Transport Model Implementation and Simulation of Microgel

ARTICLE pubs.acs.org/EF

Transport Model Implementation and Simulation of Microgel Processes for Conformance and Mobility Control Purposes Juntai Shi,†,‡ Abdoljalil Varavei,‡,* Chun Huh,‡ Mojdeh Delshad,‡ Kamy Sepehrnoori,‡ and Xiangfang Li† † ‡

MOE Key Laboratory of Petroleum Engineering, China University of Petroleum at Beijing, Beijing 102249, P.R. China The University of Texas at Austin, Austin, Texas 78712, United States ABSTRACT: Various forms of microgels, which are soft and flexible micrometer-scale globules consisting of water-soluble polymer networks, are available for near-wellbore conformance control and deep-penetrating mobility control purposes. Due to various descriptions of the nature and applicability of different microgel systems and conflicting reports regarding their effectiveness, a critical literature review is first performed on the characterization and efficacy of the colloidal dispersion gel (CDG), the preformed particle gel (PPG), the relative permeability-modifying microgel, and the temperature-sensitive and pH-sensitive microgels. We then present an implementation of a model for microgel transport and retention in our 3-D chemical-flooding simulator. In addition, preliminary simulation results on the effectiveness of the CDG and PPG hydrogels are presented. The effectiveness of various microgel systems can be quantitatively tested using the reservoir simulator with the microgel transport and retention model. A mechanistic microgel-trapping model in reservoir rock that depends on pore size distribution has also been developed and implemented in our in-house simulator. In this model, microgel globules do not enter some of the smaller pores, which are bypassed by the preceding waterflood; instead, they are adsorbed on the surface of larger pores, or are hydrodynamically trapped at the throats of larger pores. The published coreflood data were matched to obtain the model parameters. We performed preliminary simulations and identified requirements for the CDG and PPG characteristics and process conditions that will aid in their effective use.

’ INTRODUCTION AND BACKGROUND At present, decreasing the water production and increasing oil and gas recovery efficiencies for the mature oil fields is a main goal in the field of enhanced oil recovery (EOR). Not only does decreasing water production improve the oil recovery efficiency significantly over time, but it also can prolong the life of an oil field and alleviate environmental pollution. To reduce the water production and improve oil recovery efficiency, various methods have been proposed for different treatment purposes: injecting polymer, surfactant, alkali, alkali surfactant polymer (ASP), a gelling system composed of a polymer and cross-linker,1
4 preformed bulk gels,5 partially preformed gels,6 gels with polyethyleneimine (PEI) organic crosslinkers,7,8 colloidal dispersion gels (CDGs),9
22 preformed particle gels (PPGs),23
28 a temperature-triggered microgel known as BrightWater,29
31 pH-sensitive microgels,32
35 and a nontoxic soft size-controlled microgel known as STARPOL.36
42 Some of these methods are used for conformance control, and some are applied for water shutoff, while others can be used for both. Appropriate methods should be chosen and adopted based on the various situations and purposes of treatment. The following paragraph describes the benefits of each method in more detail. For conformance control, polymer can be injected to increase the water viscosity and, thus, improve the microdiversion efficiency and the macroscopic sweep efficiency. Surfactant can be injected to decrease the oil
water surface tension and increase the capillary number and, thus, improve the microdiversion efficiency. ASP can be injected not only to reduce the amount of surfactant and obtain the same effect as that obtained with surfactant injection but also to obtain the same effect as that r 2011 American Chemical Society

obtained with a polymer injection. The pH-sensitive microgels can be injected with water at a low pH condition, and then, they swell after the pH value exceeds one given value to block the very high permeability zone and to diverge the flow of water to the low permeability zones.32
35 Temperature-triggered microgels (BrightWater and polymer solution) can be injected at a low temperature condition and then swell after the temperature exceeds a given value to block very high permeability zones and divert the flow of water to the low permeability zones.29
31 CDGs or stable, soft, size-controlled microgels can be injected to increase the resistance factor of the high permeability zones and to divert the flow to the low permeability zones. CDG will also improve the macroscopic sweep efficiency, by blocking the larger pore throats and diverting the flow to the smaller pore throats, thus improving the microdiversion efficiency.17,21,22,36
42 For water shutoff, preformed bulk gels can be injected to squeeze the fractures or very high permeability streaks. The gelling system composed of a polymer and cross-linker can be injected into the porous media and form gels in situ to block the high permeability streaks. However, because gelling properties have been found to depend on many factors,43
48 such as the gelling time and, consequently, the gelling strength, the depth of the gel penetration is quite difficult to predict. PPGs can be injected to block the fracture and very high permeability zones. The pH-sensitive microgels can be injected into the very high permeability zones at a low pH condition and then swell after the pH value exceeds the critical value to obtain the purpose of water Received: June 7, 2011 Revised: September 25, 2011 Published: September 27, 2011 5063

