Transient Foam Displacement in the Presence of Residual Oil

Ind. Eng. Chem. Res. 2000, 39, 2725-2741

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Transient Foam Displacement in the Presence of Residual Oil: Experiment and Simulation Using a Population-Balance Model Timothy J. Myers† and Clayton J. Radke* Department of Chemical Engineering and Lawrence Berkeley National Laboratory, University of California, Berkeley, California 94720-1462

The population-balance framework of Kovscek, Patzek, and Radke is extended to model the flow of foam in porous media containing residual oil (Kovscek et al. Chem. Eng. Sci. 1995, 50, 3783). A mechanistic rate of coalescence due to oil, based on the recently proposed pinch-off mechanism (Myers, T. J. Ph.D. Dissertation, University of California, Berkeley, CA), is included in the model. New foam-flow experiments in the absence and presence of residual oil in 1.1- and 0.28-µm2 Berea sandstone cores 60-cm long are modeled. Results reported here are for total superficial velocities of 30 cm/day and 97% foam quality with a 0.5 wt % sodium dodecyl sulfate, 0.5 wt % NaCl aqueous solution, and a decane oil phase. Additional results at varying gas and liquid velocities are reported elsewhere. Agreement between experiment and simulation is good. Both experiment and simulation show that foam is destabilized by oil and has lower mobility in higher permeability media regardless of the presence of oil. These findings suggest that in an oil reservoir foam will preferentially flow through low-permeability oil-containing regions of the reservoir and partially block high-permeability oil-depleted regions as compared to a Newtonian fluid. Introduction In recent work, we discovered two mechanisms of rupture for foam lamellae moving across nonwetting surfaces or across portions of oil drops that enter the gas-water interface.1 A pinch-off mechanism, depicted in Figure 1, was deemed applicable to the low-velocity conditions of foam flow in porous media. In Figure 1a, two gas bubbles separated by a foam lamella move across an oil globule. In this case, a water film, called a pseudoemulsion film, separates the gas bubbles from the oil globule, and no rupture is observed. In Figure 1b, a stable water film separating the gas bubbles from the oil droplet does not exist, and accordingly the oil globule enters the gas-water interfaces of the two bubbles. If the bridging coefficient is positive, B > 0 (see Nomenclature for definition of B), this configuration is unstable1,2 and the two gas-water-oil three-phase contact lines move toward each other. When the two three-phase contact lines meet, the foam lamella pinches off from the oil globule, causing the two bubbles to coalesce into one bubble as shown in Figure 1c. From the criteria required for the pinch-off rupture mechanism to occur, we formulated a rate expression for the coalescence of foam by oil3 for use in the populationbalance foam-flow model of Kovscek, Patzek, and Radke.3,4 Here, we test the proposed rate expression by simulating new transient foam-flow experiments in the presence and absence of residual oil. We also examine the effect of porous-medium permeability on the flow of foam in porous media. Elsewhere, we study further the effects of gas and surfactant-solution velocity and oil saturation.1 * To whom all correspondence should be addressed. Telephone: 510-642-5204. Fax: 510-642-4778. E-mail: radke@ cchem.berkeley.edu. † Present address: Exponent Failure Analysis Associates, 21 Strathmore Road, Natick, MA 01760.

Figure 1. Pinch-off rupture mechanism of a foam lamella moving across an oil droplet.1

Most previous studies of foam flow in the presence of oil focus on the effect of different oils and surfactant solutions on the steady-state pressure drop across a porous medium in the presence of residual oil saturation. Comparison of the foam displacement performance from these studies,5-12 gauged by steady-state pressure drops, effective viscosity, or gas-phase mobility, with previously proposed criteria for foam stability in the presence of oil does not show universal agreement.1 However, reasonable agreement is observed between negative values of the less-common bridging coefficient, a foam-stability criterion originally proposed by Gar-

10.1021/ie990909u CCC: $19.00 © 2000 American Chemical Society Published on Web 06/24/2000

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Ind. Eng. Chem. Res., Vol. 39, No. 8, 2000

rett,2 and efficient foam displacement in the presence of residual oil.1 In the few cases where transient displacement behavior is reported,9,12,13 only the overall pressure drop versus time is measured and not the pressure and saturation profiles within the porous medium during the foam displacement. Furthermore, the experimentally measured pressure drops evolve over 10-100 fluid pore volumes (PV), and, in fact, the pressure drops do not increase significantly until the gas phase reaches the end of the initially surfactantsaturated porous media.9,12,13 This behavior is not explainable by the current population-balance model that predicts a sharp increase in pressure gradient behind a pistonlike foam displacement front with attainment of steady state in little over a single PV of injection.3,4 The apparent requirement of many PV to reach steady state has also been seen in the absence of oil and is also not currently well understood.4,14-16 The experiments of Ettinger suggest that the long-time increases in pressure are due to the propagation of a capillary end effect from the back of the core to the front of the core. Gillis also sees a sharp pressure gradient only near the core outlet for what he terms “weak” foams.16 Kovscek found behavior identical that of Ettinger in experiments where he used isopropyl or ethyl alcohol to break foam from a previous experiment.4 Kovscek states that Ettinger, Gillis, and others commonly used alcohol to break foam from previous experiments.17,18 Apparently, trace quantities of a short-chain alcohol are able to destabilize foam and prevent the initial generation of a strong foam.4 Thus, it is plausible that long-time behavior seen in foam-flow experiments where oil destabilizes foam may also be due to capillary end effects. To test our proposed mechanistic rate expression for foam coalescence by oil and to resolve the cause of the long-term behavior observed in previous foam-flow experiments, in this work we conduct a series of transient foam-displacement experiments. Using 60-cmlong cores with seven pressure taps and a scanning microwave absorption apparatus, we track transient and steady-state pressure and water saturation profiles to compare to simulation predictions and to determine spatial variations attributable to end effects at both the core inlet and exit. We restrict total superficial velocities to between 30 and 60 cm/day (1 and 2 ft/day), as a compromise between the low velocities relevant to foam flow in oil reservoirs and the experimental difficulties and time requirements associated with using lower velocities. In addition, these velocities are in the lowvelocity region where the proposed pinch-off lamella rupture mechanism is applicable.1 Experimental Apparatus A more detailed description of the experimental apparatus and procedures is found elsewhere.1 The experimental apparatus used in this research is based on the microwave-attenuation apparatus originally constructed by Sharma19 and later modified by Ettinger14,20 and Hanssen.21 A limitation of the microwave-attenuation technique is that it only measures water saturation and does not distinguish among nonaqueous phases in the core. This does not pose a problem in our experiments because, although we have three phases present, the residual oil phase is virtually immobile during our foam-flow ex-

Figure 2. Schematic of foam-flow experimental apparatus.

periments, and its saturation profile can be measured before a foam-flow experiment and taken as constant in time. The gas saturation can then be calculated from a volume balance on the pore space. Thus, the saturations of all three phases are known by measuring only the water saturation during foam-flow experiments. As depicted in Figure 2, the microwave source and detectors are mounted on a carriage that scans the length of the porous medium. The carriage movement and measurement are controlled by a computer. In our experiments, power transmitted through the core is measured at 60 equally spaced locations along the length of the core. The measurement time at each location is 4 s, providing a scan of the entire core in 4 min. At the 21.3 GHz microwave frequency used in our experiments, water absorbs microwaves much stronger than any other fluids in the system, and water saturation varies linearly with the microwave absorbance measured at each location.14,19,22 Typical errors in saturation measurements are less than 1%. The porous media are two unfired Berea sandstone cores mounted in acrylic (Acrylite GP, Cyro Industries). Core 1 was obtained from Unocal Science and Technology Division, Brea, CA, already mounted in acrylic using the apparatus and technique described by Showalter.23 The core is 59.5 cm long and has a rectangular cross section of 8.1 cm × 1.9 cm. It is centered in the acrylic coreholder measuring 66 cm × 12 cm × 4.5 cm. The pore volume is 207 cm3 yielding a porosity of 0.23. The permeability to brine is 1.1 µm2. Core 2 was obtained from American Stone, Cleveland Quarries, Amherst, OH, and was mounted in acrylic using a modified version of Showalter’s technique described by Myers (Appendix 4B).1 The dimensions of Core 2 are 60.8 cm × 7.4 cm × 1.9 cm, and it is centered in a coreholder measuring 70 cm × 12.8 cm × 5.5 cm. Porous glass frits (R & H Filter Co., XXC) of 0.5-cm thickness are placed at each end of Core 2 to distribute flow across the entire core face. The manufacturer of the frits states that the pore radii are greater than 200 µm. Core 2 has a pore volume of 158 cm3, a porosity of 0.18, and a permeability to brine of 0.24 µm2. As shown in Figure 2, flow is directed along the 60cm length of the cores and microwaves are transmitted through the 60 cm × 8 cm face of the core. Three 0.7cm-diameter flow ports are spaced across each end of the core to allow fluid to enter and exit the core. Pressure taps are evenly spaced along the length of the core, approximately 10 cm apart. Two additional pres-

