Connectivity-based hysteresis modeling of two-phase flow in porous media: extension to compressible fluids and wetting-film flow, lithology-generic parameterization, and field-scale validation

Macroscopic simulation of geological CO2 storage requires hysteretic constitutive relations — capillary pressure–saturation and relative permeability functions that depend on the history of drainage and imbibition — to predict capillary entrapment of the injected fluid. Cihan et al. (2014) introduced a connectivity-based framework in which the void-size distribution of the medium together with a drainage and a wetting connectivity function generate hysteretic scanning curves, entrapped (residual) non-wetting saturations, and hysteretic relative permeabilities, verified against core-scale data and intermediate-scale flow-cell experiments with surrogate fluids. The framework assumes incompressible fluids, omits wetting-film (corner) flow, has been parameterized only for Berea sandstone and laboratory sands, and has not been demonstrated at field scale in heterogeneous media. The open problem is to generalize the connectivity-function formulation to compressible phases and film-flow-inclusive wetting-phase conductivity, establish lithology-generic parameter identification, and validate the extended model against transient two-phase experiments (point-wise saturations, plume moments) and field-scale heterogeneous applications. A 2017 follow-up by the same group tested the model against intermediate-scale experimental data; the physical extensions and the field-scale demonstration remain open.

Background

Two-phase flow in porous media — the simultaneous movement of two immiscible fluids, such as supercritical CO2 and brine, through the connected void space of a rock — is described at the continuum scale by Darcy-type equations closed by constitutive relations: the capillary pressure $P_c$ (pressure difference between non-wetting and wetting fluid, set by the curvature of fluid–fluid interfaces in the voids) as a function of saturation $S$, and the relative permeabilities $k_{rn}$ and $k_{rw}$ (each phase's conductivity normalized by the single-phase permeability). When a medium undergoes drainage (non-wetting fluid invading) followed by imbibition (wetting fluid re-invading), these relations are not single-valued: they exhibit hysteresis, i.e., dependence on the saturation history, and imbibition leaves behind a residual (entrapped) saturation of disconnected non-wetting fluid. This capillary trapping is a primary security mechanism of geological carbon storage (GCS), because the amount of immobile CO2 remaining after injection stops controls long-term leakage risk. Traditional hysteretic constitutive models are largely empirical — for example the standard closed-form van Genuchten–Mualem capillary functions combined with Land's relationship (an empirical formula giving the residual non-wetting saturation as a function of the maximum saturation reached before flow reversal) or with independent-domain theories (which build hysteresis from a tabulated population of pore elements, each assigned fixed drainage and wetting capillary thresholds) — and, in their simplest constant-residual form, treat residual saturation as independent of the local saturation maximum reached before flow reversal.

Cihan, Birkholzer, Trevisan, Bianchi, Zhou and Illangasekare (2014, Energy Procedia; journal version of the capillary-pressure part in Water Resources Research, 2014) proposed a physically based alternative built on two medium descriptors: the void-size distribution $f(r)$, the fraction of void volume with characteristic size $r$, and two connectivity functions — the drainage connectivity $P^d(r, P_c)$ and the wetting connectivity $P^w(r, P_c)$ — which give the volumetric fraction of voids of size $r$ that remain connected to the invading phase at capillary pressure $P_c$ during drainage and wetting. From $f(r)$, $P^d$, $P^w$, and the capillary pressures at each flow reversal, recursion relations yield the saturation along arbitrary drainage–imbibition scanning curves (the intermediate hysteretic paths between the main drainage and main wetting branches) and the entrapped non-wetting saturation as a function of the maximum saturation reached before reversal; relative permeabilities follow from a random-cut-and-rejoin argument that randomly pairs fluid-filled void cross-sections on two facing surfaces (after Marshall, 1958). The connectivity functions can be estimated from pore-scale invasion-percolation simulations (network simulations in which the invading fluid advances through the accessible void with the lowest capillary entry pressure) on computer-generated porous media or fitted to measured primary drainage and main wetting curves. Under its stated assumptions — incompressible fluids, a random isotropic medium, independence of drainage and wetting path fractions, and no transport of wetting liquid along thin films and pore corners — the model reproduced residual-versus-initial saturation data for decane/water and CO2/water in Berea sandstone (a widely used laboratory sandstone), scanning curves for natural sands after fitting only the boundary curves, and quasi-two-dimensional intermediate-scale flow-cell experiments (91.4 cm × 5.6 cm × 61 cm, homogeneous and heterogeneous sand packs, with surrogate liquids standing in for supercritical CO2 and brine), the last mainly through the end-of-injection saturation distribution and the first spatial moment of the migrating plume.

Problem Statement

The problem, as posed by the source paper's statement of its own limitations, is to complete the connectivity-based hysteretic constitutive framework so that it becomes predictive under the physical conditions of geological CO2 storage. All components serve one objective — a validated, physics-based predictive capability for capillary entrapment and post-injection redistribution of CO2 in reservoirs — and a complete solution must jointly address all four of the following components: (i) generalization of the connectivity-function formulation of the hysteretic $P_c$–$S$–$k_r$ relations beyond the incompressible-fluid assumption to compressible phases, whose density varies with pressure along the drainage–imbibition history; (ii) incorporation of wetting-film and pore-corner flow contributions into the wetting-phase relative permeability, which the current model neglects entirely; (iii) a defensible identification and parameterization of the drainage connectivity function $P^d$ and wetting connectivity function $P^w$ for diverse lithologies beyond Berea sandstone and laboratory sands, from standard measurements such as primary drainage and main wetting curves or pore-scale simulation, with the resulting boundary-curve fits reported and their accuracy quantified; and (iv) demonstration of predictive capability at field scale in porous media with heterogeneous connectivity structures, not only in homogeneous or laboratory-heterogeneous intermediate-scale packs, with the model's point-wise saturation and plume-moment predictions in a heterogeneous medium compared against a reference solution or cited field observations. The extended model must be validated against transient two-phase flow experiments reporting point-wise saturation measurements and the evolution of higher-order plume moments (spatial moments of the migrating fluid distribution beyond its mean), rather than only end-state saturations or a plume's first moment; the comparison base may be drawn from the cited laboratory experiments, published intermediate- or field-scale observations, or independently generated numerical reference solutions of heterogeneous media. Partial prior discharge of the testing component at intermediate scale with surrogate fluids (Cihan et al., 2017) does not by itself satisfy this objective, since compressibility, film flow, lithology generality, and field-scale heterogeneity are explicitly part of the source question.

References

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