Accurate Subgrid Models for Two-Phase Flow in Heterogeneous Reservoirs

Abstract

Abstract Subgrid effects can have a strong influence on flow and transport in oil reservoirs. In this work a new model for the representation of subgrid terms is introduced and applied to two-phase reservoir flows. The model entails a generalized convection diffusion treatment of the saturation equation as well as an extended representation for total mobility in the pressure equation. Motivation for the form of the model is provided through a consideration of volume averaging and homogenization results. The numerical computation of the subgrid terms and the implementation of the overall method are described. The accuracy of the new subgrid representation is compared to that of coarse scale models with no subgrid treatment and to coarse models based on pseudo relative permeabilities. The new model consistently provides more accurate overall coarse scale predictions, relative to reference fine scale results, than the other coarse scale models, particularly in cases when the global boundary conditions vary in time. Introduction The effects of reservoir heterogeneity at a scale smaller than a typical simulation grid block can have a significant impact on reservoir flow. A number of different approaches for the modeling of these subgrid effects have been introduced. The most commonly applied methods involve the use of pseudo relative permeabilities and the use of specialized coarse gridding procedures. Though effective in many cases, both of these approaches are known to suffer from some drawbacks, including a potentially high level of process-dependency (in the case of pseudo relative permeabilities) and the inability to provide very high degrees of coarsening (in the case of gridding procedures). In this work, we apply an alternate approach, based in part on the use of volume averaging and homogenization, for the modeling of subgrid effects. The approach extends our previous work along these lines,1,2,3 in which we applied volume averaging procedures to generate coarse scale equations that include both averaged quantities and subgrid or fluctuating properties. The subgrid effects in these earlier formulations appeared in terms of length and time-dependent dispersivities, which are driven by the interaction between local fine scale effects and the global flow field. Here we develop a related but conceptually simpler subgrid model that is better suited for general reservoir simulation. The method entails the use of a modified coarse scale convective flux function (which is somewhat akin to a pseudo relative permeability), as well as a coarse scale diffusivity, for each grid block. This approach has the benefit of appropriately representing both small scale effects (via the coarse scale diffusivity) and larger scale effects (via the modi- fied convective flux). A specialized upscaling of the pressure equation, consistent with our convection-diffusion model for transport, is also introduced in this work. As indicated above, alternate approaches for upscaling include the use of pseudo relative permeabilities and flow-based grid generation. There have been a number of previous papers addressing the development and evaluation of methods based on pseudo relative permeabilities. Barker and coworkers4,5 and Darman et al.6 discussed and evaluated a number of the relevant methods. In most cases, these methods differ from one another in the way in which the reference fine scale results are post-processed to generate the upscaled or pseudo relative permeability curves. In many cases, a more critical issue is the boundary conditions applied to the local fine scale problem used to compute the upscaled functions. Recent papers by Wallstrom et al.7,8 address this issue via the development of "effective Flux" boundary conditions. These boundary conditions attempt to mimic the average effects of the large scale flow on the local problem and to correct the bias inherent in standard approaches, which tend to overestimate the impact of local high permeability regions in the calculation of the upscaled functions.