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

Computational analysis of hypersonic flows past elliptic-cone waveriders

A comprehensive study for the inviscid numerical calculation of the hypersonic flow past a class of elliptic-cone derived waveriders is presented. The theoretical background associated with hypersonic small-disturbance theory (HSDT) is reviewed. Several approximation formulas for the waverider compression surface are established. A CFD algorithm is used to calculate flow fields for the on-design case and a variety of off-design cases. The results are compared with HSDT, experiment, and other available CFD results. For the waverider shape used in previous investigations, the bow shock for the on-design condition stands off from the leading-edge tip of the waverider. It was found that this occurs because the tip was too thick according to the approximating shape formula that was used to describe the compression surface. When this was corrected, the bow shock became closer to attached as it should be. At Mach numbers greater than the design condition, a lambda-shock configuration develops near the tip of the compression surface. At negative angles of attack, other complicated shock patterns occur near the leading-edge tip. These heretofore unknown flow patterns show the power and utility of CFD for investigating novel hypersonic configurations such as waveriders

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