Modeling Foam Displacement with the Local Equilibrium Approximation: Theory and Experiment Verification

Abstract

The gas mobility control aspects of foamed gas make it highly applicable for improved oil recovery. Gas bubble size, often termed as foam texture, determines gas flow behavior. A population balance model has been developed previously for modeling foam texture and flow in porous media. The model incorporates pore-level mechanisms of foam bubble generation, coalescence, and transport. Here, we propose a simplified foam model to reduce computational costs. The formulation is based on the assumption of local equilibrium of foam generation and coalescence and is applicable to high and low quality foams. The proposed foam model is compatible with a standard reservoir simulator. It provides a potentially useful, efficient tool to predict accurately foam flows at the field scale for designing and managing foamed-gas applications. There are three main contributions of this paper. First, the population balance representation of foam generation by gas-bubble snap off is modified to extend the capability of the population balance approach to predict foam flow behaviors in both the so-called high-quality and low-quality regimes. Second, a simplified population-balance model is developed and implemented with the local-equilibrium approximation. Third, foam displacement experiments in a linear sandstone core are conducted to verify the proposed model. A visualization cell is employed to measure the effluent foam bubble sizes for a transient flow as well as to estimate the in-situ foam bubble sizes along the length of the core during steady flow. Additionally, the evolution of aqueous phase saturation is monitored using X-ray computed tomography (CT) and the pressure profile is measured by a series of pressure taps. Good agreement is found between the experimental results and the predictions of the simplified model, with a minor mismatch in the entrance region. Introduction Foam is a dispersion of a gas within a continuous liquid. Foamed gas has attracted tremendous research interest and effort because of its significantly reduced mobility in comparison to continuous gas mobility. The unique flow properties of foamed gas make foam highly applicable as a gas mobility control agent for enhanced oil recovery (EOR). EOR by steam or carbon dioxide injection, for example, sometimes exhibit poor sweep efficiency, gravity override, and channeling of gas through the most permeable zones of the reservoir (Green and Willhite, 1998; Lake, 1989). Foaming the injected gases increases the gas-phase resistance dramatically, thereby providing mobility control to improve the sweep efficiency and oil production. Enhanced/improved oil recovery by the use of foam has been employed during field steam-foam pilots in the late 1980's and several nonthermal applications of foam in the mid 1990's. For instance, the Mecca and Bishop steam-foam pilots in the Kern River field (Patzek and Koinis, 1990) show major oil response after 2 years of foam injection, and yield incremental oil recovery of 8.5% - 18% of the original oil in place (OOIP) over a five year period.

Equivalence Between Semi-empirical and Population-Balance Foam Models

“…The first group includes population-balance (PB) models that account for pore-level mechanisms related to lamella creation and coalescence in order to track the dynamical behavior of foam texture in porous media along with its effect on gas mobility [5,14,15,16]. Alternatively, a local steady state version of PB models [15,16,17] comes out by equating the lamella creation and coalescence rates; solving this equation yields the foam texture. Semi-empirical (SE) models, that constitute the second group, assume that an equilibrium state is attained instantaneously such that the foam texture effects can be represented implicitly through a gas mobility reduction factor that depends on fluid saturation, interstitial velocity, surfactant concentration and other factors [18,19].…”

Effect of Permeability on Foam-model Parameters and the Limiting Capillary Pressure

“…Population-balance (PB) models (e.g. Friedmann et al;1991;Kovscek et al, 1995;Kam et al, 2007;Chen et al, 2010) attempt to describe the processes that create and destroy the lamellae that separate gas bubbles, as well as the effect of bubble size on gas mobility. The second group of models represents the effect of bubble size implicitly, through a mobility-reduction factor that depends on saturations, superficial velocities and other factors (e.g.…”

Stability Analysis of Uniform Equilibrium Foam States for EOR Processes

“…There are two main approaches to modeling complex mechanisms governing foam texture: population-balance modeling Friedmann et al 1991;Kovscek et al 1995;Kam et al 2007;Kam 2008;Chen et al 2010) and local-equilibrium (LE) modeling (Fisher et al 1990;de Vries and Wit 1990;Persoff et al 1991;Ettinger and Radke 1992;Vassenden and Holt 2000;Computer Modeling Group 2009;Chen et al 2010;Ashoori et al 2010). For simplicity, LE foam models assume that foam reaches the LE bubble texture corresponding to its water saturation very fast.…”

“…Therefore, no explicit balance equation for foam bubbles is required in these models. One should keep in mind that even in population-balance models, there are large regions in the solution where the flow is at local equilibrium, as far as foam is concerned (see Chen et al 2010). …”

“…Because of the change in mechanisms for bubble creation and destruction for bubbles of roughly pore size, the model of Kam (2008) and related models simply postulate an upper limit to foam texture; if it exceeds the limit, it is reset to that limit. For an alternate approach, see Chen et al (2010). Kam et al (2007) examined the stability of the strong-, weak-foam, and intermediate states numerically by checking the growth or decay of an arbitrary perturbation in foam texture (inversely related to bubble size) and water saturation using numerical simulation on one LE state from each group.…”