Dynamic consolidation problems in saturated soils solved through u-w formulation in a LME meshfree framework

A meshfree numerical model, based on the principle of Local Maximum Entropy (LME), including a B-bar algorithm to avoid instabilities, is applied to solve axisymmetric consolidation problems in elastic saturated soils. This numerical scheme has been previously validated for purely elastic problems without water (mono phase), as well as for steady seepage in elastic porous media. Hereinafter, an implementation of the novel numerical method in the axisymmetric configuration is proposed, and the model is validated for well known theoretical problems of consolidation in saturated soils, under both static and dynamic conditions with available analytical solutions. The solutions obtained with the new methodology are compared with a finite element commercial software for a set of examples. After validated, solutions for dynamic radial consolidation and sinks, which have not been found elsewhere in the literature, are presented as a novelty. This new numerical approach is demonstrated to be feasible for this kind of problems in porous media, particularly for high frequency, dynamic problems, for which very few results have been found in the literature in spite of their high practical importance

Numerical simulation of dynamic pore fluid-solid interaction in fully saturated non-linear porous media

In this paper, a large deformation formulation for dynamic analysis of the pore fluid-solid interaction in a fully saturated non-linear medium is presented in the framework of the Arbitrary Lagrangian-Eulerian method. This formulation is based on Biot’s theory of consolidation extended to include the momentum equations of the solid and fluid phases, large deformations and non-linear material behaviour. By including the displacements of the solid skeleton, u, and the pore fluid pressure, p, a (u-p) formulation is obtained, which is then discretised using finite elements. Time integration of the resulting highly nonlinear equations is accomplished by the generalized–α method, which assures second order accuracy as well as unconditional stability of the solution. Details of the formulation and its practical implementation in a finite element code are discussed. The formulation and its implementation are validated by solving some classical examples in geomechanics

Mixed Formulations in Space and Time Discretizations for the Fixed-Stress Split Method in Poromechanics

Coupled flow and geomechanics become one of the important research topics in oil and gas industry for development of unconventional petroleum reservoirs such as gas shale, tight gas, and gas hydrates. In particular, these reservoirs are naturally born with its complex behavior, exhibiting strong non-linearity, anisotropy, and heterogeneity effects within each geomaterial and fluid by itself. In addition, the coupling between flow and geomechanics is more complicated for unconsolidated reservoirs or shale formations. Thus, it is critical to assess these complex coupled processes properly through poromechanics with forward numerical simulation and to provide more accurate solutions in order to predict the reservoir performance more precisely. The main objective of this study is to address several numerical issues that are accompanied with simulation in poromechanics. We perform in-depth analysis on mathematical conditions to satisfy for numerically stable and accurate solution, employing various mixed formulations in space and time discretization. Specifically, in space discretization, we deal with the spatial instability that occurs at early times in poromechanics simulation, such as a consolidation problem. We identify two types of spatial instabilities caused by violation of two different conditions: the condition due to discontinuity in pressure and the inf-sup condition related to incompressible fluid, which both occur at early times. We find that the fixed-stress split with the finite volume method for flow and finite element method for geomechanics can provide stability in space, allowing discontinuity of pressure and circumventing violation of the inf-sup condition. In time discretization, we investigate the order of accuracy in time integration with the fixed-stress sequential method. In the study, two-pass and deferred correction methods are to be considered for studying the high-order methods in time integration. We find that there are two different inherent constraint structures that still cause order reductions against high-order accuracy while applying the two methods. As an additional in-depth analysis, we study a large deformation system, considering anisotropic properties for geomechanical and fluid flow parameters, the traverse isotropy and permeability anisotropy ratio. Seeking more accurate solutions, we adopt the total Lagrangian method in geomechanics and multi-point flux approximation in fluid flow. By comparing it to the infinitesimal transformation with two-point flux approximation, we find that substantial differences between the two approaches can exist. For a field application, we study large-scale geomechanics simulation that can honor measured well data, which leads to a constrained geomechanics problem. We employ the Uzawa’s algorithm to solve the saddle point problem from the constrained poromechanics. From numerical parallel simulations, we estimate initial stress distribution in the shale gas reservoir, which will be used for the field development plan. From this study, we find several mathematical conditions for numerically stable and accurate solution of poromechanics problems, when we take the various mixed formulations. By considering the conditions, we can overcome the numerical issues. Then, reliable and precise prediction of reservoir behavior can be obtained for coupled flow-geomechanics problems

Mixed Formulations in Space and Time Discretizations for the Fixed-Stress Split Method in Poromechanics

