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Percolation Model for Permeability Reduction in Porous Media by Continuous-Gas Foams Randy D. Hazlett*,† and Mary J. Furr Mobil Exploration and Producing Technical Center, 13777 Midway Road, Dallas, Texas 75244
Percolation modeling concepts are invoked to construct estimates of gas permeability modification as a result of the introduction of low concentration or poorly stabilizing surface-active agents in a gas injection process in porous media. The creation of a low number of quasi-stable lamellae in porous media results in an increased stationary gas saturation and is modeled as a reduction in effective connectivity of the medium. Medium connectivity impacts the minimum free gas saturation for continuity and the associated characteristic length scale strongly correlated with permeability. The complex process of in situ foam generation and propagation is modeled through a foam efficiency parameter and, ultimately, through the assertion of a roughly constant mobile gas saturation for good foaming agents. The model is shown to adequately portray literature foam relative permeability measurements with mobile gas saturation values consistent with the reported values for similar porous media systems. In the limit where gas percolation is lost, a second mode of transport involving bubble propagation is observed. The two regimes of foam transport have been observed in visualization experiments for gas injection into a brine-saturated, matched refractive index sandpack. For many processes, the dramatic decrease in gas mobility associated with propagation of lamellae is detrimental, making process design and modeling for the continuous-gas regime important. Introduction Foam generation and propagation in porous media have been well-studied phenomena.1-19 The oil and gas industry holds a particular interest because of the large number of gas injection projects designed to increase hydrocarbon recovery. The injection of low-viscosity fluids into heterogeneous media, however, often results in poor frontal conformance. Viscous fingering and gas channeling through higher permeability strata can result in significant gas-handling problems at production wells without the anticipated benefits of increased oil production. Introduction of low volumes of surfaceactive agents offers a means to control gas mobility and improve displacement efficiency via in situ foaming action. With limited ability to increase injection pressure, due to either equipment limitations or fracturing pressure of the subsurface formation, the production of mobile foam will restrict injectivity. Kuhlman et al.20 advocated usage of dilute surfactants for CO2 foams because of surfactant propagation, mobile foam pressure gradients, and oil emulsification considerations. Whereas gas mobility reduction is desirable for increased displacement efficiency, the ability to extract hydrocarbons is often directly related to the system energy input through fluid injection. Thus, it is desirable to have modest gas mobility reduction without a significant reduction in the ability to inject gas. A foaming system that propagates rather than blocks flow will also provide an in-depth treatment instead of localized gas diversion in the vicinity of the injection location. Many foam models have been reported for the mobile foam regime.1-4 Estimates of the energy requirement * To whom correspondence should be addressed. E-mail:
[email protected]. † Present address: Potential Research Solutions, 1818 Shelmire Drive, Dallas, TX 75224.
to propagate dense discontinuous-gas phases suggest such foams are not easily mobilized.2,7-9 Considerable effort has been directed toward the mechanisms of foam generation and propagation.1-6 Several generation modes have been documented5 including snap-off, stranding, and bubble division. Likewise, many propagation schemes1-4 have been documented or proposed, such as bubble trains, break-and-reform, and birth-death models. Most models address cases where a foam block is desired. From an application perspective, high-pressure gradients typically cannot be supported on interwell distances. Also, the dependence of in situ foam generation upon a rock’s microstructure is often overlooked. Even when foam is injected into a porous medium, the foam texture is rapidly modified by the interplay among hydrodynamics, interfacial physics, and the pore-level topology. Although the discontinuous-gas regime is more interesting, it is asserted that the continuous-gas regime is more practical for design of in-depth profile or mobility modification treatments. A model for the continuous-gas regime is herein proposed on the basis of percolation theory. Percolation theory has found many applications21-23 in physics and engineering. It has provided the framework for network models17-19,24-26 to simulate the patterns of invasion seen during nonwetting phase injection into porous media. Invasion percolation models do an adequate job of portraying the governing physics for such processes. The major deficiencies of these models relate to the imposed structure of the porosity network. Hazlett has shown it is possible to make network model28 and fluid dynamics29 computations on digital representations of actual pore spaces in reservoir rocks derived from computed microtomography (CMT). Although similar computations could be developed for foaming systems, the work herein describes application of theory in a
10.1021/ie990818x CCC: $19.00 © 2000 American Chemical Society Published on Web 06/24/2000
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more general fashion, when the microporous structure is not well definedsbut characterized through more routine procedures, namely, primary drainage capillary pressure and relative permeability measurements. It is the proposed relationship of foam behavior to routinely measurable quantities that distinguishes this effort from others in the literature. Percolation Modeling theory21,30
