LBM–DEM modeling of fluid–solid interaction in porous media

Three porous media flow problems, in which the fluid mechanical interactions are critical, are studied in a mesoscopic–microscopic coupling system. In this system, fluid flow in the pore space is explicitly modeled at mesoscopic level by the lattice Boltzmann method, the geometrical representation and the mechanical behavior of the solid skeleton are modeled at microscopic level by the particulate distinct element method (DEM), and the interfacial interaction between the fluid and the solids is resolved by an immersed boundary scheme. In the first benchmark problem, the well‐known and frequently utilized Ergun equation is validated in periodic particle and periodic pore models. In the second problem, the upward seepage problem is simulated over three stages: The settlement of the column of sphere under gravity loading is measured to illustrate the accuracy of the DEM scheme; the system is solved to hydrostatic state with pore space filled with fluid, showing that the buoyancy effect is captured correctly in the mesoscopic–microscopic coupling system; then, the flow with constant rate is supplied at the bottom of the column; the swelling of the ground surface and pore pressure development from the numerical simulation are compared with the predictions of the macroscopic consolidation theory. In the third problem, the fluid‐flow‐induced collapse of a sand arch inside a perforation cavity is tested to illustrate a more practical application of the developed system. Through comparing simulation results with analytical solutions, empirical law and physical laboratory observations, it is demonstrated that the developed lattice Boltzmann–distinct element coupling system is a powerful fundamental research tool for investigating hydromechanical physics in porous media flow. Copyright © 2012 John Wiley & Sons, Ltd.     

流固耦合问题及研究进展

传统的渗流理论一般假设流体流动的多孔介质骨架是完全刚性的,即在孔隙流体压力变化过程中,固体骨架不产生任何弹性或塑性变形,这时可将渗流作为非耦合问题来研究。这种简化虽然可以得到问题的近似解,但存在许多缺陷,而且也不切合生产实际。比如:在油田开采过程中,孔隙流体压力会逐渐降低,将导致储层内有效应力的变化,使储层产生变形。近年来,流固耦合问题越来越受到人们的重视,这方面的研究涉及许多领域。该文介绍了有关工程涉及到的流固耦合问题,重点针对油、气开采问题,介绍了储层流固耦合渗流的特点及研究方法和理论进展,包括单相、多相流体渗流的流固耦合数学模型及有限元数值模型。      

Numerical convergence study of iterative coupling for coupled flow and geomechanics

In this paper, we consider algorithms for modeling complex processes in porous media that include fluid and structure interactions. Numerous field applications would benefit from a better understanding and integration of porous flow and solid deformation. Important applications in environmental and petroleum engineering include carbon sequestration, surface subsidence, pore collapse, cavity generation, hydraulic fracturing, thermal fracturing, wellbore collapse, sand production, fault activation, and waste disposal, while similar issues arise in biosciences and chemical sciences as well. Here, we consider solving iteratively the coupling of flow and mechanics. We employ mixed finite element method for flow and a continuous Galerkin method for elasticity. For single-phase flow, we demonstrate the convergence and convergence rates for two widely used schemes, the undrained split and the fixed stress split. We discuss the extension of the fixed stress iterative coupling scheme to an equation of state compositional flow model coupled with elasticity and a single-phase poroelasticity model on general hexahedral grids. Computational results are presented.     

