Abstract
The use of foam for mobility control is a promising mean to improve sweep efficiency in EOR. Experimental studies discovered that foam exhibits three different states (weak foam, intermediate foam, and strong foam). The intermediate-foam state is found to be unstable in the lab whereas the weak- and strong-foam states are stable. The model of Kam (Colloids Surf A Physicochem Eng Asp 318(1–3): 62–77, 2008) is the only mechanistic foam model that can fit a variety of steady-state experimental data including multiple steady states. This model is modified from a previous mechanistic foam model to resolve the intrinsic instability of the strong-foam state. Simple finite-difference simulations have found that an arbitrary perturbation grows for the unstable intermediate foam but diminishes for the strong- and weak-foam states. The issue of the stability of foam states, especially the strong-foam state, is a serious concern in application of foam in EOR. Instabilities may rule out one or more states and consequently have considerable effect on reservoir sweep efficiency and injection pressure. Here, for the first time the stability of the various equilibrium foam states is investigated by an analytical stability-analysis method together with numerical simulations. We demonstrate the instability of most intermediate states, consistent with the laboratory observations. However, our analysis reveals an instability of the strong-foam state. We show that the diffusion, whether introduced artificially by the finite-difference scheme or representing physical dispersion, damps this instability. We obtain good agreement with finite-element simulations with and without additional diffusion. We also prove that all states are unconditionally stable for a local-equilibrium-foam model.
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Abbreviations
- C c :
-
Foam parameter in Kam model (m−3 s−1)
- C f :
-
Foam parameter in Kam model
- C g :
-
Foam parameter in Kam model (m−3 s−1)
- f w :
-
Water fractional-flow function (excluding capillarity-driven flow)
- \({k_{\rm rg}^0}\) :
-
Gas relative-permeability in the absence of foam
- k rw :
-
Water relative-permeability
- n :
-
Foam parameter in Kam model (Appendix A)
- n D :
-
Dimensionless foam texture (= n f/n max)
- n f :
-
Foam texture (number of lamellae per unit volume) (m−3)
- n max :
-
Maximum foam texture (m−3)
- \({{\nabla}p}\) :
-
Pressure gradient in Kam model (psi ft−1)
- P c :
-
Gas-water capillary pressure (Pa)
- \({{\nabla}p_0 }\) :
-
Foam parameter in Kam model (psi ft−1)
- r 1, r 2 :
-
Eigenvectors
- r g :
-
Foam generation function (m−3 s−1)
- r c :
-
Foam-coalescence function (m−3 s−1)
- S g :
-
Gas saturation
- S gr :
-
Residual gas saturation in a water-gas system
- S w :
-
Water saturation
- S wc :
-
Connate water saturation in a water-gas system
- \({S_{\rm w}^\ast }\) :
-
Limiting water saturation
- t :
-
Time (s)
- u :
-
Total superficial velocity (m s−1)
- u g :
-
Gas superficial velocity (m s−1)
- u w :
-
Water superficial velocity (m s−1)
- x :
-
Standing coordinate for displacement in 1-D (m)
- 0 :
-
Equilibrium point
- \({\phi}\) :
-
Porosity
- K :
-
Oscillation wave number, i.e., inverse wavelength (m−1)
- λg :
-
Mobility of gas (m2 (Pa s)−1)
- λw :
-
Mobility of water (m2 (Pa s)−1)
- \({\mu _{\rm g}^f }\) :
-
Gas viscosity in the presence of foam (Pa s)
- \({\mu _{\rm g}^0 }\) :
-
Gas viscosity in the absence of foam (Pa s)
- μ w :
-
Viscosity of water (Pa s)
- ν 1, ν 2 :
-
Eigenvalues (solutions of Eq. 12)
- σ:
-
Gas-water interfacial tension (N m−1)
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Ashoori, E., Marchesin, D. & Rossen, W.R. Stability Analysis of Uniform Equilibrium Foam States for EOR Processes. Transp Porous Med 92, 573–595 (2012). https://doi.org/10.1007/s11242-011-9921-8
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DOI: https://doi.org/10.1007/s11242-011-9921-8