Ind. Eng. Chem. Res., Vol. 39, No. 8, 2000 2727

sure taps are placed in the tubing outside of the coreholder to provide pressure measurements at the core inlet and exit. Surfactant solution and brine are injected into the core at fixed flow rates with an ISCO 500D syringe pump, while oil is injected with an ISCO 260D syringe pump. The pumps provide steady oscillation-free flow at rates as low as 0.001 cm3/min corresponding to superficial velocities as low as 0.001 m/day. The nitrogen injection rate is controlled by a 0-10-sccm mass flow controller (Brooks 5850), providing superficial gas velocities ranging from 0 to 1.2 m/day. During foam-flow experiments, we investigate a gas velocity of 29.1 cm/ day and a surfactant solution velocity of 0.9 cm/day. The resulting foam quality is 0.97, where the foam quality q is the injected gas velocity divided by the total injected velocity. Velocity is reported at the core backpressure and the corresponding gas-phase density. As seen in Figure 2, the core outlet stream flows through a backpressure regulator to maintain the core at an elevated backpressure (Temco BPR-05). In all but one experiment, the backpressure is set to 790 kPa (115 psia) to reduce compressibility of the gas phase. In the remaining experiment, the backpressure was 830 kPa (120 psia) because of a faulty pressure regulator. Backpressures as high as 1.48 MPa (215 psia) are used while saturating the core with brine to increase the solubility of nitrogen in the water phase. The surfactant solution used in experiments is an aqueous solution of 0.5 wt % sodium dodecyl sulfate (SDS; Acros 41953-0010) and 0.5 wt % NaCl (Fischer Scientific S271). This is well above the critical micelle concentration of SDS at this salt concentration. Phillips reports critical micelle concentrations of 0.09 and 0.04 wt % for 0.18 and 0.58 wt % NaCl solutions, respectively.24 All surfactant solutions are made with deionized water from a Millipore Milli-Q reagent-grade water system with resistance greater than 15 MΩ‚cm. As is customary in the petroleum industry, we refer to the aqueous phase as water, regardless of its surfactant or salt content. The oil is decane (Aldrich 45,711-6). In all experiments involving decane, the injected surfactant solution is equilibrated with decane by gentle mixing for at least 24 h. In experiments without residual oil present, the surfactant solution is not presaturated with decane. Experiments elsewhere show that the foam-flow performance is not affected by equilibration of the surfactant solution with either decane or hexadecane in displacement experiments without residual oil.1 Because SDS is known to hydrolyze to dodecyl alcohol, all solutions are aged at least 24 h before use to allow the hydrolysis reaction to approach equilibrium.9,25,26 Stock-compressed nitrogen cylinders supply the nitrogen for the foam-flow experiments. Experimental Procedures Surface and Interfacial Measurements. Equilibrium surface and interfacial tensions are measured using a pendant-drop apparatus described in detail elsewhere.27 The tensions are measured at a laboratory pressure of 1 atm and a temperature of 23 °C. Measurements are made in a closed optical cell, with all three immiscible phases present. When measuring the gaswater surface tension, the water drop is placed directly above a reservoir of decane to ensure that the gaswater interface is in equilibrium with decane in the vapor phase. Aveyard et al. suggest that reported

Table 1. Interfacial Properties of Fluids (0.5 wt % SDS, 0.5 wt % NaCl/Decane/Air) property

value

95% confidence interval

σgw (mN/m) σow (mN/m) σog (mN/m) So/w (mN/m)a Eo/w (mN/m)a Laa B (mN/m)2 a

28.5 5.14 23.4 0 10.3 0.83 290

(0.13 (0.026 (0.045 (0.18 (0.18 (0.0075 (8.9

a

These quantities are defined in the Nomenclature section.

measurements of positive equilibrium spreading coefficients, which are thermodynamically impossible, arise from incomplete interface equilibration.28 In all cases, the drop is allowed to age for at least an hour before five measurements of the tension are recorded, 5 min apart. Tension values and 95% confidence intervals reported in Table 1 are for these five measurements of the tension of a single drop. The small 95% confidence intervals indicate the precision of the measurement technique. However, the error between measurements taken on different drops from different solutions is probably larger than the reported confidence interval. The entering coefficient for oil entering the gas-water interface from the water side, Eo/w ) 10.3 mN/m, the spreading coefficient for oil at the gas-water interface, So/w ) 0 mN/m, the lamella number, La ) 0.83, and the bridging coefficient, B ) 290 (mN/m)2, are also reported in Table 1. Definitions of these quantities are provided in the Nomenclature section. The reported confidence intervals of the coefficients are calculated using standard error propagation techniques. Eo/w > 0 indicates that it is energetically favorable for oil to enter the gaswater interface, and So/w ) 0 indicates that an oil drop spreads at the gas-water interface. According to the criterion of Schramm and Novosad, La < 1 predicts that the Plateau border does not emulsify oil drops.29 B > 0 predicts that a foam film bridged by an oil droplet is unstable.1,2 Thus, all parameters accept the lamella number predict that this system produces an unstable foam in the presence of residual oil. In addition, based on our observed rupture mechanism, the bridging coefficient is sufficiently large to cause foam coalescence by pinch-off in times much shorter than the time for lamellae to move across an oil drop in our experiments.1 Preparation of the Porous Medium and Experimental Procedures. All porous-medium experiments are conducted at a laboratory temperature of about 23 °C. Before each experiment, 20-30 PV of 0.5 wt % NaCl brine are pumped through the core at an elevated backpressure of 1.48 MPa (215 psia) to remove trapped gas. In experiments involving oil, 5-10 PV of decane are injected into the core after complete water saturation to bring the core to irreducible water saturation. A total of 5-10 PV of oil-saturated brine are then injected to reduce the oil saturation to waterflood residual oil. During these injections, the superficial velocity is less than 2 m/day. After the oil saturation has reached waterflood residual oil and no additional oil is produced, we saturate the core with surfactant solution. The process of injecting oil, waterflooding, and saturating the core with surfactant solution yields residual oil saturations between 0.31 and 0.37. Before foam-displacement experiments, at least 7 PV of surfactant solution are flushed through the core at superficial velocities of less than 1

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m/day to equilibrate the rock surface and oil with surfactant. The surfactant solution is equilibrated with oil before injection into the core. Next, the backpressure regulator is set to 790 kPa (115 psia). The microwave-scanning program is started, and three to five scans are recorded with only the surfactant solution flowing through the core. At this point, the core contains only the surfactant solution and oil, so the initial oil saturation can be calculated directly from the water-saturation measurement. Resaturation with brine after foam-flow experiments back to the initial profile verifies that the oil-saturation profile remains essentially constant during the experiment. Before gas and the surfactant solution enter the core, they are simply mixed by a Swagelok T-fitting, yielding gas bubbles and liquid slugs of the dimensions of the 0.32-cm inlet tubing. No foam pregeneration occurs before the core inlet. Water saturation and pressures are recorded on a regular basis throughout the experiment. At completion of the experiments, the gas inlet valve is closed, the surfactant-solution injection is stopped, and 0.5 wt % aqueous NaCl brine is injected into the core. After the core is completely resaturated with brine, saturation measurements are recorded to confirm that oil is not mobilized by foam. Population-Balance Model The population-balance model is a convenient tool for modeling foam flow because it results in continuumlevel equations similar to mass and energy balances typically solved in oil-reservoir simulators.30,31 In addition, it offers a general framework for improvement as a more detailed understanding of the governing processes develops. The model treats the number of gas bubbles per unit volume, often referred to as foam texture or bubble density, as an additional component in a multicomponent system of conservation equations. The foam texture is tracked as it changes because of pore-level bubble generation and coalescence mechanisms and then is used to calculate the mobility of the gas phase. Recently, Kovscek, Patzek, and Radke successfully modeled transient and steady-state foam-flow experiments in Boise sandstone using the populationbalance model.3,4 In this section, we extend the model of Kovscek, Patzek, and Radke to simulate foam-flow experiments in the presence of oil and in porous media of different permeabilities. We do this by including the mechanistic rate expression for foam coalescence by oil developed in recent work.1 Additionally, we use a traditional threephase flow theory to include the effects of a residual oil phase on the gas and water relative permeabilities. Kovscek, Patzek, and Radke model foam flow in two different permeability porous media, without comparison to experiments in the lower permeability medium. They assume that the only change in model parameters between the two porous media is Leverett’s scaling of capillary pressure with the average pore size.32 To match experimental results in the two different permeability porous media, however, we find that it is necessary to make additional changes to the model parameters. Population-Balance Model in Oil-Free Porous Media. Here we briefly describe the one-dimensional population-balance model used by Kovscek, Patzek, and Radke, closely following the description in the thesis of

Kovscek.4 Additional descriptions are presented elsewhere.1,3,33,34 As in traditional black-oil reservoir simulation that assumes all phases are completely immiscible,35 one-dimensional mass conservation equations for each phase are written as

