Convergence of iterative coupling for coupled flow and geomechanics

In this paper, we study solving iteratively the coupling of flow and mechanics. We demonstrate the stability and convergence of two widely used schemes: the undrained split method and the fixed stress split method. To our knowledge, this is the first time that such results have been rigorously obtained and published in the scientific literature. In addition, we propose a new stress split method, with faster convergence rate than known schemes. These results are specially important today due to the interest in hydraulic fracturing (Dean and Schmidt SPE J. 14:707–714, 2009; Ji et al. SPE J. 14:423–430, 2009; Samier and De Gennaro 2007; Settari and Maurits SPE J. 3:219–226, 1998), in oil and gas shale reservoirs.     

Convergence of iterative coupling of geomechanics with flow in a fractured poroelastic medium

We consider an iterative scheme for solving a coupled geomechanics and flow problem in a fractured poroelastic medium. The fractures are treated as possibly non-planar interfaces. Our iterative scheme is an adaptation due to the presence of fractures of a classical fixed stress-splitting scheme. We prove that the iterative scheme is a contraction in an appropriate norm. Moreover, the solution converges to the unique weak solution of the coupled problem.     

Discrete fracture model for coupled flow and geomechanics

We present a fully implicit formulation of coupled flow and geomechanics for fractured three-dimensional subsurface formations. The Reservoir Characterization Model (RCM) consists of a computational grid, in which the fractures are represented explicitly. The Discrete Fracture Model (DFM) has been widely used to model the flow and transport in natural geological porous formations. Here, we extend the DFM approach to model deformation. The flow equations are discretized using a finite-volume method, and the poroelasticity equations are discretized using a Galerkin finite-element approximation. The two discretizations—flow and mechanics—share the same three-dimensional unstructured grid. The mechanical behavior of the fractures is modeled as a contact problem between two computational planes. The set of fully coupled nonlinear equations is solved implicitly. The implementation is validated for two problems with analytical solutions. The methodology is then applied to a shale-gas production scenario where a synthetic reservoir with 100 natural fractures is produced using a hydraulically fractured horizontal well.     

A supercoarsening multigrid method for poroelasticity in 3D coupled flow and geomechanics modeling

The Galerkin finite-element discretization of the force balance equation typically leads to large linear systems for geomechanical problems with realistic dimensions. In iteratively coupled flow and geomechanics modeling, a large linear system is solved at every timestep often multiple times during coupling iterations. The iterative solution of the linear system stemming from the poroelasticity equations constitutes the most time-consuming and memory-intensive component of coupled modeling. Block Jacobi, LSOR, and Incomplete LU factorization are popular preconditioning techniques used for accelerating the iterative solution of the poroelasticity linear systems. However, the need for more effective, efficient, and robust iterative solution techniques still remains especially for large coupled modeling problems requiring the solution of the poroelasticity system for a large number of timesteps. We developed a supercoarsening multigrid method (SCMG) which can be multiplicatively combined with commonly used preconditioning techniques. SCMG has been tested on a variety of coupled flow and geomechanics problems involving single-phase depletion and multiphase displacement of in-situ hydrocarbons, CO₂ injection, and extreme material property contrasts. Our analysis indicates that the SCMG consistently improves the convergence properties of the linear systems arising from the poroelasticity equations, and thus, accelerates the coupled simulations for all cases subject to investigation. The joint utilization of the two-level SCMG with the ILU1 preconditioner emerges as the most optimal preconditioning/iterative solution strategy in a great majority of the problems evaluated in this work. The BiCGSTAB iterative solver converges more rapidly compared to PCG in a number of test cases, in which various SCMG-accelerated preconditioning strategies are applied to both iterators.     

A locally conservative finite element framework for the simulation of coupled flow and reservoir geomechanics

In this paper, we present a computational framework for the simulation of coupled flow and reservoir geomechanics. The physical model is restricted to Biot’s theory of single-phase flow and linear poroelasticity, but is sufficiently general to be extended to multiphase flow problems and inelastic behavior. The distinctive technical aspects of our approach are: (1) the space discretization of the equations. The unknown variables are the pressure, the fluid velocity, and the rock displacements. We recognize that these variables are of very different nature, and need to be discretized differently. We propose a mixed finite element space discretization, which is stable, convergent, locally mass conservative, and employs a single computational grid. To ensure stability and robustness, we perform an implicit time integration of the fluid flow equations. (2) The strategies for the solution of the coupled system. We compare different solution strategies, including the fully coupled approach, the usual (conditionally stable) iteratively coupled approach, and a less common unconditionally stable sequential scheme. We show that the latter scheme corresponds to a modified block Jacobi method, which also enjoys improved convergence properties. This computational model has been implemented in an object-oriented reservoir simulator, whose modular design allows for further extensions and enhancements. We show several representative numerical simulations that illustrate the effectiveness of the approach.     

