Poroelastic Problems Research Articles

A modular finite element approach to saturated poroelasticity dynamics: Fluid–solid coupling with Neo-Hookean material and incompressible flow

Several methods have been developed to model the dynamic behavior of saturated porous media. However, most of them are suitable only for small strain and small displacement problems and are built in a monolithic way, so that individual improvements in the solution of the solid or fluid phases can be difficult. This study shows a macroscopic approach through a partitioned fluid–solid coupling, in which the skeleton solid is considered to behave as a Neo-Hookean material and the interstitial flow is incompressible following the Stokes–Brinkman model. The porous solid is numerically modeled with a total Lagrangian position-based finite element formulation, while an Arbitrary Lagrangian-Eulerian stabilized finite element approach is employed for the porous medium flow dynamics. In both fields, an averaging procedure is applied to homogenize the problem, resulting in a macroscopic continuous phase. The solid and fluid homogenized domains are overlapped and strongly coupled, based on a block-iterative solution scheme. Two-dimensional simulations of wave propagation in saturated porous media are employed to validate the proposed formulation through a comprehensive comparison with analytical and numerical results from the literature. The analyses underscore the proposed formulation as a robust and precise modular approach for addressing dynamic problems in poroelasticity.

Development of coupled fluid-flow/geomechanics model considering storage and transport mechanism in shale gas reservoirs with complex fracture morphology

Field observations frequently demonstrate stress fluctuations resulting from the reservoir depletion. The development of reservoirs, particularly the completion of infill wells and refracturing, can be significantly impacted by stress changes in and around drainage areas. Previous studies mainly focus on plane fractures and few studies consider the influence of complex transport and storage mechanism and irregular fracture geometry on stress evolution in shale gas reservoirs. Based on the embedded discrete fracture model (EDFM) and finite-volume method (FVM), a coupled geomechanics/fluid model has been successfully developed considering the adsorption, desorption, diffusion and slippage of shale gas. This model achieves coupling simulation of natural fractures, hydraulic fractures with complex geometry, storage and transport mechanism, reservoir stress, and pore-elastic effect. The open-source software OpenFOAM is used as the main solver for this model. The stress calculation and productivity simulation of the model are verified by the classical poroelasticity problem and the simulation results of published research and commercial simulator with EDFM respectively. The simulation results indicate that σxx, σyy, σxy and Δσ changes with time and space due to the time effect and anisotropy of formation pressure depletion; Due to the influence of different mechanisms on shale gas storage and transport, the reservoir pressure and stress distribution under different mechanisms are different; Among them, the stress with full mechanisms differs the most compared to the stress without any mechanism. The reservoir with stronger stress sensitivity (smaller Biot coefficient) is less sensitive to formation pressure depletion, and the stress variation range is smaller. For reservoirs with weak stress sensitivity, formation pressure depletion is more likely to lead to stress reversal. Under the influence of fracture geometry, the pressure depletion regions caused by the three types of fracture geometry are approximately rectangular, parallelogram and square, respectively. The corresponding σxx, σyy and Δσ also have great differences in spatial distribution and values. Therefore, the time effect, shale gas storage and transport mechanism and the influence of complex fracture geometry should be considered when predicting pressure depletion induced stress under the condition of simultaneous production. This study is of great significance for understanding the evolution law of stress induced by pressure consumption, as well as the design of infill wells and repeated fracturing.

Discontinuous Galerkin Method for Nonlinear Quasi-Static Poroelasticity Problems

Discontinuous Galerkin Method for Nonlinear Quasi-Static Poroelasticity Problems

Partially explicit generalized multiscale finite element methods for poroelasticity problem

We develop a partially explicit time discretization based on the framework of constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) for the problem of linear poroelasticity with high contrast. Firstly, dominant basis functions generated by the CEM-GMsFEM approach are used to capture important degrees of freedom and it is known to give contrast-independent convergence that scales with the mesh size. In typical situation, one has very few degrees of freedom in dominant basis functions. This part is treated implicitly. Secondly, we design and introduce an additional space in the complement space and these degrees are treated explicitly. We also investigate the CFL-type stability restriction for this problem, and the restriction for the time step is contrast independent.

