Poroelastic Problems Research Articles

A locking free numerical approximation for quasilinear poroelasticity problems

This paper is devoted to a new formulation which couples the weak Galerkin method and the finite element method for approximating solutions of the equations of quasi-static poroelasticity which model flow through elastic porous media. It is assumed that the permeability of the elastic matrix depends nonlinearly on the dilatation of the porous medium. The steady-state version of the system is recast in terms of displacement, pressure, and volumetric stress, and the well-posedness of both the continuous and discrete three-field formulations is proved. Error estimates for the proposed numerical method are obtained. These show that the method is locking free . Numerical experiments presented further demonstrate the accuracy and the locking free characteristic of the proposed numerical method.

Coupled poroelastic solutions for the reservoir and caprock layers with the overburden confinement effects

Coupled theory of poroelasticity is used to develop a generic analytical solution for the time-dependent solid stress and pore fluid pressure of the subsurface rocks. A uniaxial model comprising three layers of rock formations, namely, the reservoir, caprock and burden rock is considered. The model entails different mechanical and flow properties for the considered rock layers. A linear stress–displacement formulation at the interface of the caprock and overburden defined and is calibrated to account for the confinement effect of the burden rock. The closed-form solution to the considered coupled, poroelastic problem is derived in Laplace transform space. The time-domain solution is retrieved by numerical inversion of the solution in Laplace transform space.The obtained general solution is used to assess the coupled pore fluid pressure and stress evolution of rock layers in the following applied problems. (1) Caprock integrity upon fluid injection into the reservoir. (2) Seal rock efficiency of containing fluid transport from an abnormally pressured reservoir compartment. Findings indicate that the time evolution of pore fluid transport within and in between the rock layers is influenced by the relative magnitude of both flow and mechanical properties of the reservoir and caprock. In particular, the contrast between the elastic moduli of the reservoir and caprock are found to have substantial effect on the rate and magnitude of overpressure dissipation from the reservoir. Coulomb’s shear failure criterion, together with Terzaghi’s effective stress criterion for tensile failure, is used to assess the time-dependent failure tendency of the caprock upon exposure to excess pressure of the reservoir in the case of fluid injection problem. Findings suggest that higher injection rate, stiffer overburden, thinner reservoir and stiffer reservoir rock would enhance the failure tendency of the caprock.

On the hydraulic fracture of poroelastic media

The problems of quasi-static poroelasticity for stationary and extending cracks in an isotropic homogeneous medium are considered. The crack driving forces are caused by pressure of fluid injected inside the crack by external sources. Using simple and double layer potentials of poroelasticity, the problems are reduced to systems of 2D-integral equations on the crack surface. The integral equations contain operators with various time-asymptotics at infinity. Neglecting operators rapidly vanishing with time results in an approximate formulation of the crack problems. In this formulation, the process of filtration of fluid from the crack surface into the host medium is independent on the medium deformation. For numerical solution, the integral equations are discretized with respect to spatial and time variables. Examples of solutions of the filtration problem are presented for penny shape cracks with constant and extending radii. The three parameter model is used for simulation of hydraulic fracture crack propagation in poroelastic media. Time-dependencies of the crack radius, pressure distribution, and fluid flux on the crack surface are studied for various permeability parameters of the host medium.

Robust block preconditioners for poroelasticity

In this paper we develop and analyze several preconditioners for the linear systems arising from discretized poroelasticity problems. The preconditioners include one block preconditioner for the two-field Biot model and several preconditioners for the classical three-field Biot model. We manage to analyze these different preconditioners under a same theoretical framework and show that all of them are uniformly optimal with respect to material and discretization parameters. Numerical tests demonstrate the robustness of these preconditioners.

Poroelastic model parameter identification using artificial neural networks: on the effects of heterogeneous porosity and solid matrix Poisson ratio

Predictive analysis of poroelastic materials typically require expensive and time-consuming multiscale and multiphysics approaches, which demand either several simplifications or costly experimental tests for model parameter identification.This problem motivates us to develop a more efficient approach to address complex problems with an acceptable computational cost. In particular, we employ artificial neural network (ANN) for reliable and fast computation of poroelastic model parameters. Based on the strong-form governing equations for the poroelastic problem derived from asymptotic homogenisation, the weighted residuals formulation of the cell problem is obtained. Approximate solution of the resulting linear variational boundary value problem is achieved by means of the finite element method. The advantages and downsides of macroscale properties identification via asymptotic homogenisation and the application of ANN to overcome parameter characterisation challenges caused by the costly solution of cell problems are presented. Numerical examples, in this study, include spatially dependent porosity and solid matrix Poisson ratio for a generic model problem, application in tumour modelling, and utilisation in soil mechanics context which demonstrate the feasibility of the presented framework.

