Summary Solutions obtained by the method of characteristics (MOC) provide key insights into complex foam enhanced-oil-recovery (EOR) displacements and the simulators that represent them. Most applications of the MOC to foam have excluded oil. We extend the MOC to foam flow with oil, where foam is weakened or destroyed by oil saturations above a critical oil saturation and/or weakened or destroyed at low water saturations, as seen in experiments and represented in foam simulators. Simulators account for the effects of oil and capillary pressure on foam using algorithms that bring foam strength to zero as a function of oil or water saturation, respectively. Different simulators use different algorithms to accomplish this. We examine SAG (surfactant-alternating-gas) and continuous foam-flood (coinjection of gas and surfactant solution) processes in one dimension, using both the MOC and numerical simulation. We find that the way simulators express the negative effect of oil or water saturation on foam can have a large effect on the calculated nature of the displacement. For instance, for gas injection in a SAG process, if foam collapses at the injection point because of infinite capillary pressure, foam has almost no effect on the displacement in the cases examined here. On the other hand, if foam maintains finite strength at the injection point in the gas-injection cycle of a SAG process, displacement leads to implied success in several cases. However, successful mobility control is always possible with continuous foam flood if the initial oil saturation in the reservoir is below the critical oil saturation above which foam collapses. The resulting displacements can be complex. One may observe, for instance, foam propagation predicted at residual water saturation, with zero flow of water. In other cases, the displacement jumps in a shock past the entire range of conditions in which foam forms. We examine the sensitivity of the displacement to initial oil and water saturations in the reservoir, the foam quality, the functional forms used to express foam sensitivity to oil and water saturations, and linear and nonlinear relative permeability models.

We apply adjoint-based optimization to a Surfactant-Alternating-Gas foam process using a linear foam model introducing gradual changes in gas mobility and a nonlinear foam model giving abrupt changes in gas mobility as function of oil and water saturations and surfactant concentration. For the linear foam model, the objective function is a relatively smooth function of the switching time. For the nonlinear foam model, the objective function exhibits many small-scale fluctuations. As a result, a gradient-based optimization routine could have difficulty finding the optimal switching time. For the nonlinear foam model, extremely small time steps were required in the forward integration to converge to an accurate solution to the semi-discrete (discretized in space, continuous in time) problem. The semi-discrete solution still had strong oscillations in gridblock properties associated with the steep front moving through the reservoir. In addition, an extraordinarily tight tolerance was required in the backward integration to obtain accurate adjoints. We believe the small-scale oscillations in the objective function result from the large oscillations in gridblock properties associated with the front moving through the reservoir. Other EOR processes, including surfactant EOR and near-miscible flooding, have similar sharp changes, and may present similar challenges to gradient-based optimization.The final publication is available at www.springerlink.com. Namdar Zanganeh, M., Kraaijevanger, J.F.B.M., Buurman, H.W., Jansen, J.D., Rossen, W.R., 2014: Challenges to adjoint-based optimization of a foam EOR process.

Summary Foam is a means of improving sweep efficiency that reduces the gas mobility by capturing gas in foam bubbles and hindering its movement. Foam enhanced-oil-recovery (EOR) techniques are relatively expensive; hence, it is important to optimize their performance. We present a case study on the conflict between mobility control and injectivity in optimizing oil recovery in a foam EOR process in a simple 3D reservoir with constrained injection and production pressures. Specifically, we examine a surfactant-alternating-gas (SAG) process in which the surfactant-slug size is optimized. The maximum oil recovery is obtained with a surfactant slug just sufficient to advance the foam front just short of the production well. In other words, the reservoir is partially unswept by foam at the optimum surfactant-slug size. If a larger surfactant slug is used and the foam front breaks through to the production well, productivity index (PI) is seriously reduced and oil recovery is less than optimal: The benefit of sweeping the far corners of the pattern does not compensate for the harm to PI. A similar effect occurs near the injection well: Small surfactant slugs harm injectivity with little or no benefit to sweep. Larger slugs give better sweep with only a modest decrease in injectivity until the foam front approaches the production well. In some cases, SAG is inferior to gasflood (Namdar Zanganeh 2011).

