Homogenization results for a coupled system modelling immiscible compressible two-phase flow in porous media by the concept of global pressure

A model for immiscible compressible two-phase flow in heterogeneous porous media is considered. Such models appear in gas migration through engineered and geological barriers for a deep repository for radioactive waste. The main feature of this model is the introduction of a new global pressure and it is fully equivalent to the original equations. The resulting equations are written in a fractional flow formulation and lead to a coupled degenerate system which consists of a nonlinear parabolic (the global pressure) equation and a nonlinear diffusion–convection one (the saturation) equation with rapidly oscillating porosity function and absolute permeability tensor. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we obtain a nonlinear homogenized problem with effective coefficients which are computed via a cell problem and give a rigorous mathematical derivation of the upscaled model by means of two-scale convergence.     

Numerical simulations of water–gas flow in heterogeneous porous media with discontinuous capillary pressures by the concept of global pressure

We present an approach and numerical results for a new formulation modeling immiscible compressible two-phase flow in heterogeneous porous media with discontinuous capillary pressures. The main feature of this model is the introduction of a new global pressure, and it is fully equivalent to the original equations. The resulting equations are written in a fractional flow formulation and lead to a coupled degenerate system which consists of a nonlinear parabolic (the global pressure) equation and a nonlinear diffusion–convection one (the saturation equation) with nonlinear transmission conditions at the interfaces that separate different media. The resulting system is discretized using a vertex-centred finite volume method combined with pressure and flux interface conditions for the treatment of heterogeneities. An implicit Euler approach is used for time discretization. A Godunov-type method is used to treat the convection terms, and the diffusion terms are discretized by piecewise linear conforming finite elements. We present numerical simulations for three one-dimensional benchmark tests to demonstrate the ability of the method to approximate solutions of water–gas equations efficiently and accurately in nuclear underground waste disposal situations.     

On a fully nonlinear degenerate parabolic system modeling immiscible gas–water displacement in porous media

In this paper, we are interested in the simultaneous flow of two immiscible fluid phases within a porous medium. We consider a two-phase flow model where the fluids are immiscible and there is no mass transfer between the phases. The medium is saturated by compressible/incompressible phase flows. We study the gas–water displacement without simplified assumptions on the state law of gas density. We establish an existence result for the nonlinear degenerate parabolic system based on new energy estimate on pressures.     

Slightly compressible and immiscible two-phase flow in porous media

We describe the flow of two compressible phases in a porous medium. We consider the case of slightly compressible phases for which the density of each phase follows an exponential law with a small compressibility factor. A nonlinear parabolic system including quadratic velocity terms is derived to describe compressible and immiscible two-phase flow in porous media. In one-dimensional space, we establish the existence and uniqueness of a local strong solution for the regularized system. We show also that the saturation is physically admissible. We describe the asymptotic behavior of the solutions when the compressibility factor goes to zero.     

Existence of solutions to various rock types odel model of two-phase flow in porous media

We study the flow of two immiscible and incompressible fluids through a porous media c,onsisting of different rock types: capillary pressure and relative permeablities curves are different in each type of porous media. This process can be formulated as a coupled system of partial differential equations which includes an elliptic pressurevelocity equation and a nonlinear degenerated parabolic saturation equation. Moreover the transmission conditions are nonlinear and the saturation is discontinuous at interfaces separating different media. A change of unknown leads to a new formulation of this problem. We derive a weak form for this new problem, which is a combination of a mixed formulation for the elliptic pressure-velocity equation and a standard variational formulation for the new parabolic equation. Under some realistic assumptions, we prove the existence of weak solutions to the implicit system given by time discretization.     

Homogenization of Richardsʼ equation of van Genuchten–Mualem model

This paper is devoted to the homogenization of Richards? equation of van Genuchten–Mualem model, which is a nonlinear degenerate parabolic differential equation. It is usually used to model the motion of saturated–unsaturated water flow in porous media. We firstly apply the Kirchhoff transformation to the equation and obtain a simpler equivalent equation with a linear oscillated diffusion term. Then under the real assumption for van Genuchten–Mualem model, we obtain the homogenized equation based on the two-scale convergence theory. Some results on the first order corrector are also presented.     

