Stochastic Interpolation of Aquifer Properties Using Fractional Lévy Motion

Abstract

This paper describes a new technique for conditional simulation of aquifer properties in unobserved regions. The technique utilizes a recently introduced model for heterogeneity based on the Lévy‐stable family of probability distributions to achieve a higher degree of realism than is possible with Gaussian‐based techniques. Specifically, permeability and porosity variations are modeled using fractional Lévy motion. A new algorithm for producing fractional Lévy motion conditioned to match known data is introduced. Two‐dimensional simulations using actual permeability measurements as input demonstrate that this algorithm produces random fields with the specified properties. Simulations using a subset of a detailed permeability map of an outcrop as input illustrate how the new method can be used to estimate important flow and transport properties and to quantify uncertainties in these estimates.

A Corrected and Generalized Successive Random Additions Algorithm for Simulating Fractional Levy Motions

“…For H = 1/α, the increments are independent (summing to a so-called Levy flight). For 1/α < H <1 and 0 < H < 1/α (1< α ≤ 2), V(X) has long-range positive and negative dependencies in the increments, respectively (Painter, 1996). Multidimensional fLm can also be easily defined by replacing the independent variable X, used in the above discussion, with relevant location vectors.…”

“…The statistical self-affinity (scaling) of fLm may be expressed by relating the width parameters of the increments, V(X+h)-V(X), for different lags (h) (Painter, 1996):…”

“…The fLm model utilizes the Levy-stable family of probability distributions that have a canonical role in mathematical statistics similar to that of the Gaussian distribution, but have power law tails. These slowly decaying tails lead to diverging second and higher moments, but also make these distributions useful for modeling highly heterogeneous systems (Painter, 1996).…”

“…Consequently, quantification of subsurface heterogeneity is often needed for modeling subsurface contamination and remediation problems. Previous research has shown that spatial variations of K (hydraulic conductivity), or log(K) often display scaling that has been modeled using properties of truncated fractional Levy motion (fLm) (Painter and Paterson, 1994;Painter, 1996;Molz, Liu, and Szulga, 1997;Liu and Molz, 1997). The fLm model utilizes the Levy-stable family of probability distributions that have a canonical role in mathematical statistics similar to that of the Gaussian distribution, but have power law tails.…”

“…As a result, studies of algorithms for generating fLm are very limited in the literature. Painter (1996) published an algorithm based on generalized sequential simulation to simulate truncated fLm. This method was approximate and was not based on a random field model (Painter, 1998).…”