SPE 139239

Effects of Induced Migration of Fines on Water Cut during Waterflooding Abbas Zeinijahromi, The University of Adelaide, Phillip Lemon, Santos Ltd, Pavel Bedrikovetsky, The University of Adelaide

Copyright 2011, Society of Petroleum Engineers This paper was prepared for presentation at the SPE Middle East Oil and Gas Show and Conference held in Manama, Bahrain, 25–28 September 2011. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract Permeability decline during corefloods with varying water composition, especially with low salinity water, has been widely reported in the literature. It has often been explained by the lifting, migration and subsequent plugging of pores by fine particles, which has been observed in numerous core flood tests with altered water composition. This effect can be considered to provide a relatively simple method for mobility control during waterflooding. In previous research, the Dietz model for waterflooding in a layer-cake reservoir with a constant injection and production rate was combined with a particle detachment model to investigate the effect of fines migration and induced permeability decline on reservoir sweep efficiency. In this work, the analytical model was extended to waterflooding with a given pressure drop between injection and production wells. The modelling showed that permeability decline in the water swept zone, caused by the alteration of the injected water composition and induced fines migration, may be able to improve waterflood performance by delaying water breakthrough and reducing the water cut at producing wells. Introduction Fines migration, and subsequent reduction in permeability, has been observed to occur during core flood experiments as a result of decreased water salinity, increased flow velocity and altered water pH or temperature (Mungan, 1965; Bernard, 1967; Lever and Dawe, 1984; Valdya and Fogler, 1992; Khilar and Fogler, 1998; Civan, 2010). The traditional view of fines migration is that it should be avoided because of its detrimental effect on reservoir permeability. However, during waterflooding, an induced reduction in the effective permeability to water of the reservoir in the water swept zone, caused by fines migration, may be used to provide mobility control to improve the performance of the waterflood. This effect is similar to that of other EOR mobility control techniques such as polymer flooding. Reducing the salinity of the injected water is the most practical method to implement mobility control by induced fines migration as the other parameters that control the release of fines (pH, temperature, velocity) are not easily changed. Low salinity water is also often readily available and inexpensive compared to other alternatives. Low salinity waterflooding, which is presently considered as a very prospective EOR method, has been studied previously. These investigations have largely focused on the effects of water composition on wettability, relative permeability, capillary pressure and residual oil saturation (Tang and Morrow, 1999; Jerauld et al., 2008). These effects appear to be separate phenomena from fines migration but may occur simultaneously with fines migration. Some low salinity core flood studies have reported the release of significant amounts fines (Yildiz and Morrow, 1996; Tang and Morrow, 1999; Rivet et al., 2010), while others have reported no evidence of fines migration (Bernard, 1967; Jerauld et al., 2008; Lager et al., 2008; Pu et al., 2010) even though additional oil was recovered. This paper only considers the effects of fines migration to provide mobility control and does not consider changes to the residual oil saturation or relative permeability curves as a result of injecting low salinity water. Several models describing the release and capture of particles were considered. Kinetics based approaches describing particle release (Shapiro and Stenby, 2000, 2002; Tufenkji, 2007; Yuan and Shapiro, 2010) were found to exhibit a delayed response to an abrupt velocity rise or salinity decrease, which did not agree with the near instantaneous response seen in laboratory experiments

