Gridding for Petroleum Reservoir Simulation

We consider optimization of computational grids for petroleum reservoir flow simulations. In this context grid quality is determined by two independent error sources. On the one hand there is a loss of precision caused by upscaling of geological data from the fine geological grid to the coarser computational grid, and on the other hand there are numerical errors induced by a non-regular computational grid. In this paper we discuss gridding methods addressing these problems within the restrictions of industry-standard flow simulators. Grid problems in reservoir simulations The ability to predict the performance of a petroleum reservoir is of immense importance for the petroleum industry. For obvious reasons one would like to be able to know as much as possible about production rates and total production resulting from different production strategies. To this end, numerical reservoir simulation has gained wide acceptance as an important decision-making tool. By reservoir simulation we mean the process of inferring the behaviour of a real reservoir from the performance of a mathematical model of that physical system. For our purposes, the model is a set of partial differential equations with an appropriate set of boundary conditions, which describes the significant physical processes taking place in the system. The processes of interest in petroleum reservoirs are basically fluid flow and chemical mass transfer. The model equations must take into account gravitational, capillary, and viscous forces, as well as a reservoir description with respect to permeability heterogeneity and overall geometry. This paper is concerned with the problem of generating good computational grids for reservoir simulations. Here we face the particular problems connected to heterogeneity and upscaling. The problem of upscaling One of the inputs to a full field numerical reservoir performance simulator is a reservoir description. This is a model describing a possible three-dimensional map of the geology of the field. Such geological models are often generated by geostatistical methods. The geology is modelled using stochastic simulation conditioned on well observations and other available data. The reservoir description is usually generated on a fine scale, in part reflecting the scale of the input information such as core data. This is done in the belief that the geological model should capture as much as possible of the heterogeneities for accurate predictions of fluid flow. However, to enable a manageable computation, the reservoir performance simulator has to work on a much coarser grid. As a result, one must bring the fine scale permeability data over to a coarser representation. This involves an upscaling or averaging process. This is very problematic, since permeability is a non-additive property. a Fine grid b Coarse grid Figure 1: Reservoir divided into four disjunct sections by impermeable barriers. It is intuitively clear that in problems involving flow in porous media, averaging could lead to severe errors. An example is depicted in Figure 1: Imagine a reservoir divided into four disjunct regions by impermeable walls (Fig. 1a). A simple averaging process into a coarser grid would lead to a smeared-out picture where the no-flow barriers have become low-permeable layers (Fig. 1b). Thus, in this case, the coarse model is qualitatively different from the fine scale model. The challenge is to upscale with a minimal loss of precision in the predicted reservoir performance. Several ideas have been conceived to solve this problem, ranging from the direct pressure solver method of Warren and Price [1] to renormalization group techniques [2]. Different methods yield different results, and while one method is good for one type of problems another method can be better for another. However, by recognizing that upscaling involves both the choice of a coarse