Flow In Deformable Porous Media Research Articles

Invasion patterns during two-phase flow in deformable porous media

We study the formation of viscous fingering and fracturing patterns that occur when air at constant overpressure invades a circular Hele-Shaw cell containing a liquid-saturated deformable porous medium -- i.e. during the flow of two non-miscible fluids in a confined granular medium at high enough rate to deform it. The resulting patterns are characterized in terms of growth rate, average finger thickness as function of radius and time, and fractal properties. Based on experiments with various injection pressures, we identify and compare typical pattern characteristics when there is no deformation, compaction, and/or decompaction of the porous medium. This is achieved by preparing monolayers of glass beads in cells with various boundary conditions, ranging from a rigid disordered porous medium to a deformable granular medium with either a semi-permeable or a free outer boundary. We show that the patterns formed have characteristic features depending on the boundary conditions. For example, the average finger thickness is found to be constant with radius in the non-deformable system, while in the deformable ones there is a larger initial thickness decreasing to the non-deformable value. Then, depending on whether the outer boundary is semi-permeable or free there is a further decrease or increase in the average finger thickness. When estimated from the flow patterns, the box-counting fractal dimensions are not found to change significantly with boundary conditions, but by using a method to locally estimate fractal dimensions, we see a transition in behavior with radius for patterns in deformable systems; In the deformable system with a free boundary, it seems to be a transition in universality class as the local fractal dimensions decrease towards the outer rim, where fingers are opening up like fractures in a paste.

(De)compaction of porous viscoelastoplastic media: Solitary porosity waves

AbstractBuoyancy‐driven flow in deformable porous media is important for understanding sedimentary compaction as well as magmatic and metamorphic differentiation processes. Here mathematical analysis of the viscoplastic compaction equations is used to develop an understanding of the porosity wave instability and its sensitivity to the choice of rheological model. The conditions of propagation, size, speed, and shape of the porosity waves depend strongly on the properties of the solid rock frame. Whereas most of the previous studies on porosity waves were focused on viscous or viscoelastic mode, here we consider the ability of a solid matrix to undergo simultaneous plastic (rate‐independent) and viscous (rate‐dependent) deformation in parallel. Plastic yielding is identified as a cause of compaction‐decompaction asymmetry in porous media—this is known to lead to a strong focusing of porous flow. Speed and amplitude of a porosity wave are given as functions of material parameters and a volume of a source region. Formulation is applicable to fluid flow in sedimentary rocks where viscous deformation is due to pressure solution as well as in deep crustal or upper mantle rocks deforming in a semibrittle regime.

Enriched finite elements for branching cracks in deformable porous media

In this paper, we propose and verify a numerical approach to simulate fluid flow in deformable porous media without requiring the discretization to conform to the geometry of the sealed fractures (possibly intersecting). This approach is based on a fully coupled hydro-mechanical analysis and an extended finite element method (XFEM) to represent discrete fractures. Convergence tests indicate that the proposed scheme is both consistent and stable. The contributions of this paper include: (1) a new junction enrichment to describe intersecting fractures in deformable porous media; (2) the treatment of sealed fractures. We employ the resulting discretization scheme to perform numerical experiments, to illustrate that the inclination angles of the fractures and the penetration ratio of the sealed fractures are two key parameters governing the flow within the fractured porous medium.

Cell‐centered finite volume discretizations for deformable porous media

SUMMARYThe development of cell‐centered finite volume discretizations for deformation is motivated by the desire for a compatible approach with the discretization of fluid flow in deformable porous media. We express the conservation of momentum in the finite volume sense, and introduce three approximations methods for the cell‐face stresses. The discretization method is developed for general grids in one to three spatial dimensions, and leads to a global discrete system of equations for the displacement vector in each cell, after which the stresses are calculated based on a local expression. The method allows for anisotropic, heterogeneous and discontinuous coefficients.The novel finite volume discretization is justified through numerical validation tests, designed to investigate classical challenges in discretization of mechanical equations. In particular our examples explore the stability with respect to the Poisson ratio and spatial discontinuities in the material parameters. For applications, logically Cartesian grids are prevailing, and we also explore the performance on perturbations on such grids, as well as on unstructured grids. For reference, comparison is made in all cases with the lowest‐order Lagrangian finite elements, and the finite volume methods proposed herein is comparable for approximating displacement, and is superior for approximating stresses. © 2014 The Authors. International Journal for NumericalMethods in Engineering published by John Wiley & Sons Ltd.

