Foam for Enhanced Oil Recovery: Modeling and Analytical Solutions

Foam for Enhanced Oil Recovery: Modeling and Analytical Solutions. Foam for Enhanced Oil Recovery: Modeling and Analytical Solutions. Proefschrift. ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, . op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties, . in het openbaar te verdedigen op. dinsdag 20 maart 2012 om 12:30 uur. door. Elham ASHOORI. Master of Science in Chemical Engineering, Sharif University of Technology, Iran. Geboren te Karaj, Iran.. Dit proefschrift is goedgekeurd door de promotoren:. Prof. dr. W.R. Rossen. Prof. dr. D. Marchesin. Samenstelling promotiecommissie:. Rector Magnificus, voorzitter Prof. dr. W.R. Rossen, Technische Universiteit Delft, promotor Prof. dr. D. Marchesin, Instituto Nacional de Matemática Pura e , Rio de Janeiro, promotor. Prof. dr. P. L. J. Zitha, Technische Universiteit Delft . Prof. dr. J. Bruining, Technische Universiteit Delft. Prof. dr. C. J. van Duijn, Eindhoven University of Technology Assoc. prof. dr. ir. S. Kam, Louisiana State University dr. ir. A. Andrianov, Shell Projects & Technology Prof. dr. ir. J. D. Jensen, Technische Universiteit Delft, reservelid . Keywords: Enhanced oil Recovery, Foam in porous media, Population-balance foam model, Method of characteristics.. Copyright © 2012 by Elham Ashoori. All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the author.. Cover design by Anna Peksa and Elham Ashoori (Foam propagation from an injection well to a production well). Printed in the Netherlands.. ISBN: 978-90-8570-986-2.. In the name of the Lord, Life-Creating, . The Wise One, Speech-Creating with the Tongue (Saadi, 1184-1283) . to my parents. to Behzad. . CONTENTS. Contents ........................................................................................................................... vii . 1. Introduction ............................................................................................................... 1 . 1.1 Gas EOR ............................................................................................................. 1 1.2 Foam EOR .......................................................................................................... 2 . 1.2.1 Definition of foam in porous media ................................................................ 2 1.2.2 Water mobility with foam ............................................................................... 2 1.2.3 Gas mobility with foam .................................................................................. 3 1.2.4 Strong foam vs. weak foam ............................................................................ 3 1.2.5 Foam generation and destruction .................................................................... 4 1.2.6 Steady-state strong-foam regimes ................................................................... 5 1.2.7 Foam-injection strategies ................................................................................ 6 . 1.3 Models describing foam behavior in porous media ............................................ 6 1.3.1 Population-balance models ............................................................................. 7 1.3.2 Local-equilibrium models ............................................................................... 9 1.3.3 Fractional-flow methods ................................................................................. 9 1.3.4 Issues with foam-modeling approaches ........................................................ 11 . 1.4 Long-distance foam propagation in EOR processes ......................................... 13 1.5 Research objectives ........................................................................................... 15 1.6 Outline ............................................................................................................... 16 . 2. Can Formation Relative Permeabilities Rule Out a Foam EOR Process? ........ 19 . 2.1 Introduction ....................................................................................................... 20 2.1.1 Method of Characteristics for SAG foam processes ..................................... 20 2.1.2 krw(Sw) function ............................................................................................. 24 . 2.2 Results ............................................................................................................... 26 2.2.1 No foam collapse .......................................................................................... 27 2.2.2 Foam collapses at Sw. * > Swc ........................................................................... 29 2.2.3 Smooth, continuous fw(Sw) function for Sw near Sw* ..................................... 33 . 2.3 Discussion ......................................................................................................... 36 2.4 Conclusions ....................................................................................................... 39 2.5 Nomenclature .................................................................................................... 39 . 3. Fractional-Flow Theory of Foam Displacements with Oil .................................. 41 . 3.1 Introduction ....................................................................................................... 42 3.1.1 Fractional-Flow method ................................................................................ 43 3.1.2 Foam model .................................................................................................. 44 . 3.2 Results with fractional-flow method ................................................................. 46 3.2.1 FCM gas flood with foam ............................................................................. 46 3.2.2 Foam with surfactant dissolved in gas .......................................................... 51 3.2.3 Foam for mobility control in idealized surfactant flood ............................... 59 . 3.3 Simulations ....................................................................................................... 62 . viii Contents. 3.4 Conclusions ....................................................................................................... 66 3.5 Nomenclature .................................................................................................... 67 . 4. Dynamic Foam Behavior in the Entrance Region of a Porous Medium ............ 69 . 4.1 Introduction ....................................................................................................... 70 4.2 Entrance region in foam processes ................................................................... 72 . 4.2.1 Excluding capillary pressure ......................................................................... 76 4.2.2 Including capillary pressure .......................................................................... 78 . 4.3 Foam models ..................................................................................................... 79 4.4 Results for first-order-kinetic model ................................................................. 79 . 4.4.1 Excluding capillary pressure ......................................................................... 79 4.4.2 Including capillary pressure .......................................................................... 81 . 4.5 Results for model of Kam ................................................................................. 82 4.5.1 High velocity co-injection ............................................................................. 82 4.5.2 Low velocity co-injection ............................................................................. 85 . 4.6 Conclusions ....................................................................................................... 87 4.7 Nomenclature .................................................................................................... 88 . 5. Dynamic Foam Behavior in the Traveling Wave of a Porous Medium ............. 91 . 5.1 Introduction ....................................................................................................... 92 5.2 Shock-front formulation in foam processes ...................................................... 93 5.3 Results for no gas ahead of foam ...................................................................... 98 5.4 Results for gas ahead of foam ........................................................................... 99 . 5.4.1 First-order-kinetic model .............................................................................. 99 5.4.2 Model of Kam ............................................................................................. 104 . 5.5 Discussion ....................................................................................................... 113 5.6 Conclusions ..................................................................................................... 118 5.7 Nomenclature .................................................................................................. 119 . 6. Stability Analysis of Uniform Equilibrium Foam States for EOR Processes.. 123 . 6.1 Introduction ..................................................................................................... 124 6.2 Governing equations for foam flow in porous medium .................................. 127 6.3 Stability ........................................................................................................... 128 . 6.3.1 Stability analysis of uniform states in population-balance foam flow ........ 