A novel approach to eliminate error induced by cell to node projections

In engineering practices, different numerical methods for fluid flow simulation and solid deformation/stress simulation are adopted to model fluid–structure interaction problems in porous media. Cell‐centered finite volume method is widely used in fluid flow simulation, while the solid deformation/stress simulation is usually accomplished by using the Galerkin vertex‐centered finite element method, which leads to the incompatibility between cell variables with nodal variables. Therefore, the data transfer between cell variables and nodal variables is inevitable. Consequently, this kind of transfer will lead to extra artificial error. Hence, the major concern is how to minimize the error due to cell to node projections. In this paper, a problem of pore pressure diffusion within a one‐dimensional heterogeneous porous medium is investigated. We present a new projection scheme and corresponding error formula, where the error control factor is introduced. The new projection scheme is based on piecewise linear interpolations. Results demonstrate that if the error control factor is chosen properly, the error due to the projection from cell to node can be controlled effectively, and the most desired zero error can be achieved. Finally, we analyze some practical cases in consideration of permeability contrast and mesh uniformity. Copyright © 2015 John Wiley & Sons, Ltd.     

A new computational strategy for solving two‐phase flow in strongly heterogeneous poroelastic media of evolving scales

We develop a new computational methodology for solving two‐phase flow in highly heterogeneous porous media incorporating geomechanical coupling subject to uncertainty in the poromechanical parameters. Within the framework of a staggered‐in‐time coupling algorithm, the numerical method proposed herein relies on a Petrov–Galerkin postprocessing approach projected on the Raviart–Thomas space to compute the Darcy velocity of the mixture in conjunction with a locally conservative higher order finite volume discretization of the nonlinear transport equation for the saturation and an operator splitting procedure based on the difference in the time‐scales of transport and geomechanics to compute the effects of transient porosity upon saturation. Notable features of the numerical modeling proposed herein are the local conservation properties inherited by the discrete fluxes that are crucial to correctly capture the fingering patterns arising from the interaction between heterogeneity and nonlinear viscous coupling. Water flooding in a poroelastic formation subject to an overburden is simulated with the geology characterized by multiscale self‐similar permeability and Young modulus random fields with power‐law covariance structure. Statistical moments of the poromechanical unknowns are computed within the framework of a high‐resolution Monte Carlo method. Numerical results illustrate the necessity of adopting locally conservative schemes to obtain reliable predictions of secondary recovery and finger growth in strongly heterogeneous deformable reservoirs. Copyright © 2011 John Wiley & Sons, Ltd.     

A coupling of mixed and continuous Galerkin finite element methods for poroelasticity II: the discrete-in-time case

In this paper, we formulate a finite element procedure for approximating the coupled fluid and mechanics in Biot’s consolidation model of poroelasticity. Here, we approximate the pressure by a mixed finite element method and the displacements by a Galerkin method. Theoretical convergence error estimates are derived in a discrete-in-time setting. Of particular interest is the case when the lowest-order Raviart–Thomas approximating space or cell-centered finite differences are used in the mixed formulation and continuous piecewise linear approximations are used for displacements. This approach appears to be the one most frequently applied to existing reservoir engineering simulators.     

A coupling of mixed and continuous Galerkin finite element methods for poroelasticity I: the continuous in time case

In this paper, we formulate a finite element procedure for approximating the coupled fluid and mechanics in Biot’s consolidation model of poroelasticity. Here, we approximate the pressure by a mixed finite element method and the displacements by a Galerkin method. Theoretical convergence error estimates are derived in a continuous in-time setting for a strictly positive constrained specific storage coefficient. Of particular interest is the case when the lowest-order Raviart–Thomas approximating space or cell-centered finite differences are used in the mixed formulation, and continuous piecewise linear approximations are used for displacements. This approach appears to be the one most frequently applied to existing reservoir engineering simulators.     

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.     

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

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

A fully coupled hydro‐thermo‐poro‐mechanical model for black oil reservoir simulation

A fully coupled formulation of a hydro‐thermo‐poro‐mechanical model for a three‐phase black oil reservoir model is presented. The model is based upon the approach proposed by one of the authors which fully couples geomechanical effects to multiphase flow. Their work is extended here to include non‐isothermal effects. The gas phase contribution to the energy equation has been neglected based on a set of assumptions. The coupled formulation given herein differs in several ways when compared to the earlier work and an attempt is made to link the flow based formulation and mixture theory. The Finite Element Method is employed for the numerical treatment and essential algorithmic implementation is discussed. Numerical examples are presented to provide further understanding of the current methodology. Copyright © 2001 John Wiley & Sons, Ltd.     