Coupled flow and geomechanics become one of the important research topics in oil and gas industry for development of unconventional petroleum reservoirs such as gas shale, tight gas, and gas hydrates. In particular, these reservoirs are naturally born with its complex behavior, exhibiting strong non-linearity, anisotropy, and heterogeneity effects within each geomaterial and fluid by itself. In addition, the coupling between flow and geomechanics is more complicated for unconsolidated reservoirs or shale formations. Thus, it is critical to assess these complex coupled processes properly through poromechanics with forward numerical simulation and to provide more accurate solutions in order to predict the reservoir performance more precisely. The main objective of this study is to address several numerical issues that are accompanied with simulation in poromechanics. We perform in-depth analysis on mathematical conditions to satisfy for numerically stable and accurate solution, employing various mixed formulations in space and time discretization. Specifically, in space discretization, we deal with the spatial instability that occurs at early times in poromechanics simulation, such as a consolidation problem. We identify two types of spatial instabilities caused by violation of two different conditions: the condition due to discontinuity in pressure and the inf-sup condition related to incompressible fluid, which both occur at early times. We find that the fixed-stress split with the finite volume method for flow and finite element method for geomechanics can provide stability in space, allowing discontinuity of pressure and circumventing violation of the inf-sup condition. In time discretization, we investigate the order of accuracy in time integration with the fixed-stress sequential method. In the study, two-pass and deferred correction methods are to be considered for studying the high-order methods in time integration. We find that there are two different inherent constraint structures that still cause order reductions against high-order accuracy while applying the two methods. As an additional in-depth analysis, we study a large deformation system, considering anisotropic properties for geomechanical and fluid flow parameters, the traverse isotropy and permeability anisotropy ratio. Seeking more accurate solutions, we adopt the total Lagrangian method in geomechanics and multi-point flux approximation in fluid flow. By comparing it to the infinitesimal transformation with two-point flux approximation, we find that substantial differences between the two approaches can exist. For a field application, we study large-scale geomechanics simulation that can honor measured well data, which leads to a constrained geomechanics problem. We employ the Uzawa’s algorithm to solve the saddle point problem from the constrained poromechanics. From numerical parallel simulations, we estimate initial stress distribution in the shale gas reservoir, which will be used for the field development plan. From this study, we find several mathematical conditions for numerically stable and accurate solution of poromechanics problems, when we take the various mixed formulations. By considering the conditions, we can overcome the numerical issues. Then, reliable and precise prediction of reservoir behavior can be obtained for coupled flow-geomechanics problems

Finite element modelling of fracture propagation in saturated media using quasi-zero-thickness interface elements

A new computational technique for the simulation of 2D and 3D fracture propagation processes in saturated porous media is presented. A non-local damage model is conveniently used in conjunction with interface elements to predict the degradation pattern of the domain and insert new fractures followed by remeshing. FIC-stabilized elements of equal order interpolation in the displacement and the pore pressure have been successfully used under complex conditions near the undrained-incompressible limit. A bilinear cohesive fracture model describes the mechanical behaviour of the joints. A formulation derived from the cubic law models the fluid flow through the crack. Examples in 2D and 3D, using 3-noded triangles and 4-noded tetrahedra respectively, are presented to illustrate the accuracy and robustness of the proposed methodology.Peer ReviewedPostprint (author's final draft

Coupled deformation, fluid flow and fracture propagation in porous media

Polygonal faults are non-tectonic fault systems which are layer-bound (at some vertical scale) and are widely developed in fine-grained sedimentary basins. Although several qualitative mechanisms have been hypothesised to explain the formation of these faults, there is a weak general consensus that they are formed by the coupled deformation and fluid expulsion of the host sediments (consolidation). This thesis presents a numerical framework that can be extended to investigate the role consolidation plays in the development and evolution of these faults. The method is also applicable to reservoir engineering and CO2 storage. An understanding of the coupled mechanical response and fluid flow is critical in determining compaction and subsidence in oil reservoirs and fault-seal integrity during CO2 disposal and storage. The technique uses a fracture mapping approach (FM) and the extended finite element method (XFEM) to modify the single phase FEM consolidation formulation. A key feature of FM-XFEM is its ability to include discontinuities into a model independently of the computational mesh. The fracture mapping approach is used to simulate the flow interaction between the matrix and existing fractures via a transfer function. Since fractures are represented using level set data, the need for complex meshing to describe fractures is not required. The XFEM component of the method simulates the influence of the pore fluid on the mechanical behaviour of the fractured medium. In XFEM, enrichment functions are added to the standard finite element approximation to ensure an accurate approximation of discontinuous fields within the simulation domain. FM-XFEM produces results comparative to the discrete fracture method on relatively coarse meshes. FM-XFEM has also been extended to model the propagation of existing fractures using a mixed-mode criterion applicable to geological media. Stress concentrations at the tips of existing fractures show good agreement with an analytical solution found in literature

A semi-implicit discrete-continuum coupling method for porous media based on the effective stress principle at finite strain

Abstract: A finite strain multiscale hydro-mechanical model is established via an extended Hill–Mandel condition for two-phase porous media. By assuming that the effective stress principle holds at unit cell scale, we established a micro-to-macro transition that links the micromechanical responses at grain scale to the macroscopic effective stress responses, while modeling the fluid phase only at the macroscopic continuum level. We propose a dual-scale semi-implicit scheme, which treats macroscopic responses implicitly and microscopic responses explicitly. The dual-scale model is shown to have good convergence rate, and is stable and robust. By inferring effective stress measure and poro-plasticity parameters, such as porosity, Biot’s coefficient and Biot’s modulus from micro-scale simulations, the multiscale model is able to predict effective poro-elasto-plastic responses without introducing additional phenomenological laws. The performance of the proposed framework is demonstrated via a collection of representative numerical examples. Fabric tensors of the representative elementary volumes are computed and analyzed via the anisotropic critical state theory when strain localization occurs. Keywords: Multiscale poromechanics; Semi-implicit scheme; Homogenization; Discrete-continuum coupling; DEM–FEM; Anisotropic critical stat