Smoothed Particle Hydrodynamics modelling of poroelastic media

Numerical modelling of poroelastic properties in porous media allows widely varied investigations at low costs and relatively short times. The study of porous media is of high interest at different scales, in this case we focus our analysis at meso- and macroscale which is highly relevant e.g. in geothermal explorations. We model a biphasic poroelastic media assuming incompressible fluid and solid grains and a large solid-fluid density ratio. Meshfree methods are nowadays more widely used due to the advantages that present in the simulation of large deformations. In this case we choose to employ the Smoothed Particle Hydrodynamics method (SPH), a Lagrangian method where the domain is discretized in particles. The solution is computed in parallel, which allows to simulate large domains more representative of the scale of our study cases. We validate our implementation with a classical consolidation problem and compare the simulated diffusion process with Terzaghi's analytical solution. Future work includes simulation of fractures initiation and propagation in the porous media at reservoir scale. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Single‐phase flow in composite poroelastic media

The mathematical formulation and analysis of the Barenblatt–Biot model of elastic deformation and laminar flow in a heterogeneous porous medium is discussed. This describes consolidation processes in a fluid‐saturated double‐diffusion model of fractured rock. The model includes various degenerate cases, such as incompressible constituents or totally fissured components, and it is extended to include boundary conditions arising from partially exposed pores. The quasi‐static initial–boundary problem is shown to have a unique weak solution, and this solution is strong when the data are smoother. Copyright © 2002 John Wiley & Sons, Ltd.     

Poroelastodynamic potential method for transversely isotropic fluid-saturated poroelastic media

By introduction of two scalar potentials, an analytical method is developed for the solution of poroelastodynamic boundary value problems in transversely isotropic fluid-saturated poroelastic media. The governing equations of motion are considered in the framework of Biot's complete model without any assumption or simplification. As a case of application, solutions in three dimensions for a transversely isotropic fluid saturated porous half space loaded by an arbitrary distribution of time harmonic tractions at the free surface is derived. The free surface of the half space may be considered either permeable or impermeable. As a particular solution, Green's functions for uniform vertical and horizontal circular patch loads are presented as semi-infinite integrals which may be evaluated by means of an appropriate numerical method proposed. The accuracy of the solutions is verified both analytically and numerically against the preceding solutions. Some numerical results are also presented to clarify the influence of different degrees of anisotropy and frequency of excitation on the response of the medium.     

A reduced model to locate low contrast inclusions in poroelastic media

This paper introduces a numerical method to localize inclusions having slightly different elastic coefficients than those of a fully saturated poroelastic matrix, whose detection is often difficult. This method can be used to find weakly stiffer or softer objects in saturated soils or diseased biological tissues at early stages. To this end, we propose a reduced model from the Biot’s equations by splitting the fluid pressure into two parts: one embedded into an elasticity model and the other one used as a corrector term. By applying the small amplitude homogenization method, we can successfully retrieve the position and extension of inclusions in poroelastic media employing this simplified model. Numerical results show a good agreement for the location of inclusions when the contrast is below 30% stiffer or softer than the matrix, and for a noise level up to 5% for frequencies below 50 Hz.     

Non-monotone stochastic generalized porous media equations

By using the Nash inequality and a monotonicity approximation argument, existence and uniqueness of strong solutions are proved for a class of non-monotone stochastic generalized porous media equations. Moreover, we prove for a large class of stochastic PDE that the solutions stay in the smaller L²-space provided the initial value does, so that some recent results in the literature are considerably strengthened.     

Effective equations of two-phase flow in random media

We consider the behavior of incompressible two-phase flow in heterogeneous reservoirs with randomly placed heterogeneities; that is, in media with permeabilityA and porosity which are stationary random fields. We assume both Darcy velocity and the diffusion flux being given nonlinear functions of the concentration. Using the tools of stochastic homogenization we get the nonlinear effective equations which govern the flow behavior in a homogeneous medium, being equivalent in the sense of homogenization theory, to the original one. When is small the randomly heterogeneous porous medium behaves like a deterministic medium with effective permeability tensor A^o. It is shown how to calculate the effective permeability tensor A^o by solving auxiliary stochastic problems. Using the rescaling parameter, corresponding to the characteristic scale of heterogeneities, we prove the convergence of the homogenization process for 0. Furthermore, by using regularity results for the nonlinear effective equations we construct the correctors and establish strong convergence.

---

Résumé On considère le comportement des écoulements diphasiques incompressibles dans un réservoir hétérogène avec les hétérogénéités placées aléatoirement; c'est-à-dire, dans un milieux où la permeabilitéA et la porosité sont des champs aléatoires stochastiquement homogénes. On suppose à la fois que le vecteur flux de diffusion et la vitesse de Darcy sont des fonctions nonlinéaires de la concentration. En utilisant les techniques d'homogénéisation stochastique on obtient à grande échelle des équations nonlinéaires efficaces décrivants un écoulement en milieux poreux equivalent à l'écoulement original dans le sens de la théorie de l'homogénéisation. Le milieu poreux aléatoire se comporte à grande échelle comme un milieux deterministe avec un tenseur efficace de permeabilité A ^o , pour suffisemment petit. Ce tenseur de perméabilité efficace est calculé en resolvant des problèmes stochastiques auxilliaires. Lorsque le paramétre, correspondant à l'échelle caractéristique des hétérogénéités, tend vers zero, nous montrons la convergence du processus d'homogénéisation. Finalement, en utilisant des résultats de régularité pour les équations efficaces nonlineaires obtenues, nous construisons les correcteurs et démontrons la convergence forte.