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Energy & Fuels shut-off. Temperature-triggered microgels (BrightWater and polymer solution) can be injected into the very high permeability zones at a low temperature condition and then swell after the temperature exceeds the critical value to obtain the purpose of water shut-off. CDGs or stable, soft, size-controlled microgels can be designed and made to form multilayer adsorbed microgels on high permeability rock surfaces in situ, until the adsorbed thickness exceeds the pore throat radius, thus, blocking the high permeability zones (water saturated zones). This paper focuses on the application of preformed CDGs, soft, stable, and size-controlled microgels, injected for profile control. A viscosity model of stable microgels was proposed.49 The viscosity of stable microgels, at a zero shear rate, as a function of microgel concentration and correlation between the viscosity of stable microgels and the shear rate, was stablished and validated by matching the experimental data. In this paper, these correlations were implemented into our 3-D chemical-process simulator to simulate the apparent viscosity as a function of microgel concentration and apparent shear rate. In this work, the dynamic jamming ratio, which is the ratio between the diameter of the pore throat and the adsorbed layer thickness of microgels, is proposed for investigating the permeability reduction for each grid block. The adsorbed layer thickness may be smaller than the diameter of the microgel because of the shear rate effect when microgels adsorb on the pore throat wall in a monolayer way. However, the thickness may be greater than the diameter of the microgel because of the interaction between the microgels when they adsorb on the pore throat wall in a multilayer way. Three kinds of pores are determined according to the value of the dynamic jamming ratio: the inaccessible pore, the plugged pore, and the adsorbed pore. An inaccessible pore is defined as when the diameter of the pore throat is less than or equal to the adsorbed layer thickness, the microgels cannot enter into and flow through the pore. As a result of an increase in the dynamic jamming ratio, a plugged pore is where microgels can enter into the pore but not flow through the pore throat, resulting in a partially or completely blocked pore. With a further increase in the dynamic jamming ratio, a pore is defined as an adsorbed pore when the microgels can enter into and flow through the pore throat. CDG cannot enter into an inaccessible pore; hence, permeability is not reduced by the application of CDG; that is, permeability reduction, which is the ratio of permeability before treatment to that after treatment, is equal to 1. With an adsorbed pore, CDG can flow through the porous media, so experiments can be conducted to investigate the transport property of CDG in porous media. There are some experimental data in the literature that can be used to calculate the permeability reduction. For plugged pores, the dynamic jamming ratio is between 1 and 4; CDG can enter into the pore, but may not flow through the pore throat, so the permeability reduction will be very high. However, there is no available equation published to express the permeability reduction in this range. In this work, the permeability reduction for the dynamic jamming ratio between 1 and 4 was analyzed, considering different adsorption qualities of the rock. The permeability reduction trend with the dynamic jamming ratio for the whole range is similar to log-normal distribution, through analyzing the transport mechanism of CDG. Therefore, log-normal distribution was used to match the change of permeability reduction with the dynamic jamming ratio.

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’ LITERATURE REVIEW ON VARIOUS MICROGEL METHODS Colloidal Dispersion Gel (CDG). CDG technology for conformance control was first proposed and used by Mack and Smith to decrease the water production and increase the oil recovery in the Rocky Mountain region.9 Fielding et al. also investigated the application of CDG in North Rainbow Ranch Unit.10 For experimental study, Ranganathan et al. conducted experimental study on CDG rheological properties, aggregate growth of the CDG system, as well as CDG propagation through porous media.11 Smith et al. conducted laboratory studies of in-depth colloidal dispersion gel technology for the Daqing oil field, including CDG formulation, screening, core displacement, and computer simulation studies.12 Moreover, Lu et al. performed several core displacement experiments based on the homogeneous artificial cores and designed a plan of CDG flooding pilot tests in the Daqing oil field.13 In addition, Li et al. studied the size and conformation of CDG aggregate, also known as LPC (linked polymer coils).14 The pilot test of CDG technology in the Daqing oil field was reported as successful by Chang et al.15,16 Moreover, Bjorsvik et al. conducted several experiments about CDG formation, CDG rheological properties, electrophoretic mobility, and CDG particle size.17 Also, Muruaga et al. performed a pilot project in Golfo San Jorge (GSJ) reservoir in Southern Argentina, mainly combining bulk gels and colloidal dispersion gels to improve volumetric sweep efficiency in a mature waterflood.18 In addition, Diaz et al. reported a successful application of CDG to increase oil recovery in Loma Alta Sur field, a mature heterogeneous waterflood in the Meuquen Basin of Argentina.19 In comparison, Al-Assi et al. conducted some displacement experiments in unconsolidated sandpacks with a length of 2 ft or 4 ft.20 Also, Spildo et al. applied CDG for North Sea reservoirs and conducted experiments on CDG properties at high temperatures and high salinity. They proposed a new mechanism of microscopic diversion to explain the reason for increased oil recovery as well.21 Lastly, Spildo et al. investigated the propagation of pregenerated CDGs relative to that of polymer without a cross-linker and of a cross-linker alone through Berea Cores.22 As outlined above, to improve oil recovery, many experiments and pilot tests using CDG have been conducted at different cores and oil fields. Some laboratory coreflood results showed that significant retention of CDG existed in the first part of the cores; however, there is no evidence of CDG propagation through cores to give in-depth permeability modification.11,13,50,51 Some laboratory coreflood results have shown that treatment of a 10Darcy core at an interstitial velocity of 5 ft/day using CDG is limited to about 12 feet, because of the retention of the CDG aggregates.20 Other coreflood results have shown that CDG can propagate through the core with no detectable front end loading and there is little formation damage after passing CDG through the cores.17,21,22 Oil field applications of CDG in oil fields around the world have shown that CDG can successfully increase oil recovery.9,15,16,19,51 Whether or not CDG can propagate through the porous media is highly debated.11,13,17,20
22,50,51 The reason for the discrepancy, however, is that the polymer type, the polymer concentration, the ratio of polymer to cross-linker, the CDG formation process, the salinity, or the reaction temperature is different. Hence, it is very difficult to compare different results and to obtain a certain conclusion. 5064