An abstract analysis framework for monolithic discretisations of poroelasticity with application to Hybrid High-Order methods

In this work, we introduce a novel abstract framework for the stability and convergence analysis of fully coupled discretisations of the poroelasticity problem and apply it to the analysis of Hybrid High-Order (HHO) schemes. A relevant feature of the proposed framework is that it rests on mild time regularity assumptions that can be derived from an appropriate weak formulation of the continuous problem. To the best of our knowledge, these regularity results for the Biot problem are new. A novel family of HHO discretisation schemes is also proposed and analysed, and their performance numerically evaluated.

Constraint energy minimizing generalized multiscale finite element method for nonlinear poroelasticity and elasticity

In this paper, we apply the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) to first solving a nonlinear poroelasticity problem. The arising system consists of a nonlinear pressure equation and a nonlinear stress equation in strain-limiting setting, where strains keep bounded while stresses can grow arbitrarily large. After time-discretization of the system, to tackle the nonlinearity, we linearize the resulting equations by Picard iteration. To handle the linearized equations, we employ the CEM-GMsFEM and obtain appropriate offline multiscale basis functions for the pressure and the displacement. More specifically, first, auxiliary multiscale basis functions are generated by solving local spectral problems, via the GMsFEM. Then, multiscale spaces are constructed in oversampled regions, by solving a constraint energy minimizing (CEM) problem. After that, this strategy (with the CEM-GMsFEM) is also applied to a static case of the above nonlinear poroelasticity problem, that is, elasticity problem, where the residual based online multiscale basis functions are generated by an adaptive enrichment procedure, to further reduce the error. Convergence of the two cases is demonstrated by several numerical simulations, which give accurate solutions, with converging coarse-mesh sizes as well as few basis functions (degrees of freedom) and oversampling layers.

A moving finite element framework for fast infiltration in nonlinear poroelastic media

Poroelasticity theory can be used to analyse the coupled interaction between fluid flow and porous media (matrix) deformation. The classical theory of linear poroelasticity captures this coupling by combining Terzaghi’s effective stress with a linear continuity equation. Linear poroelasticity is a good model for very small deformations; however, it becomes less accurate for moderate to large deformations. On the other hand, the theory of large-deformation poroelasticity combines Terzaghi’s effective stress with a nonlinear continuity equation. In this paper, we present a finite element solver for linear and nonlinear poroelasticity problems on triangular meshes based on the displacement-pressure two-field model. We then compare the predictions of linear poroelasticity with those of large-deformation poroelasticity in the context of a two-dimensional model problem where flow through elastic, saturated porous media, under applied mechanical oscillations, is considered. In addition, the impact of introducing a deformation-dependent permeability according to the Kozeny-Carman equation is explored. We computationally show that the errors in the displacement and pressure fields that are obtained using the linear poroelasticity are primarily due to the lack of the kinematic nonlinearity. Furthermore, the error in the pressure field is amplified by incorporating a constant permeability rather than a deformation-dependent permeability.

Coupling of meshfree peridynamics with the Finite Volume Method for poroelastic problems

Peridynamics is a non-local theory of continuum mechanics that has been developed primarily for understanding material failure due to different mechanisms, including fluid-driven crack propagation during hydraulic fracturing of subsurface reservoirs. Because of its non-local nature, this theory is computationally expensive. To improve its performance, a scheme was recently proposed for coupling Peridynamics (PD) with less expensive Finite Element Method (FEM) for static equilibrium problems. This scheme has been adapted in the current paper to couple a PD-based poroelastic model with the Finite Volume Method (FVM) for simulating problems in porous media. Coupling is implemented by dividing the computational domain into two subdomains, one of which is discretized and solved with PD and the other with FVM. The formulation is developed for porous flow involving fluid mass balance and is extended for poroelastic problems to include rock momentum balance. The coupled model is verified against the analytical solutions to classical problems. No spurious behavior is observed near the PD-FVM interface region. Improvements in computational performance over the pure PD model are demonstrated. Moreover, due to differences in the sparsity patterns and the magnitudes of PD and FVM transmissibility/Jacobian terms, it is shown that appending the PD equations after all the FV equations in the global matrix has additional computational benefits.

Generalized Multiscale Finite Element Method for the poroelasticity problem in multicontinuum media

In this paper, we consider a poroelasticity problem in heterogeneous multicontinuum media that is widely used in simulations of the unconventional hydrocarbon reservoirs and geothermal fields. Mathematical model contains a coupled system of equations for pressures in each continuum and effective equation for displacement with volume force sources that are proportional to the sum of the pressure gradients for each continuum. To illustrate the idea of our approach, we consider a dual continuum background model with discrete fracture networks that can be generalized to a multicontinuum model for poroelasticity problem in complex heterogeneous media. We present a fine grid approximation based on the finite element method and Discrete Fracture Model (DFM) approach for two and three-dimensional formulations. The coarse grid approximation is constructed using the Generalized Multiscale Finite Element Method (GMsFEM), where we solve local spectral problems for construction of the multiscale basis functions for displacement and pressures in multicontinuum media. We present numerical results for the two and three dimensional model problems in heterogeneous fractured porous media. We investigate relative errors between reference fine grid solution and presented coarse grid approximation using GMsFEM with different numbers of multiscale basis functions. Our results indicate that the proposed method is able to give accurate solutions with few degrees of freedoms.

