GEOPHYSICAL RF_•EARCH LETYERS, VOL. 18,NO.5, PAGF__• 877-880, MAY 1991 KOZENY-CARMAN RELATION FOR A MEDIUM WITH TAPERED CRACKS Jack Dvorkin,Haim Gvirtzmanand AmosNur Stanford University Abstract.We examine the permeability of a medium with thin taperedcracks to a single-phase fluid flowin the presence of immobile matter which is accumulated in the tips ofcracks. Theoriginal Kozeny-Carman relation shows an increase in permeability of sucha material relative to thecase when tips are free of accumulated matter. To re- solve this paradox we introduce a corrected version of the Kozeny-Carman relation for the case whenthe shape of a crack cross-section canbe described by a power law. This class of crackshapes includes the important cases of tri- angular cracks andspace between two contacting circular grains. The revised relation includes the original porosity • and specific surface area S of the material without ac- cumulated matter as well as the degree of fillinga crack space by accumulated matter Z. The permeability is pro- portional to goand $-2, and decreases withincreasing Z. introduction Accumulated immobile x mattel e L Figure 1. Thin taperedpore and pore with taperedap- pendix with accumulated immobile matter. The Kozeny-Carman formula has been shown to be a successful instrument relating permeability andother mea- surable properties of a porous material. Typically thisfor- mula can be derivedusing the flow model in which the comphcated flow network throughthe porespace of a ma- terial is replaced by a single representative conduit. Walsh and Brace (!984) presented this relation in the following form: ! •a k = k• SaT 2' (1) where k is permeability; (I)is porosity; S is specific surface area (the surface area per unit volume); T is tortuosity; and k• is the pore shape coefficient. Parameters (I) and S can be introduced into the Kozeny-Carman relation through the concept of the hydraulic radiusthat is defined as the ratio of the volume of a conduit to its wetted area. Tortuosity T can be defined as theactual fluidpath length relative to the apparent path length. The factor k, is a weakfunction of the pore shape of theconduit cross-section. Typically k, = 2 forcircular tubes andks= 3 for flat cracks. Berryman and Blair (1987) showed that equation (!) is expected tobe a reasonable approximation whenthe distribution of pore sizes and shapes isnarrow. In this paper we focus ontheapplicability of equation (1) toa medium with thin tapered cracks with immobi!e matter accumulated in the crack tips. This accumulation Copyright 1991 by the American Geophysical Union. Paper number 91GL01069 0'094-8534 / 91 / 91GL- O10695 3. O0 may occurdue to mineral deposition in poresor precipi- tation of salts. It may also be caused by a trapped im- mobile wetting phase. We assume that a mobile phase flitratesthroughpore space free from accumulated matter and that all free porespace is involved in the flow process. The cross-section of a representative conduit,one typical of the sample of a mediumas a whole,is shown in Figure 1. The tortuosity of the pore space occupied by the mobile fluid may change depending on the amount of accumulated immobile matter. Yet, in the caseunder consideration it is natural to assume that all cross-sections of the repre- sentative conduit are equally reduced by the accumulated matter. This meansthat the process of immobilephase accumulation will not change the actual fluid path length and T can reasonably be considered the samefor different amounts of accumulated immobile matter. Thus, the medium with accumulated immobile matter can be considered, regardingthe filtration of the mobile phase, as a porous material whichdiffers from the original (without accumulated matter) material onlyby the addi- tion of the solid. The tortuosity of sucha material is the same asthat of the original material,but porosity and spe- cific surface area will be changed by the reduction of the conduitopening. The process of accumulation of the immobile matter in tapered tips at its initial stage resultsin small reduc- tion of porosity andsignificant reduction of specific surface area. Thus,if T is constant, formula (1) will givean in- crease in permeability wherea reduction might reasonably be expected. This computed artifact is similarto the effect of surface roughness of a circular conduit. Berryman and 877