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Energy & Fuels On the basis of whether CDG is formed in situ or preformed and subsequently injected, we divided the CDG into two types: CDG formed in situ and preformed CDG. CDG formed in situ is predominantly intermolecularly cross-linked CDG, in which viscosity will increase with the reaction time following mixing. Usually, this kind of CDG should be injected into the well immediately or shortly thereafter (1 or 2 h) in case the gel forms in the vicinity of wellbore. In this case, a blocked rock surface may lead to hard injectivity and failure of in-depth treatment. We call this CDG formed in situ, because it is not stable at the surface and because it does not form a microgel or gel until it arrives at the target in the deep reservoir, as designed. Sometimes polymer molecules intercross-link so quickly that CDG aggregates will form and become larger in size in the first part of core leading to the failure of in-depth transportation. Preformed CDG is predominantly intramolecularly cross-linked CDG, in which viscosity will decrease significantly first, then decrease gradually, and finally reach a level value at least 15 days following mixing. Because the CDG is stable at the surface for at least 15 days, and the size continues to be constant when propagating through the porous media, it is referred to as preformed CDG. Preformed Particle Gel (PPG). Preformed particle gel (PPG) application for conformance control or treatment of fractures was investigated by Bai et al.23
28 Bai et al. performed some experimental studies on the behavior and characteristics of particle gel transport through porous media.24,25 Three types of flow patterns were proposed, including pass, broken and pass, and plug. Wu and Bai presented a conceptual mathematical model to simulate the PPG transportation in porous media.27 pH Sensitive Microgels. The use of pH-sensitive poly(acrylic acid) microgels as a surface-prepared gel system to improve conformance control was proposed by Al-Anazi and Sharma.32 This kind of microgel has a molecular network structure and acts similar to polymer molecules but with high molecular weight. It can be dispersed in an aqueous solution, showing a weakly acidic condition due to its carboxyl acid groups. When the pH of the solution is low, the carboxyl groups combine with the many protons available in the solution to make the microgels coil, leading to a small particle size, which results in a low apparent viscosity. However, with the geochemical reaction between the minerals and protons, the pH of the solution will increase and the amount of protons in the solution will decrease; accordingly, the amount of carboxyl groups which lose protons or cannot continue combining the protons will increase. These carboxyl groups are negatively charged, so they repulse each other, leading to a sudden swelling of microgels with water inside the molecular network. The swelling ratio can be up to 1000 times its own volume, making the apparent viscosity increase accordingly by several orders of magnitude.33,34 Because pH sensitive microgels have the above characteristics, particularly a sudden change of particle size with pH, this kind of microgel can be injected into the well with anionic surfactants in a low pH environment (manipulated by a preflush of a diluted chloride acid). It is easy to inject a polymicrogel solution into the high permeability channels, which have been flooded by water because of small size and low apparent viscosity in the acidic condition. Because a preflush is conducted, the velocity of the reaction between protons in the polymicrogel solution and carbonate and alumino-silicate components of reservoirs rock is not very high. This allows the microgel solution to go deep in the reservoir, so that a long distance placement can be achieved. After a long distance propagation of microgels, the apparent

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viscosity will drastically increase at pH values higher than a critical value and maintain a plateau value with further increase in pH.32 The microgels will block the high permeability channels; so, after the treatment, the injected water will flow through the low permeability zone and will drive oil in a low permeability zone into the producing well. If needed, the microgels can be recycled and reused, because they can be reversibly shrunk with a decreasing pH. Simply injecting the acid can make the microgels be produced from the producing well. This method is very effective to improve volume sweep efficiency and conformance. Temperature-Sensitive Microgels. An industry consortium (BP, Chevron, and Ondeo Nalco Energy Services) conducted a multicompany research project known as BrightWater. The goal of this project was to develop a time-delayed, highly expandable material (temperature-triggered microgel) that would improve the sweep efficiency of a waterflood. This kind of material was capable of popping under the influence of temperature and time. When the temperature increases to a given value and after the predesigned time, BrightWater kernel particles will pop (expand by absorbing water and become large in size), and so, the viscosity of the BrightWater will increase significantly. After popping, the expanded kernel particles can provide resistance to fluid flow in the thief zone and can divert subsequent water into unswept, oil saturated, less permeable zones, and finally contribute to increased oil recovery.29
31 Microgels for Relative Permeability Modification. Relative permeability modifier (RPM) is a kind of material that can selectively modify the relative permeability to water or oil. In matured oil fields, large water production is a serious problem, so researchers try to produce special materials to selectively reduce the water relative permeability but retain an unchanged or less changed oil relative permeability. Chauveteau et al. proposed to use soft, stable, nontoxic, and size-control microgels formed and stabilized before injection (STARPOL) as a relative permeability modifier for water shutoff and conformance control.36
42,52
54 This kind of microgel for water shutoff or conformance control can be produced by cross-linking polymers under shear rate, and the microgel size can be controlled by shear stresses. Chauveteau et al. proposed that ideal microgels designed as a RPM for water shutoff or conformance control should meet the following requirements: (1) formed microgels should be insensitive to the shear rate and the geochemical condition; (2) the size of formed microgels can be controlled to prevent the face plugging; (3) the size of formed microgels should be in a proper size to ensure that they can not only propagate deep into reservoir but also significantly reduce the water relative permeability; (4) the formed microgels should be soft enough to ensure that they can yield the capillary pressure in the presence of oil flow in order to be a disproportionate RPM; (5) the formed microgels should strongly adsorb onto the rock surface and be stable over time, even under the severe geochemical conditions in the most reservoirs; (6) the formed microgels should be nontoxic for the environment.53

