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.     

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.     

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 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.     

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 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.     

Conditional statistical inverse modeling in groundwater flow by multigrid methods

Due to the notorious lack of data, stochastic simulation and conditioning of distributed parameter fields is generally acknowledged as a major task in order to produce realistic prognoses for groundwater flow phenomena, thus honouring the maximum of information available. In this paper, a new conditioning approach is presented which treats the distributed parameters directly without projection onto lower dimensional spaces and preserves certain desired statistical properties by explicitly stating them as constraints for the conditioning optimization problem. Typically, the conditioning task must be performed very often and the conditioning optimization problems are highly dimensional. Therefore, a second main focus of the paper is on the presentation of efficient multigrid methods for the solution of the conditioning problems. Numerical results are given for a practical application problem. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

三维地质建模过程中综合地质剖面构建方法研究

综合地质剖面是三维地质建模重要的数据源,在地质构造复杂地区运用综合地质剖面构建三维地质模型的方法应用较为广泛。本文以成都市为例,详细介绍了综合地质剖面构建需要准备的地质资料、剖面的部署原则和构建方法,并解决了剖面构建过程中遇到的同时代、同层位的不同地层之间的连接问题。实践表明,通过综合地质剖面的构建,可以充分发挥地质技术人员在三维地质建模中的主观能动性,准确刻画复杂地质体在三维空间的展布情况,有效实现复杂地质体交互式半自动建模,提高三维地质建模的效率和精准度。     

A direct boundary element method for plane strain poroelasticity

This paper presents a direct boundary element method (BEM), formulated in the Laplace transform space, for plane strain poroelasticity. The paper expands on work by Cheng and Liggett by recasting the theoretical foundation of BEM within the framework of Rice and Cleary's formulation of the Biot theory of poroelasticity. Furthermore, the numerical algorithm is generalized to deal with both interior and exterior domain problems, and a method for indirectly calculating the Cauchy principal value of the singular integrals is presented. Formulae for the stress and flux inside the domain are also derived. Finally, the algorithm is validated by comparing the numerical results with the analytical solution of a borehole subject to a far-field deviatoric stress (exterior domain) and with the solution of Mandel's problem (interior domain). These two examples provide a critical test of the algorithm.     

地下水在岩体结构面系统中运移的三维数值模拟

本文从描述流体流动的基本方程出发,推导了地下水在岩体结构面系统中运移的有限元方程,并基于这一方程编制了地下水在岩体中流动的三维数值模拟程序。     

复杂地质体三维实体建模方法

依托常规GIS技术的建模手段不能满足复杂地质体三维实体建模的需求,其建模效果在真三维建模、实体模型应用等方面受限。本文根据复杂地质体的特征,将复杂地质体分为层状的连续型非倒转地质体、非连续型(断裂)地质体、倒转褶皱地质体和非层状地质体。从三维实体建模的角度,提出数据拆分、数据控制、数据简化三种建模数据处理方法,并借助三维GIS的可视化技术与GOCAD真三维建模能力,研究基于GOCAD软件的复杂地质体三维实体建模方法,详细阐述了四类复杂地质体的具体实现方法,并构建三维实体模型。     

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.     

基于实际材料图的三维断层模型动态构建方法

三维地质模型是地质调查信息表达从传统二维平面向三维空间转变的一次重大突破。中国正在开展的三维地质填图作为国家的重要基础地质调查工作部署,将调查与研究相结合,在推动三维地质建模从研究走向实践具有重大意义。三维地质填图过程中如何准确、快速地构建三维断层模型是其中的关键点及难点。通过对数字填图过程不同阶段数据源的分析,详细分析了实际材料图建立断层数据以及与野外路线数据的关系映射规则,提出并实现了由线及面的断层模型构建方法。该方法强调断层模型构建与业务流程的一致性、动态性和快速性,构建模型的数据源始于野外第一手资料,充分利用填图业务过程中不同阶段数据的关联、约束与继承关系,能够满足在填图过程中根据地质认识的深入快速、动态地重构模型的要求,同时也可根据直观的三维断层模型时时验证地质连图的合理性。算法已在实际野外三维地质填图工作进行了多次实验,获得较好的成果。实验说明,基于实际材料图的三维断层模型构建方法与实际业务过程同步,具有操作便捷性,可直接应用于三维地质填图的实际工作要求。     

