Wavelet Analysis and Filtering to Identify Dominant Orientations of Permeability Anisotropy
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Wavelet Analysis and Filtering to Identify Dominant Orientations of Permeability Anisotropy Loring Watkins · Roseanna M. Neupauer · Gilbert P. Compo
Received: 7 February 2008 / Accepted: 18 May 2009 / Published online: 17 June 2009 © International Association for Mathematical Geosciences 2009
Abstract An accurate representation of permeability anisotropy is needed to model the rate and direction of groundwater flow correctly. We develop a wavelet analysis technique that can be used to characterize principal directions of anisotropy in both stationary and non-stationary permeability fields. Wavelet analysis involves the integral transform of a field using a wavelet as a kernel. The wavelet is shifted, scaled, and rotated to analyze different locations, sizes, and orientations of the field. The wavelet variance is used to identify scales and orientations that dominate the field. If the field is non-stationary, such that different zones of the field are characterized by different dominant scales or orientations, the wavelet variance can identify all dominant scales and orientations if they are distinct. If the dominant scales and orientations of different zones are similar, the wavelet variance identifies only the dominant scale and orientation of the primary zone. In this paper, we present a combined wavelet analysis and filtering approach to identify all dominant scales and orientations in a non-stationary permeability field. We apply the method to permeability data obtained in the laboratory from the Massillon sandstone.
L. Watkins University of Colorado at Boulder, Boulder, CO, USA Present address: L. Watkins Schnabel Engineering, Greensboro, NC, USA R.M. Neupauer () Department of Civil, Environmental, and Architectural Engineering, University of Colorado at Boulder, Boulder, CO 80309, USA e-mail: [email protected] G.P. Compo CIRES/Climate Diagnostics Center, and NOAA Earth System Research Laboratory/Physical Sciences Division, University of Colorado at Boulder, Boulder, CO, USA
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Math Geosci (2009) 41: 643–659
Keywords Heterogeneity · Non-stationary · Morlet wavelet · Permeability · Anisotropy · Wavelet analysis
1 Introduction Properties of porous media, such as permeability, are spatially variable and often anisotropic. In a layered porous medium, permeability varies with direction, with the largest and smallest permeabilities in directions parallel and perpendicular to the layering, respectively. These directions are the principal directions of anisotropy. Fluid flow in porous media is described by Darcy’s law, given by ρg kxx kxy qx ∂h/∂x =− , (1) ∂h/∂y qy μ kyx kyy where qx and qy are components of specific discharge in the x and y directions, respectively, ρ is fluid density, g is the gravitational constant, μ is fluid viscosity, kij is the (i,j )th entry of the permeability tensor, and ∂h/∂x and ∂h/∂y are the hydraulic gradients in the x and y directions, respectively. The components of the permeability tensor are given by (Bear 1972) k + k ⊥ k − k⊥ + cos(2θ ), 2 2 k + k ⊥ k − k⊥
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