Deformable Porous Media Research Articles

The hydromechanical coupled numerical method in pseudo-3D axisymmetric domain with cracks extension and coalescence applies to the decompression failure problem

The stress-strain state of the saturated porous media determines the behavior of fracturing, which defines the efficiency of developing tight oil, shale, coalbed, and thermal energy fields. Therefore, reliable hydromechanical coupled simulations with destruction reconstruction are critical.The proposed innovative simulator has a strong interrelation between fluid flow and rock deformation of porous media and realizes a fully coupled pseudo-transient numerical method by high-performance computing (HPC) tools. To increase the detail of the results in the problem, a finite difference numerical algorithm was implemented in the axisymmetric cylindrical domain, which reduces from three to two dimensions without loss of precision. Highly efficient parallelization using CUDA on the GPU computes meshes of up to one billion cells, allowing the simulation of a total core sample to sub-micrometer resolution in an appropriate time. The algorithm has been validated to find the exact solution to the cylinder problem. The proposed model accounts for cracks propagation with their coalescence within a single computational static grid, which keeps timing close to the continuous model.This comprehensive implementation enables solving industrial problems, such as modeling core sample damage during rapid decompression. High-resolution simulations help reconstruct fracture propagation, analyze the initial stress state, and identify critical damage factors. The comparison with the exact solution to the cylinder problem confirmed the reliability of the algorithm. The calculation results show a strong dependence of decompression failure on the coalescence and elongation of cracks, influenced significantly by the rock's cohesion. Microcracks length and distribution play a decisive role in the decompressive destruction behavior of the rock sample. For the first time, the simulations demonstrated the decompressive destruction of a core sample during an uncontrolled, rapid core retrieval operation.

Stochastic analysis of the interaction between excess fluid flow and soil deformation in heterogeneous deformable porous media

Stochastic analysis of the interaction between excess fluid flow and soil deformation in heterogeneous deformable porous media

Existence of weak solutions to a Cahn–Hilliard–Biot system

We prove existence of weak solutions to a diffuse interface model describing the flow of a fluid through a deformable porous medium consisting of two phases. The system non-linearly couples Biot’s equations for poroelasticity, including phase-field dependent material properties, with the Cahn–Hilliard equation to model the evolution of the solid, and is further augmented by a visco-elastic regularization of Kelvin–Voigt type. To obtain this result, we approximate the problem in two steps, where first a semi-Galerkin ansatz is employed to show existence of weak solutions to regularized systems, for which later on compactness arguments allow limit passage. Notably, we also establish a maximal regularity theory for linear visco-elastic problems.

Моделирование течения воздуха в упруго-деформируемой пористой среде, аппроксимирующей легкие человека: структура модели, ее основные уравнения и разрешающие соотношения

This paper is devoted to refining the mathematical model of the human respiratory system designed to predict the occurrence of respiratory pathologies caused by the negative effects of atmospheric air pollution. It is difficult to identify individual air channels in the human lungs, consisting of small respiratory tracts, called alveoli, through which oxygen from the air passes into blood. Therefore, the human lungs are modeled as a two-phase elastically deformable saturated porous medium, in which one phase is the deformable skeleton of the medium (lung tissue) described by the model of deformable solid body, and the other phase is gas filling the porous space. We construct the model of air flow in the human lungs, which consists of the sub-model of elastic deformation of the porous lung medium, experiencing large displacement gradients, and the sub-model of air filtration through the deformable porous medium. The model takes into account the relationships between sub-models of the respiratory system, i.e. we consider the coupled problems of deformation and filtration in differential form. Since at present the analytical solution of the stated essentially nonlinear problem is unfeasible, the problem is formulated in a generalized weak form to allow subsequent application of numerical methods using the Galerkin method. For the numerical solution of the nonlinear problem of deformation of the two-phase lung medium, it is suggested to use the finite element method, for the filtration problem - the finite volume method. The resolving relations needed to solve the formulated sub-problems are derived. To solve the nonlinear problem, we developed algorithms and a set of programs. The description of these computational tools, as well as analysis of the results obtained will be presented in our the next paper.