129 6.4 Results ............................................................................................................. 131 . 6.4.1 LE Linear-stability analysis of uniform states in LE foam flow ................ 131 6.4.2 Linearized stability of LE states in Kam (2008) model .............................. 132 . 6.5 Nonlinear stability of LE states in Kam’s model ............................................ 140 6.6 Discussion ....................................................................................................... 142 6.7 Conclusions ..................................................................................................... 143 6.8 Nomenclature .................................................................................................. 144 . 7. Multiple Foam States and Long-Distance Foam Propagation in EOR Displacements ................................................................................................................ 147 . 7.1 Introduction ..................................................................................................... 148 7.2 Governing equations in foam processes ......................................................... 151 7.3 Results ............................................................................................................. 152 . Contents ix. 7.3.1 Transient dynamic-foam simulation ........................................................... 153 7.3.2 Traveling-wave solution ............................................................................. 162 . 7.4 Discussion ....................................................................................................... 168 7.5 Conclusions ..................................................................................................... 169 7.6 Nomenclature .................................................................................................. 170 . 8. Conclusions & Recommendations ....................................................................... 173 . 8.1 Conclusions ..................................................................................................... 173 8.1.1 Impact of the water relative permeability on the success of SAG process . 173 8.1.2 Fractional-flow solution of foam displacement with oil ............................. 174 8.1.3 Dynamic foam behavior in the entrance region of a porous medium ......... 175 8.1.4 Dynamic foam behavior in the traveling wave of a porous medium .......... 175 8.1.5 Stability analysis of uniform states in population-balance foam flow ........ 177 8.1.6 Multiple foam states and long-distance foam propagation in EOR displacements .......................................................................................................... 178 . 8.2 Recommendations ........................................................................................... 179 . Bibliography .................................................................................................................. 181 . Appendix A: Mass Balances at Fractional-Flow Shocks in Presence of Oil ............ 193 . Appendix B: First-Order-Kinetic Model .................................................................... 199 . Appendix C: Kam Model ............................................................................................. 203 . Appendix D: Mathematical Solution of Nonlinear System of ODEs ....................... 207 . Appendix E: Linearized Stability of LE States in Kam (2008) Model for Perturbations with Extreme Wave Lengths ............................................................... 213 . Appendix F: Derivation of Linear Stability Condition in Presence of Artificial Diffusion ......................................................................................................................... 217 . Appendix G: Derivation of Linear Stability Condition in Finite-Difference Scheme ......................................................................................................................................... 219 . Appendix H: Numerical Scheme for Finite-Difference Simulations of Strong-Foam Propagation ................................................................................................................... 221 . Scientific Contribution ................................................................................................. 225 . Summary ........................................................................................................................ 227 . Samenvatting ................................................................................................................. 231 . Acknowledgements ....................................................................................................... 235 . About the Author .......................................................................................................... 238 . 1. INTRODUCTION . 1.1 Gas EOR. A "reservoir" is a subsurface body of naturally occurring hydrocarbons (mainly oil and. gas) contained in the pores of a porous rock formation, which is confined by overlying. impermeable rock layers. Oil-gas reservoirs are generally found between 1 and 5 km. below the surface. The oil is initially produced by the natural drive mechanisms of the. pressurized reservoir, this period is referred to as primary production. So-called. secondary-recovery methods include the water or gas injection into the aquifer or gascap. aiming mainly at maintaining the reservoir pressure to keep oil and gas production rates. economical. In producing oil from a reservoir, on average 50-85% of the oil originally in. place is left behind in the reservoir after primary and secondary recovery, depending on. the production method (Lake et al. 1989). More advanced techniques employed to extract. more oil/gas from the reservoir are named Enhanced Oil Recovery (EOR) methods.. These techniques improve sweep and displacement efficiencies by altering the fluids. properties and/or their interactions with rock. . One of the most applied EOR methods is gas flooding. Gas flooding can in principle be. more effective than water flooding, as the residual oil saturation to gas is usually lower. than to water. In particular, in miscible gas drives, using for example carbon dioxide. (CO2), flue gas or reinjected field gas, at sufficiently high pressure, the fluid interface. between the oil and the gas components disappears; their interfacial tension becomes zero. and capillary forces disappear, creating a single oleic phase. Since the displacement. occurs entirely within one single hydrocarbon phase these methods can be 100% efficient. on microscopic scale, leaving no residual oil behind (Orr et al. 1982).. Actual oil recovery with miscible EOR, however, is far from 100% efficient. The highly. unfavourable mobility ratio together with density differences between the driving gas and. reservoir fluids leads to viscous fingering owing to frontal instability and gravity. override, ultimately resulting in poor oil recovery. Additionally, reservoir heterogeneities. can cause regions of the reservoir to be uncontacted by the gas due to the inherent. 2 1: Introduction. tendency of a highly mobile gas to flow through the more-permeable rock sections.. Furthermore, viscous fingering or channelling can also cause operational problems by. allowing large volumes of injected gas to shoot through the formation to the production. wells (Rossen 1996).. 1.2 Foam EOR. The application of foam, a dispersion of gas in a surfactant solution, can significantly. improve oil recovery by mitigating the adverse effects of gas fingering and channelling. and by reducing the effects of reservoir heterogeneities (Rossen 1996). The idea of using. foam for mobility control in EOR was first proposed in 1958 by Bond and Holbrook. (Bond and Holbrook 1985; Lake et al. 1992). Foam has several other applications in oil. production, including drilling (to transport cuttings to the surface), near-wellbore well-. stimulation treatments (to divert acids to improve the acid-injection profile) (Gdanski. 1993; Zhou and Rossen 1994), and remediation treatments in an aquifer (Hirasaki et al.. 2000). In this work, however, mobility-control aspects of foam in EOR are of interest.. 1.2.1 Definition of foam in porous media. The creation of foam needs surfactant, water and gas. Foam in porous media is defined. as "a dispersion of gas in liquid such that the liquid phase is interconnected and at least. some of the gas flow paths are blocked by thin liquid films called lamellae", according to. Falls et al. (1988). The lamellae are normally of order 10-100 nm thick; surfactants. adsorbed at the gas/liquid interface stabilize lamellae separating bubbles. Foam bubbles. in porous media normally are at least as large as the individual pores. If bubbles smaller. than pores existed, the smaller bubbles would merge rapidly into larger ones, due to gas. diffusion, until remaining bubbles were as large as pores (Rossen 1996).. Foam is not considered as a separate phase from water and gas in oil-field rock. Upon. entering the porous medium, most of the liquid separates from the gas. A small amount of. the liquid remains with the gas phase as lamellae (if present) and Plateau borders. (prismatic region where lamellae touch the rock or each other). The bulk of the liquid. fills the same smaller pores that it would occupy in the absence of foam, and the gas, with. any lamellae, occupies the larger pores. Thus "foam mobility" means the separate. mobilities of liquid and gas in the presence of foam.. 1.2.2 Water mobility with foam. Since most of the aqueous phase is carried through the small, completely liquid-filled. channels and only a small amount of liquid transports as lamellae and Plateau borders, the. 1: In. relat. wide. de V. redu. Non. wett. (Ros. 1.2.3. Foam. can. and. effec. main. med. drag. cons. cont. Notw. inter. keep. Figur water rheol. 