The impact of space-time adaptation technique on solute transport modeling in porous media

A wide set of physical processes involves transport of solutes in porous media. These include contamination of groundwater by inorganic and organic chemicals, petroleum generation and migration, reactive processes which can modify the properties of soil and rock formation. Transport of solute mass in the subsurface is due to advection and diffusion processes, taking place at the pore level. Due to the practical infeasibility to model pore-scale transport at typical laboratory and field scales, solute transport in porous media is typically described by means of effective models. A standard choice is to adopt a continuum-based representation of the process based on the standard advection-dispersion equation (ADE). The basic assumption underlying the ADE is that the dispersion coefficient can be described by the sum of effective diffusion and hydrodynamic dispersion,according to the so-called Fickian analogy. The advective term results from a velocity field, which is typically assumed to obey Darcy’s law. In practical applications, numerical approximation methodologies are demanded to compute the evolution of the concentration in the space-time domain of interest. Independently of the employed methodology, the results of numerical simulations are unavoidably subject to an approximation error, which is related to the selected discretization scheme. For instance, the computational error associated with Eulerian discretization methods (e.g.,finite elements, finite volumes and finite differences) is a function of the spatial grid and of the time step size used for the discretization. To this end, the first aim of this thesis is the development of a two dimensional finite element model combined with a space-time grid adaptation procedure to improve the accuracy of solute transport modeling in porous media. We employ the ADE for the interpretation of non-reactive transport experiments in laboratory-scale porous media. When compared with a numerical approximation based on a fixed space-time discretization, the proposed approach is grounded on a joint automatic selection of the spatial grid and at the time step to capture the main space and time system dynamics. The space adaptive process is driven by a suitable anisotropic recoverybased error estimator which enables us to properly select the size shape and orientation of the mesh elements. The adaptation of the time step is performed through an ad-hoc local reconstruction of the time derivative of the solution in the spirit of a recovery procedureas well. Macro-scale models, including the ADE, entail the definition of effective transport parameters, which are typically assumed to be linked to the porous media geometry. In laboratory and field scale applications these parameters are generally unknown, and need to be estimated by means of inverse modeling procedures. To this aim, multiple evaluations of the selected model are typically needed. This can be computationally costly.Therefore, as second objective of this work, we quantify the impact of the implementation of the space-time adaptive procedure on parameter estimation and uncertainty quantification. The model calibration is performed in a Maximum Likelihood (ML) framework upon relying on the representation of the ADE solution through a generalized Polynomial Chaos Expansion (gPCE). The whole proposed methodology is assessed through two-dimensional numerical tests. First a numerical convergence analysis of the spatial mesh adaptivity is performed by considering a test-case with analytical solution. Then, we validate the space-time adaptive procedure by reproducing a set of experimental observations associated with solute transport in a homogeneous and blockwise heterogeneous sand pack. The impact of the space-time adaptation methodology on the capability to estimate the key parameters of an ADE model is also assessed on the basis of experimental solute breakthrough data measurements for both homogeneous and blockwise heterogeneous sand packs. These assessments show that the space-time adaptation methodology is robust and reliable. In particular, it allows us to obtain a significant improvement of the simulation quality of the early solute arrival times at the outlet of the medium. Moreover, the proposed adaptation methodology leads to ML parameter estimates and model results of markedly improved quality when compared to classical inversion approaches based on a uniform space-time discretization. Our results suggest that the implementation of a space-time adaptive methodology has a considerable impact on global analysis and uncertainty quantification procedures. In particular, it allows understanding the influence of input uncertain parameter on the model output. Further extensions of this work may involve solute transport modeling in complex heterogeneous fields as well as field-scale data interpretation. Investigation of the impact of space-time adaptation in the presence of chemical reactions is another possible and challenging application of interest.