An existence result for a coupled system modeling a fully equivalent global pressure formulation for immiscible compressible two-phase flow in porous media

A new model describing immiscible, compressible two-phase flow, such as water-gas, through heterogeneous porous media is considered. The main feature of this model is the introduction of a new global pressure and the full equivalence to the original equations. The resulting equations are written in a fractional flow formulation and lead to a coupled system which consists of a nonlinear parabolic equation (the global pressure equation) and a nonlinear diffusion-convection one (the saturation equation). Under some realistic assumptions on the data, we show an existence result with the help of appropriate regularizations and a time discretization. We use suitable test functions to get a priori estimates. In order to pass to the limit in nonlinear terms, we also obtain compactness results which are nontrivial due to the degeneracy of the system.     

Diffusion in poro‐plastic media

A model is developed for the flow of a slightly compressible fluid through a saturated inelastic porous medium. The initial‐boundary‐value problem is a system that consists of the diffusion equation for the fluid coupled to the momentum equation for the porous solid together with a constitutive law which includes a possibly hysteretic relation of elasto‐visco‐plastic type. The variational form of this problem in Hilbert space is a non‐linear evolution equation for which the existence and uniqueness of a global strong solution is proved by means of monotonicity methods. Various degenerate situations are permitted, such as incompressible fluid, negligible porosity, or a quasi‐static momentum equation. The essential sufficient conditions for the well‐posedness of the system consist of an ellipticity condition on the term for diffusion of fluid and either a viscous or a hardening assumption in the constitutive relation for the porous solid. Copyright © 2004 John Wiley & Sons, Ltd.     

On the filtration through porous media with partially soluble permeable grains

In this paper we consider the incompressible viscous fluid flow through a porous medium whose grains are also permeable and release mass to the flow. In each component porosity and permeability depend on saturation. The flow is modelled with a nonlinear parabolic equation for the pressure, with a degenerate parabolic term, depending not only on the saturations, but also on the space variable and on time averages of the saturation. We generalize the classic approach of Alt and Luckhaus to this situation and establish existence of at least one weak solution and bounds for its absolute value. Received December 1998     

Homogenization of a three-phase flow model in porous media

We give homogenization results for an immiscible and incompressible three-phase flow model in a heterogeneous petroleum reservoir with periodic structure, including capillary effects. We consider a model which leads to a coupled system of partial differential equations which includes an elliptic equation and two nonlinear degenerate parabolic equations of convection–diffusion types. Using two-scale convergence, we get an homogenized model which governs the global behavior of the flow. The determination of effective properties require the numerical resolution of local problems in a standard cell.     

An existence result for nonisothermal immiscible incompressible 2‐phase flow in heterogeneous porous media

In the present article, we study the temperature effects on two‐phase immiscible incompressible flow through a porous medium. The mathematical model is given by a coupled system of 2‐phase flow equations and an energy balance equation. The model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy‐Muskat and the capillary pressure laws. The problem is written in terms of the phase formulation; ie, the saturation of one phase, the pressure of the second phase, and the temperature are primary unknowns. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we show the existence of weak solutions with the help of an appropriate regularization and a time discretization. We use suitable test functions to obtain a priori estimates. We prove a new compactness result to pass to the limit in nonlinear terms.     

Time of complete displacement of an immiscible compressible fluid by water in porous media: Application to gas migration in a deep nuclear waste repository

A system of evolutionary partial differential equations (PDEs) describing the two-phase flow of immiscible fluids, such as water–gas, through porous media is studied. In this formulation, the wetting and nonwetting phases are treated to be incompressible and compressible, respectively. This treatment is indeed necessary when a compressible nonwetting phase is subjected to compression during confinement. The system of PDEs consists of an evolution equation for the wetting-phase saturation and an evolution equation for the pressure in the nonwetting phase. This system is applied to the problem of unsaturated flows to assess gas migration and two-phase flow through engineered and geological barriers for a deep repository for radioactive waste. This paper is primarily concerned with the large time behavior of solutions of this system. Under some realistic assumptions on the data, we derive estimates of the speed of propagation of the gas by water in porous media. Namely, we establish estimates of time stabilization for the water saturation to a constant limit profile. The analysis is based on the energy methods whose main idea involves deriving and studying suitable ordinary differential inequalities. We show that the time of complete displacement of a gas by water may be at most infinite or finite depending essentially on the power parameters defining the capillary pressure and the relative permeabilities. This result is then illustrated with two examples in the context of gas migration in a deep nuclear waste repository. We consider Van Genuchten’s and Brooks–Corey’s models for a two-phase water–gas system.     