1.2.4. "Foa. the. mob. ntroduction. tive-permea. ely reported. Vries and W. ucing the liq. nlinearity of. tability of th. ssen 1996).. 3 Gas mo. m significan. reduce gas. Rossen 19. ctive gas p. nly reflects. dium (Kovsc. g that flowi. strictions a. tinuously a. withstandin. rtwined, for. ping the oth. re 1.1 Concep r two-phase f logy (from Dh. 4 Strong. am texture". foam-bubbl. bility is redu. ability funct. d in foam ex. Wit 1990; Fri. quid saturati. f the liquid-. he rock; fo. obility with. ntly reduce. mobility as. 995). Effect. permeability. gas trappin. cek et al. 19. ing bubbles. and because. altered by v. ng the fact t. r simplicity. her constant.. pt of weak and flow: the num holkawala et a. foam vs. we. ", defined as. le populatio. uced by foa. tion and visc. xperiments. iedman et al. ion. . -phase relat. ams have p. foam . s gas mobil. s much as 3. ts of gas m. y and effect. ng, which i. 995). The e. s face whil. e the gas/. viscous and. that the eff. , foam rheo. .. d strong foam mber of lamel l. 2007).. eak foam. s the numbe. on. The "s. am; strong f. cosity of liq. (Bernard et. l. 1991). Fo. tive-permeab. poor efficien. lity depend. 3 to 5 order. mobility are. tive gas vis. impedes som. ffective gas. le passing. /liquid inte. d capillary. fective perm. ology is ofte. ms within poro llae per unit. er of lamell. strength" of. foam is a st. quid are una. t al. 1965; S. oam indirect. bility functi. ncy in reser. ding on the. rs of magni. ascribed to. scosity. The. me of the g. s viscosity a. in direct c. erfacial area. forces (H. meability an. en represen. ous media in c volume (or, f. lae per unit. f foam des. tate at whic. affected by. Sanchez and. tly reduces l. ion is stron. rvoirs that a. size of foam. tude in the. o two facto. e effective. gas-flow pa. accounts for. contact with. a of a flo. Hirasaki and. nd viscosity. nted by vary. comparison w foam texture). t volume, is. cribes how. ch foam is f. foam. This. d Schechter. liquid mobil. ngly related. are not wat. m bubbles.. laboratory. ors: alteratio. gas perme. aths in the p. r the consid. h pore wall. owing bubb. d Lawson. y are inextr. ying one of. ith convention ) is the key t. s used to qu. w much gas-. fine-texture. 3. fact is. 1989;. lity by. to the. ter-wet. Foam. (Zhou. ons in. ability. porous. derable. ls and. ble is. 1985).. ricably. f them,. nal gas- to foam. uantify. -phase. ed (has. 4 1: Introduction. small bubbles) with a substantial increase in pressure gradient and (at steady state). relatively low-water saturation, whereas weak foam is a state in which foam is coarsely. textured with a slight reduction in gas mobility (or slight increase in pressure gradient). with relatively high water saturation (Kam 2008). Figure 1.1 contrasts conventional gas-. water flow with weak- and strong- foam flow in porous media.. 1.2.5 Foam generation and destruction. Gas mobility depends on bubble size, which in turn depends on several mechanisms. controlling the creation and destruction of the lamellae that separate bubbles. The. destruction of lamellae is directly linked to the capillary pressure in that the destruction. rate increases as capillary pressure increases; in particular, Khatib et al. (1988) report that. there is a sharp transition from full-strength foam to greatly weakened or completely. collapsed foam. They define a specific "limiting capillary pressure", or Pc *, below which. foam stability is not affected and above which foam abruptly collapses. . Lamella creation is a necessary part of the foam-generation process. Lamellae can be. created by several processes, including "snap-off" (creation of a new lamellae by. fluctuations in capillary forces in pore throats); "leave behind" (lenses of liquid stranded. in pore throats during drainage of adjacent pores saturated with surfactant solution); and. "lamella division", which occurs when a moving lamella (mobilized by a sufficient. pressure gradient) deposits new lamellae in unoccupied pore throats adjacent to the flow. path. Details of these processes can be found elsewhere (Rossen 1996). Some studies. indicate that the essential step in foam generation is the mobilization of lamellae as a. result of relatively high pressure gradients, initiating the lamella-division process, which. itself results in a reduction in gas mobility (Gauglitz et al. 2002; Kam and Rossen 2006).. The creation of new lamellae and reduction in gas mobility are interlinked and reinforce. each other cyclically; if injection rates are fixed, creation of new lamellae in turn leads to. higher pressure gradients, more mobilization and division, and eventually an abrupt jump. from weak-foam state to strong-foam state. Gauglitz et al. (2002) report the behavior. illustrated in Figure 1.2 (right) if one fixes pressure drop across the porous medium. instead. Figure 1.2 illustrates three possible steady states at some injection rates: (1) a. weak-foam state with small pressure gradient, (2) a strong-foam state with large pressure. gradient and (3) an intermediate state between them which appeared to be unstable in the. experiment.. If one fixes foam quality (injected gas volume fraction) rather than pressure drop, one. observes a jump from coarse foam to strong foam as in Figure 1.2 (left), i.e., the abrupt. 1: Introduction 5. change in state called "foam generation" (Ransohoff and Radke 1988; Rossen and. Gauglitz 1990; Tanzil et al. 2002). This jump happens at lower injection rate if the quality. of the injected fluids is reduced. The reason could be that the initial population of liquid. lenses is larger and, as a result, more lamellae are available for mobilization at higher. injected liquid fraction (Rossen and Gauglitz 1990). . Figure 1.2 Left: Schematic of coreflood experiment at fixed foam quality and increasing injection rate, where there is a jump from weak foam to strong foam, followed by the core staying at the strong-foam state when injection rate is reduced. Right: laboratory coreflood data taken at fixed foam quality and increasing pressure gradient, where multiple steady states are revealed at the same injection rate.. Figure 1.3 Pressure gradient (psi/ft) as a function of gas and water superficial velocities ug and uw, from study of strong foam in Berea sandstone, from Alverz et al. (2001). The high-quality regime corresponds to the upper-left portion of the plot, where pressure gradient is nearly independent of ug; the low-quality regime corresponds to the lower-right portion of the plot, where pressure gradient is nearly independent of uw. Dark points represent data.. 1.2.6 Steady-state strong-foam regimes. Several experimental studies indicate that steady-state strong-foam behavior falls into. two flow regimes (Osterloh and Jante 1992; Alverz et al. 2001; Rossen and Wang 2002;. 6 1: Introduction. Mamun et al. 2002): high-quality and low-quality regimes, shown in the upper-left. portion and in the lower-right portions of Figure 1.3, respectively. Capillary pressure,. close to Pc *, controls the high-quality regime. Since capillary pressure and water. saturation are related in porous media, the value of Pc * corresponds to a limiting water. saturation, Sw *. Because Pc is nearly constant near Pc. * over a range of superficial. velocities, Sw is likewise nearly constant near Sw * under these same conditions. As a. result, pressure gradient is nearly independent of gas superficial velocity when Pc is near. Pc *. Foam texture in this regime is very sensitive to injection rates and especially to Pc,. while it is thought that in the low-quality regime bubble size is fixed at some low limit,. possibly at roughly pore size. Pressure gradient is independent of water velocity in this. regime as shown in Figure 1.3.. 1.2.7 Foam-injection strategies. Co-injection of surfactant, water and gas, and SAG (Surfactant-Alternating-Gas). injection, where alternating slugs of surfactant-enriched water and gas are injected into. the formation are the main foam-injection strategies. This latter approach has the. advantage of increased injectivity over co-injection because of foam dry-out in the near-. well region during gas phase injection (Shi and Rossen 1998; Rossen et al. 1995). Also,. SAG processes benefit from reduced corrosion in surface facilities and pipes (Matthews. 1989). A third unconventional method for injecting foam is by means of dissolving. surfactant in the injected CO2. Studies show that some surfactant can be dissolved in. supercritical CO2 and foam can be generated in situ upon contact of the CO2/surfactant. solution with water already present in the reservoir (Le et al. 2008). There are many. suppliers of conventional water-soluble surfactants that are adequately soluble in CO2 for. large-scale application (Xing et al. 2010). More details on the recent findings about these. surfactants can be found in Xing et al. (2010). The benefits of this approach are three-. fold: first, since injection of water has become superfluous, it has better injectivity. compared to both SAG injection and co-injection processes; second, corrosion to surface. facilities and pipes is much less of an issue; third, alternatively, the presence of surfactant. in both phases enhances the likelihood of in-situ foam generation for a SAG process in a. heterogeneous reservoir. Le et al. (2008) claim that this method reduces the loss of. surfactant onto the rock face due to adsorption. . 