’ CDG TRANSPORT MODEL Transport Mechanism of Colloidal Dispersion Gel in Porous Media. The jamming ratio, proposed by Cozic et al.,55

represents the ratio between the mean pore diameter and the 5065

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mean diameter of microgels, determined as follows: Jr ¼

dh 2rh ¼ dm dm

ð1Þ

where dh is the mean pore diameter, dm is the mean diameter of microgels, and rh is the radius of the pore throat. The adsorbed layer thickness may be smaller than the diameter of the microgel as a result of the shear rate effect when microgels adsorb to the pore throat wall in a monolayer way. The thickness may also be greater than the diameter of the microgel as a result of the interaction between the microgels when they are adsorbed to the pore throat wall in a multilayer way. In this work, the dynamic jamming ratio, which is the ratio between the diameter of the pore throat and the adsorbed layer thickness of microgels, is proposed for investigating the permeability reduction for each grid block, and can be calculated with the following correlation: JR ¼

dh 2rh ¼ εh εh

ð2Þ

where εh is the adsorbed layer thickness of microgels. Whether or not the stable microgels (such as preformed CDG) are able to go through the pore throat in each grid block can be determined by the dynamic jamming ratio. In this investigation, three kinds of pore are determined according to the value of JR: the inaccessible pore, the plugged pore, and the adsorbed pore. When the diameter of the pore throat is less than or equal to the adsorbed layer thickness, that is, JR e 1, the microgels cannot enter into the inaccessible pore; with an increase in JR, the pore becomes partially or completely

Figure 1. Distribution of tube radius using log-normal distribution.

plugged; moreover, with a further increase in JR, the pore becomes classified as an adsorbed pore. The theoretical critical value of Jr for the transition from the plugged pore to the adsorbed pore was reported as 3 by Cozic et al.55 However, a higher Jr was needed as a result of particle-size distribution, pore throat geometry, size distributions, concentration, and velocity effects. The critical value of JR for the transition from the plugged pore to the adsorbed pore was determined to be 4 in this work. This can be described using the pore-level and capillary-bandle model. The following steps are used to evaluate the permeability reduction factor. On the basis of the distribution function of tube radius (see Figure 1 for a typical log-normal distribution function of tube radius), using the Hagen
Poiseuille equation and Darcy’s law, flow rates in tubes as a function of radius are calculated, and the permeability of model as a function of radius is evaluated (see Appendix A). After gel injection, flow rates in tubes, whose radii are less than the adsorbed thickness, are not affected by CDG, because CDG cannot enter into those tubes. On the other hand, flow rates in tubes whose radii are greater than adsorbed thickness are affected as a result of the decreasing of the radius (approximately by εh). Hence, the permeability of the model for the latter case can be calculated, as explained, using the new radius (rh
εh). After the calculation of new permeability values, the ratio of original permeability to the new permeability will be the permeability reduction factor. Figure 2

Figure 3. Comparison between the measured adsorbed layer thickness by Rousseau et al.41 and those predicted by the equation.

Figure 2. (a) Flow rate versus radius and (b) calculated permeability reduction factor versus the ratio of diameter to adsorbed thickness using the capillary bundle model. 5066

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Figure 4. Comparison between the measured adsorbed layer thickness by Rousseau et al.41 and those predicted by the equation at 10 s
1 apparent shear rate.

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adsorbed microgels will prevent CDG from passing through this pore throat. CDG cannot enter into the pore until the JR is larger than 2. When the JR is larger than 2, CDG can enter into the pore, and the permeability reduction will increase with the JR. When the JR is larger than 3, CDG can partially flow through the pore, and the permeability reduction will decrease with the JR. If the rock can adsorb CDG weakly and slowly, the JR will be smaller than 1 in the inaccessible pore. Because the rock surfaces adsorb CDG weakly—hence, the JR is larger than 1—CDG will enter into the pore, and the permeability reduction will increase with the JR until some CDG may flow through the pore throat when the JR is larger than a certain value. After that, the permeability reduction will decrease with the JR. If the rock can adsorb the microgels neutrally, the JR will be smaller than a value between 1 and 2 in the inaccessible pore. Equation for Rk when JR > 4. The adsorption of microgels on the pore wall has the advantage of reducing water permeability near the wellbore where microgels have been placed. The hydrodynamic thickness of adsorbed layers, εh, can be estimated by the following capillary model relationship:56 εh ¼ rh ð1
Rk
1=4 Þ

Figure 5. Adsorbed layer thickness at 1 s
1 shear rate versus the concentration of microgels.

where Rk is the dimensionless permeability reduction factor. CDG and STARPOL are both stable microgels; transport properties of CDG and STARPOL can both be represented by eq 3. Because few lab data are available for transport of CDG and to investigate the transport property of CDG, the experimental data for transport of STARPOL (refers to the following microgels) are applied next. Figure 3 shows the comparison between the measured adsorbed layer thickness by Rousseau et al.41 and those predicted by eq 4. The influence of the concentrations of microgels was also considered. A good agreement between the predicted and measured adsorbed layer thickness is obtained. The parameter B for all of the concentration of microgels is about
0.06. In this case, the adsorbed layer thickness at 1 s
1 apparent shear rate with the microgel concentrations of 0.0004 g/mL (400 ppm), 0.001 g/mL (1000 ppm), 0.003 g/mL (3000 ppm), and 0.006 g/mL (6000 ppm) is 1.834, 1.9853, 3.0878, and 4.406 μm, respectively. :B εh ¼ εh1γeq

Figure 6. Use of a log-normal distribution curve to match the data from the theoretical equation (eq 11).

shows the flow rate in each tube in the model and the permeability reduction factor for the selected distribution. If the rock can adsorb CDG tensely and quickly, the JR will be smaller than 2 in the inaccessible pore. Because CDG will adsorb onto the entrance of the pore and will not be able to flow, the