∂(φFiSi) ∂(Fiui) )+ Qi ∂t ∂x

(1)

where φ is the porous medium porosity, Fi is the mass density of a phase, Si is the saturation or volume fraction occupied by a phase, and ui is the superficial velocity of a phase. t and x are time and axial distance, while Qi is a net generation term used as a source or sink at boundaries. The subscript i of eq 1 is replaced with g, w, or o, to signify the gas, water, or oil phases. We assume that the water and oil phases are incompressible, whereas the gas density obeys the ideal gas law. During the foam-flow experiments in this work, the oil phase is immobile; for the oil phase we set the right side of eq 1 to zero, requiring that the oil saturation at each position remains constant with time. Because all phases and surfaces are saturated with surfactant in our experiments, we need not account for its transport. The balance on the average bubble size is similar in form to the mass balance on each phase and is written as

∂(ufnf) ∂[φ(Sfnf + Stnt)] )+ φSg(rg - rc) + Qb ∂t ∂x

(2)

where the subscripts f and t refer to the flowing and trapped portions of the gas phase. nf and nt are the numbers of bubbles per unit volume of flowing and trapped gas phases, respectively, and serve a role similar to that of the mass density in eq 1. Thus, the left side of eq 2 reflects the net rate of accumulation of flowing bubbles and trapped bubbles per unit volume of porous medium. Kovscek follows Friedmann et al.18 and assumes that the trapped and flowing foam textures are equal, that is, nf ) nt4. The superficial velocity of the flowing gas phase, uf, is identical with the superficial velocity of the gas phase, ug, because the velocity of the trapped gas is zero. The first term on the right side of the equal sign in eq 2 is simply the negative divergence of the flux of bubbles. The new term, without an analogue in eq 1, φSg(rg rc), is the net rate of production of bubbles. Kovscek defined mechanistic rates of generation, rg, and coalescence, rc, of bubbles in the absence of oil.3,4 Here we add an additional rate of coalescence due to oil. As in the mass balance, Qb is a net source term of bubbles and serves as a boundary condition. Because we never inject pregenerated foam, this term is always zero. Kovscek defines a rate of generation of foam bubbles based on snap-off of bubbles in pore throats, a mechanism observed in micromodel experiments.36 From the results of pore-level calculations of the time for liquid to accumulate in the pore throat and snap off a bubble and of the time required to displace the newly generated foam lamella or lens from the throat,4,37 he writes the rate of generation as

rg ) k1vfa vwb

(3)

where k1 is a constant and vf ) uf/φSf and vw ) uw/φSw are the interstitial flowing gas and water velocities,


respectively. The snap-off model of Kovscek and Radke predicts that a ≈ 1/3 and b ≈ 1.4,37 Values of a ) 0.33 and b ) 1 match the velocity behavior of oil-free foamflow experiments and are used in simulations.3,4 In the absence of oil, the dominant mechanism of coalescence is stretching of foam lamellae as they move from pore throats to pore bodies.36,38 The stability of foam lamellae as they stretch is greatly reduced when the capillary pressure of the porous medium approaches the rupture disjoining pressure or limiting capillary pressure of lamellae.15,38,39 This leads Kovscek to write a rate of coalescence as

rc ) k-1

(

Pcgw

)

Pcgw/- Pcgw

2

vfnf

(4)

where k-1 is a constant, Pcgw/ is the limiting gas-water capillary pressure, and Pcgw is the local gas-water capillary pressure in the porous medium.3,4 Thus, from eq 4 the coalescence rate is proportional to the flux of lamellae, vfnf, to a number of rupture sites that depends strongly on capillary pressure. Constitutive relationships are needed to relate the superficial velocity of each phase to the phase pressures and saturations. As in traditional modeling of multiphase flow in porous media,35 a modified version of Darcy’s law is used in the population-balance model. For one-dimensional flow perpendicular to the gravitational acceleration, the superficial or Darcy velocity is given as

ui ) -

( )

kkri ∂Pi µi ∂x

(5)

where k is the absolute permeability of the porous medium, kri is the relative permeability to a phase, and µi and Pi are the viscosity and pressure of a phase. Throughout this paper, we refer to the ratio kri/µi as the mobility of a flowing phase. Note that, because of the way that eq 5 is written, the pressure gradient in each phase may be different. This is because the phases are highly curved, capillary pressure is significant, and the pressure of individual phases is different. The capillary pressure is related to phase saturations through the Leverett J function.32 Kovscek uses a Leverett J function of the form

J(Sw) )

x (

Pcgw σgw

k ω ) φ Sw - 0.15

)

0.2

(6)

where ω is a constant.3,4 We fit eq 6 to Leverett’s data32 and use ω ) 0.0087. Kovscek uses a Stone-type relative permeability model40 for the gas and water two-phase relative permeabilities.4 In accordance with Stone, Kovscek argues that, because water occupies the smallest pores during foam flow, its relative permeability is solely a function of water saturation. Similarly, flowing foam travels in the largest pores, on average, and its relative permeability is a function of the flowing gas saturation only. Transport of water in foam films is arguably negligible compared to the flow in small pores and corners of larger pores. The relative permeability of trapped gas is zero because its velocity is zero.

Using Corey-type relative permeability functions,41,42 Kovscek writes the gas and water relative permeabilities as

krw ) korw

( (

krf ) korg

)

Sw - Swc 1 - Swc XfSg 1 - Swc

)

f

(7)

g

(8)

where korw is the relative permeability to water at Sw ) 1 and korg is the relative permeability to gas when Sg ) 1 - Swc. Swc is the connate water saturation. Xf is the flowing fraction of gas, Sf/Sg, and f and g are the socalled Corey exponents. Kovscek uses Corey exponents of 3 for both the gas and water phases.4 Thus, eq 8 accounts for much of the increased flow resistance of foam because the minimum flowing fraction of 0.10 used by Kovscek3,4 reduces the relative permeability by a factor of 1000 compared to a continuous flowing gas phase at the same saturation. The viscosity of flowing foam is also higher than that of flowing gas. Bretherton shows that for low capillary numbers a long bubble flowing in a circular tube experiences large rates of shear in the liquid films near the front and rear of the bubble. This produces a pressure drop that scales with the velocity to the 2/3 power.43 More recently, Wong, Radke, and Morris show that the same scaling applies for bubbles flowing in polygonal tubes having corners such as pores in a porous medium.44 Using this scaling, Kovscek writes an effective viscosity of flowing foam to be used in Darcy’s law of eq 5 as

µf ) µg +

Rnf

(9)

vf1/3

where µg is the bulk viscosity of gas and R is a constant.3,4 When the bubble density, nf, is small, the viscosity reduces to that of a bulk gas. When nf is large, the flowing foam is shear thinning in accordance with the pressure-drop scaling of Bretherton. To complete the model, a relationship for the trapped fraction of foam is needed. Kovscek proposes an empirical relation for the fraction of trapped foam, Xt, that depends solely on the foam texture:

Xt )

(

)

St βnt ) Xt,max Sg 1 + βnt

(10)

where Xt,max is the maximum trapped fraction and β is a fitting parameter.4 The form of eq 10 is similar to a Langmuir adsorption isotherm.45 Kovscek chooses β so that Xt ) 0.95Xt,max when nt ) 20 mm-3.4 Kovscek further requires that the trapped fraction never decreases, or equivalently that trapped foam does not mobilize. Because for strong foams the model predicts an essentially constant water saturation and flowing fraction, much of the behavior of foam can be understood from the steady-state foam texture and foam viscosity.4 The steady-state foam texture is obtained by equating the rates of generation and coalescence from eqs 3 and

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4. After inserting the steady-state foam texture into the flowing-foam viscosity, eq 9 yields

(

)

/ Rk1vw Pcgw - Pcgw µf ) µg + k-1vg Pcgw

2

(11)

For strong foams where µf . µg, this relation predicts that the flowing-foam viscosity is proportional to the water velocity and inversely proportional to the gas velocity. This is because increasing the water velocity increases the foam generation, making a finer textured foam. In contrast, increasing the gas velocity increases the foam coalescence, making a coarser foam. The increase of the foam generation with gas velocity is exactly compensated by the shear-thinning behavior of the viscosity. Changes in foam texture explain the experimental observation that the pressure drop of strong flowing foam is linearly dependent on the water velocity and independent of the gas velocity.3,4,14,46 Extension of the Population-Balance Model To Include Effects of Oil. Within the framework of the population-balance model, we propose three main effects of an immobile residual oil phase on foam flow. First, relative permeabilities must be adjusted to include the effect of residual oil according to traditional three-phase flow theory.40,47 Second, the trapped fraction of gas is reduced because residual oil is likely trapped in some pore spaces that also trap foam bubbles. Third, and most importantly, oil often destabilizes foam and an additional rate of coalescence due to oil must be included in the model. The proposal that oil may block bubble germination or snap-off sites13 does not appear to require special treatment. In the current populationbalance model, the rate of generation is written on a per gas-volume basis, so the occupation of some snapoff sites by oil is implicitly included. Special treatment is needed only if oil is preferentially trapped in bubble generation sites relative to other sites and if the number of generation sites per volume of gas changes with oil saturation. Foams that are insensitive to oil produce pressure gradients comparable to foam in the absence of oil so blockage of generation sites is apparently not a significant effect.9 Because in our experiments and simulations the oil phase is immobile and only the gas and water phases are flowing, relative permeability functions are only needed for the gas and water phases. Traditional Stonetype three-phase flow theories assume that in waterwet porous media the water relative permeability is only a function of the water saturation and that the gas relative permeability is only a function of the gas saturation.40,47 Thus, the relative permeability functions in eqs 7 and 8 also describe the relative permeabilities of gas and water with residual oil present. However, we find experimentally that the relative permeability to water with residual oil present is somewhat lower than that predicted by the two-phase relative permeability measured with flowing gas and water phases. This is observed when only water is flowing through a gas-free core with residual oil and during two-phase foam flow. Others have observed that the water relative permeability in water-wet sandstones is reduced from its twophase flow value when a third immobile phase is present (cf. Figure 13 of ref 48 and Figure 3 of ref 49). The residual nonwetting phase causes a greater reduction in water relative permeability than a continuous nonwetting phase at the same saturation.

