For the first time, we apply three-phase fractional-flow theory combined with the wave-curve method to better understand the mechanisms of foam displacements with oil in porous media, employing a widely used foam model. Fractional-flow theory demonstrates that oil saturation in foam-created oil banks never exceeds the upper limit for stable foam, fmoil (i.e. an oil saturation above which foam is killed); see (Tang et al., 2019c) and section 3.4 below. This constraint suggests a criterion for creating significant oil banks: for the surfactant formulation, fmoil must be far above the initial oil saturation. We identify key factors controlling foam and oil-bank propagation: fmoil, foam quality, the regime in which foam is injected and foam strength at both injection and initial states. The mechanisms of these factors are revealed through a material balance on gas: any factor increasing gas volume injected while maintaining adequate foam strength, or reducing gas saturation in the foam region, accelerates foam propagation, and vice versa. Also, an optimal foam injection strategy is identified: inject foam in the low-quality regime near the transition foam quality (Tang et al., 2019a, 2019b), at which mobility reduction is at its maximum. This rule's universality needs to be further verified. Fractional-flow solutions, free of numerical artifacts, can be used to benchmark numerical simulators and machine-learning approaches for foam processes.

Long-distance propagation of foam is one key to deep gas mobility control for enhanced oil recovery and CO2 sequestration. It depends on two processes—convection of bubbles and foam generation at the displacement front. Prior studies with N2 foam show the existence of a critical threshold for foam generation in terms of a minimum pressure gradient r pmingen or minimum total interstitial velocity vmint,gen, beyond which strong-foam generation is triggered. The same mechanism controls foam propagation. There are few data for r pmingen or vmint,gen for CO2 foam.

We extend previous studies to quantify r pmingen and vmint,gen for CO2 foam generation and, for the first time, relate r pmingen and vmint,gen to factors including injected quality (gas volume fraction in the fluids injected) fg, surfactant concentration Cs, and permeability K. In each experiment, steady pressure gradient ∇p is measured at fixed injection rate and quality, with total interstitial velocity vt increasing and then decreasing in a series of steps. The trigger for strong-foam generation features an abrupt jump in ∇p upon an increase in vt.

In most cases, the data for ∇p as a function of vt identify three regimes, which are coarse foam with low ∇p, an abrupt jump in ∇p, and strong foam with high ∇p. The abrupt jump in ∇p upon foam generation confirms the existence of r pmingen and vmint,gen for CO2 foam. We further show how r pmingen and vmint,gen scale with fg, Cs, and K. Conditions that stabilize lamellae reduce the values of the thresholds: Both r pmingen and vmint,gen increase with fg and decrease with increasing Cs or K. Specifically, r pmingen scales with fg as (fg)2 and vmint,gen scales as (fg)4, and both r pmingen and vmint,gen scale with Cs as (Cs)−0.4. The effect of K on the thresholds for foam generation is greater than the effects of fg and Cs. Our data in artificial consolidated cores show that r pmingen scales with K as K−2 for CO2 foam, in comparison with K−1 for N2 foam in unconsolidated sand/bead packs. More data are needed to verify these correlations.

It is encouraging that r pmingen in the cores with K = 270 md or greater is less than 0.17 bar/m (~0.75 psi/ft), two to three orders of magnitude less than for N2 foam. Such low r pmingen can be easily attainable throughout a formation. This suggests that limited ∇p deep in formations is much less of a restriction for long-distance propagation of CO2 foam than for N2 foam. Foam propagation could still be challenging in low-K reservoirs (r pmingen ~10 bar/m for K = 27 md). Nevertheless, formation heterogeneity and alternating slug injection of gas and liquid help foam generation and can reduce the values of r pmingen.

In stratified porous media, non-uniform velocity between layers combined with thermal conduction across layers causes spreading of the thermal front: thermal Taylor dispersion. Conventional upscaling not accounting for this heterogeneity within simulation grid blocks underestimates thermal dispersion, leading to overestimation of thermal breakthrough time. We derive a model for effective longitudinal thermal diffusivity in the direction of flow, αeff, to represent the effective thermal dispersion in two-layer media. αeff, accounting for thermal Taylor dispersion, is much greater than the thermal diffusivity of the rock itself. We define a dimensionless number, NTC, a ratio of times for longitudinal convection to transverse conduction, as an indicator of transverse thermal equilibration of the system during cold- or hot-water injection. For NTC > 5, thermal dispersion in the two-layer system closely approximates a single layer with αeff. This suggests a two-layer medium satisfying NTC > 5 can be combined into a single layer with an effective longitudinal thermal diffusivity αeff. In application to a geothermal reservoir, one can apply the model to perform upscaling in stages, i.e. combining two layers satisfying the NTC criterion in each stage. The αeff model accounting for the fine-scale heterogeneity within simulation grid blocks would enhance the prediction accuracy of thermal breakthrough time and thus thermal lifetime.