HR: 14:50h
AN: H22G-05    [PDF]
TI: Operator Fractional Motion: Permeability Fields with Anisotropic Long Range Dependence and the Effect on Plume Growth
AU: * Benson, D A
EM: dbenson@dri.edu
AF: Desert Research Institute, 2215 Raggio Parkway, Reno, NV 89512 United States
AU: Meerschaert, M M
EM: mcubed@math.unr.edu
AF: Department of Mathematics University of Nevada, Reno, 1664 North Virginia Street, Reno, NV 89512 United States
AU: Baeumer, B
EM: bbaeumer@maths.otago.ac.nz
AF: Department of Mathematics University of Otago, PO Box 56, Dunedin, 0800 80 80 New Zealand
AB: Anisotropy in the dependence structure of hydraulic conductivity {\it K} is typically handled by a dilation of one or more physical coordinates. However, aquifer {\it K} measurements usually have completely different scaling behaviour in the horizontal versus vertical directions. Mandelbrot's original definition of fractional motion is based on fractional-order differentiation or integration of a stable noise, which in a discrete setting can be represented by a field of iid random numbers with any distribution. In one-dimension, fractional integration or differentiation of the noise is handled in Fourier space by division by $p(ik)^A + q(-ik)^A$, where {\it A} = {\it H} + 1/2 is the scaling exponent ({\it H} is the Hurst coefficient) and {\it p} and {\it q} represent the relative weight on forward versus backward integration. The influence of large values may have directional dependence, i.e., the occurrence of {\it K} structures may be causal. A subset of this is isotropic, {\it p} = {\it q} = 1/2, leading to the familiar division by $ |{\it k}|^ A$. For multidimensional self-similarity, the exponent {\it A} is replaced by a matrix {\bf A}. Furthermore, the weighting of the directional integrals is described by a random measure (the mixing measure) on the unit sphere. The result is an Operator Fractional Motion (OFM) that has, what is called in hydrologic applications, "Generalized Scale Invariance." The utility of the OFM is that 1) similarity of extreme {\it K} values may be restricted to a small set of directions, as one might expect in fractured rock or fluvial depositional environments (such as braided streams), and 2) the scaling matrix A has unique eigenvalues that describe the scaling (e.g., the Hurst coefficients) in any direction. The eigenvectors of A need not be orthogonal. For sedimentary deposits, we might expect that the {\it K} field is positively correlated in the horizontal direction, but anticorrelated in the vertical. It has been suggested that a plume's continuously non-Fickian growth rate can be predicted by measuring {\it H} (therefore {\it A}) in the direction of transport. We generate synthetic {\it K} fields with identical statistical properties in the direction of transport, but different values of {\it A} in the transverse direction. The growth of plumes in these fields strongly depends on the scaling properties of the {\it K} field in the transverse direction. Low values of transverse {\it A} lead to plumes that quickly become quite Fickian.
DE: 1832 Groundwater transport
DE: 1869 Stochastic processes
DE: 3250 Fractals and multifractals
SC: Hydrology [H]
MN: 2003 Fall Meeting