On the existence of global solution of the system of equations of liquid movement in porous medium

The initial-boundary value problem for the system of one-dimensional isothermal motion of viscous liquid in deformable viscous porous medium is considered. Local theorem of existence and uniqueness of problem is proved in case of compressible liquid. In case of incompressible liquid the theorem of global solvability in time is proved in Holder classes. A feature of the model of fluid filtration in a porous medium considered in this paper is the inclusion of the mobility of the solid skeleton and its poroelastiс properties. The transition from Euler variables to Lagrangian variables is used in the proof of the theorems.


Introduction
Mathematical models of fluid filtration in a porous medium apply to a broad range of practical problems. The examples include, but are not limited to filtration near river dams, irrigation and drainage of agricultural fields, dynamics of hydraulic fracturing during oil and gas mining, methane extraction from coal and shale deposits, flow of magma in the earth's crust, etc. Parameters of these models strongly depend on the properties of fluids as well as the porous solid medium. Thus, a vast number of models are currently available. Majority of these models, however, make a simplifying assumption of a static solid porous skeleton, and treat the porosity as a given function. In this work, we try to relax this assumption to account for the mobility and poroelastic properties of a solid component. Models with a given porosity function of a solid component are based on a Muskat-Leverett filtering theory. S.N. Antontsev and V.N. Monakhov [1] developed the theory for a special case of two-phase motion of immiscible incompressible liquids in a non-deformable porous medium. A large number of papers are devoted to numerical studies (see, for example, [2]).
Tertzagi [3] was the first to develop models of poroelastic media that would take into account mobility of the skeleton and its poroelastic properties. He introduced the principle of effective stress, defined as the difference between total stress and the pressure of the liquid phase. This position reflects the fact that the liquid carries a part of the load. The relation between the deformation of the skeleton of the solid matrix and the fluid flow is of key importance here. Bio [4] further developed Tertsagi's theory: he introduced a joint deformation model of a fluidsaturated porous medium and established the theory of poroelasticity. Almost simultaneously and independently, Frenkel [5] developed a similar theory. Later, V.N. Nikolayevsky, P.P. Zolotarev, and Kh.A. Rakhmatullin [6], [7], [8] proposed analogous models in their studies.
O.B. Bocharov [9], allowed the porosity to depend on the pressure, but no deformation of the porous skeleton was considered. V. V. Vedernikov and V. N. Nikolaevskii [10] proposed a two-phase filtration model in a deformable porous medium with a solid skeleton motion being described analogous to the Tertsagi principle and the modified Hooke linear law. The model was not justified. This was done later in [11], [12], where particular solutions were derived. O.B. Bocharov et all [13] derived the properties of the solutions for a degenerate case. All filtration models are very complex both from a theoretical point of view as well as in their application to specific problems. Only a handful of studies on deformable porous media models published to date included result justification. A mainstream research is based on the classical theory of filtration, and model justification is only examined for a limited number of specific cases. Strict mathematical results are only presented in a few papers exploring the existence and uniqueness of solutions of such problems. For example, A. M. Abourabia et all in [14], [15], [16] reduced the initial system of equations to a single equation of higher order by making a number of simplifying assumptions. M. Simpson et all [16] proved a local solvability of the Cauchy problem in S.L. Sobolev spaces. Y. Geng et all [14], [15], investigated solutions of the "simple wave" type. Numerical studies of such problems were carried out, for example, in [17].