1.3 Models describing foam behavior in porous media. Foam EOR has been successfully employed during field steam-foam pilots in the late. 1980’s, several non-thermal applications of foam in the mid 1990’s (e.g., Djabbarah et al.. 1: Introduction 7. 1990; Friedmann et al. 1994), a chemical and foam flooding for EOR in Daqing Oilfield. in China (Wang et al. 2001a; 2001b), and more recently, the foam assisted water. alternating gas project (FAWAG) in the Snorre field in the North Sea (e.g., Blaker et al.. 2002). For a more general review of foam field trials see Shan and Rossen (2004). These. encouraging results highlight the potential of foam application for EOR and the. importance of predictive foam models for quick project evaluation and efficient process. design.. A variety of empirical and theoretical methods for modeling foam displacement are. available in literature. These range from complex population-balance models (Falls et al. 1988; Patzek 1988; Chang et al. 1990; Friedmann et al. 1991; Kovscek et al. 1995; Fergui. et al. 1995; Kam et al. 2007b; Kam 2008; Chen et al. 2010) to percolation models. (Rossen 1990; Rossen and Gauglitz 1990; Chou 1990) and from applying so-called. fractional-flow methods (Zhou and Rossen 1995; Kam and Rossen 2003; Shan and. Rossen 2004; Rossen et al. 1999; Dholkawala et al. 2007; Kam 2008; see also chapters 2. and 3) to simple (semi-empirical) alteration of gas-phase mobilities (Fisher et al. 1990;. Mohammadi et al. 1993; Rossen et al. 1999; Patzek and Myhill 1989). Of all these, we. contrast fully-mechanistic population-balance models with local steady-state foam. models. Next fractional-flow methods are discussed.. 1.3.1 Population-balance models. Population-balance models aim to completely describe foam dynamics by incorporating. all the dominant mechanisms influencing foam-bubble texture, which in turn controls. foam mobility. Thus besides gas mobility, the population-balance model explicitly tracks. the evolution of bubble size through a conservation equation for a number of lamellae. In. this partial-differential equation (PDE), local bubble density depends on the rate of. influx, efflux, creation, destruction and trapping of lamellae. Several investigators have. successfully applied the volume-averaged bubble population equation to describe. laboratory experiments (Falls et al. 1988; Friedmann et al. 1991; Kovscek et al. 1995;. Fergui et al. 1995; Kam and Rossen 2003; Chen et al. 2010).. Although population-balance models are mathematically complete they are hard to. implement. Population-balance simulators require numerous parameters to model lamella. creation and destruction as well as foam mobility at a given texture, though in practice. only some of these parameters can be history-matched to coreflood data. A good fit of a. multi-parameter model to data cannot verify the accuracy of all parameters used. In. particular, there is an inherent ambiguity between the terms describing lamella creation. 8 1: Introduction. and destruction in any population-balance foam simulation; the reason is that, like any. mass-balance equation with source and sink terms, an error in one can be fixed. unwittingly by adjusting the other (Zeilinger 1996). . Variations of the population-balance models differ mainly in their foam-generation terms,. not in their foam destruction terms, which are all based on an abrupt collapse of foam. around Pc *. Some population-balance models are based on lamella creation by repeated. "Roof" snap-off, governed by an aspect ratio between pore throats and adjacent pore. bodies (e.g, Kovscek and Radke 1994; Kovscek et al. 1995; Bertin et al. 1998). These. models assume that snap-off continues to create lamellae during steady-state foam flow. in homogeneous porous media. This explanation for foam generation is called to question. (Rossen 2003, 2008), and several studies have shown that in homogeneous porous media. sudden drop of gas mobility in presence of foam is more probably caused by mobilization. of lamellae and subsequent division rather than primarily by Roof Snap-off (Rossen and. Gauglitz 1990; Gauglitz et al. 2002; Tanzil et al. 2002; Rossen 2003). Moreover, caution. should be taken in judging a model by its success in following the trend of high-quality. foam data since in that region capillary pressure is dominant and therefore, a small. change in Pc results in a huge change in lamella-destruction rate, accommodating large. errors in other terms such as lamella-generation rate (Rossen 2003). . 1.3.1.1 Kam population-balance foam model . Kam and Rossen (2003) propose a population-balance model incorporating bubble. creation controlled by pressure gradient, in line with the theory of "lamella mobilization. and division" (Rossen and Gauglitz 1990). Lamella creation requires a minimum pressure. gradient; above the threshold pressure gradient creation increases rapidly with increasing. pressure gradient, reflecting lamella mobilization and division; however, the high-quality. foam regime is difficult to simulate using this model because of numerical instability.. Furthermore, a separate condition (a minimum bubble size) has to be imposed to the. bubble size found from the population-balance equation to avoid creating finer bubbles. than pore size. . Kam (2008) modified the model described above by changing the pressure dependency of. the foam-generation term with a function that levels off at high pressure gradients. It fits. a variety of local-equilibrium experimental data: foam generation above a threshold. pressure gradient (Rossen and Gauglitz 1990), multiple foam steady states at given. injection rates (Gauglitz et al. 2002), and the two flow regimes for strong foam depending. on injected water fraction (Alvarez et al. 2001). It also confirms the instability of the. intermediate state seen in experiment (Gauglitz et al. 2002). Due to all these. 1: Introduction 9. achievements, we use the foam model of Kam (2008) in Chapters 4-7; details of this. model are in Appendix C.. 1.3.2 Local-equilibrium models. A simpler alternative to population-balance modeling is to assume local equilibrium (LE). (equal generation and destruction rates) at all locations in the formation. Marfoe and. Kazemi (1987) proposed one of the first LE models, which were improved by Islam and. Farouq Ali (1990). LE models use an algebraic relation (which could be empirical or. physically based) between gas mobility and factors determining foam texture, for. example water saturation, surfactant concentration, absolute rock permeability,. superficial velocities, pressure drop, etc. LE modeling is much less complex than. population-balance modeling (Kam et al. 2007b).. Vassenden and Holt (1998) and Cheng et al. (2000) show that an LE model can predict. the two strong-foam regimes seen in experiments (Osterloh and Jante 1992; Alverz et al.. 2001; Rossen and Wang 2002; Mamun et al. 2002) 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 existence (Rossen and Bruining 2007). Of course, LE models are not adequate for. representing regions in the porous medium with net foam generation such as the entrance. region where foam is created, though the effect of the entrance region is expected to be. insignificant on field scale.. 1.3.3 Fractional-flow methods. The method of characteristics (MOC) applied to injection processes involving two-phase. flow gives rise to fractional-flow theory. The method of characteristics is a method for. solving partial differential equations by reducing the system of differential equations to a. family of ordinary differential equations along which the solution can be integrated. It. embodies and illuminates the fundamental behaviour of such flows.. Fractional-flow theory was first applied to foam displacements by Zhou and Rossen. (1994). Since then, a number of studies have further developed the theory to account for. different applications in foam EOR (e.g, Hill and Rossen 1994; Zeilinger et al 1995;. Zhou and Rossen 1995; Kibodeaux and Rossen 1997; Shan and Rossen 1994;. Dholkawala et al. 2007; Kam 2008). Assumptions of the fractional-flow method include. incompressible phases; Newtonian mobilities; 1D displacement; absence of dispersion,. gradients of capillary pressure, and viscous fingering; and immediate attainment of local. steady state. Rossen et al. (1999) shows that even when the fractional-flow assumptions. 