2 SPE 139239

(Ochi and Vernoux, 1998). Hence the maximum retention function model (Bedrikovetsky et al., 2010), which exhibits response without delay, was chosen for the current investigation. The induced formation damage, as a result of fines migration in the water swept zone, can be used for mobility control during waterflooding (Lemon et al, 2011). Introduction of the maximum retention function allowed the effects of fines migration and permeability decline to be integrated into the quasi 2D Dietz model for waterflooding in layer cake reservoir (Dietz, 1953). The Dietz model was used because it provided a relatively simple and transparent analytical solution. However, the analytical model was derived assuming a constant injection / production rate. The practical application of this solution was limited since as most waterfloods are controlled by the reservoir fracture pressure and the injection / production rates vary with time. The current paper extends the previous work, addressing the problem of modelling a waterflood with fines migration for a given pressure drop across the reservoir. First, the maximum retention function as a function of water salinity is introduced and the concepts behind the use of induced fines migration for mobility control are explained. Then an analytical model for normal waterflooding in layer cake reservoir under a given pressure drop across the reservoir is derived and then adapted to include the effects of fines migration. Last, an example application of the model is presented. The injection of low salinity water under a given pressure drop between the injection and production wells was found to increase the time until water breakthrough, decrease the water cut at the producing well and decrease the volume of injected water required while having a negligible effect on oil recovery. Fines mobilisation by alteration of injected water chemistry The results of a typical core flood experiment with a natural sandstone sample are shown on a plot of permeability versus injected water salinity (Fig.1), adapted from Lever and Dawe (1984). In this example, the core plug permeability continuously decreases from 140 md to 17 md, an 88% reduction, as the water salinity decreases from 3% w/w sodium chloride to essentially zero (distilled water). Such permeability decline may be explained by fines migration, induced by a change in the chemistry of the injected water, in this case a decrease in salinity (Khilar and Fogler, 1998). The decrease in salinity of the injected water causes a reduction in the magnitude of the electrostatic force which attaches the fine particles to surface of the rock grains, resulting in release of the fines. A released particle is transported until it encounters a small pore throat which it cannot pass. The particle lodges in this pore throat and is said to be strained (Fig.2). The detachment of a particle from within a pore body causes a negligible increase in permeability; however, plugging of pore throats by strained particles causes a significant permeability reduction. The modified particle detachment model (Bedrikovetsky et al., 2010) uses a maximum (critical) retention function: if the retained concentration of particles is less than the maximum value, particle capture continues according to the classical model of deep bed filtration, otherwise, the concentration of retained particles is equal to the maximum. The maximum concentration of retained particles primarily depends on the flow velocity, water ionic strength γ and pH. However, for the purposes of this investigation, it is assumed to only be a function of ionic strength, as it may be computed at the average velocity:

( )crσ σ γ= (1)

The maximum retention concentration is determined by the condition of mechanical equilibrium of an attached particle, which is described by the torque balance of the electrostatic, drag, lifting and gravity forces acting on the particle. Fig. 3 presents the maximum retention concentration, as a function of water salinity, as obtained from the data by Lever and Dawe (1984). The details of the recalculation of permeability as a function of ionic strength, k=k(γ) (Fig. 1), into the retained particle concentration, σ=σm(γ), are presented in Lemon et al. (2011). It was shown that the dependency (1) from Fig. 3 matched adequately with the maximum retention function as calculated from the mechanical equilibrium of a single fine particle on the wall of a cylindrical capillary. The above observations are sufficient to warrant consideration of the effects of induced fines migration on waterflooding. During a waterflood, the rapid breakthrough of water can be a significant problem, leading to high water cut at producing wells and lower volumetric sweep efficiency for a given volume of injected water. The problem is particularly pronounced for a mobility ratio significantly greater than unity or where the variation of permeability across the reservoir is significant. Mobility control techniques, such as polymer flooding, may be employed to reduce a high mobility ratio by increasing the viscosity of the injection water or decreasing the effective permeability to water of the reservoir in the water swept zone behind the flood front (Lake, 1996). Such techniques decrease the fractional flow of water in the reservoir and hence decrease the water cut at the producing wells. The volumetric sweep efficiency for a given volume of injected water is also increased. Fines release, due to the alteration of the chemistry of the injected water, and the consequent decrease in permeability may be able to provide mobility control and hence the ability to improve waterflood performance. Since the mobilisation of fines by changing the chemistry of the injected water can only take place in the water swept zone, only the effective permeability to water of the reservoir is reduced, reducing the

