Upscaling and Simulation of Waterflooding in Heterogeneous Reservoirs Using Wavelet Transformations: Application to the SPE-10 Model

Abstract

We have developed an accurate and highly efficient method for upscaling and simulation of immiscible displacements in three-dimensional (3D) heterogeneous reservoirs, which is an extension of the technique that we developed previously for 2D systems. The method utilizes wavelet transformations (WTs) to upscale the geological model of a reservoir, based on the spatial distribution of the single-phase permeabilities and the locations of the wells in the reservoir. It generates a non-uniform grid in which the resolved structure of the fine grid around the wells, as well as in the high-permeability sectors, are preserved, but the rest of the grid is upscaled. A robust uplayering procedure is used to reduce the number of the layers, and the WTs are used to upscale each layer areally. To demonstrate the method's accuracy and efficiency, we have applied it to the geological model of a highly heterogeneous reservoir put forward in the tenth Society of Petroleum Engineers comparative solution project (the SPE-10 model), and carried out simulation of waterflooding in the upscaled model. Various upscaling scenarios were examined, and although some of them resulted in efficient simulations and accurate predictions, the results when non-uniform upscaling is used based on the WT technique are in excellent agreement with the solution of the same problem in the fine grid of the SPE-10 model. Most importantly, the speed-up factors that we obtain are several orders of magnitude. Hence, the method renders it unnecessary to use massively parallel computations for such problems.

Upscaling of the permeability by multiscale wavelet transformations and simulation of multiphase flows in heterogeneous porous media

“…The difference between the effective conductivities of the two models, when the frequencies were varied over ten orders of magnitude, was at most 5%. Elsewhere [73], we have demonstrated the same levels of efficiency and accuracy by simulating water flooding in the SPE-10 model [16], in which the permeabilities vary over eight orders of magnitude in a three-dimensional reservoir, exhibiting severe channeling, while the porosities vary over three orders of magnitude, with the fine-scale grid containing over 1.1 million grid blocks. The high accuracy of the results presented in the present paper and those for the SPE-10 model testify to the power of the method described in this paper.…”

“…Therefore, we upscale only the single-phase permeabilities of the grid blocks in the geological model. While, undoubtedly, there may be cases for which some multiphase flow properties might need to be upscaled, for the cases that we study in the present paper, as well as the three-dimensional model reservoir that we have studied elsewhere [73], namely, the SPE-10 model [16], there seem to be no need for upscaling of any multiphase properties.…”

“…We consider 2D models of oil reservoirs (or other types of porous media); its extension to 3D which poses no conceptual difficulty over what we describe here. The method has been applied elsewhere [73] to upscaling of the SPE-10 model, geological model of an oil reservoir in which the single-phase permeabilities vary over eight orders of magnitude. The geological model is represented by a square grid of equal-size blocks, which may be obtained by discretizing the governing equations by the standard five-point FD approximation or from a finite-element discretization of the governing equations [60].…”

Upscaled Unstructured Computational Grids for Efficient Simulation of Flow in Fractured Porous Media

“…We consider 2D models of oil reservoirs; its extension to 3D poses no conceptual difficulty over what we describe here. In fact, the method has already been applied successfully (Rasaei and Sahimi 2008) to upscaling of the SPE-10 model, the GM of a 3D oil reservoir in which the Darcy permeabilities K vary over eight orders of magnitude.…”

“…In this paper a water-flooding process was simulated as an example of a typical two-phase flow in a fractured oil reservoir. To begin with, one combines the Darcy's law and the mass conservation equation, and ignores the effect of gravity in 2D [see, Rasaei and Sahimi (2008), for similar two-phase flow simulations in a 3D model in which the gravity is not neglected]. The result is the following set of equations for flow of oil and water in the reservoir:…”