Experimental and numerical investigations of cake baking in mold

This article deals with the in-mold cake baking process. The methodology relies on numerical and experimental investigations to validate a multiphysics developed model. This numerical model is next exploited to enhance the comprehension of the physical processes that occur during a typical cake baking phase. Based on the governing equations for heat and mass transport and under few assumptions (deformable porous medium, local thermodynamic equilibrium, ideal gas mixture …), the problem consists in solving a system of five coupled partial derivative equations. Temperature, moisture content, total gas pressure, porosity and displacement are used as state variables in this study. The swelling of the dough caused by the increase in total gas pressure is predicted by a viscoelastic model. This thermo-hydro-mechanical model is implemented in a finite element code. Moreover, an experimental laboratory set-up was developed to continually acquire temperatures, water losses and gas pressure inside the product. The browning kinetic is also studied. Furthermore, the cake deformation is tracked by camera. The numerical model is validated based on the conducted experiments. Different operating conditions are tested to verify the robustness of the model. Finally, a sensitivity analysis is performed to understand the impact of estimated parameters on the outputs of the developed model.

Electrokinetics of Erosive Seepage through Deformable Porous Media.

Channelization and branching patterns frequently appear in porous structures as a result of fluid-flow-mediated erosion, which causes spatiotemporal changes in the medium. However, most studies on electrokinetic effects in porous media focus on the overall impact of the electric field on electrical double-layer formation in micropores and its influence on ionic transport, without addressing the spatiotemporal erosive characteristics and resulting porosity distribution. In this study, we explore the interplay between flow-induced shear stress and an external electric field on the dynamic evolution of porosity in deformable porous media using semi-analytical modeling. Our numerical simulations accurately predict the differences in porosity and erosive development when the electric field aligns with or opposes the flow, highlighting the importance of the direction of the external stimulus and not just its magnitude. Our findings establish a foundation for electric-field-mediated control of porous media properties and explain electrokinetic transport by considering dynamic porosity variations as a result of erosive deformation, an aspect previously unaddressed.

A novel model for oil reservoirs considering stress sensitivity and dynamic relative permeability

ABSTRACT Stress sensitivity and dynamic relative permeability are undeniable phenomenon in oil reservoir numerical simulation. In this paper, a complex two-phase flow model was established considering the deformation of porous media and relative permeability dynamic changes during water injection. A sensitivity analysis on this new model is carried out. The results reveal that stress sensitivity is a key factor that cannot be ignored when using reservoir numerical simulation to calculate water saturation. When the stress-sensitivity coefficient is not considered, the average water saturation of the model is 51.59%. When the stress-sensitivity coefficient is 0.12MPa−1, the average water saturation of the model is 48.45%. After considering stress sensitivity, the average water saturation of the model decreases by 6.1%. The water saturation difference between the new model and the traditional model is mainly reflected in the vicinity of the water injection well. Additionally, the influence of the range of changes in reservoir permeability on the calculation of water saturation by the new model cannot be ignored.

Flow-Driven Deformation in Granular Porous Media: Dimensionless Analysis

Fluid injection can induce mechanical deformation in granular porous media due to the elevation of internal pore fluid pressure. This gains more significance when more than two immiscible fluids are involved, attributable to capillary and viscous drag forces. Such a coupled hydromechanical behavior associated with immiscible fluid flows plays an important role in injection, storage, and recovery of fluids in deformable porous media. This study presents a dimensionless map with newly proposed dimensionless parameters to predict deformation occurrence due to an immiscible fluid flow in deformable porous media. A series of hydromechanically coupled pore network simulations are first performed while varying the capillary number, mobility ratio, medium stiffness, and effective confining stress over orders of magnitudes. The compilation of simulation results with previously published Hele–Shaw experiment results is analyzed with the dimensionless parameters, such as the capillary number, mobility ratio, particle-level force ratios, and particle-level pressure ratios. Particularly, the particle-level pressure ratios include the capillary pressure ratio, defined as the ratio of capillary pressure to fracture pressure, and the viscous drag pressure, defined as the ratio of viscous drag pressure to fracture pressure. The dimensionless map based on the particle-level pressure ratios, where the capillary pressure ratio and viscous drag pressure ratio are defined as the ratios of capillary pressure and viscous drag pressure to fracture pressure, effectively delineates four deformation regimes—no deformation, capillary-induced deformation, drag-driven deformation, and mixed-mode deformation. The results demonstrate that capillary-induced deformation occurs when the capillary pressure ratio is greater than 10−1, while drag-driven deformation is observed when the viscous drag pressure ratio exceeds 10−2. The presented dimensionless map and dimensionless parameters are expected to be applicable for geological subsurface processes, including geological storage of carbon dioxide and hydrogen, and enhanced oil recovery.