SPE 139239 3

mobility ratio. Fig. 4 shows that this effect tends to make the flood front more uniform across the reservoir, slowing down propagation of the front in the higher permeability layers and allowing for more advanced displacement in the lower permeability layers. However, the main disadvantage of mobility control is that, for a constant pressure drop across the reservoir, the induced formation damage results in a continued decrease of the injection and production rates with time, potentially slowing oil production overall when compared with waterflooding without mobility control. Dietz model for waterflooding in layer-cake reservoirs The Dietz model for the displacement of oil by water in a two dimensional layer cake reservoir was used to investigate the difference in performance of a waterflood with and without fines migration. The main assumptions of the mathematical model, proposed by Dietz (1953) and adopted for two-phase displacement (Kurbanov, 1961; Hearn, 1971) are:

• piston-like displacement in each layer, i.e. connate water saturation ahead of the displacement front and residual oil saturation behind the front

• gravity and capillary effects are negligibly small compared with the viscous pressure losses • the pressure drop between the injector and producer highly exceeds the pressure variation over the vertical coordinate, i.e.

the pressure is equal in all layers across the reservoir • the injected water flows into each layer of the reservoir cross section in proportion to the permeability of the layer • end point saturations and relative permeability functions are the same for all layers

The same mathematical model is also applicable to gravity dominated, segregated waterflooding where an interface exists between the two phases (Fayers and Muggeridge, 1990; Ingsoy et al., 1994; Turta and Singhal, 2004; Turta et al., 2006; Lindeberg et al., 2009; Berg et al., 2010; Jamshidnezhad and Ghazvian, 2011). The stratified flow in Fig. 4 corresponds to either the viscous dominated case, where permeability increases with depth, or the gravity dominated case. Detailed derivations of the segregated flow models and dimensionless criteria reflecting the conditions where they are applicable can be found in the works by Kanevskaya (1988), Bedrikovetsky (1993), and Yortsos (1992, 1995). It is assumed that oil with connate water fills in the reservoir pore space ahead of the water-oil front and water with residual oil flows behind the front (Fig. 4). The position of the water-oil front at any point can be expressed via the average water saturation over the reservoir cross section:

( ) ( )(1 ),

1or wi

wiwi or

h s H h s shs sH H s s

− + − −= =

− − (2)

Integration of the micro scale Darcy’s law with respect to z across the reservoir yields the following expression for the total flow velocity:

( )( )

( )( )

,

0 ,

⎡ ⎤ ∂= − +⎢ ⎥

∂⎢ ⎥⎣ ⎦∫ ∫

h x t Hrwor rowi

w o h x t

k k pU k z dz k z dzxμ μ

(3)

Under the above assumptions, the water saturation s(x,t) averaged over the vertical section is given by the Buckley-Leverett equation, where the fractional flow function for water is determined by the permeability profile k(z) (Appendix A). The mass balance for water becomes:

( ) 0D D

f sst x

∂∂+ =

∂ ∂ (4)

where the fractional flow function is equal to:

4 SPE 139239

( )( )

( )

( )

( )

( )( )

1

0

1 1

01

−− −

−− −

−− −

=

+

∫

∫ ∫

wi

wi or

wi

wi or

wi

wi or

s ss s

rworD

ws ss s

rwor rowiD D

w o s ss s

k k Z dZf s

k kk Z dZ k Z dZ

μ

μ μ (5)

The dynamics of the water saturation are described by equations (4) and (5), which form a first order quasi-linear hyperbolic equation. The initial and boundary conditions for this equation are given in Appendix B, along with the explicit solution (B-3). The form of the fractional flow function with induced fines migration (7) is shown in Fig. 5 by dashed line.