Thermal reservoir modeling in petroleum geomechanics

Thermal oil recovery processes involve high pressures and temperatures, leading to large volume changes and induced stresses. These cannot be handled by traditional reservoir simulation because it does not consider coupled geomechanics effects. In this paper we present a fully coupled, thermal half‐space model using a hybrid DDFEM method. A finite element method (FEM) solution is adopted for the reservoir and the surrounding thermally affected zone, and a displacement discontinuity method is used for the surrounding elastic, non‐thermal zone. This approach analyzes stress, pressure, temperature and volume change in the reservoir; it also provides stresses and displacements around the reservoir (including transient ground surface movements) in a natural manner without introducing extra spatial discretization outside the FEM zone. To overcome spurious spatial temperature oscillations in the convection‐dominated thermal advection–diffusion problem, we place the transient problem into an advection–diffusion–reaction problem framework, which is then efficiently addressed by a stabilized finite element approach, the subgrid‐scale/gradient subgrid‐scale method. Copyright © 2008 John Wiley & Sons, Ltd.     

储层流固耦合的数学模型和非线性有限元方程

根据饱和多孔介质固体骨架的平衡方程和多孔介质中流体的连续性方程,建立了储层流固耦合数学模型。模型中引入了Jaumann应力速率公式描述多孔介质固体骨架的大变形效应,并考虑了地应力、初始孔隙压力、初始流体密度和初始孔隙度对耦合模型的影响。基于与微分方程等价的加权余量公式,在空间域采用有限元离散,对时间域进行隐式差分格式离散,导出了以单元节点位移和单元节点孔隙压力为未知量的储层流固耦合的非线性有限元增量方程。该模型在石油工程中有广泛的应用,为储层流固耦合的数值模拟奠定了理论基础。     

Large‐scale poroelastic fractured reservoirs modeling using the fast multipole displacement discontinuity method

An effective approach to modeling the geomechanical behavior of the network and its permeability variation is to use a poroelastic displacement discontinuity method (DDM). However, the approach becomes rather computationally intensive for an extensive system of cracks, particularly when considering coupled diffusion/deformation processes. This is because of additional unknowns and the need for time‐marching schemes for the numerical integration. The Fast Multipole Method (FMM) is a technique that can accelerate the solution of large fracture problems with linear complexity with the number of unknowns both in memory and CPU time. Previous works combining DDM and FMM for large‐scale problems have accounted only for elastic rocks, neglecting the fluid leak‐off from the fractures into the matrix and its influence on pore pressure and stress field. In this work we develop an efficient geomechanical model for large‐scale natural fracture networks in poroelastic reservoirs with fracture flow in response to injection and production operations. Accuracy and computational performance of the proposed method with those of conventional poroelastic DDM are compared through several case studies involving up to several tens of thousands of boundary elements. The results show the effectiveness of the FMM approach to successfully evaluate field‐scale problems for the design of exploitation strategies in unconventional geothermal and petroleum reservoirs. An example considering faults reveals the impact of reservoir compartmentalization because of sealing faults for both geomechanical and flow variables under elastic and poroelastic rocks. Copyright © 2015 John Wiley & Sons, Ltd.     

Accelerating the convergence of coupled geomechanical‐reservoir simulations

The pressure variations during the production of petroleum reservoir induce stress changes in and around the reservoir. Such changes of the stress state can induce marked deformation of geological structures for stress sensitive reservoirs as chalk or unconsolidated sand reservoirs. The compaction of those reservoirs during depletion affects the pressure field and so the reservoir productivity. Therefore, the evaluation of the geomechanical effects requires to solve in a coupling way the geomechanical problem and the reservoir multiphase fluid flow problem. In this paper, we formulate the coupled geomechanical‐reservoir problem as a non‐linear fixed point problem and improve the resolution of the coupling problem by comparing in terms of robustness and convergence different algorithms. We study two accelerated algorithms which are much more robust and faster than the conventional staggered algorithm and we conclude that they should be used for the iterative resolution of coupled reservoir‐geomechanical problem. Copyright © 2006 John Wiley & Sons, Ltd.     

