Mini-Workshop: Numerical Upscaling for Flow Problems: Theory and Applications

The objective of this workshop was to bring together researchers working in multiscale simulations with emphasis on multigrid methods and multiscale finite element methods, aiming at chieving of better understanding and synergy between these methods. The goal of multiscale finite element methods, as upscaling methods, is to compute coarse scale solutions of the underlying equations as accurately as possible. On the other hand, multigrid methods attempt to solve fine-scale equations rapidly using a hierarchy of coarse spaces. Multigrid methods need “good” coarse scale spaces for their efficiency. The discussions of this workshop partly focused on approximation properties of coarse scale spaces and multigrid convergence. Some other presentations were on upscaling, domain decomposition methods and nonlinear multiscale methods. Some researchers discussed applications of these methods to reservoir simulations, as well as to simulations of filtration, insulating materials, and turbulence

Upscaling and Multiscale Reservoir Simulation Using Pressure Transient Concepts

Fluid flow in subsurface petroleum reservoirs occurs on a wide range of length scales and capturing all the relevant scales in reservoir modeling is a cumbersome task. Even with the advent of modern computational resources, reservoir simulation of high resolution fine scale geologic models remain a challenge. Therefore, it is customary to use some kind of upscaling procedure to coarsen the multimillion cell geologic models to a scale feasible for practical reservoir simulation. Existing methods for upscaling of geologic models are based on steady state concepts of flow while the actual flow simulations itself is utilized for the purpose of capturing pressure and saturation transients. However, steady state or pseudo steady state limits may never be achieved for a coarse cell volume during a simulation time step in high contrast low permeability systems introducing a potentially significant bias into an upscaling or downscaling calculation. In this dissertation, a novel formulation is proposed which resolves these dynamic effects using an asymptotic pressure solution. Three principal research contributions are made in this dissertation. First, a novel construction of transmissibility in 1D is derived using pseudo steady state concepts which has the advantage of localization over steady state methods, when applied for upscaling problems. This construction is general for all grid geometries usually utilized in industry standard reservoir simulation codes (block centered, radial, corner point). A new form of pressure averaging is proposed to effectively convert a 3D pseudo steady state upscaling into a 1D calculation. Second, a pressure transient diffuse source upscaling formulation is introduced to identify well-connected sub volume that reaches pseudo steady state especially in high contrast systems. The formulation is based on transients approaching pseudo steady state in the upscaling region which can effectively identify the well-connected sub volume that contributes the flow. Third, the pressure transient diffuse source formulation developed for upscaling is extended to the multiscale framework where the large scale changes in pressure are resolved on the coarse grid while the saturations are resolved on the fine scale using downscaled coarse information. Applications are shown for both incompressible and slightly compressible flow