Biot's Consolidation Model Research Articles

The nonconforming locking-free virtual element method for the Biot's consolidation model in poroelasticity

The nonconforming locking-free virtual element method for the Biot's consolidation model in poroelasticity

A combination of physics-informed neural networks with the fixed-stress splitting iteration for solving Biot's model

IntroductionBiot's consolidation model in poroelasticity describes the interaction between the fluid and the deformable porous structure. Based on the fixed-stress splitting iterative method proposed by Mikelic et al. (Computat Geosci, 2013), we present a network approach to solve Biot's consolidation model using physics-informed neural networks (PINNs).MethodsTwo independent and small neural networks are used to solve the displacement and pressure variables separately. Accordingly, separate loss functions are proposed, and the fixed stress splitting iterative algorithm is used to couple these variables. Error analysis is provided to support the capability of the proposed fixed-stress splitting-based PINNs (FS-PINNs).ResultsSeveral numerical experiments are performed to evaluate the effectiveness and accuracy of our approach, including the pure Dirichlet problem, the mixed partial Neumann and partial Dirichlet problem, and the Barry-Mercer's problem. The performance of FS-PINNs is superior to traditional PINNs, demonstrating the effectiveness of our approach.DiscussionOur study highlights the successful application of PINNs with the fixed-stress splitting iterative method to tackle Biot's model. The ability to use independent neural networks for displacement and pressure offers computational advantages while maintaining accuracy. The proposed approach shows promising potential for solving other similar geoscientific problems.

A priori error estimates of two monolithic schemes for Biot's consolidation model

AbstractThis paper concentrates on a priori error estimates of two monolithic schemes for Biot's consolidation model based on the three‐field formulation introduced by Oyarzúa et al. (SIAM J Numer Anal, 2016). The spatial discretizations are based on the Taylor–Hood finite elements combined with Lagrange elements for the three primary variables. We employ two different schemes to discretize the time domain. One uses the backward Euler method, and the other applies the combination of the backward Euler and Crank‐Nicolson methods. A priori error estimates show that both schemes are unconditionally convergent with optimal error orders. Detailed numerical experiments are presented to validate the theoretical analysis.

A locking-free and mass conservative H(div) conforming DG method for the Biot's consolidation model

A locking-free and mass conservative H(div) conforming DG method for the Biot's consolidation model

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.

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.

A mixed element analysis of the Biot's model with Darcy–Forchheimer flow

AbstractIn this article, we consider the three‐field Biot's consolidation model, in which the flow's velocity is described by Darcy–Forchheimer law. The model is nonlinear and the unknown variables are the displacement of the solid phase, fluid flux, and pore pressure. The linear finite element is proposed to approximate the solid displacement and the RT mixed finite element is proposed to approximate the fluid flux and pore pressure. The error estimates is given for semidiscrete scheme and fully discrete scheme. When analyzing the fully discrete scheme, no time step restriction is needed. Numerical tests for this method confirm the accuracy of the P1‐RT0‐P0 method when the constraint storage coefficient is positive and is null.

Parameter-Robust Stochastic Galerkin Mixed Approximation for Linear Poroelasticity with Uncertain Inputs

Linear poroelasticity models have important applications in biology and geophysics. In particular, the well-known Biot consolidation model describes the coupled interaction between the linear respo...

Finite element method with the total stress variable for Biot's consolidation model

AbstractIn this work, semi‐discrete and fully discrete error estimates are derived for the Biot's consolidation model described using a three‐field finite element formulation. These fields include displacements, total stress and pressure. The model is implemented using a backward Euler discretization in time for the fully discrete scheme and validated for benchmark examples. Computational experiments are presented to verify the convergence orders for two finite elements with discontinuous and continuous approximations for the total stress.

Influence of Wave-Induced Uplift Forces on Upheaval Buckling of Pipelines Buried in Sandy Seabeds

This study focuses on the buckling of pipelines in shallow waters subjected to surface gravity waves. The wave-induced uplift forces on pipelines buried in sandy seabeds are investigated using Biot's consolidation model. Empathetic imperfection model proposed by Taylor and Tran (1994, “Experimental and Theoretical Studies in Subsea Pipeline Buckling," Mar. Struct., 9(2), pp. 211–257.) is used for the study. Thereafter, buckling analyses are performed on the pipeline with the combined temperature and the wave-induced loads. The differences in the critical buckling temperatures for the pipe with consideration of wave loads are analyzed within a range of sea states. The influence of wave loads is found significant for low burial depth ratios.

