Inverting Hydraulic Heads In An Alluvial Aquifer Constrained With Electrical Resistivity Tomography Data Through Multiple-Point Statistics And Probability Perturbation Method: A Case Study

Solving spatial inverse problems in the Earth Sciences remains a considerable challenge given the large number of parameters to invert for, the non-linearity of forward models and as a result the ill-posedness of the problem. Geostatistics is therefore needed to specify prior models, more particularly, information to control the spatial features of the inverse solutions.
We used multiple-point statistics (MPS) to build models of pre-defined hydrofacies: clay, sand and gravel facies constrained to geological data (hard data) and geophysical data (soft data). The electrical resistivity tomography method was chosen to bring relevant spatially distributed information on the presence of the facies, given its sensitivity to variations in lithology and porosity. The comparison of the geophysical signature of the deposits with direct observations in boreholes enables to derive the conditional probability of observing a facies given its electrical resistivity. This is used to produce probability maps for each facies and constrain stochastic simulations of the alluvial aquifer.
Then, the probability perturbation method (PPM) is used to integrate hydraulic heads data, using MPS to generate models. This process enables us to obtain calibrated models of the aquifer. The PPM algorithm will automatically seek solutions fitting both hydrogeological data and training-image based geostatistical constraints. Only geometrical features of the model are affected by the perturbation, i.e. we do not attempt to directly find the optimal value of hydrogeological parameters (chosen a priori), but the optimal spatial distribution of facies whose prior distribution is quantified in a training image.
The methodology is first tested with a synthetic benchmark. The tests performed show that the choice of the training image is a major source of uncertainty. Therefore, one first needs to select those training images consistent with the geophysical data (and hence reject the inconsistent ones). Then, we proceed with them to hydrogeological inversions. Geophysical data (soft constraints) acts as an accelerator of convergence by reducing prior uncertainty. The hydraulic conductivity of each facies is a sensitive parameter, but it can be easily optimized prior to the PPM process.
The stochastic method is then successfully applied within the context of an alluvial aquifer submitted to a pumping experiment. We show how the integration of various sources of data (borehole logs, geophysics, hydraulic heads) aids in calibrating hydrogeological models, locating high hydraulic conductivity zones and reducing uncertainty.
The developed methodology proposes a common framework (multiple-point statistics) to integrate various information sources with variable resolutions relevant for hydrogeology: geological, geophysical and hydrogeological data. The method can be extended to integrate tracer tests to enable the calibration of transport parameters as well. The originality of the method is to use geophysical data both to refine the choice of the training image and to constrain the inversion of hydrogeological models.