ð3Þ

ð4Þ

where εh is adsorbed layer thickness (μm), γ_ eq is equivalent shear rate (s
1), εh1 is adsorbed layer thickness (μm) when equivalent shear rate is 1 s
1, and B is the dimensionless matching coefficient. Figure 4 shows a comparison between the measured adsorbed layer thickness by Rousseau et al.41 and those predicted by eq 4 at 10 s
1 apparent shear rate. As illustrated, a good agreement is obtained. The effect of the concentration of microgels on the adsorbed layer thickness at 1 s
1 equivalent shear rate is shown in Figure 5. It can be seen that the adsorbed layer thickness increases linearly with the increasing concentration of microgels. Lastly, the final correlation between the concentration of microgels and the adsorbed layer thickness can be determined by fitting the experimental data, as shown in the following: εh1 ¼ d2 Cm þ e2

ð5Þ

The parameters d2 and e2 are equal to 468.42 μm/(g 3 ml
1) and 1.5804 μm, respectively, in this case. 5067

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Energy & Fuels Substitution of eq 5 into eq 4 yields the following: :B εh ¼ ðd2 Cm þ e2 Þγeq

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distribution can be expressed as

Through data matching for the adsorbed layer thickness dependent on the shear rate and the concentrations of microgels from the literature,38,40,41,55 eq 6 was acquired to represent the relationship among the adsorbed layer thickness, the shear rate, and concentration of microgels. Substituting eq 6 into eq 3 yields the following: 0 14 r h ð7Þ Rk ¼ @ : BA rh
ðd2 Cm þ e2 Þγeq where the pore throat radius can be estimated by sffiffiffiffiffiffiffiffiffi 8kave rh ¼ 1:15 ϕ

! 1 ðln X
μÞ2 , Y ¼ pffiffiffiffiffiffiffiffiffiffi exp
2σ 2 X 2πσ 2

ð6Þ

ð8Þ

and term γ_ eq is an equivalent shear rate. The in situ shear rate for phase l is modeled by the modified Blake
Kozeny capillary bundle equation for multiphase flow57,58 as : : γc juj ffi γeq ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð9Þ kave krl ϕSl where |u| is the magnitude of apparent velocity, krl is the relative permeability to phase l, Sl is the saturation of the phase l, γ_ c is equal to 3.97C, and C is the shear rate coefficient used to account for nonideal effects, such as slip at the pore walls.58,59 The appropriate average permeability kave is given by "       #
1 1 uxl 2 1 uyl 2 1 uzl 2 kave ¼ þ þ ð10Þ kx ul k y ul kz ul When eq 9 and eq 10 are used in a coreflood for the purpose of investigating the transport properties of microgels in porous media, the apparent flow velocity direction can be set in the x direction. This enables uyl and uzl to be equal to zero, the magnitude of velocity u to be equal to the magnitude of velocity in x direction ux, (i.e. |u| = uxl = Q/A (Q is the flow rate; A is the cross section of the core)), and accordingly, k = kx. Moreover, the saturation of water in the core can be set 1, and the relative permeability to water can be also set to 1. After calculating the pore throat radius and the adsorbed layer thickness, using eq 2, the dynamic jamming ratio can be calculated. If the dynamic jamming ratio is larger than 4, eq 11 (which is the same as eq 7) can be used to calculate the permeability reduction to water:  4 JR ð11Þ Rk ¼ JR
2 Equation of Rk for the Whole Range. The change of the permeability reduction with the JR for the whole range is similar to log-normal distribution, through analyzing the change of permeability reduction in the above discussion; log-normal distribution was used to match the change of permeability reduction with the JR. In probability theory, a log-normal distribution is a probability distribution of a random variable whose logarithm is normally distributed. The probability density function of log-normal

X>0

ð12Þ

where μ and σ are the mean and standard deviation of the variable’s logarithm (by definition, the variable’s logarithm is normally distributed). Modifying the probability density function of log-normal distribution to match the theoretical equation (eq 11) can give Rk ¼ 1:5 "

ðlnðJR
1Þ
μÞ2 pffiffiffiffiffiffiffiffiffiffi exp
þ 2σ 2 ðJR
1Þ 2πσ2 c

!#f ð13Þ

When μ = 0.55, σ = 1.6, c = 57, and f = 1.65, the values from eq 13 match the values calculated by eq 11 (shown in Figure 6). Equation 13 can represent the change of permeability reduction with the dynamic jamming ratio for the whole range. CDG Rk Model. eqs 2, 6, 8, 9, 10, and 13 are used to calculate the permeability reduction factor versus the permeability using the following paremeters. The permeability values in all directions, kx, ky, and kz, and the porosity j are parameters assumed to be known; d2, e2, B, μ, σ, c, and f are input parameters from matching the experimental data; d2 and e2 are parameters from the plot of adsorbed layer thickness at 1 s
1 apparent shear rate versus the concentration of CDG; B is the parameter from the plot of the adsorbed layer thickness versus the equivalent shear rate; and μ, σ, c, and f are parameters from the plot of permeability reduction versus the dynamic jamming ratio. Parameters d2, e2, B, μ, σ, c, and f are all dependent on the properties of CDG, specifically the density and interactive coefficient of CDG.