Using history matching, we decrease korw and increase Swc to match the observed water relative permeability with residual oil present during single-phase water flow before foam-flow experiments and during two-phase foam flow. We use single values of korw and Swc to simulate all experiments with residual oil present, although the measured pressure drop during water injection at residual oil saturation varies as much as 50% from the predicted value in some experiments.1 A similar effect of residual oil saturation on the relative permeability of the gas phase is difficult to observe. Experimentally, we only have gas flowing in the presence of residual oil during foam flow, when the flow resistance of the gas phase is much more strongly impacted by foam texture. Because as defined korg is the value of krg at Sg ) 1 - Swc, we lower korg as Swc is increased so that the gas relative permeability remains unchanged from its measured two-phase value. Little is known about the trapped fraction of foam with oil present. All available experimental measurements of the trapped foam fraction are conducted without oil present,16,18 and recent theoretical investigations of trapping do not address the effect of oil.50,51 Thus, we use Kovscek’s trapping function to describe the trapping of foam even when oil is present.4 However, we argue that residual oil likely traps in the same pores where foam bubbles trap when oil is not present and that the trapped oil effectively replaces a portion of the trapped gas. Measurement of the shapes of solidified residual oil droplets after dissolution of Berea sandstone cores indicates that oil globules are predominantly trapped in pores where the pore-throat radius is 2.54.0 times smaller than the pore-body radius.52 This is expected because, in these same pores, snap-off is favored during imbibition53 and the required mobilization pressure gradient is largest.54 Gas bubbles are analogous to oil drops and similarly trap in pores with small pore throats because of the large mobilization pressure gradient required to displace a foam lamella through the pore throat.50 Based on the argument that the residual oil replaces a portion of the trapped gas, (St + Sor)/(Sg + Sor) is equal to the oil-free maximum trapped-gas fraction. Equivalently, this reduces the trapped-gas fraction by (1 - Xt,max)Sor/Sg where here Xt,max is the oil-free value. Use of Kovscek’s 0.90 maximum trapped-gas fraction and average values of 0.30 and 0.40 for the oil and gas saturations in our experiments yields a trapped-gas fraction of 0.83 when residual oil is present. We use this average value of Xt,max ) 0.83 in all experiments where residual oil is present. There is experimental justification for the increase in the flowing fraction when residual oil is present. Without this increase, the population-balance model predicts that an oil-tolerant foam, a foam where the rate of coalescence due to oil is zero, produces a significantly higher pressure drop in the presence of residual oil than without residual oil present. This is because when oil is present, the gas saturation and gas relative permeability are decreased at an equivalent water saturation. However, with the increased flowing-fraction correction, the model predicts that an oil-tolerant foam produces an approximately equal pressure gradient regardless of the presence of oil. Experiments reported elsewhere1,8,9,12 agree with the increased flowing-fraction case and show


that oil-tolerant foams typically produce approximately equal pressure gradients irrespective of the presence of oil. Finally, in our analysis of the presence of oil, we propose a rate of coalescence of foam by oil, rco, based on the pinch-off rupture mechanism shown in Figure 1 as1

rco ) f1(So) f2(B,vf) f3(Pcpf/Pcpf/)vfnf

(12)

where the factors f1, f2, and f3 are functions of the variables listed in parentheses. To implement eq 12, functional forms of f1, f2, and f3 are required. f1 is the number of sites per unit volume of gas phase where gas bubbles contact oil. Several researchers observe that the pressure gradient in oil-destabilized flowing foam decreases with increasing oil saturation.12,13,55 This indicates that the flow resistance of the foam decreases and that the rate of foam coalescence increases. Generally in these studies, the pressure gradient gradually increases as oil saturation decreases from about 0.40 to 0.15 and then sharply increases to near the pressure gradient of oil-free foam as the oil saturation further decreases from 0.15 to 0.05. This suggests that the number of contacts between oil droplets and gas bubbles slowly decreases as oil saturation decreases until a critical oil saturation where the remaining oil droplets are separated from gas bubbles. While this suggests a more complicated function, we choose f1 so that the number of sites is directly proportional to the oil saturation in our study where So varies from about 0.2 to 0.4, yielding

f1(So) ) k-2So

(13)

where k-2 is a constant. f2 is the proportion of lamellae that rupture by the pinch-off mechanism in a time scale shorter than the time for a lamella to move across a residual oil blob. The time for pinch-off depends on the bridging coefficient, B, and the time for a lamella to move across an oil droplet depends on the lamella velocity, vf. Frye and Berg have calculated the time for pinch-off of foam films bridged by nonwetting solid particles as a function of contact angle.56 By relating the contact angle of a solid particle to the bridging coefficient of an oil drop, we argue that, for superficial velocities of less than 1 m/day, the time for pinch-off is much shorter than the time for a lamella to move across an oil droplet when the bridging coefficient is greater than about 10 (mN/m)2.1 The system we study has a large bridging coefficient, 290 (mN/m)2, indicating that pinch-off occurs in times much shorter than the time to move across an oil droplet. Thus, we choose f2 to be unity:

f2(B,vf) ) 1

(14)

Finally, f3 is the fraction of oil-bubble contact sites where the pseudoemulsion film is ruptured and the Plateau borders of the foam lamellae form finite contact angles with the oil drop, allowing pinch-off to occur. When the oil drops are coated by stable pseudoemulsion films, foam lamellae cannot rupture by the pinch-off mechanism. Analogous to the stability of foam films in the absence of oil, the stability of pseudoemulsion films depends on the imposed capillary pressure, Pcpf, and the critical rupture capillary pressure of the pseudoemulsion film, Pcpf/. As Pcpf f Pcpf/, pseudoemulsion films rupture.

In contrast to the rate of coalescence in the absence of oil, where the rate of coalescence becomes infinite as Pcgw f Pcgw/, the rate of coalescence remains finite as Pcpf f Pcpf/. In the presence of oil, as Pcpf f Pcpf/, all pseudoemulsion films are unstable and rupture. However, in this limit, f3 simply becomes 1 because it is the fraction of oil contact sites where pseudoemulsion films are ruptured. With this in mind, we choose a functional form analogous to a Langmuir adsorption isotherm:

Pcpf/Pcpf/

/

f3(Pcpf/Pcpf ) )

(15)

1 + Pcpf/Pcpf/

where determines the initial slope at which f3 rises. This function gives the desired behavior that, as Pcpf increases, pseudoemulsion films become unstable to smaller disturbances and a larger fraction rupture until eventually all pseudoemulsion films rupture. In the experiments of this work Pcpf/Pcpf/> 1 at all gas saturations when residual oil is present. Accordingly, our experiments do not test this functionality for low values of Pcpf/Pcpf/. However, in other work, experiments are conducted with the same oil and surfactant system in a porous medium where the permeability varies from 5 to 10 µm2.1 In these experiments, Pcpf/Pcpf/< 1, and the foam behaves as oil-tolerant, verifying the functionality given in eq 15. The imposed capillary pressure on a pseudoemulsion film, Pcpf, does not correspond to either the gas-water capillary pressure Pcgw or the oil-water capillary pressure Pcow but is a combination of the two. The imposed capillary pressure on a curved pseudoemulsion film can be derived from the augmented Young-Laplace equations for the gas bubble and oil drop and the pressure drop across the pseudoemulsion film and written in terms of Pcgw and Pcow as (cf. eq 4.10 of ref 57 and section 1.2 of ref 1)

(

Pcpf ) Pcgw

)

(

σow σgw + Pcow σgw + σow σgw + σow

)

(16)

where σgw is the gas-water surface tension and σow is the oil-water interfacial tension. Note that when Pcgw ) Pcow, the case where the pseudoemulsion film is flat, Pcpf ) Pcgw ) Pcow. At and below residual oil saturation, the oil phase exists as discontinuous drops or ganglia trapped in pores throughout the porous medium. The capillary pressure of these isolated drops, Pcow, is not described by the Leverett J function but instead is based on the characteristic pore size where the oil blobs are trapped. From the Young-Laplace equation and Leverett’s scaling of the mean pore radius, 4xk/φ, we choose an average value of Pcow as

