end{array}$$

Using weak convergences and Lebesgue’s dominated convergence theorem, passing the limit in each of the terms successively gives the following:

(a)

$\underset{\eta \to 0}{lim}{〈{\partial }_t{s}_{\eta },\phi 〉}_{L^2\left({\left(H_{{\Gamma }_e}^1\right)}^{\prime }\right),L^2\left(H_{{\Gamma }_e}^1\right)}={〈{\partial }_{t}s,\phi 〉}_{L^2\left({\left(H_{{\Gamma }_e}^1\right)}^{\prime }\right),L^2\left(H_{{\Gamma }_e}^1\right)}.$

(b)

$\underset{\eta \to 0}{lim}\underset{{\Omega }_T}{\int }f\left(s_{\eta }\right){\mathbf{u}}_n·\nabla \phi dxdt=\underset{{\Omega }_T}{\int }f\left(s\right){\mathbf{u}}_n·\nabla \phi dxdt.$

(c)

$\underset{\eta \to 0}{lim}\underset{{\Omega }_T}{\int }\alpha \left(s_{\eta }\right)\nabla s_{\eta }·\nabla \phi dxdt=\underset{{\Omega }_T}{\int }\alpha \left(s\right)\nabla s·\nabla \phi dxdt$

(d)

$\underset{\eta \to 0}{lim}\underset{{\Omega }_T}{\int }k\left(s_{\eta }\right)\mathbf{g}·\nabla \phi dxdt=\underset{{\Omega }_T}{\int }k\left(s\right)\mathbf{g}·\nabla \phi dxdt.$

(e)

$\underset{\eta \to 0}{lim}\underset{{\Gamma }_s}{\int }f\left(s_{\eta }\right)\phi q_{s}d\gamma dt=\underset{{\Gamma }_s}{\int }f\left(s\right)\phi q_{s}d\gamma dt.$

(f)

Passing to the limit in the term $\eta \underset{{\Omega }_T}{\int }\nabla s_{\eta }·\nabla \phi dxdt.$

Suppose first that $\phi \in L^2\left(0,T;H_{{\Gamma }_e}^1\left(\Omega \right)\right)\cap L^2\left(0,T;H^2\left(\Omega \right)\right)$.

Let ${\Sigma }_T=\Gamma \times \left[0,T\right]$. We can write

$$\underset{{\Omega }_T}{\int }\nabla s_{\eta }·\nabla \phi dxdt=-\underset{{\Omega }_T}{\int }{s}_{\eta }.\Delta \phi dxdt+\underset{{\Sigma }_T}{\int }{s}_{\eta }·\nabla \phi ·\mathbf{n}d\gamma dt.$$

We have $0\le s_{\eta }\le 1$ in ${\Omega }_T$ and $\left\{s_{\eta }\right\}$ bounded in $L^2\left(0,T;L^2\left(\Gamma \right)\right)$. So,

$$\underset{\eta \to 0}{lim}\underset{{\Omega }_T}{\int }\nabla s_{\eta }·\nabla \phi dxdt=0.$$

To conclude, we use the density of $L^2\left(0,T;H_{{\Gamma }_e}^1\left(\Omega \right)\right)\cap L^2\left(0,T;H^2\left(\Omega \right)\right)$ in the space $L^2\left(0,T;H_{{\Gamma }_e}^1\left(\Omega \right)\right)$.

6. Conclusions

In this work, we investigated the existence of weak solutions for an incompressible and immiscible phase flow model. We considered two states of the fluid, wet and non-wet, in a porous medium with dynamic capillary pressure.

This model is a coupled system that includes a nonlinear equivalent saturation equation and an elliptic pressure–velocity equation. In the ordered state, the existence and uniqueness of the weak solution is obtained. We also let the ordering coefficient tend to veer toward zero to prove the existence of weak solutions. In future research, we plan to apply this approach to address similar problems, incorporating additional effects.