Buoyant flow of through and around a semi-permeable layer of finite extent

Abstract

The buoyancy- and capillary-driven counter-current flow of $\text{CO}_{2}$ and brine through and around a semi-permeable layer is studied both numerically and theoretically. The continuities of the capillary pressure and the total flux at the interface between the permeable matrix and layer control the $\text{CO}_{2}$ saturation discontinuity at the interface and the balance between the buoyant and capillary diffusion fluxes on each side of the interface. This interface process is first studied in a one-dimensional (1-D) vertical column geometry using the concept of extended capillary pressure and a graphical representation of the continuity conditions in the ($S_{L}$, $S_{U}$) plane, where $S_{L}$ and $S_{U}$ are the lower and upper saturation traces at the interface, respectively. In two dimensions, we heuristically extend the two-phase gravity current model to the case where the current is bounded by a semi-permeable layer. Consequently, the current is not saturated with $\text{CO}_{2}$, and its saturation and shape are derived from the flux and capillary pressure continuity conditions at the interface. This simplified model, which depends on $\text{CO}_{2}$ saturation only, is compared to fine grid simulations in the capillary-free and gravity-dominant cases. A good agreement is obtained in the second case; the current geometrical characteristics are accurately described. In the capillary-free case, we demonstrate that the local total velocity, which is, on average, zero because the flow is counter-current, must be considered in the total flux at the interface to obtain the same level of agreement.

A Gravity Current Model with Capillary Trapping for Oil Migration in Multilayer Geological Basins

“…Leakage across the whole surface separating the geological layers is represented by simple linear contributions in [33,52]. Due to their simplicity, such models cannot reproduce oil-trapping, which requires the introduction of a nonlinear contribution, as for instance proposed by Woods & Farcas [57] or Ngo et al [44]. These two models [57,44] are closely related to ours, with differences highlighted in the forthcoming Remark 1.1.…”

“…Due to their simplicity, such models cannot reproduce oil-trapping, which requires the introduction of a nonlinear contribution, as for instance proposed by Woods & Farcas [57] or Ngo et al [44]. These two models [57,44] are closely related to ours, with differences highlighted in the forthcoming Remark 1.1. In contributions by Hesse et al [32], Juanes et al [36], Zhao et al [58], related to carbon dioxyde sequestration, capillary trapping is defined as a volume contribution that models the immobile CO 2 entrapped in pores since not connected to the main (and mobile) carbon dioxyde components.…”

“…In contributions by Hesse et al [32], Juanes et al [36], Zhao et al [58], related to carbon dioxyde sequestration, capillary trapping is defined as a volume contribution that models the immobile CO 2 entrapped in pores since not connected to the main (and mobile) carbon dioxyde components. Refined models obtained by relaxing the vertical equilibrium assumption are presented by Nordbotten, Celia and collaborators [46,37], while models with diffuse interface between fluids have been proposed for instance by Gunn et al [29], Ngo et al [44], or Choquet et al [20] in the context of coastal aquifers with seawater intrusion.…”

“…h i , which is key for the analysis of the numerical scheme proposed in Section 3, see in particular the proof of Theorem 3.3. (iii) In their paper [44], Ngo et al also consider interlayer leakage due to capillary effects. In their approach detailed in [44,Section 4], capillary diffusion is not neglected in bulk regions, so that the oil and water phases are no longer clearly segregated.…”

“…(iii) In their paper [44], Ngo et al also consider interlayer leakage due to capillary effects. In their approach detailed in [44,Section 4], capillary diffusion is not neglected in bulk regions, so that the oil and water phases are no longer clearly segregated. This yields a modified threshold value depending on the pore size distribution and on the permeability ratio κ i+1 /κ i .…”