Pcow ) 0.50σow

xφk

(17)

Traditional three-phase-flow modeling uses a functional relationship for Pcog, rather than Pcgw.35 Only two of the three capillary pressures in a three-phase system are independent. During three-phase flow in water-wet porous media, it is assumed that water coats the rock grains and occupies the smallest pores, oil flows in the intermediate pores, and gas flows in the largest pores. In this picture, gas and oil compete for pore space in

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the large and intermediate pores, and Pcog is assumed to depend solely on the gas saturation. This is clearly not the case when only immobile discontinuous residual oil is present. In the presence of residual oil, we assume that oil ganglia exclude the gas phase from occupying certain pores, and the gas phase now competes with the water phase for pore space. As during traditional twophase gas injection into water-saturated porous media, the gas phase first invades the largest available pores and then invades smaller pores as the capillary pressure increases. The fact that certain pores are blocked by oil likely changes the exact shape of the gas-water capillary-pressure function, especially at low gas saturation. However, at the high gas saturations, and equivalently at the low water saturations typically present during foam flow, we assume that the gas-water capillary pressure is virtually identical with the two-phase curve and depends only on the water saturation. Thus, we use the two-phase Leverett J function used by Kovscek,4 only now the water saturation varies between Swc and 1 - Sor rather than between Swc and 1. For the surfactant system and oil in our experiments, assuming a residual oil saturation of 0.30 yields values of Pcpf of 2.0 and 3.5 kPa in the 1.1- and 0.28-µm2 cores upon gas entry. We see later that both of these values are considerably higher than the estimated Pcpf/ value used in the simulations. Therefore, in our experiments pseudoemulsion films are unstable, and oil enters the gas-water interface. With this definition of Pcpf, we have a complete functionality for the rate of coalescence of foam by oil, near residual oil saturation:

rco ) k-2So

(

Pcpf/Pcpf/

)

1 + Pcpf/Pcpf/

v fn f

(18)

Like the rate of coalescence in the absence of oil, the rate in eq 18 varies linearly with vf. This suggests that the velocity dependence of pressure gradients should be identical with that of strong foams. This is confirmed in experiments presented elsewhere.1 Dependence of Model Parameters on PorousMedium Permeability. The possible permeability dependence of the parameters of the foam populationbalance model has not been thoroughly investigated. Kovscek, Patzek, and Radke use the population-balance model to predict the flow in heterogeneous porous media consisting of two parallel layers of different permeability.33 In these simulations, they assume that the only dependence of the model parameters on permeability is that the capillary pressure follows the scaling of the Leverett J function.32 It seems reasonable that additional parameters in the model might also depend on the absolute permeability and porosity of the porous medium, because the average pore size scales as xk/φ. These dependencies are complex, and it seems fortuitous that the only impact of permeability on the parameters of the model is the scaling of the capillary pressure. Nevertheless, capillary pressure is a critical parameter in determining the stability of foam in porous media.15,38,39,58 Thus, we include the capillary pressure scaling used by Kovscek, Patzek, and Radke. In addition, we find that it is necessary to adjust k-1 (decrease), k-2 (increase), and Swc (increase) to simulate the experiments in the 0.28-µm2 permeability core with the parameters fit to experiments in the 1.1-µm2 perme-

Table 2. Model Parameters for Core 1 two-phase flow parameters k (µm2) φ f korw g korg Swc σgw (mN/m) σow (mN/m) µw (mPa s) µg

1.1 0.23 3.0 1.0 3.0 0.8 0.20 28.5 5.1 1.0 0.018

foam population-balance parameters a b Xt,max β (cm3) Pcgw (kPa) k1 (s1/3 m-13/3) k-1 (cm-1) R (mPa s2/3 cm10/3) Pcpf/(kPa) k-2 (cm-1)

0.33 1.0 0.90 10-3 15 3.28 × 105 0.0125 6 × 10-6 1.0 5.0 32.0

Parameters Modified for the Presence of Residual Oil in Core 1 0.25 Xt,max 0.83 korw 0.66 korg 0.25 Swc

ability core. Others observe an increasing Swc with decreasing permeability in conventional multiphase flow in water-wet sandstones;59 this is a directly measured effect in our experiments. However, the required changes in k-1 and k-2, the constants in the oil-free and residualoil rates of foam coalescence, cannot be directly measured. We do not imply that k-1 and k-2 necessarily depend on permeability in the exact manner that we have adjusted them, nor are the other model parameters necessarily independent of permeability. Nevertheless, it is highly likely that the ratio k-1/k-2 depends on permeability, because adjustment of no other parameters in the model separately increases the bubble density in the absence of oil and decreases the bubble density in the presence of oil as the porous-medium permeability decreases, as we find experimentally. Furthermore, because k-2 is defined as the number of oil-gas contact sites per volume of gas, it should indeed increase as the permeability decreases because the number of residual oil globules per volume of gas increases with decreasing pore size or permeability. Numerical Method and Parameter Fitting. The FORTRAN code used to solve numerically the equations of the population-balance model is a modified version of the code “FOAM1D” contained in Kovscek’s thesis4 and is available elsewhere.1 The three fundamental variables are gas-phase saturation, gas pressure, and bubble density and are resolved in both axial position and time. Because the oil saturation is constant, only one phase saturation is independent. Using a finitedifference technique, the gas-phase saturation and water pressure are solved simultaneously, while the bubble density and phase mobilities are calculated explicitly. The discretization uses upstream weighting of phase mobilities, and the equations are solved iteratively using the Newton-Raphson technique. In all simulations, 60 grid elements and time steps of less than 1 s are used. Tripling the number of nodes and decreasing the time steps by a factor of 10 produce no significant change in the simulation results. Simulation of 1 PV of injection typically requires less than 3 min on a Silicon Graphics Indigo2 IMPACT 10000 workstation. The parameter values used to simulate experiments in Core 1 are listed in Table 2. Parameters in the left column are those needed for the simulation of conventional two-phase flow in porous media, while those in the right column are specific to the foam populationbalance model. The base parameters for simulations of foam flow in the 1.1-µm2 permeability core are listed in


the top section. In the bottom section of the table, parameters are listed that are adjusted from the base case to model foam flow in the presence of residual oil. Parameters not listed in the bottom section are identical with those used in the oil-free case. The absolute permeability, porosity, and interfacial tensions are measured directly, as reported earlier. Parameters for the relative permeability functions are fit to steady-state relative permeability data obtained on a 30-cm-long core with properties similar to those of Core 1. This core was also obtained from Unocal Science and Technology Division, Brea, CA. The bulk viscosity of the aqueous surfactant solution, measured with an Ubbelohde viscometer, is virtually identical with that of water. The viscosities of water and nitrogen at laboratory temperatures are taken from a standard textbook.60 The exponents for the phase-velocity scaling of the rate of generation and the effective viscosity and the parameters for the trapping function are the values used by Kovscek.4 The limiting capillary pressure is obtained from disjoining-pressure measurements of the identical surfactant solution that are relatively independent of equilibration with different oils.9 As Kovscek does, we adjust the ratio of the coefficients of the rates of generation and coalescence so that the steady-state foam texture for the oil-free case is about 100 mm-3, consistent with the values measured by Ettinger.20 The absolute values of the rate coefficients are set so that the region of net foam generation is restricted to approximately the first 15 cm of the core for the oilfree case. The viscosity coefficient, R, is adjusted to match the experimentally measured transient pressure gradient away from the core entrance. All of the above foam population-balance parameters are fit to a single transient foam-flow displacement experiment, without oil present. The remaining three population-balance parameters are those newly introduced to describe the rate of coalescence due to oil. These parameters are used regardless of the presence of oil, because the rate of coalescence due to oil is zero when the oil saturation is zero. The pseudoemulsion-film rupture disjoining pressure, Pcpf/, is taken from measured values of a pseudoemulsion film consisting of air, the identical surfactant solution, and tetralin.9 The parameter is chosen so that most, 83%, of the foam-oil rupture sites are uncovered when Pcpf ) Pcpf/. The parameter k-2 is adjusted to match the transient pressure gradients away from the core inlet during a single foam-displacement experiment in the presence of residual oil. The changes in korw and Swc in the presence of residual oil are necessary to match the observed relative permeability in the presence of residual oil both with and without foam and also to match the relative permeability of the lower permeability core. We stress that, because of the scaling of the relative permeability, changes in korg are necessary to maintain the same gas relative permeability function throughout all simulation conditions as Swc is changed. The reduction in Xt,max is based on the argument that residual oil traps in the same pores where foam normally traps. As described earlier, the reduction in Xt,max matches the observed pressure-drop behavior of oil-tolerant foams. In Table 3, parameters that are changed to model experiments in Core 2 of absolute permeability 0.28 µm2 are listed. Model parameters not listed in Table 3 are