Coupled Two Component Fluid Flow in Deformable Porous Media – Towards a Numerical Model for Geological Carbon Storage

Carbon Sequestration by CO2 storage into deep geological formations is a short to mid-term component for mitigatingclimate change while maintaining the stability of the world’s energy systems. This storage procedure will result in a seriesof coupled physical and chemical processes within the geological formation, which may critically affect its integrityas a storage medium. This work presents the development of a finite element model, which is to collaboratively aiddesign, monitoring and risk assessment. The current emphasis of the model development is on ensuring that the inducedgeomechanical behaviour is acceptable within a given reservoir-caprock system. It is a Biot-type model, whereby theinteractions of the flow of the fluids and the mechanical behaviour of the porous media are fully coupled. The governingequations are outlined and solved using numerical methods. For assessment, a simplified benchmark storage scenario ismodelled with realistic parametrisation.

A fully coupled multiphase flow and geomechanics solver for highly heterogeneous porous media

This paper introduces a fully coupled multiphase flow and geomechanics solver that can be applied to modeling highly heterogeneous porous media. Multiphase flow in deformable porous media is a multiphysics problem that considers the flow physics and rock physics simultaneously. To model this problem, the multiphase flow equations and geomechanical equilibrium equation must be tightly coupled. Conventional finite element modeling of coupled flow and geomechanics does not conserve mass locally since it uses continuous basis functions. Mixed finite element discretization that satisfies local mass conservation of the flow equation can be a good solution for this problem. In addition, the stabilized finite element method for discretizing the saturation equation minimizes numerical diffusion and provides better resolution of saturation solution.In this work, we developed a coupled multiphase flow and geomechanics solver that solves fully coupled governing equations, namely pressure, velocity, saturation, and geomechanical equilibrium equations. The solver can deal with full tensor permeability and elastic stiffness for modeling a highly heterogeneous reservoir system.The results of the numerical experiments are very encouraging. We used the solver to simulate a reservoir system that has very heterogeneous permeability and elastic stiffness fields and found that the numerical solution captures the complex multiphysics of the system. In addition, we obtained higher resolution of saturation solution than with the conventional finite volume discretization. This would help us make a better estimate of the numerical solution of complex multiphysics problems.

Coupling multipoint flux mixed finite element methodswith continuous Galerkin methods for poroelasticity

We study the numerical approximation on irregular domains with general grids of the system of poroelasticity, which describes fluid flow in deformable porous media. The flow equation is discretized by a multipoint flux mixed finite element method and the displacements are approximated by a continuous Galerkin finite element method. First-order convergence in space and time is established in appropriate norms for the pressure, velocity, and displacement. Numerical results are presented that illustrate the behavior of the method.

Rigorous Coupling of Geomechanics and Multiphase Flow with Strong Capillarity

SummaryWe study sequential formulations for coupled multiphase flow and reservoir geomechanics. First, we identify the proper definition of effective stress in multiphase-fluid systems. Although the average pore-pressure p¯ —defined as the sum of the product of saturation and pressure of all the fluid phases that occupy the pore space—is commonly used to describe multiphase-fluid flow in deformable porous media, it can be shown that the "equivalent" pore pressure pE —defined as p¯ minus the interfacial energy—is the appropriate quantity (Coussy 2004). We show, by means of a fully implicit analysis of the system, that only the equivalent pore pressure pE leads to a continuum problem that is thermodynamically stable (thus, numerical discretizations on the basis of the average pore pressure p¯ cannot render unconditionally stable and convergent schemes). We then study the convergence and stability properties of sequential-implicit coupling strategies. We show that the stability and convergence properties of sequential-implicit coupling strategies for single-phase flow carry over for multiphase systems if the equivalent pore pressure pE is used. Specifically, the undrained and fixed-stress schemes are unconditionally stable, and the fixed-stress split is superior to the undrained approach in terms of convergence rate. The findings from stability theory are verified by use of nonlinear simulations of two-phase flow in deformable reservoirs.

