A Recursive Method for Estimating Single and Multiphase Permeabilities

Abstract

SPE Members Abstract A method is proposed for obtaining effective permeabilities of heterogeneous porous media at many length permeabilities of heterogeneous porous media at many length scales. The real space renormalization group method (RSRG) has been modified to determine the effective properties of both uncorrelated and correlated permeability fields. The model shows excellent sensitivity to directional permeabilities and to the type of heterogeneity. In addition to two component shales and sequences, the method is extended to calculate effective properties of as many as four components. The method is also applied to obtain the 'effective' two phase relative permeability curves for porous media that are heterogeneous at the macroscopic or megascopic length scales. These effective properties are inputs into numerical simulators and can be used to interpret log and core derived properties. Introduction Heterogeneities exist at several length scales in most sedimentary structures. These are evident from data obtained from closely spaced wells, interference tests, radioactive tracer testing, bottom-hole pressure monitoring, extensive coring, specialized logging tools such as Formation Micro Scanner, mini permeameter measurements and CAT scan tomography at the laboratory scale. In line with previous work, we recognize here five distinct scales of heterogeneity:* pore scale (microscopic) * core scale * high resolution log scale (macroscopic or intra-beddingscale) * Conventional log scale (megascopic or bedding scale) * Reservoir scale (gigascopic or geophysical length scale) It is important to recognize that a complete understanding of the behavior of transport properties at any length scale requires some knowledge of the underlying structure of the heterogeneity at a smaller length scale. While the methodology for estimating porosity, water saturations and other properties using wireline methods has been very successful at high resolution and conventional log scales, no reliable measure of permeability is as yet available at these length scales. Our sources of permeability then are: core measurements (core scale), and well testing which provides log or reservoir scale permeability. Fluid flow and hydrocarbon displacement behavior depend to a very large extent on the structure of the heterogeneity at the log and high resolution log scale. Our inability to construct intra-bed and bedding scale permeability maps is a major limitation in the modelling of fluid flow in reservoirs. Reservoir simulators are usually designed to use data at megascopic length scales. These simulators require that each grid block be assigned an effective permeability and relative permeability value as input. From our earlier discussion it is evident that no matter how small the grid blocks are, there is a good chance that heterogeneities exist at a smaller length scale. These heterogeneities need to be properly accounted for in deriving an effective property for each grid block. It is this averaging procedure that this paper primarily focuses on. The procedure that this paper primarily focuses on. The new recursive averaging method discussed here is a modified real space renormalization group (RSRG) method which can be used to scale up transport properties from any given length scale to a larger length scale. As with any such method, some information regarding the structure of the heterogeneity at the smaller length scale must be available. We will limit our discussion to core scale to log scale averaging, but the method can be applied to other length scales as well. BACKGROUND A heterogeneous depositional system most commonly studied is a shale-sand sequence. If the shale streaks are transverse to the direction of bulk flow, they can offer a large resistance to flow due to the added tortuosity. In addition, at a high enough volume fraction, shales may render the sand body almost discontinuous. Warren and Price assumed a random spatial distribution of permeabilities which implied that the variation of permeability was the same in the vertical and the lateral direction. P. 127