Reduced models for linear groundwater flow models using empirical orthogonal functions

In this paper we describe two reduced models that describe the hydraulic head h within three-dimensional groundwater flow models. We defined a reduced model structure as a linear combination of a set of spatial patterns P with time-varying coefficients r. The patterns were obtained by a data-driven indentification technique (Empirical Orthogonal Functions, EOFs), and they span a subspace of model results that captures most of the relevant information of the original model. Due to those patterns, we constructed two different formulations for dr/dt, by applying different projections: (1) a State-Space Projection (SSP) that projects a state-space formulation for groundwater flow; and (2) a Galerkin Projection (GP) that substitutes h within the PDE for groundwater flow by the reduced model structure PTr, and projects the outcome onto the patterns. The SSP and GP have been both applied to a realistic case study with a negligible loss of model accuracy (Relative Mean Absolute Error < 0.5%). The dimension of r (16) was significantly reduced compared to the dimension of h (32,949) and hence we achieved a maximal reduction in computational time for the SSP ≈ 1/625 and for the GP ≈ 1/70 of the original time. Both reduced models have a promising prospect as their time reduction increases whenever the number of grid cells increases and the parameterization of the original model grows in complexity.