Modeling dynamic crack branching in unsaturated porous media through multi‐phase micro‐periporomechanics

AbstractDynamic crack branching in unsaturated porous media holds significant relevance in various fields, including geotechnical engineering, geosciences, and petroleum engineering. This article presents a numerical investigation into dynamic crack branching in unsaturated porous media using a recently developed coupled micro‐periporomechanics (PPM) paradigm. This paradigm extends the PPM model by incorporating the micro‐rotation of the solid skeleton. Within this framework, each material point is equipped with three degrees of freedom: displacement, micro‐rotation, and fluid pressure. Consistent with the Cosserat continuum theory, a length scale associated with the micro‐rotation of material points is inherently integrated into the model. This study encompasses several key aspects: (1) Validation of the coupled micro‐PPM paradigm for effectively modeling crack branching in deformable porous media, (2) Examination of the transition from a single branch to multiple branches in porous media under drained conditions, (3) Simulation of single crack branching in unsaturated porous media under dynamic loading conditions, and (4) Investigation of multiple crack branching in unsaturated porous media under dynamic loading conditions. The numerical results obtained in this study are systematically analyzed to elucidate the factors that influence crack branching in porous media subjected to dynamic loading. Furthermore, the comprehensive numerical findings underscore the efficacy and robustness of the coupled micro‐PPM paradigm in accurately modeling dynamic crack branching in variably saturated porous media.

Permeability estimation for deformable porous media with convolutional neural network

PurposeAccurately evaluating fluid flow behaviors and determining permeability for deforming porous media is time-consuming and remains challenging. This paper aims to propose a novel machine-learning method for the rapid estimation of permeability of porous media at different deformation stages constrained by hydro-mechanical coupling analysis.Design/methodology/approachA convolutional neural network (CNN) is proposed in this paper, which is guided by the results of finite element coupling analysis of equilibrium equation for mechanical deformation and Boltzmann equation for fluid dynamics during the hydro-mechanical coupling process [denoted as Finite element lattice Boltzmann model (FELBM) in this paper]. The FELBM ensures the Lattice Boltzmann analysis of coupled fluid flow with an unstructured mesh, which varies with the corresponding nodal displacement resulting from mechanical deformation. It provides reliable label data for permeability estimation at different stages using CNN.FindingsThe proposed CNN can rapidly and accurately estimate the permeability of deformable porous media, significantly reducing processing time. The application studies demonstrate high accuracy in predicting the permeability of deformable porous media for both the test and validation sets. The corresponding correlation coefficients (R2) is 0.93 for the validation set, and the R2 for the test set A and test set B are 0.93 and 0.94, respectively.Originality/valueThis study proposes an innovative approach with the CNN to rapidly estimate permeability in porous media under dynamic deformations, guided by FELBM coupling analysis. The fast and accurate performance of CNN underscores its promising potential for future applications.

Two-phase flow thermo-hydro-mechanical modeling for a water flooding field case

Simulation of subsurface energy system involves multi-physical processes such as thermal, hydraulical, and mechanical (THM) processes, and requires a so-called THM coupled modeling approach. THM coupled modeling is commonly performed in geothermal energy production. However, for hydrocarbon extraction, we need to consider multiphase flow additionally. In this paper, we describe a three-dimensional numerical model of non-isothermal two-phase flow in the deformable porous medium by integrating governing equations of two-phase mixture in the porous media flow in the reservoir. To account for inter-woven impacts in subsurface conditions, we introduced a temperature-dependent fluid viscosity and a fluid density along with a strain-dependent reservoir permeability. Subsequently, we performed numerical experiments of a ten-year water flooding process employing the open-source parallelized code, OpenGeoSys. We considered different well patterns with colder water injection in realistic scenarios. Our results demonstrate that our model can simulate complex interactions of temperature, pore pressure, subsurface stress and water saturation simultaneously to evaluate the recovery performance. High temperature can promote fluid flow while cold water injection under non-isothermal conditions causes the normal stress reduction by significant thermal stress. Under different well patterns the displacement efficiency will be changed by the relative location between injection and production wells. This finding has provided the important reference for fluid flow and induced stress evolution during hydrocarbon exploitation under the environment of large reservoir depth and high temperature.

Data-driven discovery of interpretable water retention models for deformable porous media

The water retention behavior—a critical factor of unsaturated flow in porous media—can be strongly affected by deformation in the solid matrix. However, it remains challenging to model the water retention behavior with explicit consideration of its dependence on deformation. Here, we propose a data-driven approach that can automatically discover an interpretable model describing the water retention behavior of a deformable porous material, which can be as accurate as non-interpretable models obtained by other data-driven approaches. Specifically, we present a divide-and-conquer approach for discovering a mathematical expression that best fits a neural network trained with the data collected from a series of image-based drainage simulations at the pore-scale. We validate the predictive capability of the symbolically regressed counterpart of the trained neural network against unseen pore-scale simulations. Further, through incorporating the discovered symbolic function into a continuum-scale simulation, we showcase the inherent portability of the proposed approach: The discovered water retention model can provide results comparable to those from a hierarchical multi-scale model, while bypassing the need for sub-scale simulations at individual material points.