10 1: Introduction. are not satisfied completely, fractional-flow theory is still able to capture complex. displacement mechanisms in foam EOR.. The basic mathematical principles of the Buckley-Leverett displacement are retained;. foam flow comprises spreading or shock waves of fixed phase saturation. Saturations. advance through the medium with dimensionless velocities equal to the slope of the. fractional-flow curve at that saturation. A valid, single-valued solution must have. monotonically increasing slope (velocity) from upstream to downstream state. Buckley-. Leverett shocks are discontinuities in saturation along a fractional-flow curve. They are. formed where rearward waves overtake forward waves of lower velocity. There are two. rules governing shocks: first, algebraic conditions derived from material balances on. components involved in displacement, e.g., water, surfactant and gas, must be satisfied at. the shocks. Second, the shock-front profile must construct a valid traveling wave.. Based on the fractional-flow functions, results can then be characterized in terms of. saturations. Changes in saturation (and bubble texture, here) are called waves. Depending. on their spreading characteristics, waves may be classified into the following categories. (Lake 1989; Lake and Bryant 2002): . Spreading waves (or rarefraction waves): the shape of these waves becomes more. spread in proportion to time and distance, without limit. . Self-sharpening waves: the shape of these waves becomes less diffuse on propagation in. time and distance. Shocks are the result of self-sharpening waves. In the absence of. dispersion and source terms, these waves become discontinuities even on the small scale.. With dispersion and source terms, on the small scale, the shock is a sharp but continuous. transition with a constant shape, in which the effects of dispersion or source term are. balanced by the self-sharpening effects of the wave. . Indifferent waves (or contact waves): in the absence of dispersion, these waves may be. discontinuous, like shocks. With dispersion, the discontinuity is replaced by a continuous. transition whose width grows without limit as the square root of time or distance.. Miscible fronts and, in the absence of adsorption, surfactant-concentration fronts are. indifferent waves. . The Buckley-Leverett solution for a waterflood usually consists of a shock adjacent to a. spreading wave. Lake (1989) calls it a mixed wave. . 1: Introduction 11. A sequence of rarefaction waves, shock waves and constant states that connects the. upstream and downstream states in a displacement, following so-called fractional-flow. rules, is called a "Riemann solution" (Riemann 1860).. 1.3.3.1 Three-phase-fractional-flow methods for foam. The application of fractional-flow theory to foam has mostly been limited to two-phase. flow of gas and water; in some cases oil is assumed to be present but at irreducible. saturation. Application to three-phase flow with foam is limited to a few recent studies. due to its complexity (Mayberry et al. 2008; Namdar Zanganeh et al. 2010). In the case of. miscible displacements, Walsh and Lake (1989) show how three-phase displacements can. be simplified as two-phase displacements with an aqueous phase and a non-aqueous or. oleic phase; the non-aqueous phase being either oil or gas. A miscible gas-drive with CO2. could be an example of this case. For these cases, the solution method is identical to. conventional two-phase fractional-flow theory, except that multiple fractional-flow. curves exist depending on the number of components (e.g, water, gas, oil and surfactant). . 1.3.4 Issues with foam-modeling approaches. Population-balance and fractional-flow models have been shown to produce comparable. results when modeling foam at the field scale and even at the laboratory scale (Rossen et. al. 1999; Kam 2008). Fractional-flow solutions have proved useful in highlighting key. mechanisms and strategies for improving foam performance (Zhou and Rossen 1994,. 1995; Rossen et al. 1999; Shan and Rossen 2004), and in better understanding foam. simulation models (Rossen et al. 1999; Dong and Rossen 2007). On the other hand,. population-balance models are a framework for a complete description of foam, but. computationally expensive: several PDEs, including mass balances on all the components. and a bubble-conservation equation, have to be solved simultaneously at each time step in. a population-balance simulator. Therefore, it is important to determine where the added. complexity of this method is essential in foam displacements. One strategy, described in. this dissertation, is to switch from LE models to population-balance models wherever,. due to the dominant role of non-LE effects, it is worth the increased computational cost.. Only population-balance models can represent foam dynamics where local-equilibrium. does not apply, for instance in the entrance region near the injection face where foam. generation occurs and at shock front where there is an abrupt change of state. Population-. balance models help to determine which LE foam state appears when multiple steady-. states are possible; they are also necessary to evaluate the stability of the individual. states. . 12 1: Introduction. 1.3.4.1 Entrance region. The entrance region is a region near the well or injection face of a coreflood, where gas,. water and surfactant create foam, or where finely textured pre-generated foam may come. to a coarser texture. For the former, the entrance region may be observed as a region of. greater water saturation near the entrance, measured for instance by CT imaging. (Minssieux 1974; Kovscek et al. 1995; Nguyen et al. 2003; Chen et al. 2010), or a region. of smaller pressure gradient than that downstream (Minssieux 1974; Friedmann and. Jansen 1986; Falls et al. 1989; Chou 1991; Ettinger and Radke 1992; Kibodeaux et al.. 1994; Kovscek et al. 1995; Myers and Radke 2000; Farajzadeh et al. 2009; Nguyen et al.. 2003; Kam et al. 2007b). Since net foam generation occurs in the entrance region, LE. models are unable to describe this region. On reservoir scale, however, an entrance region. of order of centimeters in length, as seen in the laboratory, is insignificant.. 1.3.4.2 Shock front (Traveling wave). On the large scale, a shock is a discontinuity in saturation and foam texture for foam. processes. On the small scale however, a discontinuity is replaced by a region where the. shock-promoting viscous pressure gradients come into balance with the spreading. properties of capillary pressure and foam kinetics to form a continuous saturation profile. that moves in pure translation, without change of shape (Courant and Friedrichs 1948;. Gelfand 1963; Rossen and Bruining 2007). The fully-established small-scale. representation of this transition is called a traveling wave. . There are two mechanisms that keep the foam traveling wave from collapsing into a. discontinuity: capillary-pressure gradients and finite foam kinetic rates. LE models are. incapable of capturing the role of non-LE foam-kinetics in the transition within the. traveling wave. Rossen and Bruining (2007) resolve the foam traveling wave using the. LE assumption within the shock, i.e., only capillary-pressure gradients are accounted for,. not foam dynamics, within the traveling wave.. Rossen and Bruining (2007) show cases for which solving for the traveling wave is. necessary to guide a correct construction of the shock in the 1D solution for a foam. displacement. Solving for the traveling wave could be useful for 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 foam. processes, most mobility control occurs within the shock itself (Shan and Rossen 2004);. in such cases success on the field scale could depend on the width and foam strength. within the traveling wave. . 1: Introduction 13. 1.3.4.3 Feasible stable LE state among multiple steady states. As discussed above, experiments show that foam can exist in three different states (weak. foam, intermediate foam, and strong foam) at the same injection rate (see Figure 1.2). In. the process of co-injection of gas and water, the key question is which of these LE states,. with orders of magnitude difference in mobility, would be observed in an experiment or. the field. One might believe that the only method that can assure which LE state would. occur in a certain injection process is to run a fully-mechanistic population-balance. simulation. Numerical simulations can introduce numerical artifacts, however. One can. construct a graphical fractional-flow solution using any possible LE state. However, for. example, to determine which LE state constructs a possible shock with a given state, the. Welge (1952) or Oleinik (1957) condition (i.e., that the shock line cannot cut through the. fractional-flow curve) might give some insights in ruling out some of the LE states. . Another prerequisite for feasibility of an LE state is its intrinsic stability. The. intermediate foam state (among the three possible LE states) is found to be unstable in the. lab, whereas the weak- and strong-foam states are stable (Gauglitz et al. 2002). It is. important to study model stability because 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; ignoring such instabilities would have. considerable effect on predictions of reservoir sweep efficiency and injection pressure. If. the model is unstable but the physical foam state stable, it indicates a serious problem for. the model.. Foam is reported to be unstable in the presence of oil (Law et al. 1992; Schramm 1994;. Rossen 1996; Mannhardt and Svorstøl 1999). "Foam stability" in that context indicates. whether an LE strong-foam state exists in the presence of a given saturation of a given. oil. This is the definition of foam stability used in chapter 3. However, it is also important. to investigate whether an LE foam state that would be stable to oil is also stable to slight. perturbations in foam texture and saturation that occur naturally in flow in the field. In. this dissertation (except for chapter 3), an LE state is called stable if the state moves back. toward the same equilibrium when it is slightly perturbed in saturation and/or bubble. texture. A fully-mechanistic population-balance model is required to investigate the. stability of an LE state because only it can test stability to non-LE perturbations.. 