A fully implicit and consistent finite element framework for modeling reservoir compaction with large deformation and nonlinear flow model. Part I: theory and formulation

Reservoir depletion results in rock failure, wellbore instability, hydrocarbon production loss, oil sand production, and ground surface subsidence. Specifically, the compaction of carbonate reservoirs with soft rocks often induces large plastic deformation due to rock pore collapse. On the other hand, following the compaction of reservoirs and failure of rock formations, the porosity and permeability of formations will, in general, decrease. These bring a challenge for reservoir simulations because of high nonlinearity of coupled geomechanics and fluid flow fields. In this work, we present a fully implicit, fully coupled, and fully consistent finite element formulation for coupled geomechanics and fluid flow problems with finite deformation and nonlinear flow models. The Pelessone smooth cap plasticity model, an important material model to capture rock compaction behavior and a challenging material model for implicit numerical formulations, is incorporated in the proposed formulation. Furthermore, a stress-dependent permeability model is taken into account in the formulation. A co-rotational framework is adopted for finite deformation, and an implicit material integrator for cap plasticity models is consistently derived. Furthermore, the coupled field equations are consistently linearized including nonlinear flow models. The physical theories, nonlinear material and flow models, and numerical formulations are the focus of part I of this work. In part II, we verify the proposed numerical framework and demonstrate the performance of our numerical formulation using several numerical examples including a field reservoir with soft rocks undergoing serious compaction.     

Hydro‐mechanical modeling of cohesive crack propagation in multiphase porous media using the extended finite element method

In this paper, a numerical model is developed for the fully coupled hydro‐mechanical analysis of deformable, progressively fracturing porous media interacting with the flow of two immiscible, compressible wetting and non‐wetting pore fluids, in which the coupling between various processes is taken into account. The governing equations involving the coupled solid skeleton deformation and two‐phase fluid flow in partially saturated porous media including cohesive cracks are derived within the framework of the generalized Biot theory. The fluid flow within the crack is simulated using the Darcy law in which the permeability variation with porosity because of the cracking of the solid skeleton is accounted. The cohesive crack model is integrated into the numerical modeling by means of which the nonlinear fracture processes occurring along the fracture process zone are simulated. The solid phase displacement, the wetting phase pressure and the capillary pressure are taken as the primary variables of the three‐phase formulation. The other variables are incorporated into the model via the experimentally determined functions, which specify the relationship between the hydraulic properties of the fracturing porous medium, that is saturation, permeability and capillary pressure. The spatial discretization is implemented by employing the extended finite element method, and the time domain discretization is performed using the generalized Newmark scheme to derive the final system of fully coupled nonlinear equations of the hydro‐mechanical problem. It is illustrated that by allowing for the interaction between various processes, that is the solid skeleton deformation, the wetting and the non‐wetting pore fluid flow and the cohesive crack propagation, the effect of the presence of the geomechanical discontinuity can be completely captured. Copyright © 2012 John Wiley & Sons, Ltd.     

Poro-viscoelasticity modelling based on upscaling quasistatic fluid-saturated solids

In this paper, quasistatic models are developed for the slow flow of compressible fluids through porous solids, where the solid exhibits fading memory viscoelasticity. Problems of this type are important in practical geomechanics contexts, for example, in the context of fluid flow through unconsolidated reservoir sands and of wellbore deformation behaviour in gas and oil shale reservoirs, all of which have been studied extensively. For slow viscous fluid flow in the poro-viscoelastic media we are able to neglect the dynamic effects related to inertia forces, as well as the dissipation associated with the viscous flows. This is in contrast to the vast body of work in the poro-elastic context, where much faster flow of the viscous fluids may give rise to memory effects associated with microflows in pores of the solid media. Such problems have been treated extensively in both the dynamic and quasistatic cases. We are taking a specific type of the porous medium subject to slow deformation processes possibly inducing moderate pressure gradients and flow rates characterised by negligible inertia effects. As the result of homogenisation of such a two-phase medium, we observe the fading memory behaviour in the Biot modulus which controls the pressure increase due to skeleton macroscopic deformation and pore fluid content. Although our derivation leads to a result which is consistent with the formal phenomenological approach proposed by Biot (J Appl Phys 23:1482–1498, 1962), we offer the reader more insight into the structure of the poro-viscoelastic constitutive relations obtained; in particular, we can show that the Biot compressibility evolves in time according to the creep function while the skeleton stiffness is driven by the relaxation function.     