Computational Multiscale Methods for Linear Heterogeneous Poroelasticity

We consider a strongly heterogeneous medium saturated by an incompressible viscous fluid as it appears in geomechanical modeling. This poroelasticity problem suffers from rapidly oscillating material parameters, which calls for a thorough numerical treatment. In this paper, we propose a method based on the local orthogonal decomposition technique and motivated by a similar approach used for linear thermoelasticity. Therein, local corrector problems are constructed in line with the static equations, whereas we propose to consider the full system. This allows to benefit from the given saddle point structure and results in two decoupled corrector problems for the displacement and the pressure. We prove the optimal first-order convergence of this method and verify the result by numerical experiments.

A generalized poroelastic model using FEniCS with insights into the Noordbergum effect

Understanding the coupled behavior of fluid flow and solid deformation in porous media is often a critical aspect of geoscientific investigations, including studies of injection-induced seismicity, geological carbon sequestration, or surface deformation from pumping operations. In 1941, Biot first outlined the standard poroelastic formulation that accounted for this coupling within an isotropic and elastic porous medium. This fully coupled system of partial differential equations typically requires the use of numerical modeling for any practical purposes, necessitating either lengthy code development or the use of a complex, prebuilt model that often lacks flexibility. However, due to recent advances in automated approaches to solving systems of differential equations, problems of poroelasticity can now be easily handled in a streamlined yet flexible manner. Here, we present a poroelastic model built within the framework of FEniCS – an open source, general purpose finite element method software – which solves the system monolithically and can produce a continuous and mass-conserving solution for specific discharge. We present both a linear model and a generalized, nonlinear model. The nonlinear model allows for variable fluid density and employs a fluid pressure and volumetric strain dependent porosity relationship. The behavior of the linear model is verified with common benchmark problems, whereas results from the nonlinear model are compared to assess the impact of including its generalizations. Finally, the model is applied to a two-dimensional problem of fluid extraction meant to replicate the well-known Noordbergum effect (in which the fluid pressure in layers adjacent to the pumped layer temporarily increases during pumping). This is often referred to as ”reverse water fluctuation”. This showcases not only the flexibility of the model but also its ability to simulate numerically challenging scenarios. The results from this simulation suggest a novel explanation of the physical mechanism generating the Noordbergum effect: strain gradients.

Finite volume discretization for poroelastic media with fractures modeled by contact mechanics

SummaryA fractured poroelastic body is considered where the opening of the fractures is governed by a nonpenetration law, whereas slip is described by a Coulomb‐type friction law. This physical model results in a nonlinear variational inequality problem. The variational inequality is rewritten as a complementary function, and a semismooth Newton method is used to solve the system of equations. For the discretization, we use a hybrid scheme where the displacements are given in terms of degrees of freedom per element, and an additional Lagrange multiplier representing the traction is added on the fracture faces. The novelty of our method comes from combining the Lagrange multiplier from the hybrid scheme with a finite volume discretization of the poroelastic Biot equation, which allows us to directly impose the inequality constraints on each subface. The convergence of the method is studied for several challenging geometries in 2D and 3D, showing that the convergence rates of the finite volume scheme do not deteriorate when it is coupled to the Lagrange multipliers. Our method is especially attractive for the poroelastic problem because it allows for a straightforward coupling between the matrix deformation, contact conditions, and fluid pressure.

Perfectly matched absorbing layer for modelling transient wave propagation in heterogeneous poroelastic media

AbstractThe perfectly matched layer (PML) has been demonstrated to be an efficient absorbing boundary for near-field wave simulation. For heterogeneous media, the property of the PML needs to be carefully specified to avoid numerical instability and artificial reflection because part of it lies at the discontinuous interface. Coupled acoustic-poroelastic (A-P) media or coupled elastic-poroelastic (E-P) media often arise in the field of geophysics. However, PMLs that appropriately terminate these heterogeneous poroelastic media are still lacking. The main purpose of this paper is to explore the application of unsplit PMLs for transient wave modeling in infinite, heterogeneous, coupled A-P media or coupled E-P media. To this end, a consistent derivation of memory-efficient PML formulations for the second-order Biot's equations, elastic wave equations and acoustic wave equations is performed based on complex coordinate transformation using auxiliary differential equations. Furthermore, the interface boundary conditions inside the absorbing layer are rigorously derived for the considered A-P and E-P cases. Finally, the weak form of PML formulations for coupled poroelastic problems is presented. The finite element method is used to validate the proposed PML based on several two-dimensional benchmarks. The accuracy and stability of weak PML formulations are investigated. In particular, for coupled acoustic-poroelastic PML, two extreme (open-pore and sealed-pore) interface conditions are considered and PML results are compared with known analytical solutions. This study demonstrates the ability of the PML to effectively eliminate outgoing bulk waves and surface waves in coupled poroelastic media.

