Stability and Monotonicity for Some Discretizations of the Biot's Model

We consider finite element discretizations of the Biot's consolidation model in poroelasticity with MINI and stabilized P1-P1 elements. We analyze the convergence of the fully discrete model based on spatial discretization with these types of finite elements and implicit Euler method in time. We also address the issue related to the presence of non-physical oscillations in the pressure approximation for low permeabilities and/or small time steps. We show that even in 1D a Stokes-stable finite element pair fails to provide a monotone discretization for the pressure in such regimes. We then introduce a stabilization term which removes the oscillations. We present numerical results confirming the monotone behavior of the stabilized schemes

Nonlinear Evolutionary Problem of Filtration Consolidation With the Non-Classical Conjugation Condition

Finite-element solutions of the initial-boundary value problem for a nonlinear parabolic equation in an inhomogeneous domain with the conjugation condition of a non-ideal contact were found. The initial boundary value problem is a mathematical model of an important technical problem of filtration consolidation of inhomogeneous soils. Inhomogeneity is considered in terms of the presence of thin inclusions, physicochemical characteristics of which differ from those of the main soil. The problem of longterm consolidation is especially pronounced in soils with low filtration coefficient. Low permeability of the porous medium causes deviation from the linear relationship between the pressure gradient and the filtration rate.Weak formulation of the problem is suggested, and the accuracy of the approximate finite element solution, its existence and uniqueness are substantiated for the case of Darcy’s nonlinear law. A test example and the effect of the nonlinear filtration law for thin inclusion on the dynamics of scattering of excess pressures in the entire area of the problem are considered

A model for coupled electro-hydro-mechanical processes in fine grained soils accounting for gas generation and transport

A theoretical and numerical model is developed for the quantitative analysis of coupled processes taking place in active waste containment systems, such as electrokinetic barriers or fences, in which alow intensity DC current is circulated across the clay barrier to move polar and non-polar contaminants. A novel feature of the proposed approach is the allowance for the presence of air in the pore space. Under unsaturated conditions, all transport coefficients involved in the electrokinetic process are strongly dependent on the degree of saturation of pore liquid. In order to assess the predictive capability of the proposed theory and to appreciate the impact of gas production at the electrodes, a series of numerical simulations of simple onedimensional electrokinetic tests have been performed. The results of the simulations compare reasonably well with data obtained from laboratory experiments performed on an illitic clayey silt. The numerical results indicate that the impact of gas production at the electrodes can be significant, even in low-intensity and short-duration treatments

From arteries to boreholes: Steady-state response of a poroelastic cylinder to fluid injection

The radially outward flow of fluid into a porous medium occurs in many practical problems, from transport across vascular walls to the pressurisation of boreholes. As the driving pressure becomes non-negligible relative to the stiffness of the solid structure, the poromechanical coupling between the fluid and the solid has an increasingly strong impact on the flow. For very large pressures or very soft materials, as is the case for hydraulic fracturing and arterial flows, this coupling can lead to large deformations and, hence, to strong deviations from a classical, linear-poroelastic response. Here, we study this problem by analysing the steady-state response of a poroelastic cylinder to fluid injection. We consider the qualitative and quantitative impacts of kinematic and constitutive nonlinearity, highlighting the strong impact of deformation-dependent permeability. We show that the wall thickness (thick vs. thin) and the outer boundary condition (free vs. constrained) play a central role in controlling the mechanics

A coupled discrete-element model of fluid-saturated rock and the results of studying of the impact of a fluid on the shear strength of a rock under combined compression and shear

Within a discrete-element model of a porous permeable elastic-plastic rock, filled with a fluid, we have studied the shear strength of a fractured interface zone (a shear band) between blocks of a geological medium subject to compression and shear. Under these conditions, a fluid pore pressure is controlled by interplay of dilation of the elastic-plastic shear band and fluid transport between the blocks and the interface. We have found that the shear strength is a unique function of a combination of parameters, which includes viscosity of a fluid, permeability of the medium, shear rate and a characteristic size of the system. Based on the simulation results we have constructed the generalized binomial dependence of the shear strength of samples on the obtained combination of parameters

A novel block non-symmetric preconditioner for mixed-hybrid finite-element-based flow simulations

In this work we propose a novel block preconditioner, labelled Explicit Decoupling Factor Approximation (EDFA), to accelerate the convergence of Krylov subspace solvers used to address the sequence of non-symmetric systems of linear equations originating from flow simulations in porous media. The flow model is discretized blending the Mixed Hybrid Finite Element (MHFE) method for Darcy's equation with the Finite Volume (FV) scheme for the mass conservation. The EDFA preconditioner is characterized by two features: the exploitation of the system matrix decoupling factors to recast the Schur complement and their inexact fully-parallel computation by means of restriction operators. We introduce two adaptive techniques aimed at building the restriction operators according to the properties of the system at hand. The proposed block preconditioner has been tested through an extensive experimentation on both synthetic and real-case applications, pointing out its robustness and computational efficiency

Poroelastic Modelling of Wavefields in Heterogeneous Media. Poroelastische Modellierung von Wellenfeldern in Heterogenen Medien

Numerical modelling of seismic waves in heterogeneous, porous reservoir rocks is an important tool in the field of reservoir engineering. A new 2-D velocity-stress finite-differences scheme is presented that allows to simulate waves and coupled diffusion processes within poroelastic media as described by Biot theory. The presented numerical methods allow to further develop rock physics theories of wave-induced fluid flow and contribute to the interpretation of new laboratory experiments