A coupled compressible flow and geomechanics model for dynamic fracture aperture during carbon sequestration in coal

This paper presents the development of a discrete fracture model of fully coupled compressible fluid flow, adsorption and geomechanics to investigate the dynamic behaviour of fractures in coal. The model is applied in the study of geological carbon dioxide sequestration and differs from the dual porosity model developed in our previous work, with fractures now represented explicitly using lower-dimensional interface elements. The model consists of the fracture-matrix fluid transport model, the matrix deformation model and the stress-strain model for fracture deformation. A sequential implicit numerical method based on Galerkin finite element is employed to numerically solve the coupled governing equations, and verification is completed using published solutions as benchmarks. To explore the dynamic behaviour of fractures for understanding the process of carbon sequestration in coal, the model is used to investigate the effects of gas injection pressure and composition, adsorption and matrix permeability on the dynamic behaviour of fractures. The numerical results indicate that injecting nonadsorbing gas causes a monotonic increase in fracture aperture; however, the evolution of fracture aperture due to gas adsorption is complex due to the swelling-induced transition from local swelling to macro swelling. The change of fracture aperture is mainly controlled by the normal stress acting on the fracture surface. The fracture aperture initially increases for smaller matrix permeability and then declines after reaching a maximum value. When the local swelling becomes global, fracture aperture starts to rebound. However, when the matrix permeability is larger, the fracture aperture decreases before recovering to a higher value and remaining constant. Gas mixtures containing more carbon dioxide lead to larger closure of fracture aperture compared with those containing more nitrogen.     

A priori error estimates for the numerical solution of a coupled geomechanics and reservoir flow model with stress-dependent permeability

In this paper we consider the numerical solution of a coupled geomechanics and a stress-sensitive porous media reservoir flow model. We combine mixed finite elements for Darcy flow and Galerkin finite elements for elasticity. This work focuses on deriving convergence results for the numerical solution of this nonlinear partial differential system. We establish convergence with respect to the L ²-norm for the pressure and for the average fluid velocity and with respect to the H ¹-norm for the deformation. Estimates with respect to the L ²-norm for mean stress, which is of special importance since it is used in the computation of permeability for poro-elasticity, can be derived using the estimates in the H ¹-norm for the deformation. We start by deriving error estimates in a continuous-in-time setting. A cut-off operator is introduced in the numerical scheme in order to derive convergence. The spatial grids for the discrete approximations of the pressure and deformation do not need be the same. Theoretical convergence error estimates in a discrete-in-time setting are also derived in the scope of this investigation. A numerical example supports the convergence results.     

An enhanced ensemble Kalman filter scheme incorporating model error in sequential coupling between flow and geomechanics

In this work, we construct a new methodology for enhancing the predictive accuracy of sequential methods for coupling flow and geomechanics while preserving low computational cost. The new computational approach is developed within the framework of the fixed-stress split algorithm procedure in conjunction with data assimilation based on the ensemble Kalman filter (EnKF). In this context, we identify the high-fidelity model with the two-way formulation where additional source term appears in the flow equation containing the time derivative of total mean stress. The iterative scheme is then interlaced with data assimilation steps, which also incorporate the modeling error inherent to the EnKF framework. Such a procedure gives rise to an “enhanced one-way formulation,” exhibiting substantial improvement in accuracy compared with the classical one-way method. The governing equations are discretized by mixed finite elements, and numerical simulation of a 2D slab problem between injection and production wells illustrate the tremendous achievement of the method proposed herein.     

