Calculation of large scale relative permeabilities from stochastic properties of the permeability field and fluid properties
R. Lenormand, Marco R. Thiele · OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information) · 1997
The paper describes the method and presents preliminary results for the calculation of homogenized relative permeabilities using stochastic properties of the permeability field. In heterogeneous media, the spreading of an injected fluid is mainly sue to the permeability heterogeneity and viscosity fingering. At large scale, when the heterogeneous medium is replaced by a homogeneous one, we need to introduce a homogenized (or pseudo) relative permeability to obtain the same spreading. Generally, is derived by using fine-grid numerical simulations (Kyte and Berry). However, this operation is time consuming and cannot be performed for all the meshes of the reservoir. We propose an alternate method which uses the information given by the stochastic properties of the field without any numerical simulation. The method is based on recent developments on homogenized transport equations (the {open_quotes}MHD{close_quotes} equation, Lenormand SPE 30797). The MHD equation accounts for the three basic mechanisms of spreading of the injected fluid: (1) Dispersive spreading due to small scale randomness, characterized by a macrodispersion coefficient D. (2) Convective spreading due to large scale heterogeneities (layers) characterized by a heterogeneity factor H. (3) Viscous fingering characterized by an apparent viscosity ration M. In the paper, we first derive the parameters D and H as functions of variance and correlation length of the permeability field. The results are shown to be in good agreement with fine-grid simulations. The are then derived a function of D, H and M. The main result is that this approach lead to a time dependent . Finally, the calculated are compared to the values derived by history matching using fine-grid numerical simulations.