is applied to a Background. Percolation network of sites linked together by bonds, typically in a well-defined pattern. Irregular networks are allowed, but linkages should be defined with respect to a given distribution function. In standard percolation theory, properties relating to cooperative effects of occupied network elements are developed on the basis of a random selection of either site or bond occupation with a probability of p. Variations, such as invasion percolation, were developed for processes using conditional probabilities. In invasion percolation,24-26 occupation is allowed for only those network elements that are connected to a predefined set of entry points in the lattice. Continuous distributions of physical bond or site attributes with probabilities defined by cumulative distribution functions can be used to set the bond or site filling order. In these cases, derivations from basic percolation theory can still be useful with random spatial distribution of elemental properties. Bond-bond, site-site, and bond-site spatial correlations can be entertained, although network properties become specific to the assumed correlation structures. The percolation theshold, pc, corresponds to the lowest fractional occupation of sites or bonds for which there exists a nonzero probability for connectivity across the system. These thresholds are lattice specific and are normally defined with respect to random selection of elements to be filled. As long as site and bond properties are randomly distributed, most random theory limits are honored or can be appropriately modified. Percolation probabilities can be obtained through either series solutions or numerical experiments.30 Certain universality conditions and critical-scaling relations make results from previous investigators useful for a variety of problems. Drainage Processes. In porous media, rock-fluid interactions can control the pore-level motions of fluidfluid interfaces. The constrictions in the medium impose energy barriers that must be overcome for interfacial advancement of a nonwetting phase. The height of the energy barrier is dictated by the Young-Laplace equation, so that smaller pore throats have larger barriers. Pore bodies are connected by the constrictions or throats. Once pore throat entry barriers are overcome, the downstream pore bodies fill spontaneously. The interface advances through all subsequent pore throats with lower energy barriers for penetration and their associated pore bodies until interfaces come to rest at smaller constrictions, provided no constraints are placed on the supply of the nonwetting phase. A drainage process is said to take place when the nonwetting phase saturation increases. The invasion of the nonwetting phase is best described as a bond percolation process because occupancy is controlled by the size and shape of the pore throats or constrictions in the network. If the bond volume can be neglected and pore bodies and throat sizes are uncorrelated, the number fraction of bonds occupied can be
Figure 1. Steady-state pressure distribution along a central slice of a three-dimensional, sinusoidally constricted tube as derived from the lattice Boltzmann simulation method, indicating the role of pore constrictions and pore bodies as viscous dissipation and near-constant pressure reservoir components, respectively.
simply related to the site volume fraction of occupation. Every bond problem can be mapped to a site problem.21 We can, in fact, construct an equivalent site percolation process for a covering lattice, generated from the original bond percolation problem. We make use of the relationship
pc(b,L) ) pc(s,Lc)
(1)
where b and s are the probabilites of bond and site occupation and Lc is the covering lattice for the original lattice, L. Properties of the covering lattice are more natural for flow in porous media problems because phase saturation is the accepted tie variable for assignment of transport properties: capillary pressure and relative permeability. Examination of commonly selected outcrop reservoir rocks for research purposes31 has shown the lack of pore body-throat correlation to be more true of Berea sandstone than of Indiana limestone, but the generalization of these observations should be avoided. Percolation and Permeability. The inflection point in a porosimetry experiment, corresponding to the pressure or pore entry size of the first considerable fluid uptake, has been associated with the filled volume fraction at the percolation threshold.34 Medium permeability, k, has been strongly correlated with the length scale associated with the onset of percolation, λc, by Katz and Thompson,34
k ) cλc2(σ/σ0)
(2)
where σ and σ0 are conductivities of brine-saturated rock and brine solution, respectively, and c is a constant on the order of 1/226. This is reasonable because this size represents the smallest pore throat that must be included in a flowing network. De Gennes2 also introduced a characteristic length scale for foam modeling, but his length scale pertained to the distance between lamellae in a description of bubble train mobilization. Our interest concerns a characteristic constriction size. With permeability as a strongly nonlinear function of pore size, the smallest constriction in the flow path will tend to dictate the flow rate-pressure drop relationship. Katz and Thomson34 take the conductivity ratio to reflect the connectedness of the medium. In essence, we have pathways through the rock with a rate-determining link captured by λc and the effective use of porosity represented in a formation factor. The negligible impact of pore bodies is clearly demonstrated in lattice Boltzmann simulations35 of steady flow in periodically constricted tubes. A map of pressure is provided in the pore spaces of one such simulation in Figure 1, where the pressure drop is seen to occur primarily at the constrictions and the pore bodies behave almost as constant pressure reservoirs. Thus, identification of the percolation theshold and how this critical pore size may shift with the addition of surface-
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active agents are the keys to understanding and modeling permeability modification. Percolation and Coordination Number. Percolation theory relates the percolation threshold to a coordination number, that is, the number of bonds connected to a site. Although much of percolation theory is dependent upon the choice of lattice, there are some invariant expressions21 relating percolation threshold, pc, coordination number, z, and dimensionality, d.