Parameters controlling pressure and fracture behaviors in field injectivity tests: A numerical investigation using coupled flow and geomechanics model

Field injectivity tests are widely used in the oil and gas industry to obtain key formation characteristics. The prevailing approaches for injectivity test interpretation rely on traditional analytical models. A number of parameters may affect the test results and lead to interpretation difficulties. Understanding their impacts on pressure response and fracture geometry of the test is essential for accurate test interpretation. In this work, a coupled flow and geomechanics model is developed for numerical simulation of field injectivity tests. The coupled model combines a cohesive zone model for simulating fluid-driven fracture and a poro-elastic/plastic model for simulating formation behavior. The model can capture fracture propagation, fluid flow within the fracture and formation, deformation of the formation, and evolution of pore pressure and stress around the wellbore and fracture during the tests. Numerical simulations are carried out to investigate the impacts of a multitude of parameters on test behaviors. The parameters include rock permeability, the leak-off coefficient of the fracture, rock stiffness, rock toughness, rock strength, plasticity deformation, and injection rate. The sensitivity of pressure response and fracture geometry on each parameter is reported and discussed. The coupled flow and geomechanics model provides additional advantages in the understanding of the fundamental mechanisms of field injectivity tests.     

Numerical modeling of strongly coupled microscale multiphase flow and solid deformation

When fluid flows in porous media under subsurface conditions, significant deformation can occur. Such deformation is dependent on structural and phase characteristics. In this paper, we investigate the effect of multiphase flow on the deformation of porous media at the pore scale by implementing a strongly coupled partitioned solver discretized with finite volume (FV) technique. Specifically, the role of capillary forces on grain deformation in porous media is investigated. The fluid and solid subdomains are meshed using unstructured independent grids. The model is applied for solving multiphase coupled equations and is capable of capturing pore scale physics during primary drainage by solving the Navier-Stokes equation and advecting fluid indicator function using volume of fluid (VOF) while the fluid is interacting with a nonlinear elastic solid matrix. The convergence of the coupled solver is accelerated by Aitken underrelaxation. We also reproduce geomechanical stress conditions, at the pore scale, by applying uniaxial stress on the solid while simultaneously solving the multiphase fluid-solid interaction problem to investigate the effect of external stress on fluid occupancy, velocity-field distribution, and relative permeability. We observe that the solid matrix exhibits elasto-capillary behavior during the drainage sequence. Relative permeability endpoints are shifted on the basis of the external stress exerted.     

基于渗流和管流耦合的管涌数值模拟

堤坝地基的渗透变形过程实际上是“土中水”转变为“水中土”的过程。在渗透变形发生的集中管涌通道区域,采用常规渗流分析理论,单纯增大管涌通道渗透系数的方法是不太合适的。在未发生渗透变形的区域,用常规渗流理论计算;在管涌通道区域,用管流理论,公共边界上两者之间水头相等、流量大小相等且方向相反,能够较好符合渗透变形的发展规律。为了适应计算过程中内部边界条件不断变化的特点,采用无网格法伽辽金法（element free Galerkin method,EFG）对渗流场进行计算。算例计算表明,这种渗流-管流耦合的方法能够模拟管涌通道绕过防渗墙等复杂的发展过程。     

Numerical modelling of coupled flow and deformation in fractured rock specimens

A dual-porosity poroelastic model is extended to represent behaviour in cylindrical co-ordinates for the evaluation of flow-deformation effects in cylindrical laboratory samples incorporating a central wellbore or non-repeating axisymmetric injection on the periphery. Nine-node quadratic elements are used to represent mechanical deformation, while eight-node linear elements are used to interpolate the pressure fields, which offers significant advantages over the behaviour of other non-conforming elements. The model presented is validated against simplified analytical results, and extended to describe the behaviour of homogeneous and heterogeneous laboratory specimens subjected to controlled triaxial state of stress and injection tests. Apparent from the results is the significant influence of stress-deformation effects over system behaviour. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical simulations for coupled rock deformation and gas leak flow in parallel coal seams

Based on the new viewpoint of interaction mechanics for solid and gas, gas leakage in parallel deformable coal seams can be understood. That is, under the action of varied geophysical fields, the methane gas flow in a double deformable coal seam can be essentially considered to be compressible with time-dependent and mixed permeation and diffusion through a pore-cleat deformable, heterogeneous and anisotropic medium. From this new viewpoint, coupled mathematical models for coal seam deformation and gas leak flow in parallel coal seams were formulated and the numerical simulations for slow gas emission from the parallel coal seams are presented. It is found that coupled models might be close to reality. Meanwhile, a coupled model for solid deformation and gas leak flow can be applied to the problems of gas leak flow including mining engineering, gas drainage engineering and mining safety engineering in particular the prediction of the safe range using protective layer mining where coal and gas outbursts can efficiently be prevented. This revised version was published online in July 2006 with corrections to the Cover Date.     

