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Uncertainty quantification of unsteady source flows in heterogeneous porous media

Published online by Cambridge University Press:  07 May 2019

Gerardo Severino Open the ORCID record for Gerardo Severino [Opens in a new window] ,
Santolo Leveque  and
Gerardo Toraldo
Gerardo Severino
Affiliation:
Department of Agricultural Sciences, University of Naples – Federico II, I-80055 Portici (NA), Italy
Santolo Leveque
Affiliation:
School of Mathematics, University of Edinburgh, Edinburgh EH8 8FQ, UK
Gerardo Toraldo
Affiliation:
Department of Mathematics and Applications, University of Naples – Federico II, Monte S. Angelo I-80126 Naples, Italy
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Abstract

Unsteady flow generated by a point-like source takes place into a $d$-dimensional porous formation where the spatial variability of the hydraulic conductivity $K$ is modelled within a stochastic framework that regards $K$ as a stationary, normally distributed random space function (rsf). As a consequence, the hydraulic head $H$ becomes also stochastic, and we aim at quantifying its uncertainty. Towards this aim, we have derived the head covariance by means of a perturbation expansion which regards the variance $\unicode[STIX]{x1D70E}^{2}$ of the zero mean rsf$\unicode[STIX]{x1D700}=1-K/\langle K\rangle$ (hereafter $\langle \rangle$ being the ensemble average operator) as a small parameter. The analytical results are expressed in terms of multiple quadratures which are markedly reduced after adopting specific autocorrelation $\unicode[STIX]{x1D70C}$ for $\unicode[STIX]{x1D700}$. This enables one to obtain simple results providing straightforward physical insight into the spatial distribution of $H$ as a consequence of the heterogeneity of $K$. In view of those applications (pumping tests) aiming at the identification of the hydraulic properties of geological formations, we have focused on a flow generated by a source of instantaneous and constant strength. The attainment of the large time (steady-state) regime is studied in detail.

JFM classification

Low-Reynolds-number flows: Porous media
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 870 , 10 July 2019 , pp. 5 - 26
Copyright
© 2019 Cambridge University Press 

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Footnotes

†

Present address: Division of Water Resources Management and Biosystems Engineering, via Universitá 100, I80055 Portici (NA), Italy. Email address for correspondence: [email protected]

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