Determining Interblock Transmissibility in Reservoir Simulators
Authors: Toronyi R.M. , Ali S.M. Farouq 
Published: December 2007

It is customary to employ in reservoir simulators a harmonic average for the interblock transmissibility (related to ki+1/2, for a one-dimensional uniform grid, where k is permeability), This may be in error if the block permeability contrast is large, and the amount of information available is limited. We shall compare the values of ki+1/2 obtained by several different methods for such cases. The question of scale enters into the analysis of an average permeability. We are interested in a scale larger than a pore (see Bear's "representative elementary volume"). In the Russian literature 2 the scale of average permeability is denoted as "integral scale," which is many time larger than the scale at which permeability is measured, denoted as "local scale." Hence, the average permeability between two points is related to the influence of local scale on points is related to the influence of local scale on the integral scale. If the local or point variations of permeability are known with certainty, then, for one dimension, an integral-scale measure of permeability is the harmonic average. Usually, however, the local-scale permeability may be known with certainty (for example, from core data) at very few points - say, at the centers of two adjacent grid blocks. The indeterministic variation of permeability between these two points can be approached by degrees or levels of certainty; i.e., lack of entropy. Complete lack of certainty will be considered as random fluctuations of local scale permeability. Added certainty is attained by considering the local measurements to be composed of both deterministic and indeterministic information (see Agterberg). Both these approaches were employed in this study. According to Law, permeability is a log-normal distributed random variable. Thus, a local-scale permeability measurement mean and variance would give permeability measurement mean and variance would give a measure of the population mean and variance of a log-normally distributed local-scale permeability. (It should be noted that permeability might follow any distribution, as shown by Yawberg, but this will not hinder the foregoing analysis.) Therefore, a simulated point variation of permeability is determined by randomly sampling a log-normal distribution with the assumed population mean and variance. This implies complete lack of certainty. Warren and Price considered permeability to behave in such a manner. Thus, given two local permeability measurements at block centers, and given the local scale (determined by core diameter and the distance between the two points, the number of local-scale permeabilities points, the number of local-scale permeabilities between the two points can be determined for the distribution assumed. This was done in the present study for 2,000 local-scale permeabilities, for various block-center permeabilities (see Table 1), and the integral-scale permeability was computed by obtaining the harmonic average. Table 1 also gives the arithmetic, harmonic, and geometric averages, based upon the two values. Notice that the differences among the averages increase as the block permeability contrast increases. permeability contrast increases. Consider, now, the addition of some certainty a trend - to the random information; i.e., the simulation of the point permeability variation is composed of a random sample from a log normal distribution, with the mean given by the regression curve fit, and with a constant variance given by the regression analysis. The last two columns of Table 1 give results for the integral scale permeability for two such cases. It is seen to remain nearly constant for a changing variance.
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