1.4 Long-distance foam propagation in EOR processes. Application of foam falls into two basic approaches: one is to plug unproductive reservoir. layers near the injection well with relatively small volumes of foam to divert flow in the. 14 1: Introduction. near-well region. The more ambitious goal is mobility control throughout the entire. reservoir, wherever oil resides. This latter goal requires that the foam created near the. well travels long distances and lasts over periods of months or years. A practical. definition of foam propagation is "the growth of the region of low gas mobility",. according to Rossen (1996).. Studies report that higher injection velocities favor the strong-foam state (Friedmann and. Jensen 1986; Rossen and Gauglitz 1990; Gauglitz et al. 2002; Simjoo et al 2011); as. superficial velocity is increased, there is a velocity at which a jump from the weak-foam. state to the strong-foam state occurs, as shown in Figure 1.2 (Gauglitz et al. 2002). It is. important to determine whether strong foam created near an injection well can propagate. to large distances from the well, where superficial velocity is much smaller.. There is no theoretical study of long-distance foam propagation in the literature that. captures the jumps between strong-foam and weak-foam states. Kovscek et al. (1997). simulate a large-scale foam displacement with a population-balance model that represents. only a strong-foam state. Friedmann et al. (1991) incorporate a trigger for foam. generation based on superficial velocity, but no foam-destruction mechanism.. There are conflicting data on the ability of foams to propagate at low superficial. velocities. Friedmann and Jensen (1986) report that foam can propagate at superficial. velocities too low for generation in a 10 d-permeability sandpack initially filled with gas. and water at the injected fractional flow. However, the rate of propagation is too slow for. an economical mobility-control foam processes, and slower than the rate of surfactant and. gas propagations. Contrary to Ettinger and Radke's results in consolidated sandstone. (1992), Friedmann and Jensen find that injected foam texture affected both foam mobility. and propagation rates in sand packs. Friedmann et al. (1991) find that foam created at. high velocity in a variable-diameter sandstone core could propagate at 20 to 30 times. lower superficial velocity than that at which it could be created from a state of no-foam. (cf. Rossen and Gauglitz 1990); foam propagated slowly, behind the surfactant front,. however, which suggests some difficulty in propagation. In contrast, Friedmann et al.. (1994) find that foam does not propagate through a cone-shaped sandpack, where. superficial velocity decreases with distance from the injection face. Shi (1996) and. Gauglitz et al. (2002) find hysteresis in their foam propagation experiments; they show. that strong foam can persist at lower velocities than those at which it can be generated in. first place. Simjoo et al. (2011) report similar results for experiments on Bentheimer. sandstone at relatively lower injection velocities.. 1: Introduction 15. Besides the effect of superficial velocity on foam propagation, the foam front can also be. slowed by a slow-moving surfactant front (Irani and Solomon 1986; Kovscek et al. 1993;. Kovscek et al. 1997), the destructive effect of oil on foam (Aarra et al. 1997; Mannhardt. and Svorstøl 1999; Vassenden et al. 1999), and/or gas tapping (Friedmann et al. 1991).. 1.5 Research objectives. The specific objectives of this research are stated as follows:.  To gain insight into the effect of the water relative-permeability function on the success of SAG foam process from fractional-flow methods..  To analytically solve for foam flow with oil that is miscible with gas using two- phase fractional-flow methods, assess the power of this method against 1-D. simulations, and finally propose optimal process designs..  To propose an approach between fully-mechanistic population-balance models and local-equilibriums models, i.e., assuming LE except for an entrance region. and shock fronts..  To solve for water saturation and bubble texture within the entrance region, to study the impact of capillary pressure and foam-kinetic rates on the length of the. entrance-region, and to investigate the possibility of ruling out some of the LE. foam states in the multiple-steady-state Kam (2008) model. .  To extend the LE-foam traveling-wave solution of Rossen and Bruining (2007) to non-LE traveling-wave solutions, and to study the influence of non-LE foam. dynamics as well as capillary dispersion within the front in foam model of Kam. (2008). .  To investigate the stability of different LE foam states using the foam model of Kam (2008), and assess the effect of diffusion, whether introduced artificially by. the finite-difference scheme or representing physical dispersion, on the stability of. foam states..  To study long-distance strong-foam propagation with finite-difference simulations and Riemann solutions, applying a population-balance foam model of Kam (2008). that represents the multiple steady states of foam. . 16 1: Introduction. 1.6 Outline. This thesis is based on six refereed journal articles published or to be published by the. author describing results in the area of foam EOR. The thesis consists of 8 chapters,. starting with Chapter 1 as the introduction.. Chapter 2 shows how the water relative-permeability function for a formation, which is. independent of the foam process, can place limits on the mobility reduction achievable. during gas injection in SAG foam processes for any conceivable foam formulation. This. limit is derived from the geometrical constraints on possible positions of a point of. tangency in gas injection in a SAG process in the fractional-flow diagram. One can. determine total mobility directly from the coordinates of this point if the relative-. permeability function is known.. Chapter 3 evaluates foam-process efficiency in one dimension (1D) with two-phase. fractional-flow methods for three examples: a first-contact-miscible gas flood with foam. injection, where oil may or may not destroy foam; a first-contact-miscible gas flood with. foam, with surfactant dissolved in the "gas" (e.g., supercritical CO2); an idealized. surfactant flood using immiscible-gas foam for mobility control. For each of these cases. we derive the fractional-flow solution for the foam displacement and find the water-. saturation distribution. We investigate what controls the mobility and velocity of the. foam bank and propose optimal process designs (in the context of idealized 1D. displacements). We compare the solutions obtained against fine-grid numerical. simulations using the same models.. Chapter 4 gives the analytical solution for steady-state water saturation and foam texture. in the entrance region accounting for foam kinetics, employing two different foam. models: a simple schematic first-order-kinetic model and a more-realistic population-. balance model with multiple steady states (Kam 2008). We show how consideration of. the entrance region can rule out some candidates for the state downstream of the entrance. region. That is, it shows that certain foam states could not be produced inside the porous. medium by injection of surfactant solution and gas at a given superficial velocity. We. also examine sensitivity of the entrance region to the kinetic parameters in the model and. fit those parameters to the length of the entrance region typical of coreflood experiments.. Chapter 5 resolves the foam shock accounting for non-equilibrium foam kinetics within. the traveling wave using the same models used in Chapter 4. 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 find a range of conditions in which the local-equilibrium. 1: Introduction 17. condition applies even within the traveling wave. We compare the traveling-wave. solution accounting for non-equilibrium foam kinetics within the traveling wave to. solutions computed assuming local equilibrium. Finally, we explain how the approach of. solving for the traveling wave can rule out some of the multiple LE states for certain. displacements.. Chapter 6 examines the stability of the various foam states (weak foam, intermediate. foam, and strong foam) in the model of Kam (2008) with 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 a slight (slowly growing) instability of the strong-foam state. We show how . diffusion, whether introduced artificially by the finite-difference scheme or representing. physical dispersion, could damp this slight instability. We compare the results with finite-. element simulations with and without additional diffusion. We also prove that all states. are unconditionally stable for any local-equilibrium foam model. . Chapter 7 seeks to understand the possible limiting role of foam dynamics on long-. distance foam propagation. We investigate strong-foam propagation applying the same. multiple-steady-state foam model with two approaches: first, we simulate foam. propagation with an upwind-finite-difference simulator; second, we find the unique. Riemann solution given the injected fractional flow and initial state of reservoir. For the. latter approach, we assume LE applies throughout foam displacement on the field scale,. with the exception of shock fronts.. Finally, the main conclusions of the thesis are summarized in Chapter 8.. 2. CAN FORMATION RELATIVE PERMEABILITIES RULE OUT A FOAM EOR PROCESS?