A fractional order theory of poroelasticity

We introduce a time memory formalism in the flux-pressure constitutive relation, ruling the fluid diffusion phenomenon occurring in several classes of porous media. The resulting flux-pressure law is adopted into the Biot’s formulation of the poroelasticity problem. The time memory formalism, useful to capture non-Darcy behavior, is modeled by the Caputo’s fractional derivative. We show that the time-evolution of both the degree of settlement and the pressure field is strongly influenced by the order of Caputo’s fractional derivative. Also a numerical experiment aiming at simulating the confined compression test poroelasticity problem of a sand sample is performed. In such a case, the classical Darcy equation may lead to inaccurate estimates of the settlement time.

A fully coupled scheme using virtual element method and finite volume for poroelasticity

In this paper, we design and study a fully coupled numerical scheme for the poroelasticity problem modeled through Biot’s equations. The classical way to numerically solve this system is to use a finite element method for the mechanical equilibrium equation and a finite volume method for the fluid mass conservation equation. However, to capture specific properties of the underground medium such as heterogeneities, discontinuities, and faults, meshing procedures commonly lead to badly shaped cells for finite element-based modeling. Consequently, we investigate the use of the recent virtual element method which appears as a potential discretization method for the mechanical part and could therefore allow the use of a unique mesh for both the mechanical and fluid flow modeling. Starting from a first insight into virtual element method applied to the elastic problem in the context of geomechanical simulations, we apply in addition a finite volume method to take care of the fluid conservation equation. We focus on the first-order virtual element method and the two-point flux approximation for the finite volume part. A mathematical analysis of this original coupled scheme is provided, including existence and uniqueness results and a priori estimates. The method is then illustrated by some computations on two- or three-dimensional grids inspired by realistic application cases.

Meshless Modeling of Flow Dispersion and Progressive Piping in Poroelastic Levees

Performance data on earth dams and levees continue to indicate that piping is one of the major causes of failure. Current criteria for prevention of piping in earth dams and levees have remained largely empirical. This paper aims at developing a mechanistic understanding of the conditions necessary to prevent piping and to enhance the likelihood of self-healing of cracks in levees subjected to hydrodynamic loading from astronomical and meteorological (including hurricane storm surge-induced) forces. Systematic experimental investigations are performed to evaluate erosion in finite-length cracks as a result of transient hydrodynamic loading. Here, a novel application of the localized collocation meshless method (LCMM) to the hydrodynamic and poroelastic problem is introduced to arrive at high-fidelity field solutions. Results from the LCMM numerical simulations are designed to be used as an input, along with the soil and erosion parameters obtained experimentally, to characterize progressive piping.

A Hybrid High-Order Discretization Method for Nonlinear Poroelasticity

Abstract In this work, we construct and analyze a nonconforming high-order discretization method for the quasi-static single-phase nonlinear poroelasticity problem describing Darcean flow in a deformable porous medium saturated by a slightly compressible fluid. The nonlinear elasticity operator is discretized using a Hybrid High-Order method, while the Darcy operator relies on a Symmetric Weighted Interior Penalty discontinuous Galerkin scheme. The method is valid in two and three space dimensions, delivers an inf-sup stable discretization on general meshes including polyhedral elements and nonmatching interfaces, supports arbitrary approximation orders, and has a reduced cost thanks to the possibility of statically condensing a large subset of the unknowns for linearized versions of the problem. Moreover, the proposed construction can handle both nonzero and vanishing specific storage coefficients.

Computational multiscale methods for linear poroelasticity with high contrast

In this work, we employ the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) to solve the problem of linear heterogeneous poroelasticity with coefficients of high contrast. The proposed method makes use of the idea of energy minimization with suitable constraints to generate efficient basis functions for the displacement and the pressure. These basis functions are constructed by solving a class of local auxiliary optimization problems based on eigenfunctions containing local information on the heterogeneity. Techniques of oversampling are adapted to enhance the computational performance. A convergence of first order is shown and illustrated by several numerical tests.

On the Solvability of Some Dynamic Poroelastic Problems

We consider the direct problems for poroelasticity equations. In the low-frequency approximation we prove existence and uniqueness theorems for the solution to a certain mixed problem. In the high-frequency approximation we establish the uniqueness of a weak solution to the mixed problem and its continuous dependence on the data in the cases of bounded and unbounded temporal intervals and for however many spatial variables.