Homogenization of Flow Equations Using the MHD Equation: Numerical ValidationzyxwvutsrqponmlkjihgfedcbaZYXWV LENORMAND Roland, Institut Français du Pétrole THIELE Marco R., Stanford University Paper presented at the 5th European Conference on the Mathematics of OH Recovery, Leohen, Austria,3-6 Sept 1996 ABSTRACT Homogenized equations capture flow mechanisms at a sub-gridblock scale and allow for more accurate transport equations to be implemented in numerical simulators .. Homogenized equations can also improve the calculation of pseudofunctions like pseudo relative per~eabilities and alpha factors. In this study we vah~ate a ne~ form of a homogenized equation previously denved by Lenormand using fine-grid numerical simulations. The MHD-equation accounts for permeability heterogeneities through the dispersion coefficient D and the heteT?geneity factor U. The viscous instability is charactenzed by an apparent viscosity ratio M. The advantage of the MHD-equation is its ability to predict th~ efficiency of a displacement with viscous effects by usmg values for the parameters D and H derived from the simpler unit-viscosity-ratio displacement. In this paper, we compare the results predicted by the MHD-equabonzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA to two-dimensional, fine-grid numerical results obtained using a streamtube and a higher-order finite difference simulator. INTRODUCTION Powerful geostatistical algorithms (Deutsch and Joumel 1992) conditioned to log and core measurements mak~ it possible to generate geological images with a pixel resolution for permeability and porosity on the orders of centimeters. Interwell distances and field dimensions, on the other hand, are on the order of hundreds of meters and kilometers respectively. Field-scale numerical modeling of fluid flow using fine-scale information would therefore result on the order of 10 12 grid blocks which is weIl beyond current computational resources. As aresuit, simulations have to be performed on larger grid blocks with average properties derived from the underlying fine-grid information. One approach consists in using upscaling algorithms to homogeneize the permeability heterogeneities, either for one-phase or two-phase flows (King 1989; PickupzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA et al. 1995; Durlovsky 1991; Kossack et al., 1990). On the other hand, improved transport equations have been derived to account for viscous fingering in homogeneous and hererogeneous media. (Koval 1963, Todd and Longstaff 1972, Fayers 1988, Blunt and Christie, 1991; GoreIl 1992, Sorbie et al. 1992, Fayers el al., 199.1,1992). A stochastic approach for denving a h.omo~eDlzed flow equation accounting for both fingering and heterogeneities has recently been presen~ by Lenormand (1995, 1996). In this paper, we vahdate the approach proposed by Lenormand numericaIly by considering first-contact miscible di~placements in 20 heterogeneous fields with 25,000 gndblocks and varying statistical properties. The principal idea underlying the homogenized flow model is to find an equation that can approximate the displacement of one fluid by another in the principal flow direction under the following assumptions: 1) At the small scale, the interface between the fluids is assumed to be sharp, and there is no mixing or capillary spreading. This assumption corresponds to the ideal case of immiscible fluids with negligible capillary pressure and a flux function of the form f=S. 2) The injection is continuous at the inlet (step injection). 3) Gravity effects are negligible. THE "MUD" EOUATION. The result of the stochastic calculation presented by Lenormand (1995, 1996) is a general transport equation with a nonlinear first-order convective term and nonlinear second-order dispersive term. as + u af = D~ (.!.zyxwvutsrqponmlkjihgfedcbaZYXWVU as) at àx àx sax (1) f= S S+(I- S)/ HM' ..................... (2) Here S is the saturation/concentration, f is the flux function, u is the mean front velocity, D is thezyxwvutsrq 475