*. oam is a promising means of increasing sweep in miscible- and immiscible-gas enhanced oil recovery. SAG (surfactant-alternating-gas) is a preferred method of injection. Numerous. studies verify that the water relative-permeability function krw(Sw) is unaffected by foam.. Studies of foam have used a variety of krw functions. This chapter shows a connection. between the krw(Sw) function and SAG foam effectiveness that is independent of the details of. how foam reduces gas mobility. For simplicity we analyze SAG processes in the absence of. mobile oil; success without oil is a precondition to success with oil, and our analysis also. applies to a miscible-gas process with oil in 1D in the absence of dispersion. Fractional-flow. methods have proved useful and accurate for modeling foam EOR processes. The success of. SAG depends on total mobility at a point of tangency to the fractional-flow curve, which. defines the shock front at the leading edge of the foam bank. One can determine total. mobility directly from the coordinates of this point (Sw, fw) if the function krw(Sw) is known.. Geometric constraints limit the region in the fractional-flow diagram in which this point of. tangency can occur. For a given krw(Sw) function, this limits the mobility reduction achievable. for any possible SAG process. We examine the implications of this limitation for different krw. functions. These implications include the following: increasing nonlinearity of the krw. function is advantageous for SAG processes, regardless of how foam reduces gas mobility.. SAG is inappropriate for naturally fractured reservoirs if straight-line relative permeabilities. apply, even if extremely strong foam can be stabilized in fractures. It is important to measure. krw(Sw) separately for any formation for which a SAG process is envisioned. . * Accepted for publication in: SPE Journal, 2011a.. F. 20 2: Can Formation Relative Permeabilities Rule out a Foam EOR Process?. 2.1 Introduction. Injected gas (CO2, steam, light hydrocarbons, or N2) can recover oil effectively on the. pore scale, but suffers from poor sweep efficiency (Lake 1989). The poor sweep. efficiency arises from permeability variations, segregation of injected gas under gravity,. and viscous instability between gas and oil. Foam can rectify these problems (Kovscek. and Radke 1994; Schramm 1994; Rossen 1996). To work, foam must reduce gas mobility. sufficiently to overcome or modify permeability differences between layers, and to. reduce gravity override (Shan and Rossen 2004; Rossen et al. 2010). Depending on the. design of the process, foam must reduce gas mobility sufficiently in the presence of oil to. divert gas flow, or reduce gas mobility in the absence of oil sufficiently to divert gas into. unswept, oil-rich layers. . Surfactant-alternating-gas (SAG) is a promising method of injection, for both operational. reasons (Matthews 1989; Heller 1994) and optimal sweep efficiency. SAG processes are. especially beneficial for field application where injection pressure is constraining,. because in a SAG process foam dries out and collapses in the immediate vicinity of an. injection well and allows high injectivity (Shan and Rossen 2004).. 2.1.1 Method of Characteristics for SAG foam processes. The method of characteristics (MOC) for two-phase flow, also known as fractional-flow. theory in petroleum engineering, is a useful tool in understanding and analysis of foam. injection processes (Zhou and Rossen 1995; Rossen et al. 1999; Shan and Rossen 2004;. see also chapter 3) as it is for other EOR processes (Lake 1989). Insights from the MOC. have guided the development of improved designs to overcome gravity override (Shan. and Rossen 2004), highlighted the key mechanisms in complex foam displacements. (Rossen et al. 1999), indicated the best conditions under which experiments should be. conducted for scale-up to the field (Rossen et al. 1999), exposed numerical artifacts in. simulations (Shan and Rossen 2004), and identified mechanisms in foam models that. introduce serious errors into simulations of foam processes (Namdar Zanganeh et al.. 2009). If entrance effects, the width of the shock front, or time for foam to reach steady. state is significant on the laboratory scale, then fractional-flow analysis based on steady-. state data is a more-reliable way to scale-up a SAG flood than conducting a dynamic. SAG coreflood (Xu and Rossen 2004). Fractional-flow methods proved accurate and. provided key insights into a SAG foam field trial in the Snorre field (Martinsen and. Vassenden 1999).. 2: Can Formation Relative Permeabilities Rule out a Foam EOR Process? 21. In a region swept of mobile oil, or in a miscible displacement of oil (chapter 3), a foam. process can be represented as a two-phase displacement. An implicit assumption of this. work is that a foam process that cannot succeed without mobile oil present cannot. succeed with mobile oil either. Three-phase MOC analysis of foam with mobile oil is. complex, and to date studies have been limited to relatively simple mobility functions and. oils that have no mutual solubility with injected gas (Mayberry et al. 2008; Namdar. Zanganeh et al. 2009). As a starting point, with one exception these studies assumed. linear relative-permeability functions in the absence of foam, including the water relative-. permeability function krw(Sw). Only one of the three-phase MOC studies of foam (Namdar. Zanganeh et al. 2009) modeled SAG foam processes, and in that study the SAG foam. processes modeled with oil would have failed the test for foam processes without oil. discussed below; that is, it would not have produced a low-mobility foam bank even in. the absence of oil.. The two-phase MOC or fractional-flow analysis of foam processes proceeds briefly as. follows. Details can be found elsewhere (Zhou and Rossen 1995; see also chapter 3). One. plots the fractional-flow function for water fw(Sw) in the presence of foam, based on. steady-state behavior determined in the laboratory. Though foam processes include. complex dynamics (Kovscek and Radke 1994; Rossen 1996; Kam et al. 2007b), it is now. recognized that in most cases steady-state behavior controls foam processes over large. distances on the field scale, and even on the laboratory scale (Rossen et al. 1999; Chen et. al. 2010; see also chapter 5). The fractional-flow function is defined as . ( ) / ( ). ( ) / ( ) / rw w w. w w f f rw w w rg w g. k S f S. k S k S.   .  . (2.1). where i is viscosity of phase i and superscript f indicates effective gas properties in the presence of foam. Strictly, it is impossible to distinguish the components of gas mobility,. i.e., effective gas relative permeability and effective gas viscosity, in foam flow (Rossen. 1992), but most models make this separation either for convenience or theoretical. reasons. If immobile, insoluble oil is present in the region of interest, then its saturation is. present everywhere; it does not change during the displacement, but its presence alters. the mobilities of gas and water and therefore must be taken into account in fw(Sw), krg f and. µg f (cf. e.g., Kloet et al. 2009).. If there is a change of surfactant concentration in the displacement, e.g., if water with. surfactant solution is injected into a formation with no surfactant initially present, there is. a second fractional-flow curve representing the formation ahead of the surfactant front. 22 2: Can Formation Relative Permeabilities Rule out a Foam EOR Process?. (Zhou and Rossen 1995; see also chapter 3). In this chapter we focus on gas injection in. SAG processes with a surfactant preflush injected ahead of the gas; therefore the entire. region of interest has the surfactant at the injected concentration in the water and only one. fractional-flow curve applies.. Next one locates the initial condition of the formation I (a specified water saturation) and. the injection condition (a specified fractional-flow of water) J on the fractional-flow. curve. For gas injection in a SAG process, I is either at zero gas saturation (if no gas has. previously been injected) or at residual gas saturation Sgr. J, representing gas injection, is. at fw = 0. One seeks a path along the fractional-flow curve fw(Sw) from J to I with. monotonically increasing (or, more precisely, non-decreasing) slope dfw/dSw. In such a. displacement, each saturation in the path advances through the formation with a. dimensionless velocity dxD/dtD equal to the slope of the function dfw/dSw at that. saturation. (Dimensionless position is the fraction of the pore volume in the formation. between the injection point and the given position. Dimensionless time is normalized by. the volume of fluids injected up to that time, divided by the pore volume of the. formation.) Each saturation implies a mobility and other properties corresponding to that. saturation. Therefore, though the MOC represents a one-dimensional (1D) displacement,. one can determine from the sequence of saturations whether viscous instability would. likely lead to fingering or channeling in 2D or 3D. . If the path from J to I along fw(Sw) does not have monotonically increasing slope, then. there is a jump from one point along the curve to another, representing a discontinuity or. shock front in saturation in the displacement. The dimensionless velocity of the shock is. fw/Sw across the jump, a condition that derives from material balances on water and gas at the shock. The shock must fit into the sequence of monotonically increasing slopes. (velocities) from J to I.. Figure 2.1 shows a typical fractional-flow function for foam (solid black curve) and no-. foam (gray curve) along with J and I for gas injection into a medium with gas initially. present at residual saturation Sgr. (If one allows for multiple steady states, the curve can. become much more complex (Kam and Rossen 2003; Kam 2008; see also chapters 4-7.). The large reduction in gas mobility caused by foam shifts the gas-water fractional-flow. functions upward and to the left on such a plot. At some water saturation, corresponding. to the "limiting capillary pressure" (Khatib et al. 1988), foam abruptly weakens or. collapses: gas mobility rises, and fw(Sw) plunges nearly vertically to a small value fw near. zero (Zhou and Rossen 1995; Dong and Rossen 2007).. 