zpc(b,L) )
d (d - 1)
d (d - 1)
(4)
where the average number of bonds per site, 〈z〉, is the relevant parameter. In the proposed model for foam in porous media, the impact of surfactant introduction and static lamella production is manifested in a lowering of the effective medium connectivity. Thus, all surfactantrelated parameters, such as concentration, snap-off probability, or critical capillary pressure, serve as modifiers to 〈z〉. In effect, the effective porous medium, as seen by gas, changes as trapped gas alters the possible flow paths through the system. Critical-Scaling Relations. Critical-scaling laws30 from percolation theory allow us to make predictions of network properties, such as permeability or conductivity, as a function of distance from the percolation threshold. We have the proportionality relationship,
keff ∝ (p - pc)µ
krg(Sw) ∝ (p - pc)µ/σλc2
(3)
For random lattices with a population of coordination numbers, it is more helpful to cast the above expression in a statistical sense,27
〈z〉pc(b,L) )
the fraction of pore throats blocked by foam. Whereas Rossen’s interest was in the critical scaling of cluster sizes for estimation of mobilization pressure gradients, this work focuses upon modeling the permeability modification that results from removing bonds in the continuous-gas regime. If we take the effective network to be the gas-filled porosity at immobile water saturation, we can obtain an expression for the gas relative permeability through combination of eqs 2 and 5
(5)
which holds for both bond and site percolation processes. The effective gas permeability, keff, as a function of water saturation, Sw, is represented by the product of the gas relative permeability, krg(Sw), and the endpoint gas permeability, kg(Swi), in the presence of immobile water saturation, Swi. The critical exponent, µ, has been estimated to be 1.7 for three-dimensional problems,32 but more recent detailed simulation results36 have suggested a value of 2.003 ( 0.047. Equation 5 describes the behavior of permeability in the neighborhood of the percolation threshold. Notably, power law curves are typically used to capture the behavior of relative permeability and electrical conductivity over the entire range of observable saturation. We choose to express the relationship of eq 5 in terms of site occupational probability of the covering lattice to recover power law representations in terms of mobile saturation. Not surprisingly, the critical-scaling exponent for conductivity is equal to the so-called “normal” Corey and Archie exponent values of 2, commonly assigned in the absence of experimental data. For multiphase flow situations, eq 4 captures the phase connectedness and how this property changes with fractional occupation or saturation.
We can use eqs 1 and 6 to generate estimates of a gas permeability correction factor, Rf, because of foam as a function of saturation, assuming we equate occupational probability with volume fraction of gas-accessible pore space.
