Geologic expertise and petrophysical relationships can be brought together to provide prior information while inverting multiple geophysical data sets. The merging of such information can result in more realistic solution in the distribution of the model parameters. We have evaluated the geophysical inverse problem in terms of Gaussian random fields with mean functions controlled by petrophysical relationships and covariance functions controlled by a prior geologic cross section, including the definition of spatial boundaries for the geologic facies. The petrophysical relationship problem is formulated as a regression problem upon each facies. The inversion of the geophysical data is performed in a Bayesian framework. We have developed the usefulness of this strategy using a first synthetic case for which we have performed a joint inversion of gravity and galvanometric resistivity data with the stations located at the ground surface. The joint inversion was used to recover the density and resistivity distributions of the subsurface. In a second step, we considered the possibility that the facies boundaries were deformable and their shapes were inverted as well. We used the level-set approach to perform such deformation preserving a priori topological properties of the facies throughout the inversion. With the help of a priori facies petrophysical relationships and the topological characteristics of each facies, we made a posteriori inference about multiple geophysical tomograms based on their corresponding geophysical data misfits. We have applied this method to a second synthetic case showing that we can recover the heterogeneities inside the facies, the mean values for the petrophysical properties, and, to some extent, the facies boundaries using the 2D joint inversion of gravity and galvanometric resistivity data.

Time-lapse joint inversion of geophysical data is required to image the evolution of oil reservoirs during production and enhanced oil recovery, [Formula: see text] sequestration, geothermal fields during production, and to monitor the evolution of contaminant plumes. Joint inversion schemes reduce space-related artifacts in filtering out noise that is spatially uncorrelated, and time-lapse inversion algorithms reduce time-related artifacts in filtering out noise that is uncorrelated over time. There are several approaches that are possible to perform the joint inverse problem. In this work, we investigate the structural crossgradient (SCG) joint inversion approach and the crosspetrophysical (CP) approach, which are appropriate for time-lapse problems. In the first case, the inversion scheme looks for models with structural similarities. In the second case, we use a direct relationship between the geophysical parameters. Time-lapse inversion is performed with an actively time-constrained (ATC) approach. In this approach, the subsurface is defined as a space-time model. All the snapshots are inverted together assuming a regularization of the sequence of snapshots over time. First, we showed the advantage of combining the SCG or CP inversion approaches and the ATC inversion by using a synthetic problem corresponding to crosshole seismic and DC-resistivity data and piecewise constant resistivity and seismic velocity distributions. We also showed that the combined SCG/ATC approach reduces the presence of artifacts with respect to individual inversion of the resistivity and seismic data sets, as well as with respect to the joint inversion of both data sets at each time step. We also performed a synthetic study using a secondary oil recovery problem. The combined CP/ATC approach was successful in retrieving the position of the oil/water encroachment front.

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