WATER RESOURCES RESEARCH, VOL. 26, NO. 8, PAGES 1843-1846, AUGUST 1990 Comment on "Effective Groundwater Model Parameter Values' Influence of Spatial Variability of Hydraulic Conductivity, Leakance, and Recharge" by J. J. G6mez-Hernfindez and S. M. Gorelick RACHID ABABOU 1 AND ERIC F. WOOD Water ResourcesProgram, Department of Civil Engineering and Operations Research, Princeton University, Princeton, New Jersey 1. INTRODUCTION AND SCOPE In a recent paper, G6mez-Herndndez and Gorelick [1989] studied the effective conductivity and the statistics of hy- draulic heads in a bounded two-dimensional unconfined aquifer with pumping wells, assuming for the most part that hydraulic conductivity is a two-dimensional random field. The objective of this paper is to present a few constructive comments on their statistical analysis of heads and effective conductivity in the presence of pumping wells. Our com- ments are divided into three parts. First, we address a technical point concerning the statistics of lognormal con- ductivity random fields and suggest a closed form expression for the p norm which would have simplified their investiga- tion of effective conductivity. Second, we comment on the behavior of head variance at the wells with respect to pumping rates and indicate how their Monte Carlo results relate to earlier theories. Third, we discuss the effective behavior of nonuniform flow systems and attempt to explain their effective conductivity results in connectionwith earlier theoretical works, plus some modest extensionsof our own. We will not consider in this discussion the effects of random recharge and river leakage, which they analyzed separately. 2. THE p NORM OF LOGNORMAL RANDOM FIELDS In their paper, G6mez-Herndndez and Gorelick [1989] assumed for the most part that hydraulic conductivity K(x) is a two-dimensionalrandom field and that the log-conductivity field In (K(x)) is itself normally distributed and stationary. The two-point covariance of In (K) was an exponentially decaying isotropic function (more on this choice later). The authors used this model to obtain via Monte-Carlo simula- tions the effective conductivity K e that "best fits" the simulated hydraulic heads in some least squaressense. Moreover, they chose to relate the resulting value of K e to the p norm of the random conductivity field, which they defined numerically as Kp = . (Ki) • (1) where p is a real number -1 -< p -< +1. Briefly, their procedure was as follows. They plotted the bias and error variance of the hydraulic heads, where the error is the difference between the heads for randomly generatedrepli- 1Now at Southwest Research Institute, San Antonio, Texas. Copyright 1990 by the American Geophysical Union. Paper number 90WR00656. 0043-1397/90/90WR-00656502. O0 1843 cates of K(x) and the heads obtained for spatially constant K = Kp. For example, in theunconditional case they found that the bias was zero and the error variance was minimal at p = -0.4. Therefore this particular p norm appears to give the effective conductivity value for the flow system at hand [see G6mez-Herndndez and Gorelick, 1989, Figure 18]. However, in order to compute the p norm Kp of lognormal random conductivity as a function of p, the authors resorted to an empirical sampling of a lognormal distribution. We quote directly from G6mez-Herndndez and Gorelick [1989, p. 415]: The values of p were systematically varied from 0.0 to -1.0 in steps of -0.1 to obtain the effective K values for different averagingprocesses.Since there was no analytical expression that provided the p norms of a lognormal distribution, they were obtained by Monte Carlo simulations. A very large sample of values (20,000) taken from the lognormal distribution of hydrau- lic conductivity was generated and several p norms computed. However, we show below that a simple analytical expression does exist for the p norm of a lognormal distribution, so that the sampling procedure described just above is unnecessary. We will seize this opportunity to briefly state some other useful identities concerning the statistics of lognormal ran- dom fields, besides their p norm. Our purpose is to compute in closed form the p norm of the random field K, where K > 0 and Y = In (K) is a normally distributed second-order stationary random field. The p norm of K is defined by Kp = (K p) •/p (2) This gives, in particular, the arithmetic mean K a for p = 1, the geometric mean Kv for p---) 0, and the harmonic mean KH for p = -1. These three means are respectively defined by K A = {K) Kc = exp ((ln (K))) (3) Km = (K-l) -1 Let us now make use of the transformed random field' Y= In (K) where Y is normally distributed with known mean, variance, and covariance functions' Y= 0'¾= {(Y-- Y) (4) yy() = + - -