( ) ( )(
λc1 ˆ g) ) Rf(S λc2
2
)
σ1 S ˆg - S ˆ cg2 µ ; σ2 S ˆg - S ˆ cg1
S ˆg ≡
Sg 1 - Swi
(7)
where S ˆ cgi represents the normalized gas saturation at the percolation threshold and subscripts 1 and 2 refer to no foam and foam cases, respectively. The gas mobility reduction by foam in the continuous-gas regime results from a moderate increase in the stationary gas saturation associated with static lamellae at various pore constrictions within the gas-invaded zone. The impact of foam is captured through a shift in the percolation threshold for gas mobility and a reduction in the effective medium connectivity. The model suggests a trapped gas fraction for a weak-to-strong foam transition as the point when gas becomes discontinuous and further gas mobility depends on lamellae propagation. Gas becomes discontinuous when the average coordination number of the effective medium falls below the critical value of 1.5 for percolation in three dimensions.27 To make use of eq 7, we must identify the critical saturation and length scale parameters from common physical measurements. At this point, we turn to measured values of capillary pressure, as capillary pressure is inversely proportional to critical pore throat entry size through the Young-LaPlace equation. The length scale and corresponding capillary pressure are taken at the point of sudden uptake of fluid. Equation 7 may be rewritten in terms of inflection point saturation, S ˆ cg1, and the capillary pressure at that saturation, ˆ cg1) from physical measurements: Πc1(S
( )(
)(
σ1 Πc2(S ˆ cg2) ˆ g) ) Rf(S c σ2 Π (S c1 ˆ g1)
2
)
S ˆg - S ˆ cg2
S ˆg - S ˆ cg1
µ
(8)
With the added assumption that σ1 ≈ σ2, once a capillary pressure curve is provided, the only free parameter of eq 8 is an estimate of the new percolation threshold in the presence of a foaming agent, S ˆ cg2. Making use of eq 4, we choose to recast this parameter as the efficiency of stable lamellae generation, 1 - η,
A Model for Foam Rossen8,9 proposed a percolation theory-based model to describe foam in which the critical parameter was
(6)
S ˆ cg2 ≈
〈z1〉S ˆ cg1 〈z2〉
)
S ˆ cg1 1-η
(9)
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Figure 2. Foam mobility reduction sample calculations for an 800-md Berea sandstone: (a) capillary pressure curve and (b) the mobility reduction factor for various values of the efficiency parameter, η.
where the reduced coordination number in the presence of foam has been expressed as
〈z2〉 ) 〈z1〉(1 - η)
(10)
In Figure 2, we provide an example computation of the mobility reduction factor for various values of the foam efficiency parameter. Note that from theoretical considerations an efficiency of unity is not necessary to produce a foam block. All geometrical snap-off criteria,5,37-39 surfactant structure and concentration effects,14-16 and critical capillary pressure limits4 serve to modify the efficiency parameter in a manner consistent with known behavior. Should relative permeability data be available, eq 5 can be eliminated in favor of actual measurements as a function of saturation. This is desirable because the critical-scaling law applies only in the neighborhood of the percolation threshold. If we choose to represent the data in a Corey-type expression, we can recast eq 5 as an equality,
ˆ g ) 1)(S ˆg - S ˆ gr)n keff ) kg(S
(11)
where the residual gas saturation, S ˆ gr, can represent either the critical gas saturation, S ˆ gc, as the threshold saturation to establish a percolating network in gaswater drainage experiments, or the sum of the critical gas saturation and the trapped gas saturation due to ˆ gc. Following the same line of development, foam, S ˆ gt + S we arrive at the remarkably similar expression for the gas permeability reduction factor,
ˆ g) ) Rf(S
(
)(
Πc(S ˆ gt + S ˆ gc) Πc(S ˆ gc)
2
)
S ˆg - S ˆ gt - S ˆ gc S ˆg - S ˆ gc
n
(12)
where the conductivity ratio has already been eliminated. Again, this expression models the change in porous medium transport properties as the stationary gas-phase shifts the critical pore throat size for percolaˆ gt tion and decreases the free gas saturation, (S ˆg - S S ˆ gc). To apply eq 12, an expression is needed that describes the trapped gas saturation, such as eq 9. Unfortunately, the efficiency parameter of eqs 9 and 10 should not be expected to be a constant. We know that snapoff probability is a function of the pore throatpore body aspect ratio. As gas saturation increases and the number of small penetrated pore throats increases, trapping should also increase dramatically because of active lamellae generation mechanisms. We choose to
Figure 3. Nitrogen foam relative permeability in a 100-140-mesh Ottawa sandpack with dilute solutions of an alkyl toluene sulfonate, Suntech IV. Data are those of Sanchez and Schechter (1989). Curves are least-squares best fits to the present model.
examine a particular form for the saturation dependence of the foam efficiency suggested by the experimental evidence of multiple authors6,10-12 regarding the relative insensitivity of foam response to fractional flow.
