Upscaling unsaturated flows in vertically heterogeneous porous layers

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.

The radial slump of a gravity current in a confined porous layer

“…[3,8,51] and with or without the influence of wetting and capillary forces, e.g. [9,16,45,47]. On the other hand, once hfalse(r,tfalse) is obtained, the pressure distribution p0false(r,tfalse) along the cap rock can actually be calculated by integrating the pressure gradient normal∂p0/normal∂r from the origin towards any arbitrary downstream location r=rm of interest.…”

“…[37][38][39][40][41][42][43][44], and the wetting and capillary forces, e.g. [9,16,[45][46][47], meanwhile, can also play important roles in the miscible region and capillary fringe, respectively, which leads to many interesting and non-trivial facets of gravity current flows. More descriptions of recent advances are summarized in a review article [48], with a focus on the influence of boundaries, such as the influence of drainage, flow confinement, and converging and deformable boundaries.…”

“…[4,10,63,64], the unsaturated flow dynamics when the invading fluid can be trapped due to wetting and capillary forces, e.g. [9,16,45,47,54,[65][66][67], and the dispersion of tracer particles that advect with the flow while redistributing themselves simultaneously due to diffusion, e.g. [68,69].…”

The interaction of gravity currents in a porous layer with a finite edge

“…Huppert & Woods 1995;Nordbotten & Celia 2006;Hesse et al 2007;MacMinn, Szulczewski & Juanes 2010;Pegler, Huppert & Neufeld 2014;Zheng et al 2015;Hinton & Woods 2018;Zheng & Neufeld 2019;Zheng 2023), the influence of wetting and capillary forces as a current becomes unsaturated (e.g. Golding et al 2011;Zheng & Neufeld 2019;Zheng 2022), the influence of radially spreading configuration for fluid injection through vertical wells (e.g. Pattle 1959;Kochina, Mikhailov & Filinov 1983;Lyle et al 2005;Nordbotten & Celia 2006;Guo et al 2016;Hinton 2020;Hutchinson, Gusinow & Worster 2023) and the influence of a lubricating fluid layer that is related to the grounding line dynamics between marine ice sheets and shelves (e.g.…”

“…Golding et al. 2011; Zheng & Neufeld 2019; Zheng 2022), the influence of radially spreading configuration for fluid injection through vertical wells (e.g. Pattle 1959; Kochina, Mikhailov & Filinov 1983; Lyle et al.…”