Table 3. Model Parameters for Core 2 two-phase flow parameters k (µm2) φ korg Swc

0.28 0.18 0.61 0.27

foam population-balance parameters k-1 (cm-1) k-2 (cm-1)

0.0050 135.0

Parameters Modified for the Presence of Residual Oil in Core 2 0.83 korw 0.53 korg 0.30 Swc

identical with the values reported in the corresponding sections of Table 2. As described earlier, slight adjustments to the relative permeability functions are necessary to match the two-phase flow behavior of the lower permeability core. In addition, it is necessary to decrease k-1 and to increase k-2 in order to match the observed foam-flow behavior in the lower permeability porous medium. To summarize our parameter-fitting procedure, in Core 1 we fit all of the parameters of the populationbalance model to one oil-free experiment and to a second single experiment with residual oil present. These same parameters have been used elsewhere to predict the performance of other experiments in Core 1 under different conditions.1 With slight adjustment to the parameters, we model the behavior of foam flow in the lower permeability Core 2, both with and without residual oil present. Results Core Entrance and Outlet Effects. In some experiments during this work, we observed an entrance region in the core with a low pressure gradient and high water saturation that extended as long as 15 cm. This region apparently indicates a region where net foam generation is low before the gas phase develops into finely divided foam. Additionally, in some experiments where the foam was weak or destabilized by oil, we observed an end effect where pressure gradients grew from the core outlet to the core entrance over several pore volumes after gas breakthrough. This entrance region and the core end effect greatly impacted our experimental program. Initially, we attempted to conduct experiments in short cores, less than 15-cm long, to reduce the large pressure drops associated with foam flow in porous media. However, foam flow in these short cores is dominated by entrance and exit effects. Because of space limitations, these effects are not described further here but are described in detail elsewhere.1 It is our opinion that many of the previous foam-flow experiments conducted in short cores are dominated by entrance and end effects. This cannot be confirmed directly because in most cases only overall pressure gradients are recorded and not the more revealing pressure and saturation profiles along the length of the core. However, in many of these experiments large pressure drops are not observed until many pore volumes of fluid have been injected.7,9,12 We believe that in these experiments the foam is initially weak because the core is shorter than the previously described entrance region and does not develop into a strong foam before reaching the core outlet. Once the foam reaches the core outlet, large pressure drops develop because of the capillary end effect. The capillary end effect is also likely the cause of the large pressure drops observed to

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Figure 3. Comparison of experimental (symbols connected by dashed lines) and model (solid lines) saturation profiles for oilfree foam displacement in Core 1.

Figure 4. Comparison of experimental (symbols connected by dashed lines) and model (solid lines) pressure profiles for oil-free foam displacement in Core 1.

build over several pore volumes after gas breakthrough in foam experiments in the presence of foam-destabilizing oil, even in long cores. Here, regardless of the core length, the foam is relatively weak, and the end effect is significant. In these experiments with oil present, the pressure drop is also observed to grow only after several pore volumes of injection.1 In experiments presented below, the end effect at the core outlet is not apparent because experiments are stopped shortly after gas breakthrough. However, entrance effects are apparent in several experiments presented here. More detailed experimental results portraying core entrance and end effects are presented elsewhere.1 Comparison of Experimental Results and Population-Balance-Model Predictions. Figures 3 and 4 portray the experimental saturation and pressure profiles used to fit the Core 1 model parameters to simulation results. In these figures and those to follow, simulation results appear as solid lines whereas the experimental data points appear as symbols connected by dashed lines. Upward arrows on three of the experimental pressure points in Figure 4 indicate that the pressure at this location may be higher than that reported. (The pressure transducer at this location reaches its maximum value during the experiment.) In all cases, the pressure must be less than the pressure reported at the core entrance. The experiment consists of gas and surfactant-solution co-injection at fixed velocities into an oil-free core

completely saturated with surfactant solution. The gas and liquid injection velocities correspond to a total superficial velocity of 30 cm/day with a 97% gas fractional flow rate, evaluated at the outlet pressure. In the first displayed time step, 0.11 PV, gas invades the core and only reduces the aqueous phase saturation to about 0.45, and the pressure drop is barely visible on the scale of the graph. The population-balance model does not predict this entrance effect but instead predicts a steep displacement front where the water saturation sharply reduces to about 0.23 behind the front. In addition, the model predicts a larger pressure drop for the first time step. Later, in both the experiment and the simulation, a pistonlike displacement process is observed, similar to the results of Kovscek.3,4,33 Sharp displacement fronts are seen where the water saturation reduces from 1 to about 0.23 over the width of the front. The sharpness of the experimental fronts agrees well with the simulation results, indicating no gas fingering. The pressure drops directly behind the front are very large. For comparison, the pressure drop across the core for surfactant solution flowing at an equivalent velocity is only 1.9 kPa. This indicates that behind the displacement front is a very strong, low-mobility foam. In fact, the pressure drop across the core became so large at 0.88 PV that the tested pressure limit of the coreholder was approached, and the experiment was stopped. These large pressure drops and low water saturations demonstrate foam’s remarkable ability to decrease the gas-phase mobility. In contrast to the efficient displacement properties of foam, an experiment of nitrogen injection into the brine-saturated core with no surfactant only reduces the water saturation to 0.80 before gas breakthrough. Compressibility of the gas phase is considerable in Figures 3 and 4. At 0.88 PV, the pressure drop across the core is greater than the 830 kPa backpressure. This indicates that the gas-phase density at the core inlet is over twice that near the foam-displacement front where the pressure is essentially the backpressure. Because the gas is significantly compressible, both in the experiment and in the simulation, differences between the experiment and simulation pressure profiles lead to differences in the volume of gas present in the core. This is reflected in the relative position of the experimental and simulation saturation fronts. Both the experimental and simulation saturation profiles in Figure 3 show an entrance region where the water saturation decreases from a higher value at the core entrance to a lower downstream value. Similarly, in Figure 4 the simulated pressure profiles and the experimental pressure profile at time 0.88 PV show a lower pressure gradient in the entrance region, indicating that the gas phase has a lower flow resistance near the core inlet than downstream. In the entrance region, a net generation of gas bubbles occurs, and the highmobility continuous gas phase evolves into a lower mobility foam. In the experiments, a higher saturation near the core inlet and a longer entrance region are observed than predicted by the simulations. Minssieux also observed a significant water-saturation entrance region extending approximately 10 cm.61 Unfortunately, it is difficult to compare the entrance region of the pressure profiles in Figure 4 because of the limited spatial resolution and accuracy of the experimental pressure measurements.


Figure 5. Model transient bubble density profiles for the simulation in Figures 3 and 4. The profile at 10 PV is the steady-state profile.

It is unclear whether the saturation discrepancy in the entrance region is due to an overprediction of the bubble density or an overprediction of the corresponding flow resistance of the low-bubble-density foam in the entrance region. Figure 5 shows that the net region of bubble generation extends much farther than the predicted saturation entrance region. The trapping function predicts that the trapped fraction is 96% of its maximum value at a bubble density of 20 mm-3. The distance where this occurs is roughly equivalent to the length of the predicted saturation entrance region. A lag in either the trapping function or bubble density lengthens this entrance region. Because we know of no method to measure in situ bubble density, we cannot resolve this issue. Beyond the entrance region, the bubble density approaches a uniform value wherever gas is present in the core. The position of the bubble-density fronts matches well the position of the gas-displacement fronts, confirming that the low water saturation and high pressure gradients behind displacement fronts indicate the presence of a finely divided foam. Before breakthrough, compressibility of the gas phase causes the bubble density to increase at each time step. The pressure in the core increases with each time step, and the gas velocity decreases, because a portion of the gas accumulates in the previously gas-filled portions of the core as the gas density increases. Calculations show that although the gas density decreases down the length of the core, in this case the velocity actually decreases down the core length. As the gas velocity decreases, the ratio of foam generation to foam coalescence increases, yielding a higher bubble density. Therefore, before gas breakthrough, the calculated bubble density increases with both time and position in Figure 5. After gas breakthrough, the model predicts a steady-state bubble density, shown at 10 PV in Figure 5, that increases in the entrance region and then decreases down the length of the core. In this case, there is no longer accumulation of gas along the length of the core, and the mass flux of gas along the length of the core is constant. Necessarily, as the pressure decreases down the length of the core, the gas volumetric flow rate and velocity increase, and the rates of bubble generation and coalescence predict that the bubble density decreases down the length of the core. Interestingly, the pressure gradients in Figure 4 are linear downstream of the entrance region, even with these compressibility effects on gas velocity and bubble density. This is because, as is shown in eq 11, the gas-phase viscosity is inversely proportional to gas

Figure 6. Comparison of experimental and model saturation profiles for foam displacement in Core 1 in the presence of residual oil.

Figure 7. Comparison of experimental and model pressure profiles for foam displacement in Core 1 in the presence of residual oil.

