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.

Factors key to the success of foam diversion processes for matrix acidization were measured in Berea cores. Foam mobility was low in high-permeability (847-md) Berea and higher in low-permeability (92 md) Berea, suggesting effective foam diversion. Injection of surfactant solution, emulating foam-compatible acid injected following foam, trapped some of the foam in place. but imputed diversion of acid was not as complete as diversion of foam itself. After a period of injection of liquid, more gas was displaced, starting from the core inlet, and mobility rose further. Using these coreflood results, fractional-flow methods predict effective acid diversion between layers differing in permeability in field application. A sensitivity study indicates that the size of the preflush and the propagation rates of foam within rock are secondary factors in the success of foam diversion in the field. Foam strength and especially gas trapping following foam injection are the keys to successful application. In a process in which gas trapping following foam injection is ineffective, or is less effective in high-permeability layers, a continuous-injection foamed-acid process would outperform a process of alternating slugs of foam and acid. Further data, especially in field cores, are needed to confirm these conclusions. Introduction Foams are widely used to divert acid to desired intervals in matrix acidization treatments. The goal of such a diversion process is to reduce the injectivity of acid into layers where less is needed and thereby divert it into layers more in need of stimulation. Foams do not directly alter the mobility of an aqueous phase such as acid in rock; relative permeability krw is the same function of water saturation Sw in the presence of foam as in its absence. The individual liquid films, or lamellae, that aerate gas bubbles in a foam do restrict the flow of gas, however. By driving down gas mobility, foam indirectly forces down Sw and thereby krw(Sw), accomplishing the process goal of lower acid injectivity into the given layer. Moreover, foams are stronger, reducing liquid mobility more, in higher-permeability layers, diverting acid into lower-permeability intervals that otherwise would not receive much acid. Whether foams are stronger in more-heavily-damaged rock than less-damaged rock is less clear. Many foams collapse in the presence of oil, which could also help divert acid into productive, oil-bearing intervals. We do not address the effects of oil or of formation damage on foam further here, however. The key to foam effectiveness in acid diversion is the ability of acid following foam to maintain low water saturation and low krw(Sw) during acid injection following foam. To accomplish this the acid slug must contain surfactant and be formulated for compatibility with foam. There are no published data on this property for acid slugs, but there are data for surfactant slugs without acid. Persoff et al. found that surfactant injected without acid or gas following foam maintained for some time the same low Sw and krw(Sw) created by the foam, evidently by trapping all the gas in the foam in place. This effect is highly desirable in foam diversion. Bernard et al. found less complete trapping of gas by surfactant solution injected after foam. Zerhboub et al. found that diversion of a surfactant slug following foam can be increased by adding a brief shut-in period following foam injection. Exactly how the shut-in period works is not yet clear. There are various injection strategies for a foam diversion process: injecting foam with or without a surfactant preflush; foaming the acid itself or alternating acid injection with foam; designing an acid formulation either to destroy or maintain foam; incorporating a shut-in period between foam and acid injection. There are few data in the literature to guide choices among these alternatives, although there is a wide body of literature on foams for diverting gas flow in enhanced oil recovery (EOR). Building on this literature, Zhou and Rossen developed a model for the foam diversion process based on fractional-flow methods. They conclude that the best foam process is one in which an optimally-sized preflush precedes foam injection and the acid slug is compatible with the foam. The preflush satisfies surfactant adsorption in the near- well region and greatly accelerates foam propagation there. Since preflush injection precedes any diversion or damage removal, most reflush by far enters the high-permeability or least-damaged layers. Therefore the preflush accelerates foam propagation most in the layer that is to be blocked, because that layer receives the most preflush. It is important to design the acid slug for compatibility with foam in order that the acid slug not immediately destroy the diversion brought about by foam. Zhou and Rossen conclude that both foamed acid and schemes of alternate injection of acid and foam can be effective in diversion. P. 347^

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.

