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Time-Lapse 3D Inversion of Complex Conductivity Data Using an Active Time Constrained (ATC) Approach
Karaoulis, M., A. Revil, D. D. WERKEMA, B. Minsley, W. F. Woodruff, AND A. Kemna. Time-Lapse 3D Inversion of Complex Conductivity Data Using an Active Time Constrained (ATC) Approach. Journal of Geophysics. Springer, New York, NY, 1:1-41, (2012).
Electrical resistivity is sensitive to salinity, porosity, saturation, pore shape, temperature, clay content, and biological activity (e.g., Waxman & Smits, 1968; Revil et al., 1998; Atekwana et al., 2004). Variability in any of these parameters can have an influence on resistivity and can be monitored by time-lapse electrical resistivity tomography (TL-ERT). In the recent literature, TL-ERT has started to be a popular method to monitor dynamic processes occurring in the shallow subsurface (typically the first hundred meters, see Legaz et al., 2009, Müller et al., 2010 and references therein). TL-ERT imaging, often involving permanent electrode installations, has proven to provide information complementary to in situ geochemical measurements. Applications of TL-ERT include monitoring of subsurface flow (e.g., Daily et al., 1992; Ramirez et al., 1993; Park, 1998; Daily & Ramirez, 2000; Nimmer et al., 2007), characterization of solute transport (e.g., Slater et al., 2000; Kemna et al., 2002; Singha & Gorelick, 2005; Looms et al., 2008), saturation and temperature (Legaz et al., 2009), and mapping of salt-water intrusion in aquifers (e.g., Nguyen et al., 2009; Ogilvy et al., 2009) just to cite few applications.
Induced polarization (more precisely the magnitude and the phase of the impedance of the subsurface) is measured using a network of electrodes located at the ground surface or in boreholes. This method yields important information related to the distribution of permeability and contaminants in the shallow subsurface. We propose a new time-lapse 3D modeling and inversion algorithm to image the evolution of complex conductivity over time. We discretize the subsurface using hexahedronal cells. Each cell is assigned a complex resistivity or conductivity value. Using the finite-element approach, we model the in-phase and out-of-phase (quadrature) electrical potentials on the 3D grid, which are then transformed into apparent complex resistivity. Inhomogeneous Dirichlet boundary conditions are used at the boundary of the domain. The calculation of the Jacobian matrix is based on the principles of reciprocity. The goal of time-lapse inversion is to determine the change in the complex resistivity of each cell of the spatial grid as a function of time. Each model along the time axis is called a "reference space model". This approach can be simplified into an inverse problem looking for the optimum of several reference space models using the approximation that the material properties vary linearly in time between two subsequent reference models. Regularizations in both space domain and time domain reduce inversion artifacts and improve the stability of the inversion problem. In addition, the use of the time-lapse equations allows the simultaneous inversion of data obtained at different times in just one inversion step (4D inversion). The advantages of this new inversion algorithm are demonstrated on synthetic time-lapse data resulting from the simulation of a salt tracer test in an heterogeneous random material described by an anisotropic semi-variogram.