基于三维流-固耦合模型的油井出砂细观机制研究

油井出砂机制的研究是提高油藏产能和石油开采成本减小的关键课题,而常规的宏观力学理论和方法不能全面反映油藏开采过程中油井出砂的发生和发展。鉴于砂岩储层的物理性质和射孔试验特征,从岩土力学的角度建立基于柱坐标系的三维颗粒流数值模型,与理论分析成果进行比较,以说明该细观数值模型可行性,有效地模拟出砂过程中的渗流及流-固耦合效应。在该基础上,综合考虑流体压力梯度力和拖曳力,基于PFC3D模型模拟流体不同运动时的砂岩性态。数值分析得到的模型宏观应力图形说明流体运动对砂岩力学特性的影响不可忽略,且在相同条件下,流量越大,砂岩的塑性区越大,形成砂岩破坏出砂的几率也越大。同时,不同工况的砂岩黏结分布和颗粒转动图形表明,相同条件下流量越大,颗粒间平行黏结破坏越多,颗粒转动越大,失去黏结约束的颗粒也越多,出砂量就越大,可见两种细观特征图形与宏观应力图形变化规律一致,该模型可用于油井出砂机制的研究,可为出砂量预测及出砂控制提供新的研究思路。     

Coupled bonded particle and lattice Boltzmann method for modelling fluid–solid interaction

This paper presents a two‐dimensional coupled bonded particle and lattice Boltzmann method (BPLBM) developed to simulate the fluid–solid interactions in geomechanics. In this new technique, the bonded particle model is employed to describe the inter‐particle movement and forces, and the bond between a pair of contacting particles is assumed to be broken when the tensile force or tangential force reaches a certain critical value. As a result the fracture process can be delineated based on the present model for the solid phase comprising particles, such as rocks and cohesive soils. In the meantime, the fluid phase is modelled by using the LBM, and the immersed moving boundary scheme is utilized to characterize the fluid–solid interactions. Based on the novel technique case studies have been conducted, which show that the coupled BPLBM enjoys substantially improved accuracy and enlarged range of applicability in characterizing the mechanics responses of the fluid–solid systems. Indeed such a new technique is promising for a wide range of application in soil erosion in Geotechnical Engineering, sand production phenomenon in Petroleum Engineering, fracture flow in Mining Engineering and fracture process in a variety of engineering disciplines. Copyright © 2016 John Wiley & Sons, Ltd.     

Response of saturated and nearly saturated porous media: Different formulations and their applicability

The response of saturated porous medium is of significant interest in many fields ranging from geomechanics to biomechanics. Biot was the first to formulate the basic equations governing the process of coupled flow and deformation in porous media. Depending on the nature of loading vis‐à‐vis the characteristics of the media, different formulations (fully dynamic, partly dynamic, quasi‐static) are possible. In this study, analytical solutions are developed for the response of saturated and nearly saturated porous media under plane strain condition. The solutions for different formulations are developed in terms of non‐dimensional parameters. The response is studied for various conditions and the regions of validity for various formulations are identified in a parametric space. An assessment of the needed formulation for few important problems is also presented. Copyright © 2008 John Wiley & Sons, Ltd.     

Fully coupled simulation of fluid injection into geomaterials with focus on nonlinear near‐well behavior

An important part of our global wealth depends on the extraction of fluids from porous media. More recently, sequestration of carbon dioxide (rmCO₂) into deep geological layers as a possible measure to mitigate climate change has increased interest in fluid injection into porous media. Sophisticated numerical models play an important role in managing the uncertainties related to the subsurface, and finite element methods are the most versatile tool allowing the coupling of fluid flow, geomechanics and other physical processes. This paper gives insight into two important aspects of fluid injection/extraction in porous media: the correct modeling of the bore hole through specification of initial stresses, which together with a fully coupled strategy allows simulation of nonlinear poromechanics, and the imposition of appropriate boundary conditions that allow the controlled injection/extraction of a total specified amount of fluid in an anisotropic porous medium, without exceeding a safe operating pressure. Copyright © 2011 John Wiley & Sons, Ltd.     