Parallel solution to ill‐conditioned FE geomechanical problems

Parallel computers are potentially very attractive for the implementation of large size geomechanical models. One of the main difficulties of parallelization, however, relies on the efficient solution of the frequently ill‐conditioned algebraic system arising from the linearization of the discretized equilibrium equations. While very efficient preconditioners have been developed for sequential computers, not much work has been devoted to parallel solution algorithms in geomechanics. The present study investigates the state‐of‐the‐art performance of the factorized sparse approximate inverse (FSAI) as a preconditioner for the iterative solution of ill‐conditioned geomechanical problems. Pre‐and post‐filtration strategies are experimented with to increase the FSAI efficiency. Numerical results show that FSAI exhibits a promising potential for parallel geomechanical models mainly because of its almost ideal scalability. With the present formulation, however, at least 4 or 8 processors are required in the selected test cases to outperform one of the most efficient sequential algorithms available for FE geomechanics, i.e. the multilevel incomplete factorization (MIF). Further research is needed to improve the FSAI efficiency with a more effective selection of the preconditioner non‐zero pattern. Copyright © 2011 John Wiley & Sons, Ltd.     

The Geomechanics of CO2 Storage in Deep Sedimentary Formations

This paper provides a review of the geomechanics and modeling of geomechanics associated with geologic carbon storage (GCS), focusing on storage in deep sedimentary formations, in particular saline aquifers. The paper first introduces the concept of storage in deep sedimentary formations, the geomechanical processes and issues related with such an operation, and the relevant geomechanical modeling tools. This is followed by a more detailed review of geomechanical aspects, including reservoir stress-strain and microseismicity, well integrity, caprock sealing performance, and the potential for fault reactivation and notable (felt) seismic events. Geomechanical observations at current GCS field deployments, mainly at the In Salah CO₂ storage project in Algeria, are also integrated into the review. The In Salah project, with its injection into a relatively thin, low-permeability sandstone is an excellent analogue to the saline aquifers that might be used for large scale GCS in parts of Northwest Europe, the U.S. Midwest, and China. Some of the lessons learned at In Salah related to geomechanics are discussed, including how monitoring of geomechanical responses is used for detecting subsurface geomechanical changes and tracking fluid movements, and how such monitoring and geomechanical analyses have led to preventative changes in the injection parameters. Recently, the importance of geomechanics has become more widely recognized among GCS stakeholders, especially with respect to the potential for triggering notable (felt) seismic events and how such events could impact the long-term integrity of a CO₂ repository (as well as how it could impact the public perception of GCS). As described in the paper, to date, no notable seismic event has been reported from any of the current CO₂ storage projects, although some unfelt microseismic activities have been detected by geophones. However, potential future commercial GCS operations from large power plants will require injection at a much larger scale. For such large-scale injections, a staged, learn-as-you-go approach is recommended, involving a gradual increase of injection rates combined with continuous monitoring of geomechanical changes, as well as siting beneath a multiple layered overburden for multiple flow barrier protection, should an unexpected deep fault reactivation occur.     

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

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

A Review of Some Rock Mechanics Issues in Geothermal Reservoir Development

Rock mechanics and geomechanical studies can provide crucial information for economic geothermal reservoir development. Although significant progress has been made in reservoir geomechanics, technical challenges specific to the geothermal area (high temps, data collection, experimentation issues) have prevented widespread use of geomechanics in geothermal reservoir development. However, as the geothermal industry moves to develop more challenging resources using the concept of enhanced geothermal systems (EGS), and to maximize productivity from conventional resources, the need for improved understanding of geomechanical issues and developing specific technologies for geothermal reservoirs has become critical. Rock mechanics research and improved technologies can impact areas related to in-situ stress characterization, initiation and propagation of artificial and natural fractures, and the effects of coupled hydro-thermo-chemo-mechanical processes on fracture permeability and induced seismicity. Rock mechanics/geomechanics research, including experimental and theoretical investigations as well as numerical and analytical solutions, has an important role in optimizing reservoir design and heat extraction strategies for sustainable geothermal energy development. A number of major areas where rock mechanics research can facilitate geothermal systems development are reviewed in this paper with particular emphasis on EGS design and management.     

A fully coupled numerical modeling to investigate the role of rock thermo-mechanical properties on reservoir uplifting in steam assisted gravity drainage