Figure 8. Model transient bubble density profiles for the simulation in Figures 6 and 7.

velocity, and thus the gas-phase flow resistance is independent of gas velocity. The results presented in Figures 6-8 are for an experiment conducted at the same flow rates as the experiment in Figures 3-5, but now residual oil is present in the core. The average oil saturation in the core is 0.33. At time 0 PV, the core contains only surfactant solution and residual oil; thus, the residual oil saturation at each position in the core is simply 1 Sw. The quantity of residual oil present in the core corresponds to the area above the 0 PV saturation curve in Figure 6 and remains essentially constant during the foam displacement. As depicted in the figure, the

2736


residual oil saturation varies along the length of the core. In all simulations with residual oil, the initial experimental oil saturation profile sets the residual oil saturation in each simulation grid block. This single experiment is used to fit the parameter k-2 in the rate expression for foam coalescence due to oil. Transient profiles are shown for times before gas breakthrough. Comparison of the pressure-drop magnitudes in Figures 4 and 7 shows that contact with a destabilizing oil greatly increases the gas-phase or foam mobility. The experimental saturation profiles in the presence of oil are much less sharp than those in the absence of oil, and now the final water saturation is only reduced to about 0.32. Both of these results imply that the gas phase has a much higher mobility and a lower bubble density. Although the positions of the experiment and simulation gas fronts agree well, the simulation predicts much sharper fronts. This is because the current model only includes a capillary-pressure dispersion term and does not include terms to account for gas fingering phenomena, such as viscous instabilities or gas override. Interestingly, from times 0.04 to 0.24 PV, the saturation fronts grow sharper as they approach the end of the core. This implies that the gas-phase mobility decreases and the bubble density increases as the front travels the length of the core. This is apparently an extended entrance effect. Although the oil saturation decreases from the core inlet to outlet, the model predicts that the variation in oil saturation is not great enough to explain this effect. The pressure drops in Figure 7 are about a factor of 10 smaller than those in Figure 4, confirming that oil significantly decreases the gas-phase flow resistance. The simulation overpredicts the experimental pressure drops at early times and near the core inlet. We again believe this corresponds to an entrance region where the gas-phase mobility remains relatively high before gas develops into a low-mobility foam. Later, and away from the entrance region, agreement between experiment and simulation is good. Although the pressure and saturation fronts indicate that the foam is greatly destabilized by oil, it still displaces water more efficiently than a simple gas injection and results in much larger pressure gradients. Simulation of gas and brine injection into the core under identical conditions, except with no surfactant present, predicts an average steadystate water saturation of 0.55 and a pressure drop of 6 kPa. The bubble density profiles in Figure 8 indicate that the foam bubble density is roughly 10 times smaller than that predicted in the absence of oil. Large peaks in the bubble density appear near the foam front. These peaks appear to be physical because, near the foam front, the water saturation is high, foam generation rates are high, and foam coalescence is low, whereas behind the foam front, the water saturation is low and the foam coalescence rate is much higher. Thus, near the front a high bubble density may be reached that subsequently decreases quickly as the front passes. No experimental measurements of in situ bubble density exist to verify these predictions. In other experiments, we observe that the pressure drop across a section of core goes through a maximum as the foam front moves through that section.1 This implies a peak in bubble density and flow resistance near the foam front. As expected, the model predicts a smaller bubble density

Figure 9. Comparison of experimental and model saturation profiles for oil-free foam displacement in Core 2.

Figure 10. Comparison of experimental and model pressure profiles for oil-free foam displacement in Core 2.

near the front of the core where the oil saturation is the highest. The same model parameters fit to the experiments in Figures 3-8 have been successfully used to predict additional experiments at different gas and surfactantsolution velocities and oil saturations. These results are not shown here, but are presented elsewhere.1 Effect of Permeability. The remaining two experiments examine the effects of porous-medium permeability on foam flow in both the presence and absence of a residual oil phase. As described earlier, to match the data of these experiments in the 0.28-µm2 permeability Core 2, we use model parameters fit to the experiments in the 1.1-µm2 permeability Core 1 with minor adjustments. We make a reasonable change to the water relative permeability function and also change the coefficients for the rates of coalescence in the absence and in the presence of oil (cf. Table 3). Unfortunately, there are no thorough experimental or theoretical analyses of how the suite of population-balance parameters scales with permeability. Thus, we can only say that our chosen parameters are consistent with our experimental results. In Figures 9 and 10, experimental and simulated saturation and pressure profiles are plotted for foam displacement in the oil-free Core 2. As in the experiment of Figures 3-5, gas and surfactant solution are coinjected into a surfactant-solution-saturated core at a total superficial velocity of 30 cm/day with a gas fractional flow rate of 97%. Simulation results agree with experimental saturation and pressure profiles quite well. Both the experiment and simulation show



sharp saturation profiles yielding a pistonlike displacement. Interestingly, a much shorter, experimental saturation entrance region is observed in Figure 9 than in Figure 3. Examination of earlier times in a replicate experiment shows that an entrance effect also occurs in this lower permeability core. However, the entrance region now only occurs over about the first tenth, or the first 6 cm of the core. It is unclear whether this is a result of the lower core permeability or the glass frit before the core that is not present in Core 1. The glass frit may serve as a pregenerator or allow a capillary end effect to occur in the small gap between the frit and the core. These effects are likely insignificant because the pores of the glass frit are quite large compared to those in the sandstone, and the frit is quite thin in the axial direction. Unlike the results in Figure 3, the saturation profile upstream from the saturation fronts is somewhat uneven. This is also observed in replicate experiments presented elsewhere.1 The uneven saturation profiles are also seen in the foam-flow experiments in the presence of oil and in simple gas and oil displacement experiments of water. Thus, it may simply imply that the gas phase is not a very low mobility foam because the smooth saturation profiles upstream of the front seem to be characteristic of very strong foams only. Indeed the pressure profiles in Figure 10 confirm that the foam has a higher mobility in the lower-permeability porous medium. If foam is simply a Newtonian fluid, the pressure drop must scale inversely with porousmedium permeability. Therefore, a 4-fold increase in pressure drop is expected compared to the experiment in Figure 4. Instead, almost a 4-fold decrease in pressure drop is observed. Hence, the gas-phase mobility is roughly 16 times higher in the lower permeability core. This confirms Kovscek’s simulations33 and recent experiments62 that also predict and show that the mobility of foam is higher in lower permeability media. This behavior can be explained by Leverett’s scaling of capillary pressure that demands, at a given saturation, a higher capillary pressure in lower permeability media.32 In a low-permeability porous medium, even when the foam has a relatively high mobility approaching that of continuous gas and the gas saturation is low, the capillary pressure and the rate of foam coalescence is high. In Figure 11, predicted bubble density profiles are displayed. The simulations predict an approximate 10fold decrease in the bubble density compared to those in Figure 5 for the 1.1-µm2 case. However, we caution

Figure 12. Comparison of experimental and model saturation profiles for foam displacement in Core 2 in the presence of residual oil.

Figure 13. Comparison of experimental and model pressure profiles for foam displacement in Core 2 in the presence of residual oil.


that while these predicted bubble densities agree with the observed experimental results, no direct measure of bubble density is made in any of the experiments. Furthermore, the scaling of model parameters with permeability is not rigorous, and other families of parameters may match the observed behavior while predicting different bubble densities. As in the previous experiment with oil present, peaks in bubble density appear near the displacement fronts. In Figures 12-14, an identical experiment is conducted but now in the presence of 0.31 residual oil saturation. As in the higher permeability core, a larger entrance region is suggested by the experimental pres-

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sure and saturation profiles in the presence of oil. Agreement of the experimental and simulation saturation profiles is quite good, both in position down the length of the core and in the saturation profile behind the displacement front. Similar to Core 1, the presence of oil results in a less efficient displacement of water from the core. The water saturation upstream of the foam front is about 0.35 in the presence of oil versus 0.3 in the oil-free case. Resaturating the core with brine after the foam experiment confirms that little oil is produced from the core during the experiment. The observed pressure gradients confirm that oil again destabilizes the foam and causes lower pressure drops across the core. Interestingly, the pressure gradient away from the entrance region is approximately twice that observed in the identical experiment in Core 1 in Figure 7. Again, if foam behaves as a Newtonian fluid, a 4-fold increase in the pressure gradient is expected. Thus, the gas-phase mobility in the lower permeability core is a factor of 2 lower than that in the higher permeability core. This is much less than the 16fold decrease in foam mobility measured in the absence of oil. However, this can be simply explained by the rate of coalescence due to oil. Because Pcpf/Pcpf/ > 1 in the high-permeability core, the increased capillary pressure in the lower permeability core has little effect on the rate of coalescence in eq 18. This is because once most of the pseudoemulsion films are already ruptured, further increases in the capillary pressure expose few additional oil-gas contacts. A greater mobility difference is expected with a surfactant-oil system where Pcpf/ is greater than the entry value of Pcpf in the higher permeability medium. Indeed, in other work, we find that this same surfactant and oil system produces an oil-tolerant foam in a yet higher permeability porous medium where Pcpf/Pcpf/< 1 except at low water saturation.1 Similar to earlier results, experiments show a larger pressure gradient near and behind the foam front that later decays with time. The simulations underpredict this phenomenon. However, later and away from the displacement front, agreement between the simulated and experimental pressure gradients is quite good. The simulated bubble densities in Figure 14 again show that oil destabilizes the foam, causing a lower bubble density. As in previous experiments, peaks in bubble density are observed near the displacement front. Conclusions Good agreement is seen between experiments and simulations of transient foam flow in the presence of oil and in porous media of different permeability using an extended version of the Kovscek, Patzek, and Radke population-balance model.3,4 The agreement is quite remarkable considering that previously the model was developed and tested to simulate rather strong foam in the absence of oil.3,4 When a new mechanistic rate of coalescence due to oil is included1 and a physically meaningful change to one parameter in the trapping model of the population-balance model is made, it is possible to predict the behavior of relatively weak foams flowing in the presence of oil at varying gas and liquid velocities. Similarly, with only slight adjustment to coefficients in the rates of foam coalescence, the population-balance model accurately describes foam flow in a porous media with a 4 times lower permeability in both the presence and absence of oil. These findings speak