Coupled Hydromechanical Model of Two-Phase Fluid Flow in Deformable Porous Media

A model of solid-water-air coupling in triphasic mixtures is compared with solid-water coupling in biphasic mixtures with an application to partially saturated porous media. Based on thermodynamics, the mathematical framework governing the behavior of a partially saturated soil is derived using balance equations, and the numerical implementation and drainage tests of a soil column are carried out to validate the obtained formulations. The role of the air phase in the hydro-mechanical behavior of triphasic mixtures can be analyzed from the interactions among multiple phases for the constitutive behavior of a solid skeleton, and the triphasic mixture model can be applied in geotechnical engineering problems, such asCO2sequestration and air storage in aquifers.

Numerical simulation of two-phase flow in deformable porous media: Application to carbon dioxide storage in the subsurface

In this paper, conceptual modeling as well as numerical simulation of two-phase flow in deep, deformable geological formations induced by CO2 injection are presented. The conceptual approach is based on balance equations for mass, momentum and energy completed by appropriate constitutive relations for the fluid phases as well as the solid matrix. Within the context of the primary effects under consideration, the fluid motion will be expressed by the extended Darcy's law for two phase flow. Additionally, constraint conditions for the partial saturations and the pressure fractions of carbon dioxide and brine are defined. To characterize the stress state in the solid matrix, the effective stress principle is applied. Furthermore, the interaction of fluid and solid phases is illustrated by constitutive models for capillary pressure, porosity and permeability as functions of saturation. Based on this conceptual model, a coupled system of nonlinear differential equations for two-phase flow in a deformable porous matrix (H2M model) is formulated. As the displacement vector acts as primary variable for the solid matrix, multiphase flow is simulated using both pressure/pressure or pressure/saturation formulations. An object-oriented finite element method is used to solve the multi-field problem numerically. The capabilities of the model and the numerical tools to treat complex processes during CO2 sequestration are demonstrated on three benchmark examples: (1) a 1-D case to investigate the influence of variable fluid properties, (2) 2-D vertical axi-symmetric cross-section to study the interaction between hydraulic and deformation processes, and (3) 3-D to test the stability and computational costs of the H2M model for real applications.

A parallel block preconditioner for large-scale poroelasticity with highly heterogeneous material parameters

Large-scale simulations of coupled flow in deformable porous media require iterative methods for solving the systems of linear algebraic equations. Construction of efficient iterative methods is particularly challenging in problems with large jumps in material properties, which is often the case in realistic geological applications, such as basin evolution at regional scales. The success of iterative methods for such problems depends strongly on finding effective preconditioners with good parallel scaling properties, which is the topic of the present paper. We present a parallel preconditioner for Biot’s equations of coupled elasticity and fluid flow in porous media. The preconditioner is based on an approximation of the exact inverse of the two-by-two block system arising from a finite element discretisation. The approximation relies on a highly scalable approximation of the global Schur complement of the coefficient matrix, combined with generally available state-of-the-art multilevel preconditioners for the individual blocks. This preconditioner is shown to be robust on problems with highly heterogeneous material parameters. We investigate the weak and strong parallel scaling of this preconditioner on up to 512 processors and demonstrate its ability on a realistic basin-scale problem in poroelasticity with over eight million tetrahedral elements.

Direct numerical simulation of saturated deformable porous media using parallel hybrid Lattice–Boltzmann and finite element method

AbstractNumerical techniques for modeling saturated deformable porous media have mainly been based on mixture theory or homogenization techniques. However, these techniques rely on phenomenological relationships for the constitutive equations along with assumptions of homogeneous and isotopic material properties to obtain closure. Direct numerical simulations of the multiphasic problem for flow in deformable porous media avoid such assumptions and thus can provide significantly accurate understanding of the physics involved. They serve as a tool to investigate the constitutive relationships in complex geometries. They also allow the validation of the existing mixture theory models and determine their limitations.In this work, a parallel hybrid method using Lattice Boltzmann Method (LBM) for fluid phase and Finite Element Method (FEM) for solid phase is used for direct numerical simulation of saturated deformable porous media. The method provides a number of unique features including scalability on distributed computing necessary for such a problem. The method has been validated for modeling fluid–structure interactions in complex geometries against a number of experimental and analytical solutions. Further some challenging problems has been chosen to show the capability of the method. Copyright © 2010 John Wiley & Sons, Ltd.

