Summary The mechanisms controlling transport and mobility of foam in porous media are complex, but in some cases the mechanisms that dominate foam properties can be represented in simple models. For instance, fractional-flow methods can reproduce the predictions of more-complex foam simulators while highlighting the mechanisms controlling foam behavior. Fractional-flow methods indicate that the effectiveness of foam processes that alternate injection of liquid and gas ("SAG" processes) depends on foam strength at extremely high foam quality, conditions difficult to control in the laboratory. A numerical simulator that incorporates the relation between capillary pressure and foam stability extends simplified foam modeling to cases where fractional-flow methods do not apply. Applications of this simulator to one-dimensional (1D) foam displacements match predictions of analytical models based on laboratory data and illustrate the numerical artifacts that challenge foam simulation. Applications to 2D flow through layers in capillary contact show that the interplay between capillary crossflow and foam collapse depends on both the dimensions of the layers and the relative magnitudes of the capillary and viscous pressure differences. Introduction Foams can improve sweep efficiency and oil recovery in miscible and steam improved-oil-recovery (IOR) processes.1–3 Effective application of foam requires accurate prediction of its performance under field conditions. Accurate prediction is difficult, however, because foam mobility depends in a complex way on bubble size, or foam texture,4,5 and texture itself depends on many factors.6 Of these factors the most important is capillary pressure Pc the difference between gas- and water-phase pressures in the porous medium. (For simplicity here, we refer to the aqueous phase, with or without added surfactant and salts, as "water.") Capillary pressure controls foam coalescence7,8 and is also important to foam generation.3,6 There are several ways to cope with this complexity. The first is to represent foam mobility as an empirical function of surfactant concentration, flow rates, and other factors.9–15 The second approach is to quantify the relation between foam mobility and texture and all the mechanisms of creation and destruction of the liquid films, or lamellae, that separate and define gas bubbles.6,16-20 This latter approach is called the "population balance." As currently applied, the population balance gives a differential equation for evolving foam texture, but the basic concept is consistent with a local-equilibrium version, in which foam texture is an algebraic function of local conditions. A third approach intermediate between the other two, the "fixed- pc∗ model,"?21 relies on the relation between capillary pressure, foam texture, and foam mobility. In essence, it is a local-equilibrium version of the population balance for strong foams under conditions where capillary pressure dominates foam texture and gas mobility. Thus it retains the simplicity of empirical foam models while recognizing the central role of foam texture. Related to the issue of representing foam mobility is the question of how to solve the differential equations for transport of the various phases and components. In principle, three-dimensional (3D) numerical simulators can incorporate arbitrarily complex transport models. Under more-restricted conditions, the much-simpler method of characteristics, or fractional-flow theory, applies.22–25 This approach can offer significant insights into process mechanisms, even in conditions under which its assumptions are not satisfied quantitatively. The relative merits of these approaches reflect the competing goals of simplicity and completeness. In this paper we show the power of a simple foam simulator based on the fixed-pc∗ model and compare its results to analytical solutions using fractional-flow methods in certain cases. These comparisons illustrate the numerical artifacts that complicate simulation of foam displacements in the laboratory and the field. The Fixed-pc∗ Model. One expects all foams to degrade at sufficiently high capillary pressure, since lamellae thin and eventually break as Pc increases. 7,8 However, for many strong foams there is a narrow range in Pc so narrow that it can be identified with a single value, the "limiting capillary pressure," pc∗, at which foam collapses abruptly.7 Since Sw is related to P c through the capillary-pressure function Pc(Sw) this means that foam remains at a given water saturation Sw∗≡Sw(pc∗) over a wide range of flow rates and foam qualities (injected gas volume fraction, usually expressed as a percent between 0 and 100). This implies that krw (Sw) is nearly constant over this same wide range of flow rates and that ? p is simply proportional to water volumetric flux uw independent of foam texture, according to Darcy's law for the water phase ∇ p = u w μ w k k r w ( S w ∗ ) , ( 1 ) where ?w is water viscosity and k is the permeability of the medium. Foam texture merely reacts as needed to maintain water saturation at Sw∗ and capillary pressure at pc∗.21 The "fixed-pc∗ model,"21,25,26 in which Sw∗ does not vary with flow rate or foam quality, fits many strong foams at the relatively low flow rates and high foam qualities27–29 typical of many IOR processes, away from a narrow region near the inlet of the porous medium where the injected foam attains its steady-state texture.5,20 Where it does apply, the fixed- pc∗ model retains both the simplicity of empirical foam models and the accuracy of the much-more-complex population balance.

A finite-difference, equation-of-state (EOS), compositional simulator has been used to study C02 flooding. First, unstable first-contact-miscible (FCM) displacements were simulated with a fine mesh to investigate the transition from gravity override to viscous fingering. Next, a direct comparison was made for FCM and multiple-contact-miscible (MCM) displacements under the same conditions to investigate effects of phase behavior on the growth of viscous fingers. Then, the effects of gravity, physical dispersion, capillary pressure, phase behavior, and heterogeneity were combined and simulated for C02 flooding on a field scale with stochastic permeability fields.