If the rate of the injection / production is given, the real time t can be transformed into dimensionless time tD using formula (A-1) and the pressure drop between injector and producer can be calculated from Darcy’s law for overall flux (3) using solution (B-3) (Lemon et al., 2011). In the current work, the pressure drop is given, so the rate U(t) is unknown. The detailed derivations of an analytical model are performed in Appendix C. Expressions for pressure drop before and after water breakthrough (C-3) and (C-9) and the definition of dimensionless time tD (C-5) yields an ordinary differential equation for the overall flux and implicit expressions (C-7, C-11) for time t=t(tD). Formulae for times before and after the breakthrough allow for the recalculation of the accumulated injected volume tD into real time t. The expressions for the volume of injected water can be slightly simplified for the case of a constant pressure drop across the reservoir, allowing for explicit formulae for real time versus the injected water volume. For the case before breakthrough (C-7) becomes:

2

2o D

DL tt t G M

k p⎛ ⎞= +⎜ ⎟Δ ⎝ ⎠

μ φ (6)

For the case after breakthrough:

( ) ( )( )( )

( )12 2

21,

1, 1,2

orD

Dwi D

stD Dwi w or

wi D D Do o rwor Dt s t

f z dzt t Dk p t t t dt f s tL k z t

−⎛ ⎞′′⎛ ⎞⎛ ⎞−Δ ′⎜ ⎟− = − = ⎡ ⎤⎜ ⎟⎜ ⎟ ⎣ ⎦⎜ ⎟Λ⎝ ⎠⎝ ⎠ ⎝ ⎠∫ ∫

μμ φ μ

(7)

Waterflood with fines migration Let us consider waterflood in layer-cake formation accounting for particle release due to salinity alternation and particle straining with consequent permeability decline. The fines release is described by the maximum retention function (1). It is assumed that the presence of a small residual oil phase does not change the process of fines release or the maximum retention function. The free run length of released particles in natural reservoirs, the distance a particle travels before being captured, is the reciprocal to the filtration coefficient and typically varies from 0.01-10 m (Pang and Sharma, 1987). Therefore, it is assumed that a released particle is instantly strained at the reservoir scale and the permeability decline occurs on the water front. Following Pang and Sharma (1994), Bachman et al. (2003) and Mojarad and Settari (2007), it is assumed that the inverse to normalised permeability k/k0 is a linear function of retained particle concentration:

( )= +0k 1 βσ

k σ (8)

where β is the formation damage coefficient. The formation damage coefficient for straining is assumed to be much greater than that for attachment, i.e. the detachment of fines causes a negligibly small permeability increase while the plugging of pore throat results in a significant decrease of permeability. So, σ in (8) corresponds to the concentration of strained particles.

SPE 139239 5

Due to the assumption of instant straining of released particles, the concentration of strained particles is equal to the concentration of detached particles minus the concentration of suspended particles produced at the core effluent. This allows for recalculation of the curve “stabilized permeability versus salinity” from Fig. 1 into the maximum retention function (1) (Fig. 3). Incorporating the expression for formation damage (8) into the fraction flow function (5) yields an expression for the fractional flow function with fines migration:

( )

( )( )

( )

( )( )

( )

( )( )

1

0

1 1

01

1

1

−− −

−− −

−− −

+=

++

∫

∫ ∫

wi

wi or

wi

wi or

wi

wi or

s ss s

Drwor

ws ss s

Drwor rowiD

w o s ss s

k Zk dZZ

f s

k Zk kdZ k Z dZZ

μ βσ

μ βσ μ

(9)

Retention in each layer was calculated as follows. The average pore radius in each layer was calculated from permeability and porosity (Amyx et al., 1960):

( ) kr Z 5 φ= (10)

