GEOPHYSICAL RESEARCH LETTERS, VOL. 22,NO. 12, PAGES 1529-1532, JUNE15, 1995 Pore-scale heterogeneity, energy dissipation and the transportproperties of rocks Yves Bemab6 and Andr6 Revil Ecole et Observatoire de Physique du Globe, Universit6 LouisPasteur, Strasbourg, France Abstract. The pore structure of rocksis highly complex, with wide variations in poresize andshape. In this work, pore-scale heterogeneity was simulated by distributing spheres, tubesand cracks with variable dimensions on a square lattice. The transport properties of 100 such networkrealizations, covering 11 orders of magnitude in permeability, werecalculated.Seeking the appropriate averaging procedure to calculate the permeability and electrical conductivityfrom the local pore parameters, we computed theenergy locallydissipated in each bond during fluid or electric flow and the energyglobally dissipated in the whole network. By equatingthe latter to the sum of the former, we obtained averaging expressions exactlypredicting the transport properties of the networkrealizations. Since these relations hold on a wide variety of heterogeneous networkscoveringa broad range of permeabilities andelectrical conductivities, we propose that they should also be valid on rocks. We can thus gain insights into how pore-scale heterogeneity affects the transport properties of rocks. Introduction Although the pores in rocks aretremendously variable in size and shape, the rock transport properties appear to depend on a very small number of parameters (Scheidegger[1974]). In particular, permeability seems to be controlled by a single length scale (in addition to a few dimensionless parameters such as porosity and tortuosity' Gudguen and Palciauskas [1992]). It is thusreasonable to assume that these parameters can be derived from the local pore geometry through some unspecified averaging process. The search for the appropriate averaging procedure hasyielded rather disappointing results sofar, leading some people to explore entirely different approaches such as percolation theory (e.g. KatzandThompson [ 1986]). Among the averaging models,one of the simplest and mostpopular is the Kozeny-Carman (KC) model (e.g.Walsh and Brace [1984]) which considers the hydraulic radius (i.e. ratio of the average pore volume to the average fluid/solid surface area) as the appropriatelength scale. The KC model uses arithmetic averaging with equal weights given to all pores regardless of how well they conduct, a procedure which should not be adequate in the presence of significant pore-scale heterogeneity. Recently, Johnson and Sen (1988) proposed an improved averaging schemegiving reduced weights to poorly conducting pores, based on a theory of electrical conduction in saturated rocks. In the present study, following the work of Schwartz,Sen and Johnson (SSJ), we derive a new averagingrule by comparing Copyright1995 by the AmericanGeophysical Union. Paper number95GL01418 0094-8534/95/95GL-01418503.00 global and local evaluations of the energydissipated in the porous material during hydraulic or electrical flow (note thatour averaging procedure coincides with the SSJone in the case of electrical conduction). Our approach is to treatpore-scale heterogeneity explicitly by studying the transport properties of networks of tubes, cracks and spheres with randomdimensions (see references in Bernabd [ 1995] for applications of networks to a broadrangeof physical properties). The energy principle used to develop our averaging procedure is easily written on suchnetworksand leads to exact predictions of the transport properties on the entire domain covered (i.e. 11 orders of magnitudein permeability). Our contention is that the proposed averaging procedure is alsovalid in real porous media, at least those characterized by a single pore spacing scale (i.e. the network cell size). The Numerical Procedures This study continues the work alreadyrelatedin Bernabd (1995). As explainedin that article, 100 network realizations were constructed by filling the nodes of a 20X20 two- dimensional square latticewith spheres andthebonds with tubes and/or cracks, according to three independent, variable probabilities. For each filled node and each filled bond (indicated by the index i), the corresponding sphere radiusrs(i), tube radius r t(i), crack aperture ac(i), andcrack lateral dimension dc(i) (see Figure 1), were then randomlychosen. For each networkrealization, the hydraulicand electrical flow equations (i.e. Poiseuille's or Ohm'slaw and Kirchoff'scontinuity relation) were solved with the following boundary cor/ditions: constant pressure or voltage imposed along therightand left edges and no flow across theupper andlower boundaries. Knowing the total flow through the network,the permeability k andthe electrical formation factor F (i.e. the ratio •f/o ofthe fluid bulk conducti- vity to the rock conductivity) were calculated. The results of these calculations were reported and discussed in Bernabd (1995) and are again used here. Moreover, we treated a moregeneral electrical conduction problem including surface conduction. We describe thisnew work in detailin the next paragraphs because it provides a remarkable illustration of theimportance of pore-scale heterogeneity. Surface conduction was introduced by simply assuming that the tubes and/or cracks are coated with a conducting layer of infinitesimal thickness andconstant conductivity Os (note that Os has the dimension of a conductance). The surface and bulk conduction paths arelocallyparallel andOhm's law caneasily be written for each bond, the conductances being nrt(i)(ofrt(i)+2Os)/L(i) for atube and dc(i)(ofac(i)+2Os)/L(i) for a crack (L(i) is the length of the tube or crack). For a bond occupied by bothelements, the tubeandthe crackare connected in parallel and their conductances must be added. As in Bernabd (1995), the resistance of the nodal spheres was assumed to be 1529