2: Can Formation Relative Permeabilities Rule out a Foam EOR Process? 23. In this chapter we focus on the gas-injection cycle of SAG. In principle, mobility can be. reduced during liquid injection in SAG, and in 1D, sufficiently far from the well,. conditions approximate continuous injection (see, e.g., Faisal et al. 2009). However, if. gas mobility is not reduced during gas injection, we believe, in 2D or 3D the entire SAG. process is likely to fail; gas could disappear to the override zone or into viscous fingers. before the next liquid slug is introduced. With sufficiently small slugs, SAG. approximates continuous injection, at least in 1D: both the advantages and disadvantages. of SAG are reduced. We focus on gas injection in SAG processes with relatively large. slugs, designed to take advantages of SAG in injectivity and fighting gravity override. (Shan and Rossen 2004; Kloet et al. 2009).. As illustrated in Figure 2.1, for gas injection in a SAG process, dfw/dSw is not. monotonically increasing from J to I. Instead, there is a shock from I to a point of. tangency to the fractional-flow curve at a small value of fw. The properties of foam for all. values of fw larger than the value at the point of tangency are irrelevant to the. displacement except within the narrow traveling wave at the shock front (Rossen and. Bruining 2007; see also chapter 5). The total mobility at the saturation corresponding to. the point of tangency is critical to the success of the process, because mobility only. increases from this point back to point J. For any foam process foam must collapse. completely at connate water saturation Swc, where capillary pressure becomes extremely. large (Khatib et al. 1988). Therefore, if there is insufficient mobility reduction at the point. of tangency, then the foam process would fail (Zhou and Rossen 1995). This failure can. be masked in computer simulations (Shan and Rossen 2004).. Figure 2.1 Foam fractional-flow curve, with construction of shock front formed when injecting gas following surfactant preflush. The fractional-flow curve is nearly vertical at Sw. *. . 0 0.2 0.4 0.6 0.8 1 0. 0.2. 0.4. 0.6. 0.8. 1. S w. f w. I. J S. w *. Foam f w. Foam-free f w. Shock line. 24 2: Can Formation Relative Permeabilities Rule out a Foam EOR Process?. The total relative mobility rt at this point of tangency, or at any point in the fractional- flow plot, is a simple function of the coordinates of the point and the relative-. permeability function krw(Sw):. ( ) / ( , ) rw w wrt w w. w. k S f S. f.   (2.2). which derives from the definition of fw(Sw). Equivalently, one could define the effective. viscosity of foam at this location f, in effect representing foam as a single-phase fluid, as. 1. ( ) f w w. rt rw w. f. k S.  .   . (2.3). Foam is useful as a mobility-control agent provided that the total mobility of foam is. lower than the mobility of the swept bank ahead of foam, or, equivalently, if the foam. effective viscosity is greater than the effective viscosity of the bank ahead of it. The exact. minimum required effective viscosity of foam thus depends on the phase to be swept;. e.g., it could vary from less than 1 cp for a light oil to much greater viscosities for heavier. oils. . 2.1.2 krw(Sw) function. It is widely reported that the krw(Sw) function is unaffected by the presence or strength of. foam (Bernard et al. 1965; Sanchez and Schechter 1989; de Vries and Wit 1990;. Friedman et al. 1991). In other words, it is an inherent property of the formation, and not. altered by a foam process (as long as the formation is initially water wet - if the formation. is initially oil-wet, surfactant itself may reverse the wettability to water-wet (Sanchez and. Hazlett 1992)).. Different studies of foam have used different krw(Sw) functions: e.g., Chierichi, Burdine. and Corey functions (Vassenden and Holt 2000). Most use Corey-type functions (Corey. 1954) for krw(Sw): . ( ) , 0 1. B. w wc rw w w wc w wc. wc gr or. S S k S A for S S for S S. S S S.          . (2.4). where Equation (2.4) allows for a uniform saturation of inert, residual oil, Sor, within the. region of interest. For the remainder of this chapter we assume for simplicity Sor = 0. within the foam-swept zone. The Corey-type relative-permeability function is widely. 2: Can Formation Relative Permeabilities Rule out a Foam EOR Process? 25. accepted for its simplicity. It requires limited input data and it is fairly accurate for. consolidated porous media with intergranular porosity (Honarpour et al. 1986). . Jones (1966) proposes a Corey-type krw function with exponent 3 for water-gas systems.. Johnson (1968) reports an approximate exponent B of 4 for water relative permeability. for consolidated rocks. He reports that the exponent has a value of 3 for rocks with. perfectly uniform pore-size distribution. Several other authors propose similar water. relative-permeability equations with different exponents for other types of porous media.. Values of 3 (Irmay 1954) and 3.5 (Averganov 1950) are proposed for unconsolidated. sands with a single grain size, which may not be absolutely uniform in pore size but. should have a narrow range of pore sizes. Frick’s Petroleum Production Handbook. suggests exponent B=3 for well-sorted unconsolidated sand, 3.5 for poorly-sorted. unconsolidated sand and 4 for cemented sandstone (oolitic limestone) for water relative. permeability in drainage. Honarpour (personal communication) recommends values of B. from 5 to 8 for water-wet, relatively clean sandstones.. For foam application, earlier studies in our group (e.g., Zhou and Rossen 1995; Rossen et. al. 1999; Kloet et al. 2009) use a Corey exponent B of 4.2, based on a fit to relative-. permeability data of Persoff et al. (1991). Kovscek et al. (1995) use a cubic function. (B=3). To date, most three-phase MOC studies of oil with foam for simplicity use linear. relative permeabilities (B = 1) for all phases in the absence of foam (Mayberry et al.. 2008), though Namdar Zanganeh et al. (2009) extend their study to some nonlinear. functions. In naturally fractured reservoirs, linear relative-permeability functions are. sometimes assumed within the fracture network (Romm 1996, van Heel et al. 2008;. Zhang et al. 2010), but not always (Saidi et al. 1979; Horne et al. 2000; Firoozabadi. 2000; Rangel-German et al. 2006). Kibodeaux and Rossen (1997) and Xu and Rossen. (2004) measured krw(Sw) in foam corefloods in sandstone cores, but over a too narrow. range in Sw to allow reliable fitting to Corey-type functions. Xu and Rossen (2004) fit a. Corey-type krw to their limited data for two experiments, in effect assuming negligible Swc. and Sor. Their fit results in large exponents (A=170, B=10, and A=20 and B=7.2) not. intended to apply outside the narrow range of their data. . In this chapter, we show that the krw(Sw) function for a formation, which is independent of. the foam process, can place limits on the mobility reduction achievable during gas. injection in SAG foam processes for any conceivable foam formulation. In some cases,. then, one might consider co-injection of gas and surfactant solution instead of SAG, or. SAG with slugs so small that they mix near the well and give nearly constant mobility. some distance away (Faisal et al. 2009). Simulation studies show a strong advantage of. 26 2: Can Formation Relative Permeabilities Rule out a Foam EOR Process?. SAG with extremely large slugs that give low mobility at the displacement front during. gas injection (Shan and Rossen 2004; Kloet et al. 2009), so a limit on this mobility is a. serious constraint on the process. This limit derives from Equation (2.2) and the. geometrical constraints on possible positions of a point of tangency in gas injection in a. SAG process. This shows that the choice of krw(Sw) function for a formation is not simply. an arbitrary choice, but an important constraint in design of a foam process for a field. application. It is important to measure this function directly in the laboratory for each. formation for which a SAG process is envisioned.. 2.2 Results. In this study, we examine three cases with increasingly restrictive but more-realistic. assumptions. In the first case, it is assumed that foam can survive in any water saturation.. In the second case we restrict foam existence to water saturations greater than a "limiting". water saturation Sw* that corresponds to the "limiting capillary pressure" Pc* at which. foam collapses (Khatib et al. 1988; Zhou and Rossen 1995). In the third case we assume a. smooth, continuous foam-fractional-flow function near the limiting water saturation.. We consider five different Corey-type relative-permeability functions (Equation (2.4)): .  Linear (A=B=1). Linear relative-permeability functions are sometimes assumed to apply to fractured reservoirs (Romm 1996; van Heel et al. 2008; Zhang et al.. 2010). Moreover, initial MOC studies of foam with oil included linear relative-. permeability functions (Mayberry et al. 2008, Namdar-Zanganeh et al. 2009)..  Quadratic (A=0.8, B=2). This is similar to the fit Kam and Rossen (2003) make to data for gas-water flow in an unconsolidated sandpack in Collins (1961)..  Cubic (A=0.7, B=3). This function is taken from that in Kovscek et al. (1995). .  Quartic (A=0.2, B=4.2): This is taken from curve-fitting of gas-water relative- permeability data in sandstone (Persoff et al. 1991) and used in a number foam. studies (e.g., Zhou and Rossen 1995; Rossen et al. 1999, Kloet et al. 2009))..  Quintic (A=0.5, B=5): Tak