ICIES'2020
are introduced. Then the specific pore volume (porosity, fraction of the volume of the medium happen to the voids) can be stated as = , where the total volume is = + . Darcy flow, which describes the fluid velocity relative to the solid velocity, is defined as ⃗ = ( ⃗ − ⃗ ), where ⃗ , ⃗ are velocities of fluid and porous skeleton respectively [18].
Mass conservation laws for liquid and solid phases in absence of phase transitions look like [19] ( ) +▽⋅ ( ⃗ ) = 0, where is time, is density of liquid, is density of ) is gradient operator, Laws of conservation of mass can be wrote in terms of material derivative: ( = + ⃗ ⋅▽). Then we obtain = − ▽⋅ ( ( ⃗ + ⃗ )), By motion of fluid in deformable medium it is assumed that [19], [20]: 1. the deviator of stress tensor in liquid phase is neglected ( = 0), because the fluid viscosity is much lower than the skeleton shear viscosity. ) is deviatoric strain rate tensor, is the skeleton shear viscosity, , are stress tensor and pressure of liquid phase.
Terzaghi's principle states that deformation of the solid matrix is determined by an effective stress defined as [3] = + . Then, for a fully saturated, lowporosity media, the effective dynamic pressure is = − [21]. Notice, that If density of solid phase is constant, then = 0 and = . From equation (1) we obtain The assumption that porosity is function of effective pressure was used in work [22] = ( ), in particular: = 0 exp{− }. In approach, used in [22], bulk compressibility of two-phase medium is defined as relativity summary change of volume, responding on changing applied effective dynamic pressure: Equation (2) in this case can be written as . Volumetric compressibility is also function of porosity, for example [23]: ( ) = , where is the bulk compressibility, is positive constant = −1/ ( / ). Thus the temporal variation of the porosity owing to mechanical compaction can be written as [24]: The constitutive creep law can be written as [25], [26] where corresponds to a bulk viscosity. This formulation is analogous to a creep-controlled viscous compaction law used in studies dealing with magma transport in the Earth's mantle [27].
The bulk viscosity as rule depends on , for example: ( ) = , where is positive constant [28].
Thus the rheological law combining mechanical and viscous compaction can be expressed as [23], [25] The conservation of momentum for the fluid can be stated as Darcy's law [18], [29] where is the hydraulic conductivity, = ( ′ )/ , ′, are the permeability and the fluid dynamic viscosity, is the density of the mass forces ( ⃗ = (0,0, − )), is the excess fluid pressure, defined as the difference between the fluid pressure and the hydrostatic pressure: = − ℎ . In this way, we have In some cases coefficients ′, , could be determined in experiment another way. In particular they can be determined as: = , = / , ′ = , where is permability, = 1/2, ∈ [0,2], = 3 [23]. The laws of momentum conservation for each of phase can be written Adding these equations, we obtain the equation of conservation of momentum of the system "solid matrixpore fluid" [19], [20], [24], namely: the equation of incompressible deformation of the solid skeleton matrix taking into account the effect of the pore fluid pressure: is average density of the compressible medium. In expanded form, the previous equation can be written In some applications, the force balance equation is written as [19], [30]. −∇ + ⃗ = 0. Thus, the equations of model in the absence of phase transitions have the form [19], [20], [23], [30]: (7) This quasilinear composite type system describes the spatial unsteady isothermal motion of a compressible fluid in a viscoelastic medium. Here ( ), 1 ( ), 2 ( ) are parameters of poroelastic medium. Numerical studies of various initial-boundary value problems for the system of equations (3)-(7) were carried out in the works [20], [23], [31]. Questions of justification in these papers were not considered. In some particular cases, the issues of justifying this model are discussed in [13], [32], [33], [34]. The the main difference this work from [34] is the taking into account the compressibility of liquid and additional term in the equation of conservation of momentum of the system on the whole. Work [33] is the short announcement of results of this work.
The system of equations describing the onedimensional unsteady motion of a compressible fluid in a viscous porous medium in the domain ( , ) ∈ = Ω × (0, ), Ω = (0,1), is as follows [20], [30]: The problem is written in the Eulerian coordinates , . The real density of the solid particles are assumed constant, and Clapeyron dependence is taken for fluid = , = > 0. At the boundary of the region Ω, the velocities of the phases , are set, and at the initial moment of time the density 0 ( ) and porosity 0 ( ) are set. The following conditions are considered for the system (8)- (11): (12) Following [1], [34], [35], [36] we rewrite the system (8)- (11) in Lagrangian variables: (17) We will represent the equation (17) as where 0 ( ) is the some time dependent function. Taking into account equation (16), we obtain a representation for : We represent the equation (16) as After integrate this equation by from 0 to 1, taking into account the submission for , we get Converting the original system of equations we get system for finding porosity and density (using equation (14), taking into account equation (15) and using equation (13), taking into account equation (16): Where and the function is defined as follows:
Here ( ) = From the representation (23) it follows that smoothness is determined by the smoothness of functions ̅ , ̅, 0 , 0 and . Therefor there exists a value 1 = 1 ( 0 , 0 , 1 , 1 ), such that for all 0 ≤ 1 the following inequality holds Using (21), ̅ and ̅( , ), we find the function ( , ) as a solution of the problem (here and elsewhere, we assume that the initial and boundary conditions are matched): The equation for ( , ) is uniformly parabolic. In view of the properties of ̅( , ) and 0 ( ) the problem (25) has a classical solution [37]. In addition, we have the following estimate: Under the additional condition smallness for the value of the time interval the following statement holds. Lemma 1. There exists such a t 2 , that when t 0 ≤ min(t 1 , t 2 ), the classical solution of problem (25) satisfies the following inequality in Q t 0 : 0 < m 1 ≤ ρ(x, t) + ρ 0 (x) ≤ M 1 < ∞.

The Case of Incompressible Medium
If the density of the liquid phase is constant ( = ) the system of equations (18), (19) can be reduced to one equation for the porosity of in Lagrange variables [34]:  (28), (29) is the function , ∈ 2+ ,1+ ( ), such that 0 < < 1. This functions satisfy the equation (28) and the initial and boundary conditions (29) and regarded as continuous functions in . Theorem 2. Suppose that the data of problem (28), (29) satisfies the following conditions: 1) the functions ( ), ( ) and their derivatives up to the second order are continuous for ∈ (0,1) and satisfy the conditions
Whence it follows that the difference − ̅ satisfies the condition − ̅ = ∫ (∫ ( ) ) + 3 + 4 ≥ 3 + 4 , 0 < ≤ , where 3 , 4 are arbitrary constants. If < , we represent the equation for the difference and ̅ in the form  The proof of Lemma 4 and all subsequent reasoning fully follows [34]. Then we obtain Hlder estimates for and [34]. Further we use the theory of elliptic equations for the function set forth in [37]. Theorem 3 is proved.

Conclusion
In this article, we described the system of equations of one-dimensional movement of a viscous liquid in a deformable viscous porous medium. We proved the local solvability of this problem in case of compressible liquid and global solvability in Holder classes in case of incompressible liquid. In future we want to take into account dependences pressure of temperature and density.
The work of the first author was supported by the Russian Science Foundation (Project No. 19-11-00069).