Efficient block preconditioners for the coupled equations of pressure and deformation in highly discontinuous media

SUMMARYLarge‐scale simulations of flow in deformable porous media require efficient iterative methods for solving the involved systems of linear algebraic equations. Construction of efficient iterative methods is particularly challenging in problems with large jumps in material properties, which is often the case in geological applications, such as basin evolution at regional scales. The success of iterative methods for this type of problems depends strongly on finding effective preconditioners. This paper investigates how the block‐structured matrix system arising from single‐phase flow in elastic porous media should be preconditioned, in particular for highly discontinuous permeability and significant jumps in elastic properties. The most promising preconditioner combines algebraic multigrid with a Schur complement‐based exact block decomposition. The paper compares numerous block preconditioners with the aim of providing guidelines on how to formulate efficient preconditioners. Copyright © 2010 John Wiley & Sons, Ltd.

Numerical Modelling of Geomechanical Response of Sandy-Shale Formation in Oil Sands Reservoir During Steam Injection

Abstract In this study, we conducted numerical simulations of steam injection into a horizontal well. A sandy-shale layer with thickness of 1 metre is located 2 metre above a horizontal injection well. The numerical simulations intend to study the variations of the stress state and the permeability in the sandy-shale layer during steam injection, and to provide useful effective stress paths for laboratory experiments to follow when testing material properties of the sandy-shale. A self-developed coupled geomechanics and thermal multi-phase flow simulator was employed to conduct the simulation cases. A strain-induced permeability model was used to describe the anisotropic permeability change of the sandy-shale caused by steam injection. The simulation results demonstrate that the permeability of 5 and 50 md for the sandy-shale in two simulation cases is sufficient to allow pore pressure dissipation to occur, therefore, preventing tensile or shear failure of sandy-shale as a result of high thermal-induced pore pressure. The stress paths within the sandy-shale are approaching the failure envelope as steam injection goes on indicating simultaneous influence of the temperature and the pore pressure on the stress state change. Introduction Sandy-shale formations of several metres in thickness frequently exist in oil sands deposit. Their material properties, sedimentary characteristics and distribution in a steam-assisted gravity drainage (SAGD) reservoir are important factors that determine SAGD performance and strategies of positioning horizontal well-pairs. It has been confirmed that sandy-shale formations are usually very sandy and discontinuous; therefore, they are expected to be permeable to varying degrees during SAGD operations. This is completely different from the interbedded shales that have extremely low permeability and can be considered as impermeable. Analysis of stress state change and anisotropic permeability variation in the sandy-shale requires conducting coupled simulation of geomechanics and thermal reservoir flow. Numerical modelling of the coupled processes is historically carried out in the areas of geomechanics modelling and the reservoir simulation. Gutierrez and Lewis(1) extend Biot's theory to multi-phase fluid flow in deformable porous media. Based on their formulation, they conclude that the coupling between the geomechanics and the multi-phase flow occurs simultaneously. Thus, fully coupled system equations of deformations, multi-phase flow and heat transfer should be solved simultaneously.

Coupled Geomechanics Reservoir Simulation of UTF Phase A Project Using a Full Permeability Tensor