Some numerical experiences on convergence criteria for iterative finite element solvers

Several popular convergence criteria which are frequently used in practical finite element computations are investigated for two kinds of systems: the symmetric positive definite linear system and the symmetric indefinite system involving two distinct variables (displacement and pore fluid pressure). For the first system, the relative residual norm and the relative improvement norm are satisfactory as long as boundary fixities are handled appropriately. For the second system, the relative improvement norm must be adopted with greater care. It was further shown numerically that decoupled relative residual norms can be attractive alternates to the current global stopping criterion.     

煤层气产出过程中气—水两相流与煤岩变形耦合数学模型研究

气－水二相流和煤岩变形耦合作用是煤层气产出过程中一种复杂的物理现象，为准确描述这一现象，本文建立了气－水二相流和煤岩变形的微分方程，并用有限元分别将它们进行离散化，然后讨论了煤岩变形模型和气－水二相流模型进行耦合数值求解的方法。     

堆石料单轴流变试验的颗粒流模拟

岩石断裂力学的亚临界裂缝扩展理论认为微裂缝扩展可导致岩石破碎,即岩石颗粒破碎具有时间效应。根据亚临界裂缝扩展理论,提出了考虑微裂缝扩展导致堆石颗粒破碎时间效应的数值流变模拟新方法,并进行了考虑颗粒不同典型破碎模式的单轴流变颗粒流数值试验。在对比数值与室内流变试验曲线的基础上,分析了数值流变过程中颗粒破碎情况与颗粒体内部结构发展过程等。研究成果表明,两种试验手段得到的堆石流变的发展趋势基本一致,由微裂缝扩展引起的颗粒延时破碎是堆石流变的主要原因之一,深化了对堆石料变形机制的认识。     

Numerical modeling of coupled fluid flow and thermal and reactive biogeochemical transport in porous and fractured media

Subsurface contamination problems of metals and radionuclides are ubiquitous. Metals and radionuclides may exist in the solute phase or may be bound to soil particles and interstitial portions of the geologic matrix. Accurate tools to reliably predict the migration and transformation of these metals and radionuclides in the subsurface environment enhance the ability of environmental scientists, engineers, and decision makers to analyze their impact and to evaluate the efficacy of alternative remediation techniques prior to incurring expense in the field. A mechanistic-based numerical model could provide such a tool. This paper communicates the development and verification of a mechanistically coupled fluid-flow thermal-reactive biogeochemical-transport model where both fast and slow reactions occur in porous and fractured media. Theoretical bases, numerical implementations, and numerical experiments using the model are described. A definition of the “rates” of fast/equilibrium reactions is presented to come up with a consistent set of governing equations. Two example problems are presented. The first one is a reactive transport problem which elucidates the non-isothermal effects on heterogeneous reactions. It also demonstrates that the rates of fast/equilibrium reactions are not necessarily greater than that of slow/kinetic reactions in the context of reactive transport. The second example focuses on a complicated but realistic advective–dispersive–reactive transport problem. This example exemplifies the need for innovative numerical algorithms to solve problems involving stiff geochemical reactions. It also demonstrates that rates of all fast/equilibrium reactions are finite and definite. Furthermore, it is noted that a species-versus-time curve cannot be used to characterize the rate of homogeneous fast/equilibrium reaction in a reactive transport system even if one and only one such reaction is responsible for the production of this species.     

成矿多过程耦合模型数值模拟研究态势

成矿作用是构造变形-热量传递-流体流动-质量转移等四种性质完全不同过程的耦合。数值模拟是通过建立多过程耦合模型来刻画其物理和化学规律。由于这些耦合模型过于复杂而难以得到其解析解，因而数值模拟被用来讨论这些复杂动力学过程。针对成矿预测领域主要的耦合模型：形变-流动模型、形变-热-流动模型、热-流动，质模型、变形-热-流动-质模型，通过这些耦合模型应用实例分析得出，印证了高差和温度变化是流体对流、矿物沉淀发生的重要机制；详释了构造控制流体输运的方式及过程，高渗透率断裂是流体汇聚的有利场所，从而渗透率等控制成矿的关键参数成为成矿预测的主要指标。     

On multicomponent gas diffusion and coupling concepts for porous media and free flow: a benchmark study

Computational Geosciences - Multicomponent gas transport in porous media and at the interface between porous media and free flow occurs in a wide range of technical and environmental systems....