The flow velocity U in each layer is proportional to permeability, so

( ) ( )= DU Z U k Z (11)

where <U> is the average Darcy velocity in the reservoir. The concentration of retained particles in each layer was calculated using the micro scale model for the torque balance of a particle on the wall of a capillary (Bedrikovetsky and Siqueira, 2010) using the above mentioned values for r(Z) and U(Z). The form of the fractional flow function with induced fines migration (7) is shown in Fig. 5 by continuous line. The 1D problem (4,9) subject to initial and boundary conditions (C-1,C-2) allows for exact analytical solution (C-3). The expression for the fractional flow curve is the only difference between the analytical models for normal and alternated salinity water floods. Fig. 4 shows the water-oil interface h(xD,tD=1) after injection of one pore volume for low salinity waterflooding (continuous curve) and for conventional waterflood (dashed curve) as calculated from saturation distribution for lognormal permeability distribution with Cv=0.6. Decrease of permeability in each reservoir point after passing the low salinity water front reduces the permeability variation coefficient up to Cv=0.3. The model (6-8) assumes that salinity alteration with consequent permeability change occurs at the moment of passing the reservoir point by the displacement front, i.e. the lag between the displacement and “tracer” fronts due to the presence of connate water is ignored (Bedrikovetsky, 1993; Lake, 1996). Yet, usually the ratio between the tracer and waterfront speeds is 1.1-1.4, which justifies the assumption of equal velocities for the purposes of rough oil recovery evaluation. Results and discussions The performance of waterfloods with normal and induced fines migration were compared, for a given pressure drop, using the analytical model that was developed previously. All parameters except the injection water salinity were kept constant. The retained concentration in each layer was calculated using the model σ=σm(γ) adjusted from the experimental data by Lever and Dawe (1984), with corrections (10) and (11) for each layer. Hence, it was assumed that the reservoir rock and fines had the same properties as those used in the laboratory tests: sandstone with average permeability k=140 md, porosity φ=0.12 and fines primarily comprised of clay. The normal and low salinity waterfloods were assumed to use water salinities of 30 g/L and 0.25 g/L respectively. Suitable assumptions for the other parameters, mainly related to determining the electrostatic forces, required to calculate the retained concentration function were made. As required by the model, the relative permeability end points were constant for all layers and set to typical values for sandstone: swi=0.2, sor=0.3, krwor=0.2, krowi=0.5. The vertical distribution of permeability across the reservoir was assumed to be lognormal with a mean of 140 md. The effect of different degrees of vertical heterogeneity was investigated by considering three different values for the coefficient of variation, Cv=0.3, 0.6 and 1.5. These

