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.     

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.     

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 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 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 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.     

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.     

Discrete fracture model for simulating waterflooding processes under fracturing conditions

In our study, we develop a model for simulating fracturing processes in a poroelastic medium. The proposed approach combines the discrete fracture model enriched with contact plane mechanics. The model captures mechanical interactions of fractures and a deformable medium, fluid, and heat transfer in fractures and in a porous medium. Both effects of poroelasticity and thermoelasticity are accounted in our model. Mass and heat conservation equations are approximated by the finite volume method, and mechanical equilibrium equations are discretized by means of the Galerkin finite element approach. Two‐dimensional grid facets between 3‐dimensional finite elements are considered as possible fracture surfaces. Most of these facets are inactive from the beginning and are activated throughout the simulation. A fracture propagation criterion, based on Irwin's approach, is verified on each nonlinear iteration. When the criterion is satisfied, additional contact elements are added into finite element and discrete fracture model formulations respectively. The proposed approach allows modeling of existing natural and artificially created fractures within one framework. The model is tested on single‐ and multiple‐phase fluid flow examples for both isothermal and thermal conditions and verified against existing semianalytical solutions. The applicability of the approach is demonstrated on an example of practical interests where a sector model of an oil reservoir is simulated with different injection and production regimes.     

A coupled model for solid deformation and gas leak flow

From the viewpoint of interaction mechanics of solid and gas, a coupled mathematical model is presented for solid coal/rock‐mass deformation and gas leak flow in parallel deformable coal seams. Numerical solutions using the strong implicit procedure (SIP) method to the coupled mathematical model for double parallel coal seams are also developed in detail. Numerical simulations for the prediction of safety range using protection layer mining are performed with experimental data from a mine with potential danger of coal/gas outbursts. Analyses show that the numerical simulation results are consistent with the measured data on the spot. The coupled model shows a positive future for applications in a wide range of gas‐leak‐flow‐related problems in mining engineering, gas drainage engineering and mining safety engineering. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

A model for nonlinear flow behavior in two-dimensional fracture intersections and the estimation of flow model coefficients

Hydrogeology Journal - There are multiple flow paths with different flow directions in fracture intersections. A general flow model synthetically describing the nonlinear flow behavior of multiple...     

地质力学磁力模型试验原理及其在工程中的应用

地质力学磁力模型试验借鉴土工离心模型试验中利用离心力模拟重力的思想。在均匀梯度磁场中,掺有铁磁材料的相似材料会受到均匀的磁力作用,由此可以利用均匀的磁力场模拟重力场。根据电磁学基本原理和磁路设计原理,设计并制造了一台2 000匝线圈的均匀梯度磁场发生装置。该装置在220 V电压情况下,能够产生0～16 A电流,在1.2 m×1.2 m×1.0 m试验空间内,可将磁性相似材料的重度提高10～30倍。以锦屏一级电站左岸高陡边坡为例,进行地质力学磁力模型试验。试验结果表明,在一定的电流作用下,可以使模型所受体力的提高倍数达到相似比的要求,并与数值模拟结果进行了对比,试验结果与计算结果基本一致。在增加通电电流时,试验设备可以模拟边坡的超载过程,进而可以求出边坡的超载安全系数。     

渗流-水平井流耦合数学模型和数值模拟

在有侧向流入(流出)的水平井流运动和连续性方程的基础上,建立了渗流与水平井流耦合数学模型,主要针对混合水头损失问题,进行了渗流与水平井流耦合数值计算,分析了常用的摩擦系数修正方法对计算结果的影响。数值模拟计算表明:层流或者光滑紊流流态时,不同的摩擦系数修正方法对结果影响较小;粗糙紊流流态时,不同的摩擦系数修正方法对结果影响较大。当水平井流是层流-光滑紊流流态时,使用了一种新的流态分界点识别方法对水平井流态进行判断,计算结果证实了方法的可靠性、合理性。     

A constitutive-relationship model for film flow on rough fracture surfaces

Film flow on fracture surfaces may be an important mechanism for fast flow in unsaturated fractured rocks. Incorporating this mechanism into a numerical model requires knowledge of constitutive relationships for film flow. Based on fractal concepts and a conceptual argument of Tokunaga et al. that water films could be treated as analogues to water in unsaturated porous media, a simple constitutive-relationship model has been developed. The validity of the model is supported by excellent agreements between calculation results and experimental observations for two different fracture surfaces.

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Resumen El flujo pelicular en la superficie de las fracturas puede ser un mecanismo importante para el flujo rápido en rocas fracturadas no saturadas. Incorporarlo en un modelo numérico requiere conocer las relaciones constitutivas del flujo pelicular. Se ha desarrollado un modelo de una relación constitutiva sencilla partiendo de conceptos fractales y de un argumento conceptual de Tokunaga et al., según el cual las películas de agua pueden ser tratadas como análogos del agua en medios porosos no saturados. La validez del modelo es corroborada por los excelentes ajustes entre los resultados numéricos y las observaciones experimentales en dos superficies de fracturas diferentes.

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Résumé Lécoulement en film sur les surfaces de fracture peut être un mécanisme important pour lécoulement rapide dans les roches fracturées non saturées. Lincorporation de ce mécanisme dans un modèle numérique nécessite la connaissance des relations fondamentales pour lécoulement en film. Basé sur des concepts fractals et sur un argument conceptuel de Tokunaga et al. selon lequel des films deau peuvent être considérés comme des analogues de leau en milieu poreux non saturé, un modèle simple des relations fondamentales a été développé. La validité de ce modèle est confirmée grâce à une excellente concordance entre les résultats du calcul et les observations expérimentales pour deux surfaces différentes de fractures.