Hydro‐mechanical erosion models for sand production

Sand production is a complex physical process that depends on the external stress and flow rate conditions as well as on the state of the material. Models developed for the prediction of sand production are usually solved numerically because of the complexity of the governing equations. Testing of new sand production models can very well be performed through calibration with laboratory experiments, which by construction possess geometric symmetry facilitating explicit mathematical analysis. We introduce an erosion model that is built upon the physics (poro‐mechanical coupling of the fluid‐solid system) usually incorporated in erosion models for the prediction of sand production. Around this model, we set up a mathematical framework in which sand production models because of erosion can be tested and calibrated without having to resort to complex numerical work or specialised software. The model is validated by data of volumetric sand production from a hollow cylinder test on synthetic sandstone. Generalisations of the model, which are naturally incorporated in the same framework and have useful phenomenological features, are discussed. Copyright © 2015 John Wiley & Sons, Ltd.     

General coupling extended multiscale FEM for elasto‐plastic consolidation analysis of heterogeneous saturated porous media

This paper presents a general coupling extended multiscale FEM (GCEMs) for solving the coupling problem of elasto‐plastic consolidation of heterogeneous saturated porous media. In the GCEMs, the numerical multiscale base functions for the solid skeleton and fluid phase of the coupling system are all constructed on the basis of the equivalent stiffness matrix of the unit cell, which not only contain the interaction between the solid and fluid phases but also consider the time effect. Furthermore, in order to improve the computational accuracy for two‐dimensional problems, a multi‐node coarse element strategy for the GCEMs is proposed, and a two‐scale iteration algorithm for the elasto‐plastic consolidation analysis is developed. Some one‐dimensional and two‐dimensional homogeneous and heterogeneous numerical examples are carried out to validate the proposed method through the comparison with the coupling multiscale FEM and standard FEM. Numerical results show that the newly developed GCEMs can almost preserve the same convergent property as the standard FEM and also possesses the advantages of high computational efficiency. In addition, the GCEMs can be easily applied to other coupling multifield and multiphase transient problems. Copyright © 2014 John Wiley & Sons, Ltd.     

Cell to node projections: An assessment of error

Reservoir simulators typically use cell‐centered finite volume schemes and do not model directly the coupling of the flow processes with the geomechanics. Coupling of geomechanics with fluid flow can be important in many cases, but introducing fully coupled geomechanical effects in those simulators is not a trivial issue, because the geomechanics is better done by using the Galerkin vertex‐centered finite element methods by which the solid displacements are computed at the vertices of the cells. This creates difficulties in interfacing cell variables with nodal variables. Uncoupled or loosely coupled models are used by many researchers/practitioners by which a reservoir model is coupled to a geomechanical model by staggering in‐time flow and deformation via a sophisticated interface that repeatedly calls first flow and then mechanics. The method therefore requires projection of the reservoir cell variables onto the nodes of the geomechanics Galerkin finite element mesh. In this note, we attempt to quantify the errors associated with cell to node projection operations. For that purpose, we use a simple model of the pressure equation for a heterogeneous medium in one dimension. We are able to derive the exact analytical solution for this problem for both nodal and cell pressures. This allows us to compute the errors due to projection analytically, function of meshing refinement and permeability field variations. We compute upper and lower bounds for the errors, and analyze their magnitude for a variety of cases. We conclude that, in general, cell to node projection operations lead to substantial errors. Copyright © 2010 John Wiley & Sons, Ltd.     