6 SPE 139239

values represent essentially homogeneous, mildly heterogeneous and very heterogeneous reservoirs respectively (Jensen et al., 1997). Variation with viscosity ratios of 5, 10 and 50, with the water viscosity assumed to be 1 cp, was also considered. The first significant difference between a normal waterflood and one with induced fines migration is evident from the fractional flow curves (Fig. 5), calculated from equations (5) and (9) using the mildly heterogeneous case and a viscosity ratio of 10. As expected, the low salinity curve is less than the normal curve for all values of water saturation as the induced fines migration has reduced the effective permeability to water at residual oil saturation. Hence for a given injected pore volume, the average water saturation at breakthrough, and consequently the sweep efficiency, is greater for the low salinity case. However, as the injection rate is not constant the results must be viewed versus real time. For all three cases of reservoir heterogeneity, the water cut with fines migration is lower than that for the normal waterflood (Fig. 6 and Table 1). Initially the benefit is quite significant, due to an increase in the time to breakthrough with fines migration, but this reduces in later time. For example, for the very heterogeneous case, the time to breakthrough increases from 10 to 57 days and the water cut after 6 months of injection is reduced by 19%. However, the reduction in water cut decreases to only 7% after one year. The relative increase in the breakthrough time and the decrease in water cut both become more significant as the reservoir heterogeneity increases. For a constant injection rate, inducing fines migration increases the sweep efficiency at a given time but increases the pressure drop between an injector and producer (Lemon et al., 2011). For a waterflood with a constant pressure drop, there is a negligible increase in recovery factor with fines migration. Fig. 7 shows that the curves of recovery factor versus real time almost coincide for all cases of heterogeneity. This shows that for the waterflood under a given pressure drop, the positive effect on sweep increase is annihilated by the negative effect of induced formation damage. Since there is a negligible difference in recovery factor with and without fines migration, a plot of real time versus pore volumes injected (Fig. 8 and Table 1) allows for a comparative estimate of the volume of injected required for a given volume of produced oil. For all levels of heterogeneity, the number of pore volumes injected at a given time is less for the waterflood with fines migration. The difference is greatest for the case with the highest heterogeneity. For example, after 6 months of injection for the very heterogeneous reservoir, the volume of water injected is 0.64 PVI and 0.3 PVI for the normal and low salinity waterfloods respectively for approximately the same volume of oil recovered. Hence the low salinity waterflood has the advantage of requiring less water at a lower injection rate. Similar results were found when the viscosity ratio was varied for the mildly heterogeneous reservoir (Figs. 9-11 and Table 2). The increase in breakthrough time and reduction in water cut became less significant as the viscosity ratio increased as the very high viscosity ratio controlled the performance of the waterflood (Fig. 9). Again, there was a negligible increase in the recovery factor with induced fines migration (Fig. 10) and the production of a given volume of oil required less injection water (Fig. 11). As the viscosity ratio increased, the volume of injection water required for the low salinity waterflood injection water compared to the normal waterflood decreased. The limitations of the model mean the results of this analysis are indicative only. Fundamentally, these results are based on data from only one core flood with some significant assumptions. The study of data from additional core floods is required to confirm these results. Also, only the vertical performance of a layer-cake reservoir was considered, neglecting areal performance and heterogeneity between the injector and producer. Hence the analysis presented above represents an upper estimate of the possible advantages of induced fines migration when waterflooding. A better estimate of the advantages of waterflooding with induced fines migration could be obtained by incorporating the fines migration model into a dynamic reservoir simulation. This would allow for an analysis with full three dimensional flow and more representative heterogeneity. The model did not consider other effects resulting from the injection of low salinity water, including the alteration of relative permeability and capillary pressure. For example, the injection of low salinity water can decrease the residual oil saturation, resulting in a higher oil recovery than for a normal waterflood (Mungan, 1965; Bernard, 1967), presently considered the main benefit of low salinity waterflooding. Hence the results of this analysis, obtained under the assumption of constant residual oil saturation, underestimate the total benefit of low salinity waterflooding. To get a more complete understanding, the combined effects would have to be captured by the same model. One of the key assumptions of the model is that the maximum retention function is independent of oil saturation and equivalent to the case with single phase water. This is believed to be reasonable for completely water wet rock with high water saturation, since

SPE 139239 7

the release of fines only occurs near the pore walls. A literature search found that the release and capture of fine particles in a porous medium has only been investigated in the presence of single phase water (Lever and Dawe, 1984; Valdya and Fogler, 1992; Khilar and Fogler, 1998; Civan, 2010). For oil wet and mixed wet rocks, where the wetted pore surface area depends on saturation, the saturation dependency of the maximum retention function must be accounted for. Even for the completely water wet case, the effect of residual oil saturation should be investigated. Conclusions The modified particle detachment model with the maximum retained particle concentration as a function of water salinity was validated for single phase flow of water by comparison with laboratory test data. Nevertheless, laboratory validation of this hypothesis for two phase flow of immiscible water and oil is still required. The effects of fines release and capture due to changes in the composition of injected water can be included in the Dietz model for waterflooding in a layer cake reservoir by changing the effective permeability to water in the water swept zone using the maximum retention function. Although only one example of decreasing the salinity of the injected water was considered, the same process could be followed for the alteration of any property of the injection water that results in the release of fines, such as pH. The main effects of induced fines migration on waterflooding with a constant pressure drop between injectors and producers are an increased time to breakthrough, decreased water cut with time and a decreased volume of injection water required. The effects were more pronounced for more heterogeneous reservoirs. For the quasi 2D examples investigated, the time to breakthrough increased 2-5 times, water cut after 6 months of injection decreased by 20-40% and the volumes of injected water after 6 month injection decrease 1.4-2.1 times. In reality, benefit from induced fines migration alone is likely to be less than these ranges. For the examples investigated, altering the injected water composition and inducing fines migration had a negligible effect on oil production versus real time for waterflooding with constant pressure drop between injectors and producers. This means that the negative effects of induced formation damage may be compensated by the positive effect of improved sweep efficiency. Nomenclature Latin letters