     

Superconvergence analysis for Maxwell's equations in dispersive media

In this paper, we consider the time dependent Maxwell's equations in dispersive media on a bounded three-dimensional domain. Global superconvergence is obtained for semi-discrete mixed finite element methods for three most popular dispersive media models: the isotropic cold plasma, the one-pole Debye medium, and the two-pole Lorentz medium. Global superconvergence for a standard finite element method is also presented. To our best knowledge, this is the first superconvergence analysis obtained for Maxwell's equations when dispersive media are involved.

     

Infinite-dimensional porous media equations and optimal transportation

In this paper, we study a class of nonlinear diffusion equations in a Hilbert space X, $\partial_t\mu_t -\nabla\cdot\left(\nabla (L\circ\rho_t)\gamma\right)=0 \quad\mbox{\rm in}\,X\times(0,+\infty)$ with respect to a log-concave reference probability measure γ. We obtain existence, uniqueness and stability properties, in the framework of gradient flows in spaces of probability measures.     

Entropy solutions for stochastic porous media equations

We provide an entropy formulation for porous medium-type equations with a stochastic, non-linear, spatially inhomogeneous forcing. Well-posedness and $L_1$-contraction is obtained in the class of entropy solutions. Our scope allows for porous medium operators $\mathrm{\Delta }(|u{|}^{m?1}u)$ for all $m\in (1,\infty )$, and Hölder continuous diffusion nonlinearity with exponent 1/2.     

Abstract kinetic equations relevant to supercritical media

The abstract Hilbert space equation $\text{(T?)′(x) = ?(A?)(x)}$, x∈$\text{R}$₊, is studied with a partial range boundary condition $\text{(Q}{}_{\mathrm{+}}\text{?)(0) = ?}{}_{\mathrm{+}}\text{?}\text{Ran}\text{Q}{}_{\mathrm{+}}$. Here T is bounded, injective and self-adjoint, A is Fredholm and self-adjoint, with finite-dimensional negative part, and Q₊ is the orthogonal projection onto the maximal T-positive T-invariant subspace. This models half-space stationary transport problems in supercritical media. A complete existence and uniqueness theory is developed.     

Weak solutions to stochastic porous media equations

A stochastic version of the porous medium equation is studied. The corresponding Kolmogorov equation is solved in a space where is an invariant measure. Then a weak solution, that is a solution in the sense of the corresponding martingale problem, is constructed.     

Stochastic generalized porous media equations with reflection

A non-negative Markovian solution is constructed for a class of stochastic generalized porous media equations with reflection. To this end, some regularity properties and a comparison theorem are proved for stochastic generalized porous media equations, which are interesting by themselves. Invariant probability measures and ergodicity of the solution are also investigated.     

Linear elliptic equations in composite media with anisotropic fibres

Linear elliptic equations in composite media with anisotropic fibres are concerned. The media consist of a periodic set of anisotropic fibres with low conductivity, included in a connected matrix with high conductivity. Inside the anisotropic fibres, the conductivity in the longitudinal direction is relatively high compared with that in the transverse directions. The coefficients of the elliptic equations depend on the conductivity. This work is to derive the Hölder and the gradient $L^p$ estimates (uniformly in the period size of the set of anisotropic fibres as well as in the conductivity ratio of the fibres in the transverse directions to the connected matrix) for the solutions of the elliptic equations. Furthermore, it is shown that, inside the fibres, the solutions have higher regularity along the fibres than in the transverse directions.     

Entire solutions of sublinear elliptic equations in anisotropic media

We study the nonlinear elliptic problem −Δu=ρ(x)f(u) in RN (N?3), lim|x|→∞u(x)=?, where ??0 is a real number, ρ(x) is a nonnegative potential belonging to a certain Kato class, and f(u) has a sublinear growth. We distinguish the cases ?>0 and ?=0 and prove existence and uniqueness results if the potential ρ(x) decays fast enough at infinity. Our arguments rely on comparison techniques and on a theorem of Brezis and Oswald for sublinear elliptic equations.     

Large deviations for stochastic generalized porous media equations

The large deviation principle is established for the distributions of a class of generalized stochastic porous media equations for both small noise and short time.     

Space-time discontinuous galerkin method for maxwell equations in dispersive media

In this paper, a unified model for time-dependent Maxwell equations in dispersive media is considered. The space-time DG method developed in [29] is applied to solve the underlying problem. Unconditional L²-stability and error estimate of order O(τ^r+1+h^k+1/2) are obtained when polynomials of degree at most r and k are used for the temporal discretization and spatial discretization respectively. 2-D and 3-D numerical examples are given to validate the theoretical results. Moreover, numerical results show an ultra-convergence of order 2r+1 in temporal variable t.