η)1-
S ˆ gc S ˆg - C
(13)
This is actually a statement that the free gas saturation is a constant. We will now examine the merit of this functional form in modeling actual experiments. Modeling Literature Data Seldom are relative permeabilities measured for foam flow in porous media. These laborious measurements have given way to simpler foam-screening tests in which the pressure drop and flow rate relationship is examined in the absence of in situ saturation information. We test our proposed model against the literature data suites of Sanchez and Schechter10 and Huh and Handy.12 These authors lumped the effects of foaming agent addition into a foam relative permeability and were thorough enough to include baseline relative permeability data and capillary pressure. Supplemental data for the analysis of the data of Sanchez and Schechter10 were documented by Sanchez.11 Figure 3 shows nitrogen foam relative permeability data of Sanchez and Schechter10 as a function of surfactant concentration and model results with the free gas saturation as the only free parameter. Figure 4 shows the same authors’ data from steam foam experiments with best fit model predictions. Figure 5 indicates the free gas saturations required to fit the data of Figures 3 and 4 for steam versus nitrogen as a function of surfactant concentration. Given that the porous medium was the same 100140-mesh Ottawa sandpack for both nitrogen and steam cases, these results quantify the enhanced stability of nitrogen foam over its steam counterpart. Friedmann et al.6 performed tracer tests in conjunction with nitrogen foam experiments at 150 °C, which indicated a flowing fraction of 0.15 in Berea sandstone with 1 wt % Chaser SD1000. These authors interpreted their results as a flowing foam phase, but their assessment of a flowing fraction is in line with the quantities
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Figure 4. Steam foam relative permeability in a 100-140-mesh Ottawa sandpack with dilute solutions of an alkyl toluene sulfonate, Suntech IV. Data are those of Sanchez and Schechter (1989). Curves are least-squares best fits to the present model.
Figure 6. Unsteady-state foam relative permeability data in Berea sandstone of Huh and Handy (1986) and the best fit model curves with the free gas saturation as the adjustable parameter for four surfactant concentrations: 0.0%, 0.02%, 0.2%, and 1% alkyl toluene sulfonate solutions.
Experimental Section
Figure 5. Best fit model parameters for the data represented in Figures 3 and 4 for steam and nitrogen foam flow in a sandpack as a function of surfactant concentration.
predicted by matching the literature data of Sanchez and Schechter.10 Much earlier, Huh and Handy12 used the same surfactant, only at higher concentration levels, using 400-600-md fired Berea sandstone and nitrogen as the gaseous phase. These authors found significant differences between steady- and unsteady-state relative permeability experiments with foam. Figure 6 represents an attempt to model the unsteady-state relative permeability data of Huh and Handy12 for three different surfactant concentrations. The free gas saturation used to fit the data for each surfactant concentration is annotated in Figure 6. The values of free gas saturation for these room-temperature experiments are expected to be lower than the 0.15 flowing fraction measured by Friedmann et al.6 for a high concentration of a good foaming surfactant in Berea at elevated temperature. They are quite compatible with the experimental measurements of a trapped gas fraction using a dual gas tracer technique by Radke and Gillis,13 who reported values of a trapped gas fraction between 0.72 and 0.99 for nitrogen foam in Berea sandstone. The magnitudes are also comparable with the range in free gas saturation used to model the sandpack nitrogen foam experiments of Sanchez and Schechter.10
Weak versus Strong Foam. To illustrate the benefit and mechanisms of foam generation in porous media, a synthetic, unconsolidated porous medium was constructed using crushed and sorted cryolite (Na3AlF6) sand. Naturally occurring crystalline cryolite was selected on the basis of the ability to match the refractive index using a NaCl brine solution. The coarse-grained (80-120-mesh) fraction was slurry packed in a 1/2-in. internal diameter Plexiglass cell, fitted with interior pressure ports every 5 in. along the 25-in. length. When saturated with 3.4 wt % brine, the porous medium became transparent. Porosity was gravimetrically determined to be 46%, while single-phase permeability was measured at 30.1 darcies. The cell was backlit and monitored by a time lapse video. The Rosemount pressure transducers were calibrated to 0-30 psi and monitored by a digital data acquisition system and strip chart recorder. Gas was introduced via a mass flow controller. Initially, gas was introduced at 8 cm3/h (at STP) into the brine-filled system in the absence of surfactant to log the system response without foam. A gravity-induced channel at the top of the horizontal sandpack due to slumping was quite evident. Gas breakthrough occurred at 0.1 pore volume (PV) injected without any gas invasion into the lower 75% of the sandpack. This is evident in Figure 7a in the upstream portion of the cryolite pack. Following breakthrough, the gas flow induced a pressure gradient of no more than 0.02 psi/ in. Gas and water were co-injected at a 4:1 volumetric ratio. The co-injection scheme was believed to simulate water-alternating gas (WAG), a common field practice to cut gas injection volume and slow gas channeling. Water and gas appeared to flow in separate channels, as no appreciable diversion of gas was observed at this ratio and rate of fluid injection. The pressure transducer response, however, was cyclic, following roughly a 4:1 cycle time between average pressure gradients of 0.005 and 0.125 psi/in. with a period of roughly 5 min. This behavior resulted in a time-averaged pressure gradient of 0.03 psi/in. A 1 wt % dodecylbenzene sulfonate (Alcolac Siponate DS-10) brine was substituted for the aqueous phase at
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Figure 7. Evidence of weak and strong foam generation in a match-refractive index sandpack: (a) Introduction of surfactant generated a low-pressure-drop sweep improvement over the no-foam case as seen by the near-piston-like displacement of a weak foam in a pack clearly showing channeling of gas prior to surfactant introduction. (b) A mobile or strong foam is observed following continued introduction of nitrogen at constant mass flow conditions. Unlike the so-called weak foam, the strong foam is characterized by large, sometimes orders of magnitude, reduction in gas mobility.