f fractional flow of water H reservoir thickness, L, m h position of oil-water interface, L, m k absolute permeability, L2 ,mD

k average permeability, L2 ,mD kD dimensionless permeability krowi oil end point relative permeability krwor water end point relative permeability p pressure, ML-1T-2 , Pa S water saturation

s average water saturation sor residual oil saturation swi connate water saturation t time, T, s tD dimensionless time, PVI U total velocity of the flux, LT-1, m/s <U> average velocity, LT-1, m/s X position of oil-water interface, L, m xD dimensionless coordinate Z dimensionless depth z vertical position of oil-water interface, L, m

8 SPE 139239

Greek letters γ brine ionic strength, molL-3, mol/lit μo oil dynamic viscosity, ML-1T-1,CP μw water dynamic viscosity, ML-1T-1,CP β formation damage coefficient ρ water density, ML-3, kg/m3

σ volumetric concentration of captured particles, L-3, 1/m3

σcr critical volumetric concentration of captured particles, L-3, 1/m3 φ porosity Abbreviations BT Breakthrough PVI Pore volume injected RF Recovery factor WF Waterflooding EOR Enhanced oil recovery Acknowledgement

Authors thank N. Lemon, T. Rodrigues, I. Abbasy, K. Boyle (Santos Ltd, Australia) and F. Machado, A.L. S. de Souza (Petrobras, Brazil) for detailed discussions of the field applications, for support and encouragement. PB is grateful to Prof. P. Currie (Delft University of Technology), Prof. A. Shapiro (Technical University of Denmark) and Prof A. Kotousov (The University of Adelaide) for long-time co-operation in formation damage.

The work is sponsored by Santos Ltd and Discovery grant of Autralian Research Council. References Amix, R. Bass, A. Whiting, A. 1964. Applied Reservoir Engineering. NY: McGraw Hill Book Co. Bedrikovetsky, P. 1993. Mathematical Theory of Oil and Gas Recovery-with Applications to ex-USSR Oil and Gas Fields, Vol. 4, ed G. Rowan,

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Khilar, K., Fogler, H. 1998. Migrations of Fines in Porous Media. Dordrecht/London/Boston : Kluwer Academic Publishers. Kurbanov, A.K. 1961. On Some Generalization of the Equations of Flow of a Two-Phase Liquid in Porous Media. Collected Research Papers on

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Induced Fines Migration. SPE-140141, SPE International Symposium on Oilfield Chemistry, Woodlands, Texas, USA, 11- 13 April. Lever, A., Dawe, R. 1984. Water-Sensitivity and Migration of Fines in the Hopeman Sandstone. Journal of Petroleum Geology 7 (1): 97-107.

