An Implicit Treatment of Upscaling in Numerical Reservoir Simulation

Abstract

Summary This paper presents a multiscale computational model for multiphase flow that implicitly treats upscaling without using pseudofunctions. The model overcomes some practical difficulties related to use of traditional pseudocurves by executing the upscaling and solution processes in one step and taking into account changes in the numerical model in an adaptive manner. Introduction Numerical simulation of petroleum reservoirs is associated with intensive use of computational resources. Advances in petroleum reservoir descriptions have provided an amount of data that cannot be used directly in flow simulations. Geostatistical techniques are able to generate descriptions of heterogeneous reservoirs with great detail in a very fine scale. This detailed geological information must be incorporated into a coarser model during multiphase-fluid-flow simulation by use of some upscaling technique. In numerical simulation processes, equations are discretized and the flow domain is divided into blocks with associated rock properties. This step requires that the geological description be transferred from the flow-properties model to the reservoir simulation. Because of computational limitations, it is not possible to run multiphase-flow simulations at such a scale; therefore, properties must be scaled up and the problem must be solved on a coarser grid. In single-phase flow, the most important parameter to scale up is absolute permeability, and methods for this are well established. When multiphase flow occurs, however, it is also necessary to adjust the phase flow through the connections of the coarse grid. In such cases, the most widely used upscaling technique uses pseudorelative permeabilities. The Kyte and Berry1 method is the most common approach applied to calculate pseudocurves. Their procedure requires two steps:generation of pseudocurves for each block of the coarser grid andsimulation of the model considering such functions. In addition to these generation steps, limitations associated to these pseudocurves restrict their use in a more general way. The procedure we propose uses parameters generated from numerical flow simulation in some regions of the domain to create an equivalence between the description and the simulation scales. By solving a sequence of local problems on the more refined scale, it is possible to achieve good agreement between a coarse and a fine grid without expensive computations on a fine-grid model of the whole reservoir. This procedure does not use multiphase pseudofunction concepts and avoids the computational cost of solving the fine grid. The examples presented here consider 2D, two-phase flow (oil and water) and a black-oil formulation. Results of flow simulation considering homogeneous and heterogeneous porous media are presented. These results are also used to compare this approach with commercial upscaling software. Upscaling Techniques With Pseudofunctions Use of pseudofunctions is the traditional way to perform upscaling. This consists of replacing original saturation-dependent functions on a certain scale by fictitious ones that represent the same physical process in a coarser solution mesh. Simplified numerical and analytical models can be used to construct pseudofunctions. Analytical methods are suitable when simplified assumptions are valid. Coats et al.2 derived pseudofunctions for vertical-equilibrium conditions based on gravity/capillary equilibrium. Hearn3 extended this procedure to noncommunicating layers. Jacks et al.4 introduced space- and time-dependent functions to overcome the rate limitations of the vertical pseudos. To obtain these dynamic functions for each coarse block, it is necessary to run numerical models in a section of the reservoir. Jacks et al. proposed a method based on simulation of 2D cross sections that generates a set of pseudorelative permeability curves representing each column and runs the final model in a 2D areal model. Emanuel and Cook5 extended the pseudorelative-permeability-function concepts to fit vertical performance of individual wells. Using a similar technique, they proposed well pseudos for well completions in the coarse grid. With examples of different completion schemes, they showed that the proposed procedure works for almost all cases presented. Kyte and Berry1 proposed the most common method to calculate dynamic pseudocurves. They developed a method based on Darcy's law to calculate pseudofunctions that is considered to be an extension of Jacks et al.'s4 work and includes pseudocapillary pressure curves. Despite the fact that their method is popular and used as a reference, it does not give good results in strongly heterogeneous media and some inconsistencies, such as negative or infinite values of relative permeability, can occur. On the basis of the Kyte and Berry approach, Lasseter et al.6 presented a multiscale upscaling method suitable for heterogeneous reservoirs. Using some particular reservoir permeability distributions, they showed how reservoir heterogeneities at small, medium, and large scales influence ultimate recovery and how they affect the multiphase behavior. Lasseter et al.'s proposed pseudofunction-generation process begins at the laboratory scale, and the next largest scale can be achieved by replacing effective properties determined at the previous scale. Starley7 presented a procedure based on material balance to derive pseudorelative permeability curves with application to 2D problems. The method is similar to that of Jacks et al.4 and focuses on matching fluid fluxes between interfaces of a reference fine-grid model and a coarse-grid areal model. Their method includes a dispersion-control scheme to offset numerical dispersion and works only for the specific displacement process for which it was derived. Kossack et al.8 described a multistep scaleup process to consider several scales of heterogeneities in two-phase, incompressible displacements. Three geological descriptions of heterogeneities (homogeneous, layered, and random) and various groups of fluid-flow regimes were tested. They ran extensive numerical experiments to verify the effects of different flow regimes on the pseudofunction curves in the three geological descriptions. As in Lasseter et al.,6 Kossack et al.'s pseudofunctions did not consider gridblocks adjacent to wells and they used the Kyte and Berry1 method in their derivation. Stone9 was the first to use the average total mobility to avoid calculating phase potential on the coarser grid (as required by the Kyte and Berry method). He introduced a fractional-flow formula instead of calculating the flow terms by Darcy's law. His method can be applied even to noncommunicating layers. Hewett and Berhens10 and Beier11 proposed other methods based on averaged total mobility. Alabert and Corre12 presented an approach, covering three-phase flow, to deal with 3D models of geological heterogeneity generated by geostatistical conditional simulation techniques.