Modeling of subgrid effects in coarse‐scale simulations of transport in heterogeneous porous media

Abstract

Abstract. A methodology for incorporating subgrid effects in coarse-scale numerical models of flow in heterogeneous porous media is presented. The method proceeds by upscaling the deterministic fine-grid permeability description and then solving the pressure equation over the coarse grid to obtain coarse-scale velocities. The coarse-grid saturation equation is formed through a volume average of the fine-scale equations and includes terms involving both the average and fluctuating components of the velocity field. The terms involving the fluctuating components are subgrid effects that appear as length -and time-dependent dispersivities. A simplified model for the coarse-scale dispersivity, in terms of these subgrid velocity fluctuations, is proposed, and a numerical scheme based on this model is implemented. Results using the new method are presented for a variety of twodimensional heterogeneous systems characterized by moderate permeability correlation length in the dominant (horizontal) flow direction and small correlation length in the vertical direction. The new method is shown to provide much more accurate results than comparable coarse-grid models that do not contain the subgrid treatment. Extensions of the overall methodology to handle more general systems are discussed. Using the second, Monte Carlo approach, multiple realizations of the heterogeneous formation are generated, and each is then simulated deterministically [Hewett and Behrens, 1990]. In this approach, there is no uncertainty associated with the formation properties or the flow results for any single realization (other than errors due to numerical discretization). The probability distribution associated with the flow problem is determined through the simulation of multiple realizations of the formation. The complication in this case is that each realization must be highly detailed in order to capture all of the relevant scales of heterogeneity. This renders the simulation of even a single realization demanding. The handling of multiple realizations represents an even greater challenge.Upscaling procedures are commonly used to coarsen these highly detailed geostatistical realizations to scales more suitable for flow simulation. A variety of such approaches exists, and there are advantages and disadvantages associated with the different methodologies (for recent reviews, see Christie [1996] and Wen and Gomez-Hernandez [1996]). In general, there is somewhat of a trade-off between robustness and computational efficiency. Robustness in this context refers to process independence, meaning the same coarse-scale description is applicable for different flow processes; for example, different boundary and initial conditions. Less robust approaches typically achieve higher degrees of coarsening, but their use for problems which significantly deviate from the base case (used 2031

Virtual Telemetry for Dynamic Data-Driven Application Simulations

“…Our previous approaches on this issue were mainly in two directions. First approach that is based on perturbation techniques models subgrid effects as a nonlocal macrodispersion term [5]. This approach takes into account the long range interaction in the form of diffusion term that grows in time.…”

Upscaled modeling in multiphase flow applications

“…Carrying out the calculations in an analogous manner to the ones performed in [17], we can easily obtain the following coarse scale saturation equation: 15) where D(x, t) is the macro-diffusive tensor, whose entries are written as…”

“…Note that the diffusion coefficient is a correlation between the velocity perturbation and the displacement. This is different from [17], where the diffusion is taken to be proportional to the length of the coarse scale trajectory. Using our upscaling methodology for the pressure equation, we can recover the small scale features of the velocity field that allows us to compute the fine scale displacement.…”

“…Previous approaches for the coarse scale modeling of transport in heterogeneous oil reservoirs include the use of pseudo relative permeabilities [3,7,31,32], the application of nonuniform or flow-based coarse grids [12], and the use of volume averaging and higher moments [10,16,17]. Our methodology for subgrid upscaling of the hyperbolic (or convection dominated) equations uses volume averaging techniques.…”

“…In addition to the above described coarse model, we will also revisit a coarse model on the saturation proposed by [17], which uses λ(S) = 1 and f (S) = S. This model was derived using perturbation argument for (2.3), in which the saturation, S, and the velocity, v, on the fine scale are assumed to be the sum of their volume-averaged and fluctuating components,…”