the same 4:1 gas-liquid ratio with immediate, pronounced diversion of gas over the full cross section. This is also shown in Figure 7a as a piston-like front progression down the length of the sandpack. Although significant gas diversion took place, the displacement was characterized by a pressure gradient of 0.05 ( 0.03 psi/in. The period of fluctuations was reduced to 15-20 s. The frontal advance rate was roughly half that of the injected rate, indicating that the gas thief channel was still active. The gas saturation behind the weak foam front is estimated to be between 35% and 40%. Depending upon whether the presumed gas flow cycle or time-averaged behavior is taken to represent gas flow characteristics for WAG, a mobility reduction factor between 1.7 and 10 can be estimated. This kind of estimate, however, ignores perhaps the most significant differencesthe ability to effect saturation change. Although the weak foam front traveled in plug flow fashion on a macroscopic scale, close examination of the visual record revealed the formation of temporary fingers. A portion of gas would finger ahead of the main front and then be suppressed. The gas would “fill in” the intervening pore space and then fingering would recommence. Backflow was commonly observed during this process. After the weak foam had progressed roughly 20 in. down the pack, a second front was observed forming in upstream sections. Although the foam regime was clearly observed by sight and documented in Figure 7b, this “strong” foam was easily monitored by a transducer response. Each transducer climbed linearly with the front position until the passage of the pressure port location. Whereas the weak foam response was characterized by pressure gradients of ≈0.05 psi/in., the strong foam generated a pressure gradient for this low-rate coinjection sequence of 5.7 psi/in. despite local gas compression effects. Gas compression reduced the injected foam quality from an initial 80% to about 40%. The pulse response of a sodium iodide tracer as measured by an ion-selective probe suggested an in-situ gas saturation of 70%. Role of Surfactant. The pack was flushed with brine at 10 cm3/min to remove injected Siponate DS-10. Again, gas was injected at 8 cm3/h and seen to migrate swiftly across the upper high-permeability channel. Brine coinjection at the same 4:1 ratio yielded only slight visual response in saturation distribution. Then, 1 PV of 1 wt % Alipal CD-128 (GAF Chemicals, 58% active) was injected without any nitrogen injection. Nitrogen was then injected at 8 cm3/h (STP) without accompanying
surfactant solution. Following breakthrough, the pressure drop across the sandpack was less than 0.5 psi/ft, yet the water saturation was reduced to 25%. Considerable saturation redistribution was effected without the generation of the high-pressure gradients observed in the first series. Co-injection was initiated with no noticeable change in pressure response. A final experiment was performed to clarify the role of surfactant versus the role of co-injection. The pack was flushed with 24 PV of brine at 300 cm3/h to remove Alipal CD-128 and resaturate the medium. Then, 1 PV of 1 wt % Siponate DS-10 was injected, followed by nitrogen injection alone at 8 cm3/h. A pressure response in the range of 4 psi/in. was recorded, indicating the in situ generation of a “strong foam” with Siponate DS-10 in both co-injection and single-phase alternating sequences for foaming solution placement. Thus, although the mode of chemical introduction has an affect, the choice of surfactant plays a significant role in the characteristics of foam-generated in situ. Role of Porous Medium. Although strong foams were spontaneously formed with nitrogen in the unconsolidated sandpack, prominent applications of industrial foam in situ generation involved mobility alteration of carbon dioxide flooding operations in carbonate reservoirs. Two factors became apparent: (i) foams created with carbon dioxide above the critical point were inherently much less stable than comparable nitrogen foams and (ii) the porous medium microscopic topology dictated in situ foam generation potential. Foaming agent performance screening did not involve exhaustive studies of saturation-dependent effective relative permeability or viscosity with and without surfactant at various concentration levels. Rather, less elaborate tests were constructed to identify promising candidate chemicals and modes of application. One such test was a simple mobility reduction factor experiment. In this test, a reservoir core plug was mounted in a Hassler-type core holder in an oven at 125 °F. Brine and carbon dioxide were co-injected into the core against a back pressure of 1900 psi. The steady pressure drop for two-phase flow was recorded for an 85:15 mixture of gas and brine at reservoir conditions. Although multiple flow rates were tested, 10 and 20 ft/day were chosen for benchmarking. The brine-containing surfactant was then substituted for the aqueous phase; again the steady pressure drop response of the brine-surfactant-CO2-core system was recorded. The ratio of the two pressure gradients was recorded as a mobility reduction factor, although no accompanying internal