to the strength and generality of the population-balance model. Because it is based on physical mechanisms, it can describe phenomena under a variety of conditions and can readily be extended to model more complicated systems by incorporating additional physically meaningful mechanisms to the model. For foam-flow experiments and simulations in surfactant-saturated 1.1- and 0.28-µm2 permeability Berea sandstone porous media at velocities ranging from 30 to 60 cm/day with gas fractional flow rates ranging from 97 to 98.5%, we make the following conclusions. Experiments show that a significant entrance region exists both in the presence and in the absence of oil where water saturation remains high and pressure gradients are quite shallow. This entrance region of high gas mobility implies a low bubble density and is underestimated both in magnitude and in length by the current population-balance model. We observe this entrance region to extend up to 15 cm in the absence of oil and over 20 cm in the presence of oil. The entrance region appears to be shorter in experiments in the lower permeability porous medium and at higher gas fractional flow rates. Because the entrance region is of comparable length and in some cases longer than porous media used previously in foam-flow experiments, we believe a significant number of previous experimental results are dominated by entrance effects. For experiments in the presence of residual oil and in the 0.28-µm2 permeability porous medium where foam is relatively weak, we observe an end effect where pressure gradients build from the core outlet toward the core inlet with time. In most cases, the pressure gradient caused by this end effect is several times larger than the pressure gradient before gas breakthrough. As the pressure gradients grow, a matching decrease in water saturation is observed to move toward the core inlet. This end effect seems to be caused by a capillary end effect as previously described by Kovscek.4 While we do not study the end effect in detail, it may be an important effect in oil reservoirs when there is flow across fractures or large permeability contrasts. We believe that this capillary end effect is responsible for the long-term increases in pressure drop observed in experiments in short porous media or in experiments where foam is relatively weak. At identical superficial velocities in the 1.1-µm2 permeability porous medium, the presence of an average 0.33 residual oil saturation causes a decrease in the pressure gradient of foam flow by a factor of 10. This is explained by coarsening of the foam due to the rate of foam coalescence by oil. Because the surfactant and oil system has positive bridging and entering coefficients and the pseudoemulsion-film rupture disjoining pressure is exceeded in both porous media, our mechanistic rate expression correctly predicts that foam is destabilized by oil. Experiments in the lower permeability 0.28-µm2 core show a decreased foam mobility by a factor of 16 in the absence of oil and a factor of 2 in the presence of residual oil. Thus, in the absence of oil, the pressure gradient at identical superficial flow rates in both cores is 4 times lower in the lower permeability core, although it would be 4 times higher for a Newtonian fluid. Simulations show that this is due to a coarsening of foam texture caused by the larger capillary pressure in the lower permeability porous medium at equivalent water saturation. When residual oil is present, the effect of capillary pressure is not as significant, and the pressure gradient is actually a factor of 2 larger in the lower


permeability porous medium than in the higher permeability porous medium at identical flow rates and oil saturation. This is because the rupture disjoining pressure of the pseudoemulsion films is already exceeded at all gas saturations in the higher permeability core, and so raising the capillary pressure has little effect on the rupture of pseudoemulsion films. The rate of coalescence predicts a more significant dependence of foam mobility on porous-medium permeability when the rupture disjoining pressure is larger than the entry capillary pressure of the higher permeability porous medium. This is confirmed by other experiments.1 Although we are able to model successfully foam-flow experiments in media of different permeability, a theoretical analysis of the scaling of population-balancemodel parameters with permeability is sorely needed. The experiments and simulations demonstrate foam’s remarkable ability to exhibit a lower mobility in higher permeability porous media. This offers great potential in improved oil recovery where foams can be used to block previously swept high-permeability regions of an oil reservoir and divert flow to low-permeability regions still containing high oil saturation. In this scenario, our extended population-balance model predicts an even lower mobility in the high-permeability region than in the low-permeability region because of lower oil saturation in the previously swept high-permeability region. The decreased sensitivity to porous-medium permeability in the presence of oil observed in our experiments demonstrates an important point. To design foams to divert foam from high-permeability regions to lowpermeability regions efficiently, the rupture disjoining pressure of the foam and pseudoemulsion films must be matched to the permeabilities so that the foam is indeed significantly more stable in the high-permeability regions. We observe the largest disagreements between foamflow experiments and population-balance-model predictions in the entrance region of the core and near foamdisplacement fronts. While these regions can be argued to be quite small compared to oil reservoir dimensions where they are likely unimportant, they are difficult to avoid at the laboratory length scale. This is complicated with weak foams where we are limited to observing transient behavior in order to avoid the capillary end effect that arises when the foam front reaches the end of the core. When foam fronts reach the end of a 60cm-long core, it is questionable whether the region behind the foam front can be considered at steady state. One alternative is to study even longer porous media. Hanssen offers a similar suggestion when studying gasblocking foams and states “the longest laboratory porous medium available should be used, and all data from a less than 1-m long media should be treated with great caution”.63 Indeed, foam has been studied in continuous sandpacks of almost 10 m long.64 However, using cores much longer than 1-2 m presents considerable experimental difficulties such as increased experiment time, large absolute pressure, and incompatibility with standard saturation scanning devices. The second alternative is to study the entrance region and regions near foam-displacement fronts in more detail and to improve the population-balance model so that they are adequately described. As observed in our experiments, this requires improved spatial resolution and accuracy of the pressure measurements.

Acknowledgment This work was supported by the Assistant Secretary for Fossil Energy, Office of Oil, Gas and Shale Technologies of the U.S. Department of Energy under Contract DE-AC03-76FS00098 to the Lawrence Berkeley National Laboratory of the University of California. Nomenclature A ) microwave absorbance B ) σgw2 + σow2 - σοg2 , bridging coefficient, (N/m)2 Eo/w ) σgw + σow - σog, entering coefficient for oil at the gas-water interface, N/m f1, f2, f3 ) functions in the rate of coalescence due to oil J ) Leverett J function k ) absolute permeability, m2 k1, k-1, k-2 ) bubble generation and coalescence rate constants, s1/3 m-13/3, m-1, m-1 kri ) relative permeability of phase i L ) core length, m La ) lamella number, 0.15σgw/σow nf, nt ) flowing and trapped bubble densities, m-3 Pi ) pressure of phase i, N/m2 Pcij ) capillary pressure between phases i and j, N/m2 Pcpf ) capillary pressure on a pseudoemulsion film, N/m2 q ) ug/(us + uw), foam quality Qi ) net source of phase i or bubbles, kg/m3 s, m-3 s-1 rg, rc, rco ) rates of generation and coalescence of bubbles, m-3 s-1 Si, Swc ) saturation of phase i, connate water saturation So/w ) σgw - σow - σog, spreading coefficient for oil at the gas-water interface, N/m t ) time, s ui ) superficial or Darcy velocity of phase i, m/s vi ) ui/φSi, interstitial velocity of phase i, m/s x ) axial distance, m Xf, Xt ) Sf/Sg, St/Sg, flowing and trapped gas functions Greek Symbols R ) constant in the foam viscosity expression, Pa s-2/3 m10/3 β ) constant in the foam trapping function, cm3 φ ) porosity, void fraction of porous medium µi ) viscosity of phase i, Pa s ω ) constant in Leverett J function ) constant in the foam rate of coalescence Fi ) density of phase i, kg/m3 σij ) surface or interfacial tension between phases i and j, N/m Superscripts o ) gas or water relative permeability at connate water saturation or gas-free water saturation, respectively / ) rupture capillary pressure on a foam or pseudoemulsion film a, b ) scaling coefficients in the foam rate of generation f, g ) Corey exponents in relative permeability functions Subscripts b ) bubble f, t ) flowing, trapped gas phase i, j ) dummy indices g, o, w ) gas, oil, and water phases

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Received for review December 20, 1999 Revised manuscript received March 22, 2000 Accepted April 18, 2000 IE990909U

Transient Foam Displacement in the Presence of Residual Oil

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