WATER RESOURCES RESEARCH, VOL. 33, NO. 4, PAGES 907-908, APRIL 1997 Comment on "Evidence for non-Gaussian scaling behavior in heterogeneous sedimentary formations" by Scott Painter Hui Hai Liu and Fred J. Molz Environmental Systems Engineering Department,Clemson University, Anderson, SouthCarolina For some time hydrologists haveaccepted, more or less asa rule of thumb rather than as an absolute fact, that log (K) measurements in manysedimentary formations are distributed lognormally.Now that more detailed K measurements are becomingavailable through the use of borehole flowmeters and other geophysical loggingequipment,it is important to examine the lognormal assumption in greaterdetail.This ques- tion is important also whenselecting a fractal-based procedure for generating K or other property distributions based on a set of relatively small-scale measurements [Hewett, 1986; Molz and Boman, 1993, 1995;Painter and Paterson,1994; Painter, 1996b; Liu and Molz, 1996]. Painter[1996a]is to be commended for his excellent paper wherein he describes evidence for non-Gaussian behavior in hydraulic conductivity and porosity data. As an alternative to the Gaussian model he proposes the L6vy-stable probability distribution, a distribution with tailsthat decrease according to P(Ixl >- u) • u-", which is much more slowly than the exponentialdecrease associated with the Gaussiandistribu- tion. This causes the L6vy-stable distribution to have second and higher moments that are infinite [Samorodnitsky and Taqqu, 1994]. As shown in (1), i f0 © (ax) da -- exp(-Iekl cos f(x) = rr the L6vy-stable process is characterized by two parameters, a so-called L6vy index, a, and a width parameter,C. The distri- bution converges to a Gaussian distribution for a = 2 [Man- tegna, 1994]. The purposeof this commentis not to disagree with the main conclusions of Painter[1996a]concerning non-Gaussian scaling.Rather, we would like to present some interesting results from a brief analysis of the macrodispersion experiment (MADE) dataset,a collection of several thousand K measure- ments made using a borehole flowmeter at ColumbusAir Force Base, Mississippi [Boggs et al., 1992, 1993].This data set is more extensive than that examined by Painter[1996a]. Two methods were used to estimate a values for probability density functions of increments of In (K). The first is a curve- fitting method basedon (1), wherein a and C valueswere estimated by numerically minimizing the following mean square deviation: = • [f(xj) -- f*(xj)] 2 j=l (2) where f* (x i) is the observed histogram value when an incre- mentof In (K) is equal to xi. The second is the method of Copyright 1997by the AmericanGeophysical Union. Paper number96WR03788. 0043-1397/97/96WR-03788509.00 Fama and Roll [1972],used by Painter[1996a]. Shown in Fig- ures1 and 2 are comparisons between the observed histograms and L6vy-stable distributions with parametervalues estimated from the firstmethod. Figure 1 is for a lag, Az, of 0.15 m, which is the same as the measurement interval at the MADE site. Figure 2 corresponds to a Az of 2.4 m. Apparently, the calcu- lated L6vy-stable distributions matchthe observed histograms very well. However, we found that a values, estimatedwith both methods, are not constant but are lag-dependent (Figure 3). The a value consistently increases with lag and converges approximately to 2, corresponding to the Gaussian distribu- tion, when the lag getslarge. The scale-variant behavior of a may be a very important phenomenon in heterogeneous sedimentary formations, al- thoughfurther research is needed to confirm this finding. It implies that a distribution of In (K) at a smaller scale is more heterogeneous relativeto that at a larger scale. Physically, this makes sense, because an abruptchange in a subsurface prop- erty often occurs over a smallvertical interval. The observed behavior also seems to indicate that the Gaussian distribution may be applicable abovea certainscale. Mechanisms behind this scale-variant behavior of a are not very clear at this stage. However, the followinganalysis of an idealized case, in which increments of In (K) are assumed to be independent for smalllags, may shedsome light on the mech- anisms. An increment of W -- In (K) at a relatively large lag nAz, where n is an integerlarger than 1, can be expressed as n •)n • W(Z1 q- I'IAZ) -- W(z1)= Z [W(zi) - W(zi-1)] i=2 (3) 10 0 0. t 1 10-2 o oo 10-• -10 ß -5 0 5 10 increment of In(K) Figure 1. Comparison betweenthe histogram of increments of In (K) for lag Az = 0.15 m and a L6vy-stable distribution with a = 0.8 and C = 0.36, estimated with the curve-fitting method. 907