Full weak Galerkin finite element discretizations for poroelasticity problems in the primal formulation

This paper develops novel finite element solvers for linear poroelasticity problems on quadrilateral meshes. These solvers are based on the primal formulations of linear elasticity and Darcy flow. Specifically, the fluid pressure and solid displacement are approximated by scalar- or vector-valued polynomials of degree k≥0 separately in element interiors and on edges. The discrete weak gradients of these shape functions are established in the broken (vector- or matrix-version) Arbogast–Correa spaces for approximations of the classical gradients in the variational forms. These weak Galerkin spatial discretizations are combined with the implicit Euler or Crank–Nicolson temporal discretizations to develop locking-free numerical solvers that have optimal order (k+1) convergence rates in pressure, velocity, displacement, stress, and dilation. Rigorous analysis is presented and illustrated by numerical experiments on popular test cases.

Robust block diagonal preconditioners for poroelastic problems with strongly heterogeneous material

AbstractThis paper focuses on the analysis and the solution of the saddle‐point problem arising from a three‐field formulation of Biot's model of poroelasticity, discretized in time by the implicit Euler method. A block diagonal‐preconditioner, based on the Schur complement, is analyzed on a functional level and compared with two other block‐diagonal preconditioners having a similar structure. The problem is discretized in space using mixed finite elements and solved with appropriate iterative solvers, incorporating the investigated preconditioners. The solvers are tested on numerical examples inspired by geotechnical practice, with particular attention devoted to the solvers' robustness concerning strong heterogeneity in permeability.

Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Three-Dimensional Media

Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Three-Dimensional Media

A space-time multigrid method for poroelasticity equations with random hydraulic conductivity

In this work, we propose a space-time multigrid method for solving a finite difference discretization of the linear Biot’s model in two space dimensions considering random hydraulic conductivity. This method is an extension of the one developed by Franco et al. [1] and which was applied to the heat equation. Particularly for the poroelasticity problem, we need to use the Tri-Diagonal Matrix Algorithm (TDMA) for systems of equations with multiple variables (block TDMA) solver on the planes xt and yt with zebra-type wise-manner (the domain is divided into planes and solve each odd plane first and then each even plane independently [2]). On the planes xy, the F-cycle multigrid with fixed-stress smoother is improved by red-black relaxation. A fixed-stress smoother is compound solver, where the mechanical and flow parts are addressed by two distinctive solvers and symmetric Gauss-Seidel iteration. This is the basis of the proposed method which does not converge only with the standard line-in-time red-black solver [1]. This combination allows the code parallelization. The use of standard coarsening associated with line or plane smoothers in the strong coupling direction allows an application in real world problems and, in particular, the random hydraulic conductivity case. This would not be possible using the standard space-time method with semicoarsening in the strong coupling direction. The spatial and temporal approximations of the problem are performed, respectively, by using Central Difference Scheme (CDS) and implicit Euler methods. The robustness and excellent performance of the multigrid method, even with random hydraulic conductivity, are illustrated with numerical experiments through the average convergence factor.

Bayesian decision making using partial data for fractured poroelastic media

Partial data is a common challenge in many applied problems, where we can measure only the part of the solution. This paper considers the poroelasticity problem in fractured media, where only partial measurements of fracture pressures are available. The poroelasticity models have applications in many engineering and scientific fields, such as geomechanics, hydrology, and reservoir simulation. This paper aims to develop a Bayesian decision making approach using partial data and consider its application to the poroelasticity problem in fractured media. The methodology of the proposed approach combines Proper Orthogonal Decomposition, Discrete Empirical Interpolation Method, Bayesian Inversion Approach, and Bayesian Decision Theory. It allows us to compute posterior probabilities and decide according to the optimal decision rule based on the partial data. Moreover, we apply partially explicit temporal discretization to solve forward problems efficiently. For testing the proposed approach, we consider a two-dimensional model problem. We simulate the cases when we need to make a single decision about the reservoir or decisions about the elements of the set of reservoirs. The results show that the proposed approach can efficiently compute the posterior probabilities and make a decision in the presence of only partial data for both cases.