Method of characteristics (MOC) provides key insights into complex foam enhanced oil recovery (EOR) displacements and the simulators that represent them. Most applications of MOC to foam have excluded oil. We extend MOC to foam flow with oil, where foam might be weakened or destroyed by oil saturations above a critical oil saturation and/or weakened or destroyed at low water saturations, as seen in experiments and represented in foam simulators. Simulators account for the effects of oil and capillary pressure on foam using algorithms that bring foam strength to zero as a function of oil or water saturation, respectively. Different simulators use different algorithms to accomplish this. We examine SAG (surfactant-alternating-gas) and continuous foam injection (process of co-injection of gas and surfactant solution) processes in one dimension (1D), using both MOC and numerical simulation. We find that the way simulators express the negative effect of oil or water saturation on foam can have a large impact on the nature of the displacement. For instance, for gas injection in a SAG process, if oil is assumed to completely destroy foam above some critical saturation, and initial oil saturation is above that value, foam has nearly no effect on the displacement. On the other hand, if oil saturation weakens foam abruptly but smoothly, then successful mobility control is possible. The resulting displacements can be complex. One may observe, for instance, foam propagation predicted at residual water saturation, with zero flow of water. In other cases, the displacement jumps in a shock past the entire range of conditions in which foam forms. We examine the sensitivity of the displacement to initial oil and water saturations in the reservoir, the fraction of water in the injected foam and the functional forms used to express foam sensitivity to oil and water saturations. Introduction Foam is a means of improving sweep efficiency in gas EOR (Schramm 1994; Rossen 1996) and surfactant EOR (Li et al. 2008). Foam is also used routinely for improving the injection profile of acid in well-stimulation treatments (Gdanski 1993; Zhou and Rossen 1994) and on a pilot basis to improve the sweep of surfactant solutions in aquifer remediation (Hirasaki et al. 2000). Fractional-flow theory for two mobile phases (water-oil) was first developed by Buckley and Leverett (1942). Certain aspects of this theory were extended to multicomponent, two-phase systems by Helfferich (1981) under the name coherence theory (Pope 1980; Lake 1989; Falls and Schulte 1992a,b). Rarefaction wave theory, also known as coherence theory, is a subset of the method of characteristics (MOC); more mathematical details can be found in Lake (1989), Rhee et al. (2001), Juanes and Patzek (2004), Helfferich and Klein (1970), and Dafermos (2005) for general systems. Marchesin and Plohr (2001) review the recent progress on mathematical theory of immiscible three-phase flow in the Appendix of their paper for the models with non-linear permeabilities. Method of characteristics (MOC) has proven useful in highlighting key mechanisms and strategies for improving foam performance (Zhou and Rossen 1994, 1995; Rossen et al. 1999; Shan and Rossen 2004; Mayberry et al. 2008) and better understanding foam simulation models (Rossen et al. 1999; Dong and Rossen 2007). Previous application of MOC to foam has been mostly limited to two-phase flow; oil, if present, has been assumed to be immobile, at its residual saturation. Mayberry et al. (2008) applied three-phase MOC to foam, where foam strength does not depend on oil or water saturation and gas is completely immiscible with water and oil. They examined three cases, in which foam reduces gas mobility greatly, moderately and not at all; the reduction in initial oil saturation to its residual is much more rapid when gas mobility is reduced greatly by foam. Rosman and Kam (2009) extended this model to layered formations. Ashoori et al. (2009) examine first-contact-miscible gas floods with foam, where there is only one non-aqueous phase present at any location.

fax 01-972-952-9435. AbstractHomogenization is a powerful upscaling technique, which has been successfully applied to a variety of problems of interest, such as reactive contaminant transport and two phase flow in layered and fractured media. It has several advantages over other upscaling techniques, such as REV averaging. It does not use intuitive closure equations and it explicitly shows the dependence of the upscaled equations on the characteristic dimensionless numbers of interest. It is based on a well defined procedure, which precludes ad hoc assumptions. A disadvantage is that the underlying assumptions of this procedure have not received sufficient attention in the existing literature.This paper examines the existing homogenization models for two phase flow in fractured media with the purpose to clarify the underlying physical assumptions. We give the derivation for a specific case for which we discuss the validity of the assumptions. Finally, we discuss an example to show the applicability of the ensuing model equations.