Abstract In this research study, we conducted a coupled thermal-stress-fluid flow numerical model on the UTF Phase A SAGD project in order to investigate the fundamental geomechanical behaviour involved in the SAGD process, and to gain insight into the reservoir response to temperature and pore pressure changes. The numerical simulation is carried out by using a self-developed coupled finite element model which incorporates our proposed strain-induced permeability model. The obtained simulation results were compared with the measured data. Introduction Thermal recovery processes involve coupling between heat transfer, multiphase flow and stress/deformation, which has become an increasingly important subject in the petroleum field(1). Particularly, the coupling is crucial in problems such as borehole stability, hydraulic fracturing and injection/production induced deformation of the ground surface during the thermal recovery process in heavy oil or oil sand reservoirs. Numerical modelling of the coupled processes is historically carried out in the areas of geomechanics modelling and reservoir simulation. The former is to compute the stress-strain behaviour; therefore, the deformation. The latter is to essentially model the multiphase flow and heat transfer in porous media. Each of these disciplines simplifies the part of the problem that is not of primary interest. These approaches are unacceptable in situations where the coupling is strong and the changes of porosity and permeability cannot be accounted for by rock compressibility alone. Gutierrez and Lewis(2) extend Biot's theory to multiphase fluid flow in deformable porous media. Based on their formulation, they conclude that the coupling between the geomechanics and the multiphase flow occurs simultaneously. Thus, fully coupled system equations of deformations, multiphase flow and heat transfer should be solved simultaneously. Development of such kinds of fully coupled geomechanics-multiphase flow-heat transfer simulators needs tremendous effort, since the existing FEM geomechanics codes and FDM reservoir simulators cannot be used. Settari and Mourits(3) present an approach to couple the stress-strain behaviour to multiphase flow-heat transfer using porosity as a coupling parameter. The geomechanics module and the thermal reservoir simulator are used in a staggered manner. Pore pressure and temperature changes are calculated from the thermal reservoir simulator and transferred to the geomechanics module. The stress and the displacement changes are then calculated in the geomechanics simulation. An iterative algorithm is used to ensure that the porosity calculated from the geomechanics module is the same as that from the thermal reservoir simulator. The staggered technique employed to solve the coupled system equations allows for the use of the existing geomechanics codes in conjunction with a standard reservoir simulator. Currently, most of the commercial coupled geomechanics-multiphase flow-heat transfer simulators are developed in this way. The disadvantage of these kinds of coupled simulators is that the thermal reservoir module, usually developed using the finite difference method (FDM), cannot accommodate the full permeability tensor, since they adopt the standard discretization scheme, such as 5-spot for 2D problems and 7-spot for 3D problems. In this paper, the development of a coupled geomechanics-multiphase flow-heat transfer simulator using the finite element method (FEM) is described with the use of Galerkin's least squares (GLS) technique(4) to stabilize the saturation equation.

The effects of buoyancy on shear-induced melt bands in a compacting porous medium

It has recently been shown [Holtzman, B., Groebner, N., Zimmerman, M., Ginsberg, S., Kohlstedt, D., 2003. Stress-driven melt segregation in partially molten rocks. Geochem. Geophys. Geosyst. 4, Art. No. 8607; Holtzman, B.K., Kohlstedt, D.L., 2007. Stress-driven melt segregation and strain partitioning in partially molten rocks: effects of stress and strain. J. Petrol. 48, 2379–2406] that when partially molten rock is subjected to simple shear, bands of high and low porosity are formed at a particular angle to the direction of instantaneous maximum extension. These have been modeled numerically and it has been speculated that high porosity bands may form an interconnected network with a bulk, effective permeability that is enhanced in a direction parallel to the bands. As a result, the bands may act to focus mantle melt towards the axis of mid-ocean ridges [Katz, R.F., Spiegelman, M., Holtzman, B., 2006. The dynamics of melt and shear localization in partially molten aggregates. Nature 442, 676–679]. In this contribution, we examine the combined effects of buoyancy and matrix shear on a deforming porous layer. The linear theory of Spiegelman [Spiegelman, M., 1993. Flow in deformable porous media. Part 1. Simple analysis. J. Fluid Mech. 247, 17–38; Spiegelman, M., 2003. Linear analysis of melt band formation by simple shear. Geochem. Geophys. Geosyst. 4, doi:10.1029/2002GC000499, Article 8615] and Katz et al. [Katz, R.F., Spiegelman, M., Holtzman, B., 2006. The dynamics of melt and shear localization in partially molten aggregates. Nature 442, 676–679] is generalized to include both the effects of buoyancy and matrix shear on a deformable porous layer with strain-rate dependent rheology. The predictions of linear theory are compared with the early time evolution of our 2D numerical model and they are found to be in excellent agreement. For conditions similar to the upper mantle, buoyancy forces can be similar to or much greater than matrix shear-induced forces. The results of the numerical model indicate that bands form when buoyancy forces are large and that these can significantly alter the direction of the flow of liquid away from vertical. The bands form at angles similar to the angle of maximum instantaneous growth rate. Consequently, for strongly strain-rate dependent rheology, there may be two sets of bands formed that are symmetric about the direction of maximum compressive stress in the background mantle flow. This second set of bands would reduce the efficiency with which melt bands would focus melts towards the ridge axis.