An energy-efficient GMRES–multigrid solver for space-time finite element computation of dynamic poroelasticity

We present and analyze computationally Geometric MultiGrid (GMG) preconditioning techniques for Generalized Minimal RESidual (GMRES) iterations to space-time finite element methods (STFEMs) for a coupled hyperbolic–parabolic system modeling, for instance, flow in deformable porous media. By using a discontinuous temporal test basis, a time marching scheme is obtained. Higher order approximations that offer the potential to inherit most of the rich structure of solutions to the continuous problem on computationally feasible grids increase the block partitioning dimension of the algebraic systems, comprised of generalized saddle point blocks. Our V-cycle GMG preconditioner uses a local Vanka-type smoother. Its action is defined in an exact mathematical way. Due to nonlocal coupling mechanisms of 348 unknowns, the smoother is applied on patches of elements. This ensures damping of higher order error frequencies. By numerical experiments of increasing complexity, the efficiency of the solver for STFEMs of different polynomial order is illustrated and confirmed. Its parallel scalability is analyzed. Beyond this study of classical performance engineering, the solver’s energy efficiency is investigated as an additional and emerging dimension in the design and tuning of algorithms on the hardware.

Shear Banding and Cracking in Unsaturated Porous Media through a Nonlocal THM Meshfree Paradigm

The mechanical behavior of unsaturated porous media under non-isothermal conditions plays a vital role in geo-hazards and geo-energy engineering (e.g., landslides triggered by fire and geothermal energy harvest and foundations). Temperature increase can trigger localized failure and cracking in unsaturated porous media. This article investigates the shear banding and cracking in unsaturated porous media under non-isothermal conditions through a thermo–hydro–mechanical (THM) periporomechanics (PPM) paradigm. PPM is a nonlocal formulation of classical poromechanics using integral equations, which is robust in simulating continuous and discontinuous deformation in porous media. As a new contribution, we formulate a nonlocal THM constitutive model for unsaturated porous media in the PPM paradigm in this study. The THM meshfree paradigm is implemented through an explicit Lagrangian meshfree algorithm. The return mapping algorithm is used to implement the nonlocal THM constitutive model numerically. Numerical examples are presented to assess the capability of the proposed THM mesh-free paradigm for modeling shear banding and cracking in unsaturated porous media under non-isothermal conditions. The numerical results are examined to study the effect of temperature variations on the formation of shear banding and cracking in unsaturated porous media.

Thermodynamically consistent numerical modeling of immiscible two‐phase flow in poro‐viscoelastic media

AbstractNumerical modeling of immiscible two‐phase flow in deformable porous media has become increasingly significant due to its applications in oil reservoir engineering, geotechnical engineering and many others. The coupling between two‐phase flow and geomechanics gives rise to a major challenge to the development of physically consistent mathematical models and effective numerical methods. In this article, based on the concept of free energies and guided by the second law of thermodynamics, we derive a thermodynamically consistent mathematical model for immiscible two‐phase flow in poro‐viscoelastic media. The model uses the fluid and solid free energies to characterize the fluid capillarity and solid skeleton elasticity, so that it rigorously follows an energy dissipation law. The thermodynamically consistent formulation of the pore fluid pressure is naturally derived for the solid mechanical equilibrium equation. Additionally, the model ensures the mass conservation law for both fluids and solids. For numerical approximation of the model, we propose an energy stable and mass conservative numerical method. The method herein inherits the energy dissipation law through appropriate energy approaches and subtle treatments for the coupling between two phase saturations, the effective pore pressure and porosity. Using the locally conservative cell‐centered finite difference methods on staggered grids with the upwind strategies for saturations and porosity, we construct the fully discrete scheme, which has the ability to conserve the masses of both fluids and solids as well as preserve the energy dissipation law at the fully discrete level. In particular, the proposed method is an unbiased algorithm, that is, treating the wetting phase, the non‐wetting phase and the solid phase in the same way. Numerical results are also given to validate and verify the features of the proposed model and numerical method.