’ CDG VISCOSITY MODEL The viscosity model of preformed CDG as a function of CDG concentration and apparent shear rate can be expressed as follows:49 μ0CDG ¼ μs ð1 þ ½ηCCDG e½ηKH CCDG Þ : μCDG
μs ¼ ½1 þ ðλγeq Þ2 ðn
1Þ=2 μ0CDG
μs

ð14Þ

λ ¼ aebCCDG

’ RESULTS In this work, two cases of coreflood history-matching were performed to validate the effectiveness of simulator for simulation of the CDG processes for conformance control. The two cores are from the coreflood experiment performed by Spildo et al.21 on fresh cores from a North Sea sandstone oil field at 85 °C with a backpressure of 2 MPa. In the work of Spildo et al.,21 CDG was formed by adding an aluminum citrate cross-link in a 600 ppm polymer solution prepared in synthetic seawater (SSW) at a polymer aluminum ratio of 20:1. The polymer was Flopaam 3630S from SNF Floerger, France, a synthetic polyacrylamide with a hydrolysis of 25
30% and an approximate molecular weight of 20 million 5068

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Table 1. Properties of Cores A and B param

core A

core B

length (cm)

17.3

16.6

A (cm2)

11.2

11.0

j (fraction)

0.33

0.32

kw (μm2)

0.9

0.5

Swi

0.06

0.18

SorCDG

0.2

0.17

Table 2. Fluid Viscosities at 85 °C fluid

viscosity (mPa 3 s)
1

reservoir STO (at 200 s ) SSW

3.6 0.29

CDG prepared in SSW

0.9

Figure 7. Comparison of oil recovery history between the experimental data and those estimated by the polymer model in our simulator for core A.

Table 3. Core Simulation Models for Cores A and B param

core A

core B

Table 6. Comparison of Pressure Drops, RF, and Rk between the Experimental Data and Those Estimated by the Polymer Model in Our Simulator for Core B param

Nx*Ny*Nz Dx (ft)

20*1*4 0.02838

20*1*3 0.02723

Dy (ft)

0.1098

0.1088

Dz (ft)

0.02745

0.03627

k1, k2, k3, k4 (mD)

80, 3050, 40, 40

1400, 55, 18

exptl

est

ΔP1 (psi)

0.43125

0.1

ΔP2(psi)

0.8723

0.83

ΔP3 (CDG)

2.7622

10.3

ΔP4

1.2357

3.75

RF

3.167

12.4

Rk

1.4

4.5

Table 4. Corey Relative Permeability Parameters for Cores A and B param

core A

core B

Swi

0.06

0.18

Sor

0.2

0.17

krwmax

0.2

0.28

kromax

1

1

nw

2.2

2

no

1.7

2

Table 5. Comparison of Pressure Drops, RF, and Rk between the Experimental Data and Those Estimated by the Polymer Model in Our Simulator for Core A param

exptl

est

ΔP1 (psi)

0.36345

0.08

ΔP2(psi)

0.7269

0.75

ΔP3 (CDG)

5.0883

5.09

ΔP4

1.89

1.87

RF Rk

7 2.6

6.8 2.5

Dalton. Five cores were used with different values of permeability, irreducible water saturation, and residual oil saturation, which are in the ranges 0.9
0.1 μm,2 0.06
0.24, and 0.51
0.25, respectively. The series of flooding was first a low-rate waterflood (0.1 mL/min) and then a high-rate waterflood (1 mL/min)

Figure 8. The comparison of oil recovery history between the experimental data and those estimated by the polymer model in our simulator for core B.

before CDG injection at a rate of 1 mL/min. After injecting CDG, the residual oil saturation decreased to 0.17
0.26, and the reduction in the residual oil saturation after CDG injection was in the range 0.61
0.19. Incremental oil recovery after CDG injection was measured for all the cores, but the data of the oil recovery in percent of original oil in place (OOIP), as a function of injected volume, were available just for two cores with higher permeability. Then, these two cores were chosen to perform the coreflood simulation. The properties of the two cores and the fluid viscosities in the work of Spildo et al.21 are shown in Tables 1 and 2. 5069

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Table 7. Corey Relative Permeability Parameters for Cores A and B param

core A

core B

Swi

0.06

0.18

Sor krwmax

0.2 0.18

0.17 0.28

kromax

1

1

nw

2.2

1.5

no

1.3

1.6

Table 8. Parameters for the CDG Viscosity Model and the CDG Transport Model param

value

CCDG (g/mL)

0.0006 (600 ppm)

[η] (mL/g) KH (dimensionless)

409.18 1.0177

a (s)

0.004621

b (1/ g 3 ml
1)

996.9207

n

0.859

d2 (μm/ g 3 ml
1)

470

e2(μm)

5.8804

B (dimensionless)


0.05

μ (dimensionless) σ (dimensionless)

0.55 1.6

c (dimensionless)

57

f (dimensionless)

1.65

Figure 9. Comparison of oil recovery history between the experimental data and those estimated by the stable microgel model in our simulator for core A.

Table 9. Comparison of Pressure Drops, RF, and Rk between the Experimental Data and Those Estimated by the Stable Microgel Model in Our Simulator for Core A param

exptl

est

ΔP1 (psi)

0.36345

0.11

ΔP2(psi)

0.7269

0.79

ΔP3 (CDG)

5.0883

5. 04

ΔP4

1.89

1.81

RF Rk

7 2.6

6.4 2.3

Two core models with multilayers for cores A and B were built to test whether the polymer model in the simulator can simulate the CDG process accurately. The properties of core models are shown in Table 3. Corey relative permeability parameters for cores A and B are listed in Table 4 (see Appendix B). For core A, the comparison of pressure drops; RF, where RF is the resistance factor, which is the ratio of pressure drop during treatment to that before treatment; and Rk between the experimental data and those estimated by the polymer model in our simulator are shown in Table 5. Also, ΔP1 represents the pressure drop during the first waterflood with lower flow rate, ΔP2 is the pressure drop during the second waterflood with higher flow rate, ΔP3 is the pressure drop during CDG treatment, and ΔP4 is the pressure drop after CDG treatment. The comparison of oil recovery history between the experimental data and those estimated by the

Figure 10. Injected and produced concentrations of CDG for core A.