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Figure 8. Mobility reduction experiments were characterized by injecting 85:15 mixtures of CO2 and brine solution with and without surfactant at 1900 psi back pressure and 125 °F for cores containing these three characteristic pore structures. Individual mobility reduction factors are indicated on each figure, illustrating some reservoir rocks are good, marginal, and poor in situ foam-generating materials. The bar represents 0.5 mm.
change in saturation was monitored. This closely resembles the mode in which the foaming agent would be introduced under a field application scenario. Mobility reduction factors for three different carbonate core plug samples are provided in Figure 8. The 60md, 28% porosity limestone core plug pore system represented in Figure 8a is composed of large, dissolved ooids connected through a system of much smaller pore throats. The 4.2-md, 9.9% porosity dolomite pore system in Figure 8b consisted of elongated pores with considerable precipitation of mineral species between matrix grains. The pores in the 80-md, 21% porosity limestone sample of Figure 8c resemble those of sandstone in that this is a sedimentary pore-grain structure. Although often overlooked, the arrangement of porosity on the microscopic scale strongly influenced the ability to generate quasi-stable lamellae and gas mobility reduction for equivalent surfactant concentrations.
channeling condition without generating large viscous dissipation forces. A strong foam was also observed to form and propagate under constant mass inflow conditions. The strong foam is believed to contain discontinuous-gas bubbles that propagated with lamellae as a pseudo-phase, best modeled with a high apparent viscosity. Mobility reduction factor experiments were performed with carbon dioxide foams in carbonate core plugs with different microscopic pore topologies. The influence of pore structure was evident as some pore systems seemingly could only support weak foams, while others generated effective gas-blocking foams with high characteristic mobility reduction factors. Pore throat-pore body relationships and connectivity are noticeably different between the three porosity types examined for foaming action. Literature Cited
Conclusions A simple percolation-based model was proposed for modeling continuous-gas foams whereby quasi-stable, static lamellae generated by the interaction of the fluids and medium lower the effective connectivity of the porelevel porosity network. The effective coordination number directly impacts the percolation threshold and the smallest pore constriction which must be included in the flowing gas-filled network of pores. The impact is captured by two terms: (i) a shift in the characteristic length scale of the pore system, λc, and (ii) a shift in the free gas saturation and associated critical scaling of permeability in terms of distance from the percolation threshold saturation. Continuous-gas foams provide gas diversion ability with low-to-moderate changes in overall gas mobility. The influences of surfactant, surfactant concentration, medium foam in situ generation potential, and so forth are all captured in a foam efficiency, η. Modeling of foam relative permeability data from the literature suggests that the mobile gas saturation is low and roughly constant. Foaming characteristics marking the progress of weak to strong foam, thought to reflect the transition from continuous to discontinuous-gas, are forecast by the model. At the point of gas percolation loss, if gas is to flow, lamellae must be propagated through the pore system, giving rise to very large pressure gradients. Field conditions can seldom support discontinuous-gas mobilization requirements, making modeling and design of projects in the continuous-gas regime important. The transition from weak to strong foam was confirmed through experiments in a matched refractive index sandpack. A weak foam was created upon introduction of surfactant, which mitigated a severe gas-
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Received for review November 15, 1999 Revised manuscript received May 1, 2000 Accepted May 4, 2000 IE990818X