The parameter inversion in coupled geomechanics and flow simulations using Bayesian inference

In many situations, uncertainty about the mechanical properties of surrounding soils due to the lack of data and spatial variations requires tools that involve the study of parameters by means of random variables or random functions. Usually only a few measurements of parameters, such as permeability or porosity, are available to build a model, and some measurements of the geomechanical behavior, such as displacements, stresses, and strains are needed to check/calibrate the model. In order to introduce this type of modeling in geomechanical analysis, taking into account the random nature of soil parameters, Bayesian inference techniques are implemented in highly heterogeneous porous media. Within the framework of a coupling algorithm, these are incorporated into the inverse poroelasticity problem, with porosity, permeability and Young modulus treated as stationary random fields obtained by the moving average (MA) method. To this end, the Metropolis–Hasting (MH) algorithm was chosen to seek the geomechanical parameters that yield the lowest misfit. Numerical simulations related to injection problems and fluid withdrawal in a 3D domain are performed to compare the performance of this methodology. We conclude with some remarks about numerical experiments.

Parameter-robust preconditioners for Biot’s model

This work presents an overview of the most relevant results obtained by the authors regarding the numerical solution of the Biot’s consolidation problem by preconditioning techniques. The emphasis here is on the design of parameter-robust preconditioners for the efficient solution of the algebraic system of equations resulting after proper discretization of such poroelastic problems. The classical two- and three-field formulations of the problem are considered, and block preconditioners are presented for some of the discretization schemes that have been proposed by the authors for these formulations. These discretizations have been proved to be well-posed with respect to the physical and discretization parameters, what provides a framework to develop preconditioners that are robust with respect to such parameters as well. In particular, we construct both norm-equivalent (block diagonal) and field-of-value-equivalent (block triangular) preconditioners, which are proved to be parameter-robust. The theoretical results on this parameter-robustness are demonstrated by considering typical benchmark problems in the literature for Biot’s model.

Displacement and Pressure Reconstruction from Magnetic Resonance Elastography Images: Application to an In Silico Brain Model

Magnetic resonance elastography is a motion-sensitive image modality that allows to measure in vivo tissue displacement fields in response to mechanical excitations. This paper investigates a data assimilation approach for reconstructing tissue displacement and pressure fields in an in silico brain model from partial elastography data. The data assimilation is based on a parametrized-background data-weak methodology, in which the state of the physical system—tissue displacements and pressure fields—is reconstructed from the available data assuming an underlying poroelastic biomechanics model. For this purpose, a physics-informed manifold is built by sampling the space of parameters describing the tissue model close to their physiological ranges to simulate the corresponding poroelastic problem and computing a reduced basis via proper orthogonal decomposition. Displacements and pressure reconstruction are sought in a reduced space after solving a minimization problem that encompasses both the structure of the reduced-order model and the available measurements. The proposed pipeline is validated using synthetic data obtained after simulating the poroelastic mechanics of a physiological brain. The numerical experiments demonstrate that the framework can exhibit accurate joint reconstructions of both displacement and pressure fields. The methodology can be formulated for an arbitrary resolution of available displacement data from pertinent images. It can also inherently handle uncertainty on the physical parameters of the mechanical model by enlarging the physics-informed manifold accordingly. Moreover, the framework can be used to characterize, in silico, biomarkers for pathological conditions by appropriately training the reduced-order model. A first application for the noninvasive estimation of ventricular pressure as an indicator of abnormal intracranial pressure is shown in this contribution.

Poroelastic problem of a non-penetrating crack with cohesive contact for fluid-driven fracture

A new class of unilaterally constrained problems for fully coupled poroelastic models stemming from hydraulic fracturing is introduced and studied with respect to its well-posedness. The poroelastic medium contains a fluid-driven crack, which is subjected to non-penetration conditions and cohesion forces between the crack faces. Existence of solution for the governing elliptic–parabolic variational inequality under the unilateral constraint with a small cohesion is established using the incremental approximation based on Rothe’s semi-discretization in time.

Weak Galerkin finite element method for linear poroelasticity problems

In this paper, we develop a weak Galerkin (WG) finite element method for a linear poroelasticity model where weak divergence and weak gradient operators defined over discontinuous functions are introduced. We establish both the continuous and discrete time WG schemes, and obtain their optimal convergence order estimates in a discrete H1 norm for the displacement and in H1 and L2 norms for the pressure. Finally, we present some numerical experiments on different kinds of meshes to illustrate the theoretical error estimates, and furthermore verify the locking-free property of our proposed method.

