We establish the existence of a solution to a non-linearly coupled elliptic-parabolic system of PDEs describing the single-phase, miscible displacement of one incompressible fluid by another in a porous medium. We consider a velocity-dependent diffusion-dispersion tensor and model the action of injection and production wells by measures on the domain. We obtain the solution by passing to the limit on problems with regularised well data.
In this paper we study the existence and partial regularity of weak solutions to an elliptic-parabolic system that models the single-phase miscible displacement of one incompressible fluid by another in a porous media. The system is singular and involves quadratic nonlinearity. A partial regularity result is obtained via the Fefferman-Stein inequality.
This article proves the existence of solutions to a model of incompressible miscible displacement through a porous medium, with zero molecular diffusion and modelling wells by spatial measures. We obtain the solution by passing to the limit on problems indexed by vanishing molecular diffusion coefficients. The proof employs cutoff functions to excise the supports of the measures and the discontinuities in the permeability tensor, thus enabling compensated compactness arguments used by Y. Amirat and A. Ziani for the analysis of the problem with $L^2$ wells [\emph{Z. Anal. Anwendungen}, 23(2):33…
We consider the Muskat problem describing the viscous displacement in a two-phase fluid system located in an unbounded two-dimensional porous medium or Hele-Shaw cell. After formulating the mathematical model as an evolution problem for the sharp interface between the fluids, we show that Muskat problem with surface tension is a quasilinear parabolic problem, whereas, in the absence of surface tension effects, the Rayleigh-Taylor condition identifies a domain of parabolicity for the fully nonlinear problem. Based upon these aspects, we then establish the local well-posedness for arbitrary larg…
A PDE system consisting of the momentum balance, mass balance, and energy balance equations for displacement, capillary pressure, and temperature as a model for unsaturated fluid flow in a porous viscoelastoplastic solid is shown to admit a solution under appropriate assumptions on the constitutive behavior. The problem involves two hysteresis operators accounting for plastic and capillary hysteresis.
We study the linear stability of the displacement of three Stokes fluids with constant viscosity in a porous medium when the middle fluid is contained in a bounded region. We use the Hele-Shaw model. The eigenfunctions of the stability system are the amplitudes of the linear perturbations. These amplitudes must be small. We get unbounded eigenfunctions. So the stability problem has no physical sense.
According to Biot's paper in 1956, by using the Lagrangian equations in classical mechanics, we consider a problem of the filtration of a liquid in porous elastic-deformation media whose mechanical behavior is described by the Lam'e system coupled with a hyperbolic equation. Assuming the null surface displacement on the whole boundary, we discuss an inverse source problem of determining a body force only by observation of surface traction on a suitable subdomain along a sufficiently large time interval. Our main result is a Hölder stability estimate for the inverse source problem, which is pro…
We study the displacement of three immiscible Stokes fluids with constant viscosities in a porous medium. The middle-layer fluid is contained in a bounded region. We give an analysis of the linear stability of this process. This stability problem has no solution (in general).
In the present paper, a continuum model is introduced for fluid flow in a deformable porous medium, where the fluid may undergo phase transitions. Typically, such problems arise in modeling liquid-solid phase transformations in groundwater flows. The system of equations is derived here from the conservation principles for mass, momentum, and energy and from the Clausius-Duhem inequality for entropy. It couples the evolution of the displacement in the matrix material, of the capillary pressure, of the absolute temperature, and of the phase fraction. Mathematical results are proved under the add…
We compute the complete spectrum of the displacement Hessian operator, which is obtained from the confined porous medium equation by linearization around its stationary attractor, the Barenblatt profile. On a formal level, the operator is conjugate to the Hessian of the entropy via similarity transformation. We show that the displacement Hessian can be understood as a self-adjoint operator and find that its spectrum is purely discrete. The knowledge of the complete spectrum and the explicit information about the corresponding eigenfunctions give new insights on the convergence and higher order…
In this paper we investigate the interaction of fluid flow with a thin porous elastic layer. We consider two fluid-filled bulk domains which are separated by a thin periodically perforated layer consisting of a fluid and an elastic solid part. Thickness and periodicity of the layer are of order $ε$, where $ε$ is small compared to the size of the bulk domains. The fluid flow is described by an instationary Stokes equation and the solid via linear elasticity. The main contribution of this paper is the rigorous homogenization of the porous structure in the layer and the reduction of the layer to …
A novel regularized fracture model for crack propagation in porous media is proposed. Our model is obtained through homogenization theory and formal asymptotic expansions. We start with a regularized quasi-static fracture model posed in a periodically perforated domain obtained by periodic extension of a re-scaled unit cell with a hole. This setup allows us to write two separated minimality conditions for the primary (displacement) and secondary variables plus a balance of energy relation. Then we apply the usual asymptotic expansion matching to deduce limit relations when the re-scaling param…
Accurate numerical simulations of interaction between fluid and solid play an important role in applications. The task is challenging in practical scenarios as the media are usually highly heterogeneous with very large contrast. To overcome this computational challenge, various multiscale methods are developed. In this paper, we consider a class of linear poroelasticity problems in high contrast heterogeneous porous media, and develop a mixed generalized multiscale finite element method (GMsFEM) to obtain a fast computational method. Our aim is to develop a multiscale method that is robust wit…