polymer model in the simulator is shown in Figure 7. It can be seen that good agreements were obtained for pressure drops, RF, and Rk; however, the oil recovery history estimated by the polymer model in the simulator was far below the experimental data during the CDG flood, even though a higher adsorption was applied (an example displaying the effect of adsorption on the oil recovery is shown in Appendex C). Other parameters for the polymer model were tuned to match RF, Rk, and oil recovery history at the same time; however, we were not able to obtain a good match (see Appendix D). For core B, the comparison of pressure drops, RF, and Rk between the experimental data and those estimated by the polymer model in the simulator are shown in Table 6. The comparison of oil recovery history between the experimental data and those estimated by the polymer model in the simulator is shown in Figure 8. It can be seen that good agreements were not obtained for pressure drops, RF, Rk, and the oil recovery history, even though a higher adsorption was applied, as shown in Figure 8. The above results indicate the polymer model in the simulator cannot accurately simulate the CDG process. CDG model implemented in the simulator is used to match the experimental 5070

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Table 10. Comparison of Pressure Drops, RF, and Rk between the Experimental Data and Those Estimated by the Stable Microgel Model in Our Simulator for Core B param

exptl

est

ΔP1 (psi)

0.43125

0.11

ΔP2(psi)

0.8723

0.81

ΔP3 (CDG)

2.7622

8.98

ΔP4

1.2357

1

RF Rk

3.167 1.4

11 1.23

Figure 11. Comparison of oil recovery history between the experimental data and those estimated by the stable microgel model in the simulator for core B.

Figure 12. Injected and produced concentrations of CDG for core B.

data; the core models are the same as those for the polymer shown in Table 3. Corey relative permeability parameters for cores A and B are changed by fitting the pressure drop and oil recovery in the waterflood before the CDG flood, as shown in Table 7 (see Appendix B). The parameters for the CDG viscosity model and the CDG transport model are shown in Table 8. For core A, the comparison of pressure drops, RF, and Rk between the experimental data and those estimated by the simulator are shown in Table 9. The comparison of oil recovery history between the experimental data and those estimated by the simulator is shown in Figure 9. It can be seen that good agreements were obtained for pressure drops, RF, Rk, and oil

recovery history. The injected and produced concentrations of CDG profiles are shown in Figure 10 (see Appendix D). For core B, the comparison of pressure drops, RF, and Rk between the experimental data and those estimated by simulator are shown in Table 10. The comparison of oil recovery history between the experimental data and those estimated by the simulator is shown in Figure 11. Although there are still some differences between the experimental resistance factor and those estimated by simulator, a good agreement was obtained for the oil recovery history. The injected and produced concentrations of CDG profiles are shown in Figure 12.

’ SUMMARY AND CONCLUSIONS (1) A critical literature review was first performed on the characterization and efficacy of the colloidal dispersion gel (CDG), preformed particle gel (PPG), the relative permeability-modifying microgel, as well as the temperature-sensitive and pH-sensitive microgels. We think that the following will affect the CDG propagation through the porous media: (1) the polymer type, (2) the polymer concentration, (3) the ratio of polymer to cross-linker, (4) the CDG formation process, (5) the salinity, and/or (6) the reaction temperature. On the basis of the stability of CDG with time at the surface, we divided the CDG into two kinds of CDGs: CDG formed in situ and preformed CDG. (2) The dynamic jamming ratio (JR), which is the ratio between the diameter of the pore throat and the adsorbed layer thickness of microgels, is proposed for investigating the permeability reduction for each grid block. Three kinds of pores are determined according to the value of the dynamic jamming ratio: the inaccessible pore, the plugged pore, and the adsorbed pore. The critical value of the JR for the transition from the plugged pore to the adsorbed pore was selected at 4 because of the bridge effect. (3) The permeability reduction for the dynamic jamming ratios between 1 and 4 was analyzed, considering different adsorption quality of the rock. The trend of the change of the permeability reduction with the dynamic jamming ratio for the whole range is similar to log-normal distribution, through analyzing the transport mechanism of CDG. Hence, log-normal distribution was used to match the change of permeability reduction with the dynamic jamming ratio. (4) The novel CDG viscosity model and transport model were implemented into our 3-D chemical-process simulator to simulate the CDG processes for conformance and mobility control purposes. Preliminary simulation runs, which included two cases of coreflood history-matching, were performed to validate the effectiveness of the simulator. Good agreements were obtained for the simulation of CDG processes for conformance control. ’ APPENDIX A We used the following distribution function, nr: πr πr 2 nr ¼ exp
2rb 4rb 5071

! ðA1Þ

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Figure D1. Relationship between permeability reduction factor and permeability. Figure C1. Adsorption impact on oil recovery for core B, using the polymer option in the simulator.

The flow rate can be calculated using Hagen
Poiseuille and Darcy’s law: Q ¼

Q ¼

Z r 0

Z r 0

nr ðrÞðπr ∇P=8μÞ dr 4

nr ðrÞðKπr 2 ∇P=μÞ dr

Table D1. Conversion Factors conversion factors  1.0  1.0

E
03 E+03

= kg/m3 = kg/m3

cp

 1.0

E
03

ft

 3.048

E
01

= Pa 3 s =m

 9.869233

E
01

= 10
3μm2

 6.894757

E
03

= MPa

ppm g/mL

ðA2Þ

ðA3Þ

millidarcy 2

lbf/in. (psi)

The model permeability can be calculated by the following equation: Z r nr ðrÞðπr 4 ∇P=8μÞ dr 0 Kmodel ¼ Z r ðA4Þ nr ðrÞðπr 2 ∇P=μÞ dr 0