Forchheimer law derived by homogenization of gas flow in turbomachines

A general formulation of the homogenization problem of compressible fluid flow through a periodic porous material in turbomachines is presented here. This formulation is able to derive a Forchheimer law with a mean velocity dependent permeability as equivalent macroscopic behavior. To specify this permeability, additional flow problems are defined on the unit cell and solved by a mixed stabilized finite element discretization. The application of the Galerkin least-square (GLS) method requires the introduction of two stabilization terms with appropriate parameters. The mixed finite element discretization of these unit cell problems is finally outlined.     

Asymptotic Analysis of a Nonlinear Parabolic Problem Modelling Miscible Compressible Displacement in Porous Media

We study the asymptotic behavior, with respect to high Peclet numbers, of a model describing a compressible and miscible displacement in a porous medium. The transport of mass is then described by a nonlinear, fully coupled and degenerate parabolic system. Using non-classical estimates and renormalization tools, we prove existence of relevant weak solutions for the limit problem.      

The method of mixed volume element‐characteristic mixed volume element and its numerical analysis for three‐dimensional slightly compressible two‐phase displacement

Numerical simulation of oil‐water two‐phase displacement is a fundamental problem in energy mathematics. The mathematical model for the compressible case is defined by a nonlinear system of two partial differential equations: (1) a parabolic equation for pressure and (2) a convection‐diffusion equation for saturation. The pressure appears within the saturation equation, and the Darcy velocity controls the saturation. The flow equation is solved by the conservative mixed volume element method. The order of the accuracy is improved by the Darcy velocity. The conservative mixed volume element with characteristics is applied to compute the saturation, that is, the diffusion is discretized by the mixed volume element and convection is computed by the method of characteristics. The method of characteristics has strong computational stability at sharp fronts and avoids numerical dispersion and nonphysical oscillation. Small time truncation error and accuracy are obtained through this method. The mixed volume element simulates diffusion, saturation, and the adjoint vector function simultaneously. By using the theory and technique of a priori estimates of differential equations, convergence of the optimal second order in norm is obtained. Numerical examples are provided to show the effectiveness and viability of this method. This method provides a powerful tool for solving challenging benchmark problems.     

非牛顿幂律流体球向不定常渗流

本文研究了弱压缩非牛顿幂律流体球向不定常渗流,导出了抛物型偏微分非线性方程.球向扩散方程是其特殊情况.用Laplace变换的方法,找到了线性化后方程的解析解和渐近解.用影响半径的概念和平均值方法求得了近似解.渐近解和近似解的结构是相似的,从而丰富了非牛顿流体一维不定常渗流的理论.     

Homogenization of a model for water–gas flow through double‐porosity media

This paper presents a study of immiscible compressible two‐phase, such as water and gas, flow through double porosity media. The microscopic model consists of the usual equations derived from the mass conservation laws of both fluids, along with the standard Darcy–Muskat law relating the velocities to the pressure gradients and gravitational effects. The problem is written in terms of the phase formulation, that is, where the phase pressures and the phase saturations are primary unknowns. The fractured medium consists of periodically repeating homogeneous blocks and fractures, where the absolute permeability of the medium becomes discontinuous. Consequently, the model involves highly oscillatory characteristics. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. We obtain the convergence of the solutions, and a macroscopic model of the problem is constructed using the notion of two‐scale convergence combined with the dilatation technique. Copyright © 2015 John Wiley & Sons, Ltd.     

On a multiscale analysis of an inverse problem of nonlinear transfer law identification in periodic microstructure

This paper is devoted to the multiscale analysis of a homogenization inverse problem of the heat exchange law identification, which is governed by parabolic equations with nonlinear transmission conditions in a periodic heterogeneous medium. The aim of this work is to transform this inverse problem with nonlinear transmission conditions into a new one governed by a less complex nonlinear parabolic equation, while preserving the same form and physical properties of the heat exchange law that it will be identified, based on periodic homogenization theory. For this, we reformulate first the encountered homogenization inverse problem to an optimal control one. Then, we study the well-posedness of the state problem using the Leray–Schauder topological degrees and we also check the existence of the solution for the obtained optimal control problem. Finally, using the periodic homogenization theory and priori estimates, with justified choise of test functions, we reduce our inverse problem to a less complex one in a homogeneous medium.     

Discontinuous finite volume method for compressible miscible displacement problems in porous media

The compressible miscible displacement problem in porous media is modeled by a coupled system of two nonlinear partial differential equations: the pressure equation and the concentration equation are parabolic equation. In this article, we present discontinuous finite volume method for the concentration equation and the pressure equation. The optimal order error estimates for pressure and concentration are obtained in a mesh dependent norm.     

Propagation of Perturbations of Solutions for a Doubly Degenerate Nonlinear Parabolic Equation

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