     

A coupled model for prediction of settlement and gas flow in MSW landfills

Prediction of long‐term settlement and control of gas pollution to the environment are two principle concerns during the management of municipal solid waste (MSW) landfills. The behavior of settlement and gas flow in MSW landfills is complicated due to the combined effect of mechanical deformation of the solid skeleton and continuous biodegradation of the waste. A one‐dimensional settlement and gas flow model is presented in this paper, which is capable of predicting time evolution of settlement as well as temporal and spatial distribution of gas pressure within multi‐layered landfills under a variety of operating scenarios. The analytical solution to the novel model is evaluated with numerical simulation and field measurements. The resulting efficiency and accuracy highlight the capability of the proposed model to reproduce the settlement behavior and gas flow in MSW landfills. The influences of operating conditions and waste properties on settlement and gas pressure are examined for typical MSW landfills. Copyright © 2009 John Wiley & Sons, Ltd.     

基于离散裂隙网络模型的裂隙水渗流计算

离散裂隙网络模型(Discrete Fracture Network（DFN）)是研究裂隙水渗流最为有效的手段之一。文章根据裂隙几何参数和水力参数的统计分布,利用Monte Carlo随机模拟技术生成二维裂隙网络,基于图论无向图的邻接矩阵判断裂隙网络的连通,利用递归算法提取出裂隙网络的主干网或优势流路径。基于立方定律和渗流连续性方程,利用数值解析法建立了二维裂隙网络渗流模型,分析不同边界条件下裂隙网络中的流体流动。结果表明,该方法可以模拟区域宏观水力梯度和边界条件下,裂隙网络水力梯度方向总的流量,以及节点的水位、节点间的流量和流动方向的变化特征,为区域岩溶裂隙水渗流计算提供了一种实用、可行的方法。      

多相流传输THM全耦合数值模型及程序验证

我国首个GCS示范工程神华多储层场地出现了单储层吸气量剧增的现象,在其原设计方案下,压缩后变冷的CO₂被注入至深部高温含水层中,引起首层含水层中流体压力和温度应力急剧变化,从而导致大量裂隙产生,增加了单储层的可注入性的同时,降低了系统总体封存能力,并带来了泄露风险。本文基于TOUGH-FLAC三维多相多组分THM耦合数值模拟程序,开发了场地尺度岩体开裂模块来研究CO₂注入方案对目标含水层耦合特性和开裂特性的综合影响,并设计了定速率、先增速后定速、间歇定速、间歇变速、二次变速等多类型注入方案,分别计算分析了储层岩体的热力学特性、多相流特性与开裂情况。结果表明：设计方案下含水层产生了较多的开裂现象,是导致其可注入性增大的根本原因,持续注入CO₂引起含水层岩体中有效应力大幅度降低,渗透率增加,定速率方案产生的温度应力最小,在设计各类注入方案中,定速率注入方案下储层的裂缝发育最少。     

A fracture mapping and extended finite element scheme for coupled deformation and fluid flow in fractured porous media

This paper presents a fracture mapping (FM) approach combined with the extended finite element method (XFEM) to simulate coupled deformation and fluid flow in fractured porous media. Specifically, the method accurately represents the impact of discrete fractures on flow and deformation, although the individual fractures are not part of the finite element mesh. A key feature of FM‐XFEM is its ability to model discontinuities in the domain independently of the computational mesh. The proposed FM approach is a continuum‐based approach that is used to model the flow interaction between the porous matrix and existing fractures via a transfer function. Fracture geometry is defined using the level set method. Therefore, in contrast to the discrete fracture flow model, the fracture representation is not meshed along with the computational domain. Consequently, the method is able to determine the influence of fractures on fluid flow within a fractured domain without the complexity of meshing the fractures within the domain. The XFEM component of the scheme addresses the discontinuous displacement field within elements that are intersected by existing fractures. In XFEM, enrichment functions are added to the standard finite element approximation to adequately resolve discontinuous fields within the simulation domain. Numerical tests illustrate the ability of the method to adequately describe the displacement and fluid pressure fields within a fractured domain at significantly less computational expense than explicitly resolving the fracture within the finite element mesh. Copyright © 2013 John Wiley & Sons, Ltd.     

A coupled surface/subsurface flow model accounting for air entrapment and air pressure counterflow

This work introduces the soil air system into integrated hydrology by simulating the flow processes and interactions of surface runoff, soil moisture and air in the shallow subsurface. The numerical model is formulated as a coupled system of partial differential equations for hydrostatic (diffusive wave) shallow flow and two-phase flow in a porous medium. The simultaneous mass transfer between the soil, overland, and atmosphere compartments is achieved by upgrading a fully established leakance concept for overland-soil liquid exchange to an air exchange flux between soil and atmosphere. In a new algorithm, leakances operate as a valve for gas pressure in a liquid-covered porous medium facilitating the simulation of air out-break events through the land surface. General criteria are stated to guarantee stability in a sequential iterative coupling algorithm and, in addition, for leakances to control the mass exchange between compartments. A benchmark test, which is based on a classic experimental data set on infiltration excess (Horton) overland flow, identified a feedback mechanism between surface runoff and soil air pressures. Our study suggests that air compression in soils amplifies surface runoff during high precipitation at specific sites, particularly in near-stream areas.