Wavelet Reconstruction of Geologic Facies From Nonlinear Dynamic Flow Measurements

The discrete wavelet transform (DWT) that is widely used in compressing natural images is considered for an effective representation of the geological facies in subsurface flow and transport inverse modeling problems. The inference of the heterogeneous hydraulic rock properties from the scattered dynamic measurements of the flow rates and pressures is a frequently encountered ill-posed inverse problem in subsurface characterization. To better pose this inverse problem, the original grid-based description of the spatial facies maps is replaced with a small number of DWT coefficients that are estimated from indirect nonlinear dynamic measurements. The compressed description of the facies in the wavelet domain after removing the unresolvable high-frequency components leads to an inverse problem with fewer parameters to resolve and improved geologic facies continuity. The main difficulty in the application of the DWT to inverse problems is the lack of sufficient data to resolve higher frequency detail coefficients. Prior information and sensitivity of the flow response to variation in the DWT coefficients are used to infer the location and value of the significant DWT coefficients. The results suggest that the large-scale geologic facies description that control the global flow pattern can be successfully inferred from the dynamic measurements in a reduced wavelet domain. While the flow data may contain information about significant DWT coefficients, a limited observability in ill-posed inverse problems may not allow the identification of these coefficients and the corresponding local spatial features. Therefore, an effective exploitation of the space-frequency localization advantage of the wavelets over the Fourier bases may not be available in solving ill-posed inverse problems.