Abstract: A Multiscale Finite Volume (MSFV) method is provided to efficiently solve large heterogeneous problems; it is usually employed for pressure equations and delivers conservative flux fields to be used in transport problems. It relies on the hypothesis that the fine-scale problem can be described by a set of local solutions coupled by a conservative coarse-scale problem. In numerically challenging cases, a more accurate localization approximation is used to obtain a good approximation of the fine-scale solution. According to an embodiment, a method is provided to iteratively improve the boundary conditions of the local problems, and is responsive to the data structure of the underlying MSFV method and employs a Krylov-subspace projection method to obtain an unconditionally stable scheme and accelerate convergence. In one embodiment the MSFV operator is used. Alternately, the MSFV operator is combined with an operator derived from the problem solved to construct the conservative flux field.

Abstract: A multi-scale finite volume method for simulating a fine-scale geological model of subsurface reservoir is disclosed. The method includes providing a fine-scale geological model of a subsurface reservoir associated with a fine-scale grid, a coarse-scale grid, and a dual coarse-scale grid. A coarse-scale operator is constructed based on internal cells, edge cells, and node cells on the fine-scale grid that are defined by the dual coarse-scale grid. Pressure in the dual coarse-scale cells is computed using the coarse-scale operator. Pressure in the primary coarse-scale cells is computed using the computed pressure in the dual coarse-scale cells. A display is produced using the computed pressure in the primary coarse-scale cells. An iterative scheme can be applied such that the computed pressure in the primary coarse-scale cells converges to the fine-scale pressure solution and mass balance is maintained on the coarse-scale.

Abstract: A Multiscale Finite Volume (MSFV) method is provided to efficiently solve large heterogeneous problems; it is usually employed for pressure equations and delivers conservative flux fields to be used in transport problems. It relies on the hypothesis that the fine-scale problem can be described by a set of local solutions coupled by a conservative coarse-scale problem. In numerically challenging cases, a more accurate localization approximation is used to obtain a good approximation of the fine-scale solution. According to an embodiment, a method is provided to iteratively improve the boundary conditions of the local problems, and is responsive to the data structure of the underlying MSFV method and employs a Krylov-subspace projection method to obtain an unconditionally stable scheme and accelerate convergence. In one embodiment the MSFV operator is used. Alternately, the MSFV operator is combined with an operator derived from the problem solved to construct the conservative flux field.

Abstract: A multi-scale finite-volume (MSFV) method simulates nonlinear immiscible three-phase compressible flow in the presence of gravity and capillary forces. Consistent with the MSFV framework, flow and transport are treated separately and differently using a fully implicit sequential algorithm. The pressure field is solved using an operator splitting algorithm. The general solution of the pressure is decomposed into an elliptic part, a buoyancy/capillary force dominant part, and an inhomogeneous part with source/sink and accumulation. A MSFV method is used to compute the basis functions of the elliptic component, capturing long range interactions in the pressure field. Direct construction of the velocity field and solution of the transport problem on the primal coarse grid provides flexibility in accommodating physical mechanisms. A MSFV method computes an approximate pressure field, including a solution of a course-scale pressure equation; constructs fine-scale fluxes; and computes a phase-transport equation.

Abstract: A multi-scale finite volume method for simulating a fine-scale geological model of subsurface reservoir is disclosed. The method includes providing a fine-scale geological model of a subsurface reservoir associated with a fine-scale grid, a coarse-scale grid, and a dual coarse-scale grid. A coarse-scale operator is constructed based on internal cells, edge cells, and node cells on the fine-scale grid that are defined by the dual coarse-scale grid. Pressure in the dual coarse-scale cells is computed using the coarse-scale operator. Pressure in the primary coarse-scale cells is computed using the computed pressure in the dual coarse-scale cells. A display is produced using the computed pressure in the primary coarse-scale cells. An iterative scheme can be applied such that the computed pressure in the primary coarse-scale cells converges to the fine-scale pressure solution and mass balance is maintained on the coarse-scale.

Abstract: A multi-scale finite-volume (MSFV) method simulates nonlinear immiscible three-phase compressible flow in the presence of gravity and capillary forces. Consistent with the MSFV framework, flow and transport are treated separately and differently using a fully implicit sequential algorithm. The pressure field is solved using an operator splitting algorithm. The general solution of the pressure is decomposed into an elliptic part, a buoyancy/capillary force dominant part, and an inhomogeneous part with source/sink and accumulation. A MSFV method is used to compute the basis functions of the elliptic component, capturing long range interactions in the pressure field. Direct construction of the velocity field and solution of the transport problem on the primal coarse grid provides flexibility in accommodating physical mechanisms. A MSFV method computes an approximate pressure field, including a solution of a course-scale pressure equation; constructs fine-scale fluxes; and computes a phase-transport equation.