Foam mechanisms are many and complex, but in some cases the mechanisms that dominate foam properties can be represented in simple models. For instance, fractional-flow methods can reproduce the predictions of much-more-complex foam simulators while highlighting the mechanisms controlling behavior. Fractional-flow methods indicate that the effectiveness of foam processes that alternate injection of liquid and gas ("SAG" processes) depends on foam strength at extremely high foam quality, conditions difficult to control in the laboratory. Ironically, surfactant formulations made with lower surfactant concentration, that form weaker foams in steady-state flow, can foam stronger, longer-lived foams in dynamic SAG processes than foams made with higher surfactant concentration, because water drains more slowly from the "weaker" foams in the SAG process. A numerical simulator that incorporates the relation between capillary pressure and foam stability extends simplified foam modeling to cases where fractional-flow methods do not apply. Applications of this simulator to one-dimensional (ID) foam displacements match predictions of analytical models based on laboratory data. Applications to 2D flow show that the ability of foam to correct gravity override is controlled both by the dimensions of the reservoir and the allowable rise in injection-well pressure. In flow through layers in capillary contact, the interplay between capillary cross-flow and foam collapse depends on both the dimensions of the layers and the relative magnitudes of the capillary and viscous pressure differences.

A compositional equation-of-state simulator has been used to make comparisons of CO2 process performance with several combinations of horizontal and vertical wells and various alternative reservoir descriptions. This is the first time that the performance of CO2 flooding using horizontal wells has been reported. We have used the reservoir and fluid properties of an actual West Texas carbonate reservoir that is currently being waterflooded. The phase behavior and properties of the CO2 used in these simulations are typical of those of multiple-contact-miscible field conditions of West Texas. A layered-type reservoir description was initially used to investigate the effect of water-alternating-gas (WAG) ratio with various combinations of well types. We then investigated the effect of vertical permeability, length, and position of the horizontal injector using the same reservoir description and a horizontal injector-vertical producer combination. Similar investigations using an unconditioned stochastic permeability field having a Dykstra Parsons coefficient (VDP) of 0.81 were made. Our final three-phase flow simulation was performed using a stochastic permeability field conditioned with core data. Since three nonaqueous phases can occur at low temperatures typical of West Texas floods resulting in up to four-phase flow with mobile water, we investigated the impact of four-phase flow in one of these simulations. This is the first time that three-dimensional, four-phase flow reservoir simulations have been reported with either vertical or horizontal wells. Introduction During the last five years the petroleum industry has experienced a rapid increase in the number of horizontal wells being drilled and completed, primarily because of the following reasons. First, recent advances in drilling technology, which have resulted in substantially reduced drilling and completion costs, have made the drilling of horizontal wells an economically viable alternative. A large U.S. independent has reported an average cost of about $1.25 million per well (average measured depth of 13,400 ft), which is comparable to the average cost of a vertical well. The second important factor that has contributed to the recent upsurge in horizontal well activity is because horizontal wells may be able to deliver two to ten times the performance of conventional wells due to their larger surface area. Therefore, the higher productivity coupled with favorable drilling and completion costs appears to have considerably increased the potential of horizontal wells to produce more oil. Finally, horizontal wells offer solutions to the problem of producing oil or gas in reservoirs where conventional technology either fails or is uneconomic, for example,reservoirs where conventional wells have low productivity,reservoirs with vertical fractures,oil reservoirs where recovery is limited by water or gas coning, andthick continuous heavy oil and bitumen reservoirs where steam-assisted gravity drainage is practical. The current target of horizontal wells is mainly primary oil, although there do exist some applications of horizontal wells in the recovery of heavy oil by steam injection. Because U.S. domestic oil reserves are declining, and the chances of discovering large fields are diminishing, various forms of enhanced oil recovery (EOR) processes, for example, carbon dioxide (CO2) miscible floods. have recently gained considerable attention as a means of increasing the reserves base. Because of the better sweep efficiencies and higher injectivities possible with horizontal wells, all EOR methods should benefit by their use. P. 753^

Coreflood experiments in naturally heterogeneous sandstone outcrop cores were conducted 1ijld simulated. Displacements included waterfloods and polymer floods containing tracers for oil or water during single-phase and two-phase flows. Displacements of viscous polymer solution were unstable. The cores were characterized by air permeability measureIllentson each square centimeter of each face and by computerized tomography (CT) scans of core cross sections spaced 1 cm apart along the core length. Characterization data were used for fine-grid simulations. The agreement between the experiments and simulations is good. Sipmlations with coarser-grid physical descriptions then were made by use of effective properties..