Modeling flow and deformation in porous media from pore-scale to the Darcy-scale

In this paper we address the connections between the computational models of coupled flow and mechanical deformation in soils at the Darcy-scale and pore-scale. At the Darcy scale the Biot model requires data including permeability which is traditionally provided by experiments and empirical measurements. At the pore-scale we consider the Discrete Element Method (DEM) to generate physically realistic assemblies of the particles, and we follow up with the Stokes flow model. Next we apply upscaling to obtain the permeabilities which we find dependent on the deformation. We outline the workflow with its challenges and methods, and present results which show, e.g., hysteretic dependence of the permeability and porosity on the load. We also show how to incorporate the deformation dependent permeability in a nonlinear Biot model, and illustrate with computational results.

Influence of thickness on porous lining when the inclined channel walls are bounded by a deformable porous layer

This paper deals with the viscous fluid flow in an inclined channel bounded by a deformable porous layer. The coupled phenomenon of fluid movement and solid deformation in the porous layer was considered. The governing equations were obtained for the fluid velocity, solid displacement and temperature distribution in the deformable porous medium. Analytical solutions were obtained for solid displacement, fluid velocity, velocity in the free flow region and temperature distributions. The results were analysed for different physical parameters using graphs. The results revealed that the effect of the thickness of the porous layer was significant on solid displacement and fluid velocity. The solid displacement decreased by increasing the volume fraction of the fluid and the thickness of the porous layer. Since the drag coefficient was the ratio of fluid viscosity to the permeability of the porous material, it follows that the enhancement in drag reduces the permeability of the bounding porous layer. The flow in the blood vessels (capillaries) is surrounded by a tissue layer which is modelled as a deformable porous medium.

Ion-Induced Swelling Behavior of Articular Cartilage due to Non-Newtonian Flow and Its Effects on Fluid Pressure and Solid Displacement.

In this study, we examine the behavior of articular cartilage equilibrated in a salt (NaCl) solution during non-Newtonian fluid flow that follows an Ostwald-de Waele model. A linearly elastic and isotropic rectangular strip of cartilage is considered for analysis. A continuum theory of mixtures has been employed to develop a coupled system of partial differential equations for the solid displacement and the fluid pressure by considering the important factor of the ion concentration by assuming the cartilage as a deformable porous media. The coupled system of partial differential equations is solved using the numerical method named method of lines. In most cases, shear-thinning fluid is compared to the shear-thickening fluid to magnify the difference. Graphical results show that shear-thickening fluids bring more solid deformation and shows less fluid pressure in comparison to the shear-thinning fluids.

Разрешимость одномерной задачи движения жидкости в пороупругой среде с проницаемыми границами

The initial-boundary value problem of a one-dimensional viscous fluid flow in a deformable viscous porous medium with permeable boundaries is considered. The governing equations are the equations of mass conservation for each phase, the equation of momentum conservation for a liquid phase in terms of Darcy’s law, the equation of momentum conservation for the whole system, and the rheological equation for porosity. The original system of equations in the Lagrange variables is reduced to a third-order equation for the porosity function. The first part of this paper presents the formulation of the problem, the definition of the classical solution to the considered problem, and the existence and uniqueness theorem for the problem of Hölder classes. In the second part of this paper, the local theorem of existence and uniqueness for the problem of Hölder classes is proved for an incompressible fluid using the Tikhonov-Schauder fixed-point theorem. The physical principle of the maximum porosity function is determined.

On the Constitutive Equations for Coupled Flow, Chemical Reaction, and Deformation of Porous Media

AbstractDeformation, chemical reactions, fluid flow in geological formations, and many engineering materials are coupled processes. Most existing models of chemical reactions coupled with fluid transport assume the dissolution‐precipitation process or mineral growth in rocks. However, these models have limitations, such as predicting restricted reaction extent due to pore clogging or disregarding porosity changes resulting from mineral growth. Recent studies indicate mineral replacement involves coupled dissolution‐precipitation, maintaining porosity while altering the solid volume. This has multiple practical implications for natural geological processes and within petroleum and environmental engineering. We present a novel model for reaction‐driven mineral expansion that preserves porosity and allows solid volume change. First, we look at fluid‐rock interaction at the pore scale and derive effective rheology of a reacting porous media. On a larger scale, we adopt a two‐phase continuum medium approach to investigate the coupling between reaction, deformation, and fluid flow. Our micromechanical model based on observations assumes that rock or cement consists of an assembly of solid reactive grains, initially composed of a single, pure phase. The reaction occurs at the fluid‐solid contact and progresses into the solid grain material. We approximate the pores and surrounding solid material as an idealized cylindrical shell to simplify the problem and obtain tractable results. We derive macroscopic stress‐strain constitute laws that account for chemical alteration and viscoelastic deformation of porous rocks. Our model explains the possibility of achieving a complete reaction, preservation of porosity during chemical reactions, and dependence of mechanical rock properties on fluid chemistry.