Permeability of deformable soft aggregated earth materials: From single pore to sample cross section

Fluid flow in deformable porous media is of interest in many areas of hydrology, geophysics, and environmental engineering at scales ranging from individual pore to field scale. With increasing interest in pore‐scale hydraulics, there is also the need to connect microscale with bulk material properties by suitable upscaling. In this study, fluid permeability through a stack of deforming spherical aggregates is described using analytical and finite element (FE) analyses. The permeability of an entire sample cross section was constructed from analytical estimates of individual pore permeabilities and from numerical solution to steady flow through the entire cross section of the sample. Experimental results of pore deformation within a stack of modeling clay aggregates were compared to FE calculations for two‐dimensional cross sections. Results showed that pores within a cubic pack of aggregates deform isotropically even under anisotropic stress conditions. Therefore aggregate arrangement seems to be as important as stress anisotropy to predicting pore shape evolution. Observations of cross sections through a porous sample under compaction showed reduction in both the mean and variance of pore permeability with increasing compaction. Despite predominantly vertical compaction, sample permeability remained nearly isotropic (with only a slight additional decrease in the vertical direction). The Aissen analytical approximation for pore permeability was very similar to numerical results for the entire range of sample deformation, thereby providing a useful tool for estimating sample permeability from images of complex pore cross‐sectional areas.

Decoupling of the coupled poroelastic equations for quasistatic flow in deformable porous media containing two immiscible fluids

The study of the poroelastic behavior of sedimentary materials containing two immiscible fluids in response to either applied stress or pore pressure change in a quasistatic limit, i.e., negligible second time-derivatives, is of great importance to many hydrogelogical problems, e.g., land subsidence caused by withdrawal of subsurface fluids. The poroelasticity models developed for analyzing these problems feature partial differential equations that are coupled in the terms describing viscous damping and strain field. To determine closed-form analytical solutions for induced volumetric strain (dilatation) of the solid framework and its interaction with fluid flows, the choice of normal coordinates whose transformation can be performed to decouple these poroelastic equations is highly desirable. In this paper, we show that normal coordinates for decoupling these equations are real-valued and equal to three different linear combinations of the dilatations of the solid and the fluids (or equivalently, three different linear combinations of two individual fluid pressures and solid dilatation). In contrast to fully saturated porous media, it is found that the viscous damping effect must be represented in normal coordinates in the presence of the second fluid. The resulting decoupled equations representing independent motional modes are a Laplace equation and two diffusion equations, which can be solved analytically under a variety of initial and boundary conditions. Thus, after inverse transformation of normal coordinates is performed, the closed-form analytical solutions for induced solid volumetric strain and excess pore fluid pressures can be obtained simultaneously from our decoupled partial differential equations.

Linear analysis of melt band formation by simple shear

Recent experiments by Holtzman et al. [2003] demonstrate that partially molten aggregates of many mantle materials undergoing simple shear (from ∼100–500% strain) will spontaneously develop localized melt‐rich bands. These bands develop at small strains but persist at low angles to the plane of shear (∼15–20°) even at large shear strains. The melt‐rich bands also appear to form localized weak regions that act as strain guides for the solid matrix flow. These experiments provide an opportunity to test the equations governing flow in deformable porous media developed by McKenzie [1984] and others. Here we present linear analysis of these equations for a solid with a porosity weakening viscosity undergoing simple shear. This analysis calculates the growth in porosity of a plane wave perturbation that initiates at an angle θ0 to the plane of shear after some amount of finite strain. For a strain of 300%, the maximum growing melt band initiates at 16.8° (but rotates to ∼70° in the linear analysis). We also calculate the additional solid shear induced by the localized weak regions and show that it develops a sense of shear consistent with observations only for melt bands less that 45°. These two results suggest that only low angle bands are favored under shear, consistent with the observations. The linear analysis, however, does not allow the porosity growth and shear to interact which causes the initially low angle bands to be rotated to higher angles by the background simple shear. To follow the further development of the bands requires solutions of the full nonlinear equations. Nevertheless, the linear analysis provides useful insights into the development of the shear bands and suggests that the theory is useful for describing partially molten systems.

Miscible and immiscible multiphase flow in deformable porous media

In this paper, two coupled models are proposed for porous media containing pore fluids: an immiscible and a miscible one. Emphasis is placed on the comparison between them at different steps during development and application. Even if their microscopic basis may be rather similar, the macroscopic formulation is different, as must be expected due to the different physical processes modelled. Each model has its particular domain of applicability and validity.