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Grid convergence for numerical solutions of stochastic moment equations of groundwater flow

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Abstract

We provide qualitative and quantitative assessment of the results of a grid convergence study in terms of (a) the rate/order of convergence and (b) the grid convergence index, GCI, associated with the numerical solutions of moment equations (MEs) of steady-state groundwater flow. The latter are approximated at second order (in terms of the standard deviation of the natural logarithm, Y, of hydraulic conductivity). We consider (1) the analytical solutions of Riva et al. (Transp Porous Med 45(1):139–193, 2001) for steady-state radial flow in a randomly heterogeneous conductivity field, which we take as references; and (2) the numerical solutions of the MEs satisfied by the (ensemble) mean and (co)variance of hydraulic head and fluxes. Based on 45 numerical grids associated with differing degrees of discretization, we find a supra-linear rate of convergence for the mean and (co)variance of hydraulic head and for the variance of the transverse component of fluxes, the variance of radial fluxes being characterized by a sub-linear convergence rate. Our estimated values of GCI suggest that an accurate computation of mean and (co)variance of head and fluxes requires a space discretization comprising at least 8 grid elements per correlation length of Y, an even finer discretization being required for an accurate representation of the second-order component of mean heads.

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Acknowledgements

This work was partially supported by National Nature Science Foundation of China (Grant No. 41530316; 51709127) and by Natural Science Foundation of Guangdong Province, China (Grant No. 2017A030310172). Part of the work was developed while Prof. A. Guadagnini was at the University of Strasbourg with funding from Région Grand-Est and Strasbourg-Eurométropole through the ‘Chair Gutenberg’. Chuan-An Xia was supported by International Young Researcher Development Project of Guangdong Province, China.

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Correspondence to Alberto Guadagnini or Bill X. Hu.

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Xia, CA., Guadagnini, A., Hu, B.X. et al. Grid convergence for numerical solutions of stochastic moment equations of groundwater flow. Stoch Environ Res Risk Assess 33, 1565–1579 (2019). https://doi.org/10.1007/s00477-019-01719-6

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