基于实际材料图的三维断层模型动态构建方法

三维地质模型是地质调查信息表达从传统二维平面向三维空间转变的一次重大突破。中国正在开展的三维地质填图作为国家的重要基础地质调查工作部署,将调查与研究相结合,在推动三维地质建模从研究走向实践具有重大意义。三维地质填图过程中如何准确、快速地构建三维断层模型是其中的关键点及难点。通过对数字填图过程不同阶段数据源的分析,详细分析了实际材料图建立断层数据以及与野外路线数据的关系映射规则,提出并实现了由线及面的断层模型构建方法。该方法强调断层模型构建与业务流程的一致性、动态性和快速性,构建模型的数据源始于野外第一手资料,充分利用填图业务过程中不同阶段数据的关联、约束与继承关系,能够满足在填图过程中根据地质认识的深入快速、动态地重构模型的要求,同时也可根据直观的三维断层模型时时验证地质连图的合理性。算法已在实际野外三维地质填图工作进行了多次实验,获得较好的成果。实验说明,基于实际材料图的三维断层模型构建方法与实际业务过程同步,具有操作便捷性,可直接应用于三维地质填图的实际工作要求。     

Iteratively coupled solution strategies for a four‐field mixed finite element method for poroelasticity

In this paper, we consider numerical algorithms for modeling of the time‐dependent coupling between the fluid flow and deformation in elastic porous media. Here, we employ a four‐field formulation which uses the total stress, displacement, flux, and pressure as its primary variables and satisfies Darcy's law and linear elasticity in mixed weak form. We present four different iteratively coupled methods, known as drained, undrained, fixed‐strain, and fixed‐stress splits, in which the diffusion operator is separated from the elasticity operator and the two subproblems are solved in a staggered way while ensuring convergence of the solution at each time step. A‐priori convergence results for each iterative coupling which differs from those found when using a traditional two‐field or three‐field formulation are presented. We also present some numerical results to support the convergence estimates and to show the accuracy and efficiency of the algorithms. Copyright © 2016 John Wiley & Sons, Ltd.     

Multiscale modeling of large deformation in geomechanics

Large deformation soil behavior underpins the operation and performance for a wide range of key geotechnical structures and needs to be properly considered in their modeling, analysis, and design. The material point method (MPM) has gained increasing popularity recently over conventional numerical methods such as finite element method (FEM) in tackling large deformation problems. In this study, we present a novel hierarchical coupling scheme to integrate MPM with discrete element method (DEM) for multiscale modeling of large deformation in geomechanics. The MPM is employed to treat a typical boundary value problem that may experience large deformation, and the DEM is used to derive the nonlinear material response from small strain to finite strain required by MPM for each of its material points. The proposed coupling framework not only inherits the advantages of MPM in tackling large deformation engineering problems over the use of FEM (eg, no need for remeshing to avoid mesh distortion in FEM), but also helps avoid the need for complicated, phenomenological assumptions on constitutive material models for soil exhibiting high nonlinearity at finite strain. The proposed framework lends great convenience for us to relate rich grain-scale information and key micromechanical mechanisms to macroscopic observations of granular soils over all deformation levels, from initial small-strain stage en route to large deformation regime before failure. Several classic geomechanics examples are used to demonstrate the key features the new MPM/DEM framework can offer on large deformation simulations, including biaxial compression test, rigid footing, soil-pipe interaction, and soil column collapse.     

A continuous/discontinuous Galerkin framework for modeling coupled subsurface and surface water flow

We consider conjunctive surface-subsurface flow modeling, where surface water flow is described by the shallow water equations and ground water flow by Richards’ equation for the vadose zone. Coupling between the models is based on the continuity of flux and water pressure. Numerical approximation of the coupled model using the framework of discontinuous Galerkin (DG) methods is formulated. In the subsurface, the local discontinuous Galerkin (LDG) method is used to approximate ground water velocity and hydraulic head; a DG method is also used to approximate surface water velocity and elevation. This approach allows for a weak coupling of the models and the use of different approximating spaces and/or meshes within each regime. A simplified LDG method based on continuous approximations to water head is also described. Numerical results that investigate physical and numerical aspects of surface–subsurface flow modeling are presented. This work was supported by National Science Foundation grant DMS-0411413.     

一种基于钻孔地质数据的快速递进三维地质建模方法

三维地质建模是地质数据可视化和空间分析的关键技术之一。针对传统建模方法建模速度慢、建模效率和精度较低的问题,提出了一种直接基于钻孔的点→线→面→体快速递进三维地质建模方法。该方法先对钻孔地层信息进行人机交互对比构成地质剖面,再按钻孔实际坐标在三维空间中还原地质剖面的实际位置,接着通过对剖面间的地层连线进行Kriging插值,形成一系列地层面模型,然后以此为基础构建三维地质框架模型,最后利用BSP矢量剪切技术来裁剪模型边界,形成研究区三维地质模型。该方法在福州市上街镇三维城市地质模型构建的实际应用中得到了验证。实践结果表明,该方法不仅可以实现三维地质模型的快速构建,还可以显著提高三维地质模型的精度。