One of the crucial consequences of steam assisted gravity drainage (SAGD) process is abnormal reservoir uplifting under thermal steam injection, which can significantly influence the reservoir rock deformation, specifically thin bed reservoirs and causes intensive failures and fractures into the cap rock formations. A thorough understanding of the influences of rock thermo-mechanical properties on reservoir uplifting plays an important role in preventing those aforementioned failures within design and optimization process in SAGD. In addition, coupling of reservoir porous medium and flowing of specific fluid with temperature as an additional degree of freedom with initial pore pressure and in-situ stress condition, are also very challenging parts of geomechanical coupled simulation which would be clearly explained. Thus, a fully coupled thermo-poro-elastic geomechanical model with finite element codes was performed in ABAQUS to investigate the role of rock thermo-mechanical parameters on reservoir vertical uplift during steam injection. It is clearly observed that, any increase in rock thermo-mechanical properties specifically rock’s thermal properties such as specific heat, thermal expansion, and formation’s thermal conductivity, have significant influences on reservoir uplift. So by coupling the temperature as an additional degree of freedom with the coupled pore-fluid stress and diffusion finite element model of SAGD process, the more realistic simulation will be conducted; hence, the errors related to not having heat as an additional degree of freedom will be diminished. In addition, Young’s modulus and specific heat are the rock thermo-mechanical parameters which have the maximum and minimum effects on the reservoir uplift, respectively.     

A new locally conservative numerical method for two-phase flow in heterogeneous poroelastic media

We construct a new class of locally conservative numerical methods for two-phase immiscible flow in heterogeneous poroelastic media. Within the framework of the so-called iteratively coupled methods and fixed-stress split algorithm we develop mixed finite element methods for the flow and geomechanics subsystems which furnish locally conservative Darcy velocity and transient porosity input fields for the transport problem for the water saturation. Such hyperbolic equation is decomposed within an operator splitting technique based on a predictor–corrector scheme with the predictor step discretized by a higher-order non-oscillatory finite volume central scheme. The proposed scheme adopts an inhomogeneous dual mesh with variable cell size ruled by the local wave speed of propagation to compute numerical fluxes at cell edges. In the limit of small time steps the central scheme gives rise to a semidiscrete formulation for the water saturation capable of incorporating heterogeneous porosity fields and generalized flux functions including the water transport due to the solid phase velocity. Numerical simulations of a water-flooding problem in secondary oil recovery are presented for different realizations of the input random fields (permeability, Young modulus and initial porosity). Comparison between the accuracies of the proposed approach and the traditional one-way coupled hydro-geomechanical formulation are presented. The effects of the cross-correlation between the input random fields and compaction drive mechanism upon finger growth and breakthrough curves are also analyzed. A notable feature of the formulation proposed herein is the accurate prediction of the influence of geomechanical effects upon the unstable movement of the water front, whose evolution is dictated by rock heterogeneity and unfavorable viscosity ratio, without deteriorating the local conservative character of the numerical schemes.     

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.     

An explicitly coupled hydro‐geomechanical model for simulating hydraulic fracturing in arbitrary discrete fracture networks

Modeling hydraulic fracturing in the presence of a natural fracture network is a challenging task, owing to the complex interactions between fluid, rock matrix, and rock interfaces, as well as the interactions between propagating fractures and existing natural interfaces. Understanding these complex interactions through numerical modeling is critical to the design of optimum stimulation strategies. In this paper, we present an explicitly integrated, fully coupled discrete‐finite element approach for the simulation of hydraulic fracturing in arbitrary fracture networks. The individual physical processes involved in hydraulic fracturing are identified and addressed as separate modules: a finite element approach for geomechanics in the rock matrix, a finite volume approach for resolving hydrodynamics, a geomechanical joint model for interfacial resolution, and an adaptive remeshing module. The model is verified against the Khristianovich–Geertsma–DeKlerk closed‐form solution for the propagation of a single hydraulic fracture and validated against laboratory testing results on the interaction between a propagating hydraulic fracture and an existing fracture. Preliminary results of simulating hydraulic fracturing in a natural fracture system consisting of multiple fractures are also presented. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical modelling of regional faults in land subsidence prediction above gas/oil reservoirs

The stress variation induced by gas/oil production may activate pre‐existing regional faults. This may enhance the expected land subsidence due to the generation of mechanically weak points close to the producing field. A class of elasto‐plastic interface elements (IE), specifically designed to address the mechanical behaviour of faults over a regional scale, is integrated into a finite element (FE) geomechanical model and used to investigate the role exerted by active faults in anthropogenic land subsidence. The importance of regional faults depends on a variety of factors including depth of the depleted reservoir, fault number, orientation and size, geomechanical properties of porous medium, pore pressure drawdown induced by fluid production, etc. With the aid of some representative examples, a useful indication is provided as to where and how fault activation may influence both magnitude and extent of the land subsidence bowl above producing gas/oil reservoirs, pointing to a generally limited impact on the ground surface. The simulation of a real faulted gas reservoir in a complex 3‐D setting shows that the proposed IE can be simply and efficiently incorporated into a FE geomechanical model, thus improving the quality of the stress and displacement prediction. Copyright © 2007 John Wiley & Sons, Ltd.