Recently, a mixed pressure displacement [u, P] formulation based on Biot's poroelasticity equations has been presented for porous materials. This model leads to a reduction of the number of degrees of freedom required for the modeling of three-dimensional porous media in comparison to classical displacement-displacement [u, U] formulations. In this paper, an extension of the [u, P] formulation based on hierarchical elements is presented. First, a variant of the weak integral form of the [u, P] formulation is presented and its numerical implementation using hierarchical elements is detailed, together with the application of boundary and loading conditions. Numerical results are presented to show the accuracy and performance of the present approach. In particular, the importance of correctly capturing the coupling effects between the two phases is highlighted.

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We introduce a new finite-element-based scheme for the fast nonlinear inversion of large 3D geoelectric data sets acquired around isolated objects or across the earth’s surface. The principal novelty of this scheme is the combination of a versatile finite-element approach with (1) a method involving minimization of an objective function using a conjugate-gradient algorithm that includes an adjoint-field technique for efficiently establishing the objective-function gradient and (2) parabolic interpolation for estimating suitable inversion step lengths. This scheme is capable of handling large volumes of data acquired using diverse electrode configurations located around or across 3D structures. Only three solutions to the forward problem are required for each iteration. Computation of the Jacobian matrix, which might require computers with a large amount of memory, is not necessary. To minimize artificial irregularities in the inverted models, particularly near the electrodes, we smooth the model parametersafter each iteration. By including the influence of a reference model in the objective function, a priori information can be incorporated in the inversion process. Our new scheme is tested successfully on synthetic data generated for current and potential electrodes distributed around the surface of a complex object of finite extent. We also demonstrate the utility of the new scheme on geoelectric data acquired around a laboratory-scale object. Tomographic inversion of the 52,272 simulated voltage values in terms of an 8775-element model requires less than 45 minutes on a relatively slow Sun workstation. For the inversion of the 1016 observed voltage values in terms of an 81,480-element model, approximately 60 minutes of computer time is required. The rapid and flexible inversion scheme opens up new possibilities for resistivity imaging in geology, hydrology, engineering, nondestructive testing, and even biology and medicine, fields of study in which finite-element models are already used to represent complicated targets.