An H(div)-conforming Finite Element Method for Biot's Consolidation Model.

In this paper, we develop an H(div)-conforming finite element method for Biot's consolidation model in poroelasticity. In our method, the flow variables are discretized by an H(div)-conforming mixed finite elements. For relaxing the H 1-conformity of the displacement, we approximate the displacement by using an H(div)-conforming finite element method, in which the tangential components are discretized in the interior penalty discontinuous Galerkin framework. For both the semi-discrete and the fully discrete schemes, we prove the existence and uniqueness theorems of the approximate solutions and derive the optimal convergence rate for each variable.

On the role of compressibility in poroviscoelastic models.

In this article we conduct an analytical study of a poroviscoelastic mixture model stemming from the classical Biot's consolidation model for poroelastic media, comprising a fluid component and a solid component, coupled with a viscoelastic stress-strain relationship for the total stress tensor. The poroviscoelastic mixture is studied in the one-dimensional case, corresponding to the experimental conditions of confined compression. Upon assuming (i) negligible inertial effects in the balance of linear momentum for the mixture, (ii) a Kelvin-Voigt model for the effective stress tensor and (iii) a constant hydraulic permeability, we obtain an initial value/boundary value problem of pseudo-parabolic type for the spatial displacement of the solid component of the mixture. The dimensionless form of the differential equation is characterized by the presence of two positive parameters γ and η, representing the contributions of compressibility and structural viscoelasticity, respectively. Explicit solutions are obtained for different functional forms characterizing the boundary traction. The main result of our analysis is that the compressibility of the components of a poroviscoelastic mixture does not give rise to unbounded responses to non-smooth traction data. Interestingly, compressibility allows the system to store potential energy as its components are elastically compressed, thereby providing an additional mechanism that limits the maximum of the discharge velocity when the imposed boundary traction is irregular in time.

Application of Genetic Algorithms for the Estimation of Hydraulic Conductivity

Abstract In the study described here model calibration was performed employing the inverse analysis using genetic algorithms (GA). The objective of analysis is to determine value of the coefficient of hydraulic conductivity, k. The commonly used method for the determination of coefficient of hydraulic conductivity based on Terzaghi consolidation leads to an underestimation of the value of k as the Terzaghi model does not take into account the deformation of soil skeleton. Here, an alternative methodology based on genetic algorithms is presented for the determination of the basic parameters of Biot consolidation model. It has been demonstrated that genetic algorithms are a highly effective tool enabling automatic calibration based on simple rules. The values of the coefficient of hydraulic conductivity obtained with GA are of at least one order smaller than values obtained with the Terzaghi model.

Upscaling of the Coupling of Hydromechanical and Thermal Processes in a Quasi-static Poroelastic Medium

We undertake a formal derivation of a linear poro-thermo-elastic system within the framework of quasi-static deformation. This work is based upon the well-known derivation of the quasi-static poroelastic equations (also known as the Biot consolidation model) by homogenization of the fluid-structure interaction at the microscale. We now include energy, which is coupled to the fluid-structure model by using linear thermoelasticity, with the full system transformed to a Lagrangian coordinate system. The resulting upscaled system is similar to the linear poroelastic equations, but with an added conservation of energy equation, fully coupled to the momentum and mass conservation equations. In the end, we obtain a system of equations on the macroscale accounting for the effects of mechanical deformation, heat transfer, and fluid flow within a fully saturated porous material, wherein the coefficients can be explicitly defined in terms of the microstructure of the material. For the heat transfer we consider two different scaling regimes, one where the Péclet number is small, and another where it is unity. We also establish the symmetry and positivity for the homogenized coefficients.

Parameter-robust stability of classical three-field formulation of Biot's consolidation model

This paper is devoted to the stability analysis of a classical three-field formulation of Biot's consolidation model where the unknown variables are the displacements, fluid flux (Darcy velocity), and pore pressure. Specific parameter-dependent norms provide the key in establishing the full parameter-robust inf-sup stability of the continuous problem. Therefore, stability results presented here are uniform not only with respect to the Lam\'e parameter $\lambda$, but also with respect to all the other model parameters. This allows for the construction of a uniform block diagonal preconditioner within the framework of operator preconditioning. Stable discretizations that meet the required conditions for full robustness and guarantee mass conservation, both locally and globally, are discussed and corresponding optimal error estimates proven.