Frictionless contact formulation for dynamic analysis of nonlinear saturated porous media based on the mortar method

A finite element algorithm for frictionless contact problems in a two‐phase saturated porous medium, considering finite deformation and inertia effects, has been formulated and implemented in a finite element programme. The mechanical behaviour of the saturated porous medium is predicted using mixture theory, which models the dynamic advection of fluids through a fully saturated porous solid matrix. The resulting mixed formulation predicts all field variables including the solid displacement, pore fluid pressure and Darcy velocity of the pore fluid. The contact constraints arising from the requirement for continuity of the contact traction, as well as the fluid flow across the contact interface, are enforced using a penalty approach that is regularised with an augmented Lagrangian method. The contact formulation is based on a mortar segment‐to‐segment scheme that allows the interpolation functions of the contact elements to be of order N. The main thrust of this paper is therefore how to deal with contact interfaces in problems that involve both dynamics and consolidation and possibly large deformations of porous media. The numerical algorithm is first verified using several illustrative examples. This algorithm is then employed to solve a pipe‐seabed interaction problem, involving large deformations and dynamic effects, and the results of the analysis are also compared with those obtained using a node‐to‐segment contact algorithm. The results of this study indicate that the proposed method is able to solve the highly nonlinear problem of dynamic soil–structure interaction when coupled with pore water pressures and Darcy velocity. Copyright © 2015 John Wiley & Sons, Ltd.     

Analytical solutions of a finite two‐dimensional fluid‐saturated poroelastic medium with compressible constituents

An analytical solution is proposed for transient flow and deformation coupling of a fluid‐saturated poroelastic medium within a finite two‐dimensional (2‐D) rectangular domain. In this study, the porous medium is assumed to be isotropic, homogeneous, and compressible. In addition, the point sink can be located at an arbitrary position in the porous medium. The fluid–solid interaction in porous media is governed by the general Biot's consolidation theory. The method of integral transforms is applied in the analytical formulation of closed‐form solutions. The proposed analytical solution is then verified against both exact and numerical results. The analytical solution is first simplified and validated by comparison with an existing exact solution for the uncoupled problem. Then, a case study for pumping from a confined aquifer is performed. The consistency between the numerical solution and the analytical solution confirms the accuracy and reliability of the analytical solution presented in this paper. The proposed analytical solution can help us to obtain in‐depth insights into time‐dependent mechanical behavior due to fluid withdrawal within finite 2‐D porous media. Moreover, it can also be of great significance to calibrate numerical solutions in plane strain poroelasticity and to formulate relevant industry norms and standards. Copyright © 2014 John Wiley & Sons, Ltd.     

A finite element method for modeling coupled flow and deformation in porous fractured media

Modeling the flow in highly fractured porous media by finite element method (FEM) has met two difficulties: mesh generation for fractured domains and a rigorous formulation of the flow problem accounting for fracture/matrix, fracture/fracture, and fracture/boundary fluid mass exchanges. Based on the recent theoretical progress for mass balance conditions in multifractured porous bodies, the governing equations for coupled flow and deformation in these bodies are first established in this paper. A weak formulation for this problem is then established allowing to build a FEM. Taking benefit from recent development of mesh‐generating tools for fractured media, this weak formulation has been implemented in a numerical code and applied to some typical problems of hydromechanical coupling in fractured porous media. It is shown that in this way, the FEM that has proved its efficiency to model hydromechanical phenomena in porous media is extended with all its performances (calculation time, couplings, and nonlinearities) to fractured porous media. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical simulation of fully saturated porous materials