Parameter‐robust mixed element method for poroelasticity with Darcy‐Forchheimer flow

AbstractIn this paper, the Biot's consolidation model is considered, and Darcy‐Forchheimer equation is used to describe the relationship between the fluid velocity and pressure. The model is nonlinear and the unknown variables are the solid displacement, the fluid velocity and pressure. The Bernardi‐Raugel element is used for displacement and the RT mixed element is used for the fluid velocity and pressure. The parameter‐robust BR‐RT0‐P0 method presented here is uniformly stable not only with respect to the Lamé constant , but also with respect to the constrained specific storage coefficient . We demonstrate the well‐posedness of the BR‐RT0‐P0 finite element solutions, and give the error estimates of the finite element approximations using the monotonicity of the nonlinear Forchheimer term. The error estimates are divided into two cases, that is, is positive and is nonnegative. Numerical experiments are presented to verify theoretical analysis. Moreover we focus on the variation of pressure by several poroelasticity problems.

Дослідження поронапруженого стану півнескінченного порожнистого циліндру під дією осесиметричного навантаження

The exact solution of the poroelasticity problem for a semi-infinite hollow cylinder under the axisymmetric load is derived in the paper. The original problem was reduced to a one-dimensional problem by applying the integral Fourier transform. The one-dimensional boundary value problem in the transform domain is formulated in a vector form, its solution is found using the matrix differential calculation apparatus. The derived analytical formulas allow to investigate the change of displacements, stresses and pore pressure depending on the type of porous material, the applied load, the size of outer and inner radii of the cylinder.

Partial learning using partially explicit discretization for multicontinuum/multiscale problems. Fractured poroelastic media simulation

The poroelasticity problem plays a vital role in various fields of science and technology. For example, it has applications in the petroleum industry, agricultural science, and biomedicine. The mathematical model consists of a coupled system of differential equations for mechanical displacements and pressure.In this paper, we propose a new approach for solving the poroelasticity problem in a fractured medium based on hybrid explicit–implicit learning (HEI). For spatial approximation, we use the finite element method with standard linear basis functions. We consider a poroelastic medium with large hydraulic fractures and use the Discrete Fracture Model. For the time approximation, we apply the explicit–implicit scheme. The explicit scheme is used for the porous matrix, which is a low-conductive continuum. While for fractures, we apply the implicit scheme due to its high conductivity. To facilitate the calculations, we use the fixed strain and fixed stress splitting schemes. The main idea of the proposed method lies in partial learning. Full training of the numerical solution is extremely difficult due to a large number of degrees of freedom, so we propose to train a part of the solution. Large hydraulic fractures have few degrees of freedom, but they are highly conductive and require a more costly implicit scheme. Therefore, we train the neural network to generate pressure in the fractures at several time points and then interpolate for other times. Thus, we treat the fracture pressure as a known function and use it to find the pressure in the porous matrix. Numerical results for a two-dimensional model problem are presented.

A mixed virtual element method for Biot's consolidation model.

In this paper, we shall present a weak virtual element method for the standard three field poroelasticity problem on polytopal meshes. The flux velocity and pressure are approximated by the low order virtual element and the piecewise constant, while the elastic displacement is discretized by the virtual element with some tangential polynomials on element boundaries. A fully discrete scheme is then given by choosing the backward Euler for the time discretization. With some assumptions on the exact solutions, we prove that the convergence order is of order 1 with respect to the meshsize and the time step, and the hidden constants are independent of the parameters of the problems. Some numerical experiments are given to verify the results.

Arnold Verruijt (1940-2022).

Arnold Verruijt (1940-2022).

Numerical analysis via MEF of poroelastic problems

This paper presents a fully coupled hydro-mechanical formulation, based on the finite element method in terms of the displacement and pore-water pressure, taking into account the effect of relative compressibility between the grains and the solid skeleton. Its main goal is to provide a clear formulation and reference examples to facilitate reproduction by anyone which intend to develop their own computer program to solve poroelasticity problems. The numerical model presented is verified with analytical solutions of problems in plane strain condition and others authors numerical results. The influence of material properties on the pore-water pressure dissipation process and, the performance of the time integration scheme are presented. Higher the Poisson’s coefficient and lower the grain compressibility faster is the process of dissipation highlighting the importance of taking into account the relative compressibility between the grains and the solid skeleton on the coupled analyses. The Crank Nicholson’s algorithm presents a good performance when comparing with other time integration scheme presented by other authors.