Accurate Subgrid Models for Two-Phase Flow in Heterogeneous Reservoirs
- Yalchin R. Efendiev (Texas A&M U.) | Louis J. Durlofsky (Stanford U. and ChevronTexaco ETC)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- June 2004
- Document Type
- Journal Paper
- 219 - 226
- 2004. Society of Petroleum Engineers
- 5.5 Reservoir Simulation, 5.6.3 Deterministic Methods, 4.1.2 Separation and Treating, 4.3.4 Scale, 5.1.1 Exploration, Development, Structural Geology, 5.5.3 Scaling Methods, 5.1.5 Geologic Modeling, 5.3.1 Flow in Porous Media
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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 pseudorelative 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.
The effects of reservoir heterogeneity at a scale smaller than a typical simulation gridblock 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 pseudorelative 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 pseudorelative permeabilities) and the inability to provide very high degrees of coarsening (in the case of gridding procedures).
In this work, we investigate 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,1-3 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 pseudorelative permeability), as well as a coarse-scale diffusivity, for each gridblock. This approach has the benefit of appropriately representing both small-scale effects (by the coarse-scale diffusivity) and larger-scale effects (by the modified 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 pseudorelative permeabilities and flow-based grid generation. There have been a number of previous papers addressing the development and evaluation of methods based on pseudorelative 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 pseudorelative 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 by 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.
Other techniques for upscaling are flow-based grid generation procedures, such as nonuniform coarsening,9 and the use of multiscale approaches, such as multiscale finite-element or finite-volume methods.10,11 For transport problems, these methods in essence attempt either to minimize subgrid effects through use of an "optimal" grid (grid generation techniques) or to reconstruct the full fine-grid velocity field using the subgrid representation (multiscale methods). Though both approaches are effective in many cases, they do have some limitations. Specifically, flow-based methods are only capable of achieving moderate levels of coarsening, which may not be sufficient when coarsening very highly detailed models. Methods that require velocity reconstruction, as currently implemented, rely on the use of two grids, which can lead to complications in data structures and the mapping of variables between grids.
Lenormand and coworkers previously introduced coarse-scale models of transport in heterogeneous porous media based on the use of convection-diffusion equations.12,13 These implementations involved a stochastic analytical approach, in which 2D cross-sectional simulation results were represented in terms of a 1D convection-diffusion model. Accurate results were demonstrated for a number of examples. In recent work, we presented an implementation of a generalized convection-diffusion subgrid model for transport in 2D systems.14 We discussed the mathematical foundations of the model and applied it to cases involving time-invariant velocity fields (as occur in constant total mobility displacements). Upscaled models for the pressure equation were not considered. Our approach differs from Lenormand's in that it is a numerical procedure applied to deterministic systems, with the goal of developing an upscaled simulation model of the same dimensionality, but with fewer gridblocks than the original model.
This paper proceeds as follows. We first present the fine-scale equations and then discuss various forms for the coarse-scale model, including the forms for pseudorelative permeability and nonlocal dispersivity models. The nonlocal dispersivity model, as well as other considerations, motivates the form for our generalized convection-diffusion model. We present the general model and describe its numerical implementation. A new form for the coarse-scale pressure equation is then introduced. Next, the overall method is applied to a number of example cases involving different permeability fields and varying boundary conditions. In all cases, improved overall results relative to those of standard procedures are attained using the new subgrid model.
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