Simulated geomechanical responses to marine methane hydrate recovery using horizontal wells in the Shenhu area, South China Sea

Abstract The recovery of natural gas hydrate (NGH) from marine sediments faces challenging due to not only the gas productivity itself but also the possible geohazards, such as seafloor subsidence, submarine landslide, wellbore instability and possible sand production. The coupled thermal-hydrodynamic-mechanical (THM) processes during NGH recovery are generally complex, and numerical simulation tools are needed to assess related geomechanical responses. We have extended the Biot consolidation model in our previous simulator TOUGH2Biot, and incorporated into the existing TOUGH + hydrate code, resulting in a THM simulator for the NGH recovery. The THM simulator is used to assess the geomechanical responses to gas recovery from an unconfined hydrate-bearing sediment (HBS) in the Shenhu area, South China Sea. We investigated depressurization using constant bottom-hole pressure through a horizontal well. Results show that methane production quickly reaches a stabilized state and the water rate increases linearly. The drawdown of pore pressure around the well controls the increase in effective stress. Subsidence becomes significant after depressurization due to the quickly propagation of pore pressure. Depressurization in early stage could contribute to more than half of the total subsidence. The decreasing production pressure leads to an increase in the methane production rate but deterioration in subsidence. A decrease in intrinsic permeability of overlying and underlying layer is undesirable due to its decrease in methane production rate and the worse of subsidence. A balance between gas productivity and related geomechanical response must be achieved. The methods and preliminary results presented in this study could help us to understand the geomechanical behaviors during NGH recovery and to design trial production schemes under similar conditions.

A block-diagonal preconditioner for a four-field mixed finite element method for Biot's equations

In this paper, we explore an efficient preconditioning method for the saddle point system resulting from a four-field mixed finite element method applied to Biot's consolidation model. The proposed preconditioner is a block diagonal preconditioner based on the Schur complement. We obtain bounds on the eigenvalues of the preconditioned matrix that are clustered away from 0. To reduce the computational expense, this preconditioner is inverted approximately. Some numerical results are provided to show the efficiency of our preconditioning strategy when applied to a poroelasticity problem in a layered medium.

Parameter-Robust Discretization and Preconditioning of Biot's Consolidation Model

Biot's consolidation model in poroelasticity has a number of applications in science, medicine, and engineering. The model depends on various parameters, and in practical applications these parameters range over several orders of magnitude. A current challenge is to design discretization techniques and solution algorithms that are well-behaved with respect to these variations. The purpose of this paper is to study finite element discretizations of this model and construct block diagonal preconditioners for the discrete Biot systems. The approach taken here is to consider the stability of the problem in nonstandard or weighted Hilbert spaces and employ the operator preconditioning approach. We derive preconditioners that are robust with respect to both the variations of the parameters and the mesh refinement. The parameters of interest are small time-step sizes, large bulk and shear moduli, and small hydraulic conductivity.

Carleman estimate for Biot consolidation system in poro‐elasticity and application to inverse problems

In this paper, we consider a coupled system of mixed hyperbolic–parabolic type, which describes the Biot consolidation model in poro‐elasticity. We establish a local Carleman estimate for Biot consolidation system. Using this estimate, we prove the uniqueness and a Hölder stability in determining on the one hand a physical parameter arising in connection with secondary consolidation effects λ∗ and on the other hand the two spatially varying densities by a single measurement of solution over ω × (0,T), where T > 0 is a sufficiently large time and a suitable subdomain ω satisfying ∂ω⊃∂Ω. Copyright © 2016 John Wiley & Sons, Ltd.

Robust Error Analysis of Coupled Mixed Methods for Biot’s Consolidation Model

We study the a priori error analysis of finite element methods for Biot's consolidation model. We consider a formulation which has the stress tensor, the fluid flux, the solid displacement, and the pore pressure as unknowns. Two mixed finite elements, one for linear elasticity and the other for mixed Poisson problems are coupled for spatial discretization, and we show that any pair of stable mixed finite elements is available. The novelty of our analysis is that the error estimates of all the unknowns are robust for material parameters. Specifically, the analysis does not need a uniformly positive storage coefficient, and the error estimates are robust for nearly incompressible materials. Numerical experiments illustrating our theoretical analysis are included.