Fully coupled, porous solid–fluid formulation, implementation and related modeling and simulation issues are presented in this work. To this end, coupled dynamic field equations with u?p?U formulation are used to simulate pore fluid and soil skeleton (elastic–plastic porous solid) responses. Present formulation allows, among other features, for water accelerations to be taken into account. This proves to be useful in modeling dynamic interaction of media of different stiffnesses (as in soil–foundation–structure interaction). Fluid compressibility is also explicitly taken into account, thus allowing excursions into modeling of limited cases of non‐saturated porous media. In addition to these features, present formulation and implementation models in a realistic way the physical damping, which dissipates energy. In particular, the velocity proportional damping is appropriately modeled and simulated by taking into account the interaction of pore fluid and solid skeleton. Similarly, the displacement proportional damping is physically modeled through elastic–plastic processes in soil skeleton. An advanced material model for sand is used in present work and is discussed at some length. Also explored in this paper are the verification and validation issues related to fully coupled modeling and simulations of porous media. Illustrative examples describing the dynamical behavior of porous media (saturated soils) are presented. The verified and validated methods and material models are used to predict the behavior of level and sloping grounds subjected to seismic shaking. Copyright © 2008 John Wiley & Sons, Ltd.     

Coupled finite‐element simulation of injection well testing in unconsolidated oil sands reservoir

This paper presents a finite‐element (FE) model for simulating injection well testing in unconsolidated oil sands reservoir. In injection well testing, the bottom‐hole pressure (BHP) is monitored during the injection and shut‐in period. The flow characteristics of a reservoir can be determined from transient BHP data using conventional reservoir or well‐testing analysis. However, conventional reservoir or well‐testing analysis does not consider geomechanics coupling effects. This simplified assumption has limitations when applied to unconsolidated (uncemented) oil sands reservoirs because oil sands deform and dilate subjected to pressure variation. In addition, hydraulic fracturing may occur in unconsolidated oil sands when high water injection rate is used. This research is motivated in numerical modeling of injection well testing in unconsolidated oil sands reservoir considering the geomechanics coupling effects including hydraulic fracturing. To simulate the strong anisotropy in mechanical and hydraulic behaviour of unconsolidated oil sands induced by fluid injection in injection well testing, a nonlinear stress‐dependent poro‐elasto‐plastic constitutive model together with a strain‐induced anisotropic permeability model are formulated and implemented into a 3D FE simulator. The 3D FE model is used to history match the BHP response measured from an injection well in an oil sands reservoir. Copyright © 2013 John Wiley & Sons, Ltd.     

Time‐scale analysis in unsaturated porous media under external wave loads

Extreme waves caused by tsunamis and storm surges can lead to soil failures in the near‐shore region, which may have severe impact on coastal environments and communities. Multiphase flows in deformable porous media involve several coupled processes and multiple time scales, which are challenging for numerical simulations. The objective of this study is to investigate the roles of the various processes and their interactions in multiphase flows in unsaturated soils under external wave loading, via theoretical time‐scale analysis and numerical simulations. A coupled geomechanics–multiphase flow model based on conservation laws is used. Theoretical analysis based on coupled and decoupled models demonstrates that transient and steady‐state responses are governed by pore pressure diffusion and saturation front propagation, respectively, and that the two processes are essentially decoupled. Numerical simulations suggest that the compressibility of the pore fluids and the deformation of the soil skeleton are important when the transient responses of the media are of concern, while the steady‐state responses are not sensitive to these factors. The responses obtained from the fully coupled numerical simulations are explained by a simplified time‐scale analysis based on coupled and decoupled models. Copyright © 2010 John Wiley & Sons, Ltd.     

Modeling the interaction between fluid flow and particle movement with discontinuous deformation analysis (DDA) method

Coupled hydro‐mechanical processes in granular media represent the interaction between solid particles movement and fluid flow during external and/or internal loading. The processes attract attention in geotechnical engineering since they cause many disasters such as landslide, slope collapse, boiling/quicksand and soil liquefaction, etc. This study presents a numerical method combining discontinuous deformation analysis (DDA) for mechanical calculation and finite element method for fluid flow simulation to model the interaction between solid particles' movement and fluid flow from microscopic point of view. The term of hydraulic pressure was newly formulated and introduced to the original DDA. The simplified examples are given to verify the new method, and the computational results correlate well with the theoretical calculations, further development is also considered. Copyright © 2010 John Wiley & Sons, Ltd.