This paper describes a fluid system developed to build integrity continuously to prevent lost returns while drilling. The primary attributes of the fluid that enable this are high solids content and extremely high filtration rates, as reflected in API fluid loss tests. It is referred to here as a drill and stress fluid (DSF). In field applications, DSF water-based systems appear to be effective over a wide range of conditions. Circulating pressures have been sustained that exceed integrity at the bit by 1.0 to 3.0 ppg without detectable losses in depleted formations with permeability ranging from 1 to 1,000 md and pay zones of 50 to 700 ft (165 to 2,300 m) in length. The mechanism through which the DSF is believed to arrest the growth of lost returns fractures and build near-wellbore stress is described. Generalized design criteria for a DSF system are presented and the assumed relationship between the design parameters and fluid performance is discussed. The results of the application of DSF are presented for eight wells, including post treatment evaluation logs of the drilling-induced fractures created while building stress. Operational practices that facilitate the safe use of an extremely high fluid loss system with overbalance exceeding 2,000 psi are also discussed. Introduction In the mid 1990's, the operator developed a family of practices for managing lost returns that are referred to as the Fracture Closure Stress (FCS) Operational Practices (Dupriest 2005). These practices are now applied uniformly worldwide by operator's affiliates, and the success rate in permeable formations is very high (Dupriest 2005). However, the FCS Practices are discrete treatments as are the majority of industry concepts. Drilling operations must be interrupted to position a discrete pill at the loss zone and non-productive rig time accumulates as the operation is performed. The drill and stress fluid concept is built on many of the basic principles proven in the development of the discrete FCS Practices. However, in the DSF process, the integrity and increase in near-wellbore stress are built continuously through specific attributes of the drilling fluid so that rig time is not lost. The rig time cost savings from continuous treatment is significant. Even greater costs may be incurred if it is necessary to run a string of casing immediately above a low integrity formation to prevent underground flow or borehole collapse following lost returns. When losses occur, the bottomhole pressure immediately declines to equal the fracture closure stress (FCS) of the induced lost returns fracture, which approximately equals the far field stress. The fracture essentially acts as a pressure relief valve and it is not possible to elevate the bottomhole pressure by filling the annulus continuously, regardless of fluid density used. If the FCS in a depleted reservoir is less than the pore pressure in shallower formations, underground flow occurs downward to the loss zone. If the FCS is less than the pressure required to stabilize the exposed shale intervals, the borehole will collapse and the string becomes stuck. Continuous treatment is also needed if long intervals of low stress formations must be penetrated and equivalent mud weight (EMW) cannot be cut below the FCS so that losses occur in every increment of new hole drilled. The operator's discrete practices have been effective in stopping these losses, but they return when drilling resumes and new, low stress, formation is exposed. This is a common issue in high angle, extended reach wells due to the long measured depth required to traverse even relatively thin sands.

Recent studies of foam-diversion processes for matrix acidization identify gas trapping during post-foam liquid injection as the key to foam effectiveness. Laboratory studies identify two degradation processes during this period: a rapid increase in mobility throughout the core, followed by a slower further rise in mobility starting at the core inlet. New coreflood results indicate that pressure gradient VP during the first process is insensitive to a shut-in period after foam injection but depends weakly on liquid flow rate. A small fall in gas saturation accounts for the first rise in mobility. which appears to depend on mobilization of foam bubbles (i.e., on P) rather than foam collapse (i.e., on capillary pressure). The second rise in mobility appears to be due to gas dissolution in injected liquid. Therefore this transition, which is harmful to field performance, can be avoided by including a small amount of gas with the injected acid. Process modeling illustrates that some designs that appear successful in laboratory linear corefloods can perform poorly in the field due to the geometry of radial flow. The fractional-flow approach, together with coreflood pressure data for multiple sections along the core, provides a uniquely simple and insightful framework for interpretation of laboratory results and extrapolation to the field. The simple diversion model of Hill and Rossen 1 can successfully model foam processes in which the secondary degradation of foam during liquid injection is avoided by including gas with the acid. Introduction Foam is used in enhanced oil recovery to reduce gas injectivity and improve sweep efficiency and in well stimulation to divert matrix acid treatments into low-permeability or more-damaged layers. During production operations foams can reduce coning of a gas cap. In matrix acidization foam diverts acid from higher-permeability (or less-damaged) layers to lower-permeability (or more-damaged) layers. The injection sequence can be either (1) surfactant preflush followed by alternating slugs of foam and foam-compatible acid, (2) surfactant preflush followed by alternating slugs of foam and foam-incompatible acid, or (3) continuous injection of foamed acid. Zerhboub et al. report that in laboratory studies of slug processes a brief shut-in time after the foam injection helps diversion of acid. To divert acid, foam must reduce the relative permeability of the liquid phase. Foams do not directly alter water viscosity or the relation between water relative permeability krw and water saturation [1] where krwf is water relative permeability in the presence of foam, krwo is water relative permeability in the absence of foam; henceforth we refer simply to water relative permeability krw, which applies in both situations. However, foams o reduce gas mobility greatly, and as a result Sw and krw are forced down. The reduction of gas mobility stems from two mechanisms:trapping of up to 80-99% of the gas phase even as foam flows at high pressure gradient P.increased effective viscosity of the gas that does flow. The two effects are intimately related, because both depend on the capillary forces on gas bubbles, on bubble size and on pressure gradient. As a result, any distinction between gas relative permeability and as viscosity in a flowing foam is ambiguous. Some report gas mobility with foam to be shear-thinning, and others find it to be nearly Newtonian. Gas trapping during fluid-injection following foam is essential to successful acid diversion. means that effective diversion depends on mechanism (1) above, but the gas saturation and the amount of gas trapped depends on the complex processes of creation, destruction and trapping of bubbles during the preceding foam displacement. These bubble-generation and -destruction mechanisms are not completely understood, but capillary pressure Pc is an important factor in the interplay that determines bubble size. The effect of capillary pressure explains foam's ability to divert flow between the layers differing in permeability, for instance, as the effect of higher Pc in the low-permeability layer weakening the foam there. Modeling Foam Diversion Equation (1) suggests that foam modeling is simple] if one knows water saturation in the presence of foam; one can then compute pressure gradient as a function of flow rates using Darcy's law and (1) without concern for the complexity of gas-phase mobility. One can model some foam displacements remarkably well, for instance, by assuming that Pc and Sw are held fixed within the foam. This model is called the "fixed-limiting capillary pressure" (fixed-Pc*) model. More generally, if one measures Sw or P experimentally, one can determine the other using Eq. (1) if the function krw(Sw) is known. P. 781