Colloids and Surfaces A: Physicochemical and Engineering Aspects
Roles of transient and local equilibrium foam behavior in porous media: Traveling wave
Graphical abstract
Research highlights
▶ We solve for traveling wave at the shock accounting for capillary dispersion and foam kinetics. ▶ We show that if there is no gas ahead of the foam, foam texture is everywhere at local equilibrium (LE) within the shock. ▶ If there is gas initially in the formation, slow foam dynamics can narrow the shock from that assuming LE or lead to oscillations near the shock. ▶ The approach of solving for the traveling wave can rule out some of the foam multiple steady states seen in experiment.
Introduction
Foam is a dispersion of a gas within a continuous liquid. Foam is used as a mobility-control agent for enhanced oil recovery (EOR), gas blocking and acid diversion during matrix stimulation. The prediction of foam performance relies on modeling. Foam modeling approaches include population-balance models [1], [2], [3], [4], [5], [6] and local-equilibrium models [6], [7], [8], [9], [10], [11], [12]. With some additional assumptions, one-dimensional displacements with local-equilibrium models can be represented with fractional-flow methods, an application of the method of characteristics (MOCs) [13], [14], [15], [16].
In the population-balance approach, foam texture (inversely related to bubble size) is modeled explicitly, using a balance equation for the lamellae (liquid films) that separate bubbles. This equation is similar to the mass balance for surfactant or water. The mechanisms for lamellae creation and destruction are represented explicitly in the balance on bubble texture. An alternative to population-balance modeling is to assume local equilibrium (LE) (equal generation and destruction rates) at all locations in the formation. In this approach, foam texture could be represented explicitly, or implicitly, in a gas-mobility-reduction factor. Cheng et al. [17] and Vassenden and Holt [12] show that a LE model can predict the two strong-foam regimes seen in experiments [18], [19] without explicitly representing bubble texture. In the context of an LE model, one can allow for abrupt jumps between steady states (i.e., abrupt foam generation or foam collapse) when one state reaches the limit of its stability [20].
The fractional-flow method based on LE models is an approach that provides useful insights and ease of use. This approach includes some simplifying assumptions: incompressible phases; Newtonian mobilities; one-dimensional flow; absence of dispersion, gradients of capillary pressure, and viscous fingering; immediate attainment of local equilibrium.
Rossen et al. [14] show that in some cases population-balance simulations can be modeled nearly equivalently with fractional-flow methods that assume LE. Chen et al. [6] show examples where simulation assuming LE closely match simulation results with the full population balance for foam texture.
LE simulation is much less complex than population-balance modeling [6]. It is important therefore to determine the conditions under which this added complexity is required.
A shock is a discontinuity in saturation and other properties on the large scale; the small-scale representation of this transition is called a traveling wave. Rossen and Bruining [20] show cases for which solving for the traveling wave is necessary to guide correct shock construction in the 1D solution for a foam displacement. Solving for the traveling wave could be useful for some other reasons: if the wave is wide on the scale of laboratory experiments, then such experiments would not be appropriate for representing foam dynamics on the large scale. In some SAG (surfactant-alternating-gas) foam processes, most mobility control occurs within the shock itself [15]; in such cases success on the field scale could depend on the width and foam strength within the traveling wave.
In two papers we reconsider whether and when non-equilibrium effects are important. We assume LE applies everywhere except at shocks and an entrance region [11]. In this paper, we concentrate on shocks: we account for non-equilibrium foam kinetics within the traveling wave that resolves the shock. Shocks can also arise from jumps in surfactant concentration [13], but we do no deal with such jumps in this paper. In a companion paper [21] we solve for the entrance region using the same models used in this paper. We employ two foam models: a simple schematic first-order-kinetic model and a more-realistic population-balance model with multiple steady states [5]. We investigate the effect of foam kinetic rates on the behavior of the traveling wave for different states upstream and downstream of the shock. We compare the traveling-wave solution accounting for non-equilibrium foam kinetics within the traveling wave to solutions computed assuming local equilibrium.
Section snippets
Shock-front formulation in foam processes
The governing equation for an immiscible two-phase (gas–water), two component incompressible displacement in rectilinear flow through a porous media is given by the Rapoport–Leas equation [22], [23]:or equivalently,where Sw and fw are water saturation and fractional flow, φ is porosity, u is total superficial velocity, x is position, t is time, and λw and λg are the mobilities of water and gas. Besides the dependency of fw on Sw in normal
No gas ahead of foam
Many papers introducing or discussing a population-balance foam model illustrate it with simulation of foam or gas injection into a medium with surfactant solution, but no gas, initially present [4], [5], [6], [24], [31], [32]. Chen et al. [6] show close agreement between simulation of such a case assuming local equilibrium and with full foam dynamics. If there is no gas ahead of the foam (foam injection into a fully water-saturated medium), then the initial state I is at (Sw, fw) = (1, 1). For
Discussion
Above we showed that with no gas ahead of the foam bank, as is common in published simulations, foam is at LE even within the traveling wave. In this section we discuss some of the implications of this conclusion and of the Welge condition if there is no gas initially in the formation or core.
Fig. 17 shows a series of LE water-fractional-flow curves in presence of foam at different total superficial velocities u for the multiple steady-state foam model presented in Dholkawala et al. [41]. The
Conclusions
We derive the equations that describe the traveling wave at the shock in a foam process, accounting for non-equilibrium between foam bubble creation and destruction in addition to capillary-pressure gradients. We describe mathematical tools for help in solving the equations for the traveling wave. We reach the following conclusions.
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If there is no gas ahead of the foam, as is common in published simulations, foam texture is everywhere at local equilibrium within the shock, regardless of the foam
Acknowledgment
We thank S.I. Kam for useful discussions of the population-balance foam model used in this paper. The authors also thank Delft University of Technology for support for the visits of DM to TU Delft. This work was supported in part by a grant from Research Centre Delft Earth, a program of Delft University of Technology; and by CNPq under Grants 301564/2009-4, 472923/2010-2,490707/2008-4; FAPERJ under Grants E-26/ 110.972/2008, E-26/102.723/2008, E-26/112.220/2008, E-26/110.310/2007,
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