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Upscaling unsaturated flows in vertically heterogeneous porous layers

Published online by Cambridge University Press:  19 October 2022

Zhong Zheng Open the ORCID record for Zhong Zheng [Opens in a new window]
Zhong Zheng*
Affiliation:
State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China MOE Key Laboratory of Hydrodynamics, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China
*
†Email addresses for correspondence: [email protected], [email protected]
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Abstract

Characterising interfacial and unsaturated flows in heterogeneous porous layers is of both fundamental and practical interest. Under the assumption of vertical gravitational–capillary equilibrium, we present a theoretical model to describe one-dimensional flows in a porous layer with vertical variations in average pore size, porosity, intrinsic permeability and capillary pressure jump between invading and displaced fluids. The model leads to asymptotic solutions for the saturation distribution and outer envelope of the invading fluid, and for the background pressure drop across the porous layer. Eight dimensionless parameters are recognised after appropriate non-dimensionalisation of the governing equations, the influence of which is demonstrated through a series of example calculations. In particular, four asymptotic regimes are identified, representing unconfined sharp-interface flows, confined sharp-interface flows, unconfined unsaturated flows and confined unsaturated flows. Finally, in the context of flow upscaling, analytical solutions are derived for the effective relative permeability curves on the basis of exact solutions of the saturation field and interface shape, shedding light on the subtle influence of competition between injection/pumping and gravitational forces, wetting and capillary effects, viscosity contrast between the invading and displaced fluids and vertical heterogeneity of the porous layer.

JFM classification

Geophysical and Geological Flows: Gravity currents
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 950 , 10 November 2022 , A17
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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