’ APPENDIX B In this appendix, the reason for using different Corey relative permeability parameters (as shown in Table 4 and Table 7) to match the CDG process by using polymer option and CDG option in the simulator is described. When using the polymer option or CDG option in the simulator to match Rk, RF, and oil recovery for CDG treatment, we first match the waterflood to obtain the original Corey relative permeability parameters and then match the CDG flood through modifying the parameters for CDG and the original Corey relative permeability parameters to obtain a better result. When we used the polymer option in the simulator to match Rk, RF, and oil recovery for CDG treatment, a good agreement between lab data and those estimated by the simulator was not obtained no matter how the polymer parameters and Corey relative permeability parameters were changed. Therefore, we used the original Corey relative permeability parameters in this work (as shown in Table 4). However, when we used the CDG option in the simulator to match Rk, RF, and oil recovery for CDG treatment, a good agreement between the lab data and those estimated by the simulator was obtained by modifying the Corey relative permeability parameters; hence, we used the modified Corey relative permeability parameters (as shown in Table 7).

’ APPENDIX C When we simulate the CDG process by using the polymer option in our simulator, a good agreement cannot be obtained for the oil recovery history no matter how the adsorption parameters are changed. Figure C1 is an example, in which a lower value for AD4 corresponds to lower adsorption. In oil fields, AD4 = 3 corresponds to average adsorption, and AD4 = 30 corresponds to very high adsorption. ’ APPENDIX D For the purpose of matching the data, we focused on the average permeability reduction factors and the average RFs for cores A and B; however, permeability reduction factors and RFs are different in different layers for cores A and B, as shown in this section. In this study, the permeabilities for the four layers of core A are 80, 3050, 40, and 40 mD, respectively. The corresponding permeability reduction factors for each layer estimated by the CDG option in the simulator are 1, 8.752, 1, and 1, respectively. The permeability for the three layers of core B are 1400, 55, and 18 mD, respectively. The corresponding permeability reduction factors for each layer estimated by the CDG option in the simulator are 23.03, 1, and 1, respectively. The permeability reduction factor of layer 1 indicates that CDG cannot flow into this layer. The RFs for layer 2 of core A and layer 1 of core B estimated by the CDG option in the simulator are about 18 and 46, respectively. Figure D1 shows the relationship between the permeability reduction factor and permeability, in which the interstitial velocity from 0.143 to 14.3 ft/day includes all the possible interstitial velocities in the oil field. The permeability reduction factor will increase dramatically first and then decrease gradually 5072

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’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT We give our gratitude to the China Scholarship Council for supporting Juntai Shi’s work at The University of Texas at Austin. We also recognize the support of MOE Key Laboratory of Petroleum Engineering in China University of Petroleum (Beijing) and the sponsors the Reservoir Simulation Joint Industry Project at The University of Texas at Austin. ’ NOMENCLATURE a = parameter by fitting the plot of λ versus concentration, s. A = cross area of flow, cm2. b = parameter by fitting the plot of λ versus concentration, 1/g 3 ml
1. B = coefficient from plot of the adsorbed layer thickness versus the equivalent shear rate, dimensionless. c = coefficient from the plot of permeability reduction versus the dynamic jamming ratio, dimensionless. C = shear rate coefficient accounting for nonideal effects, dimensionless. CCDG = concentration of CDG, g/mL. Cm = concentration of microgels, g/mL. d2 = coefficient from plot of the adsorbed layer thickness at 1 s
1 shear rate versus concentration of microgels, μm/sB, ppm. dh = diameter of pore throat, μm. dm = diameter of microgels, μm. e2 = coefficient from plot of the adsorbed layer thickness at 1 s
1 shear rate versus concentration of microgels, μm/sB. f = coefficient from the plot of permeability reduction versus the dynamic jamming ratio, dimensionless. Jr = jamming ratio, dimensionless. JR = dynamic jamming ratio, dimensionless. kave = average permeability, μm2. KH = Huggins constant, dimensionless. krl = relative permeability to phase l, dimensionless. kx = permeability at x direction, μm2. ky = permeability at y direction, μm2. kz = permeability at z direction, μm2. n = parameter by fitting the plot of log (μr/μro) versus shear rate, dimensionless. Q = flow rate, mL/min. RF = resistance factor, dimensionless. rh = radius of pore throat, μm. Rk = permeability reduction factor, dimensionless. Sl = saturation of phase l, dimensionless. SorCDG = residual oil saturation after CDG flood, dimensionless. Swi = irreducible water saturation, dimensionless. |u| = magnitude of apparent velocity, cm/min. ux = apparent velocity in the x direction, cm/min. uy = apparent velocity in the y direction, cm/min. uz = Apparent velocity in the z direction, cm/min. X = random variable, dimensionless. Y = probability density function, dimensionless. γ_ eq = equivalent shear rate, s
1.

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εh = adsorbed layer thickness, μm. εh1 = adsorbed layer thickness when equivalent shear rate is 1 s
1, μm. [η] = zero-shear intrinsic viscosity, 1/(g 3 ml
1). λ = parameter by fitting the plot of log (μr/μro) versus shear rate, s. μ = mean value of the variable’s logarithm, dimensionless. μ0CDG = viscosity at zero shear rate, cp. μCDG = viscosity, cp. μr = relative viscosity, dimensionless. μro = relative viscosity of CDG at zero shear rate, dimensionless. μs = solvent viscosity, cp μspr = Newtonian-specific reduced viscosity, 1/(g 3 ml
1). σ = the standard deviation of the variable’s logarithm, dimensionless. j = porosity, fraction.

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dx.doi.org/10.1021/ef200835c |Energy Fuels 2011, 25, 5063–5075