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Formation Damage Control Symposium, Lafayette, Louisiana,USA, 14-15 February. doi: 10.2118/31119-MS. Ochi, J., Vernoux, J.-F. 1998. Permeability Decrease in Sandstone Reservoirs by Fluid Injection. Hydrodynamic and Chemical Effects. Journal

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Technical Confrence and Exhibition, Florence, Italy, 19-22 September. doi: 10.2118/134042-MS. Rahman, S., Arshad, A., Chen, H. 1994. Prediction of Critical Condition for Fines Migration in Petroleum Reservoirs. Paper SPE 28760

presented at the SPE Asia Pacific Oil and Gas Conference, Melbourne, Australia, 7-10 November. doi: 10.2118/28760-MS. Rivet, S., Lake, L., Pope, G. 2010. A Coreflood Investigation of Low-Salinity Enhanced Oil Recovery. Paper SPE 134297 presented at the SPE

Annual Technical Confrence and Exhibition, Florence, Italy, 19-22 September. doi: 10.2118/134297-MS. Rousseau, D., Latifa, H., Nabzar, L. 2008. Injectivity Decline from Produced-Water Reinjection: New Insights on in-Depth Particle-Deposition

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10 SPE 139239

Appendix A. Quasi 2D model for waterflooding in layer-cake reservoirs Following Hearn, 1971, and Yortsos, 1992, 1995, the segregated flow equations for waterflooding in a stratified reservoir are derived as follows. The traditional dimensionless variables are introduced (Lake, 1989):

1

0

( ), , , , ( ) , ( ) ,= = = = = = ∫D D Do

x z Ut k k zx Z t P p k z k k z dzL H L UL kφ μ

(A-1)

The position of the water front (Fig. 4) is:,

( ),D Dz h x t= (A-2)

It is assumed that oil with connate water fills the pore space behind the front and ahead of the front is water with residual oil (2). This allows for the position of the water front in (3) to be expressed in term of average saturation and vice versa.

( )

( )

( )

( )

1 1

01

( ) ( )

1

−− −

−− −

Λ = +

∂= −Λ

∂

∫ ∫

wi

wi or

wi

wi or

s ss s

orwor D rowi D

w s ss s

D

s k k Z dZ k k Z dZ

Psx

μμ

(A-3)

Here Λ(s) is the dimensionless total mobility. Calculating the dimensionless total mobility at the end points yields:

( )1

0

1 ( )o oor rwor D rwor

w w

s k k z dz kμ μμ μ

Λ − = =∫ (A-4)

( )1

0

( )wi rowi D rowis k k z dz kΛ = =∫ (A-5)

Introduction of the fractional flow (5) as the ratio between the water and overall fluxes and its substitution into the mass balance equation for water results in eq (4). For the case of a constant flow rate, the equation for unknown saturation in (5) is separated from the equation for pressure (B-4) (Lemon et al., 2011). Appendix B. Analytical solution for waterflooding model Consider the waterflooding of a virgin reservoir, i.e. the initial water saturation throughout the reservoir is equal to the connate water saturation:

:= =D wit 0 s s (B-1)

The boundary condition on the injection well is that only water is injected:

( ):= =Dx 0 f s 1 (B-2)

The solution of the initial-boundary problem (4, B-1, B-2) is self-similar and can be expressed by an explicit formula:

SPE 139239 11

,

( , ) , '( , )

,

⎧<⎪

⎪⎪

= =⎨⎪⎪

>⎪⎩

Dor or

D

DD D D D

D

Dwi wi

D

xs Dt

xs x t s f x tt

xs Dt (B-3)

where:

'( ) , '( )= =or or wi wif s D f s D (B-4)

Welge’s method presents an expression for the average water saturation behind the water front and the recovery factor:

( )1

wi

wi or

s t sRFs s−

=− −

(B-5)

1 ( )( ) ( )'( )

D

D

x f ss t st f s

−= + (B-6)

Appendix C. Pressure drop and rate expressions before water breakthrough The following solves the problem of segregated flow for waterflood in layer-cake reservoir, including determining the injection/production rate U(t) for a given pressure drop Δp(t) across the reservoir. Expressing the pressure gradient from (2) and integrating in xD along the reservoir from injector to producer:

1

0D

D

PP dxx

⎛ ⎞∂Δ = −⎜ ⎟∂⎝ ⎠

∫ (C-1)

the pressure drop between injector and producer is obtained:

( )

( )

( )

1

11 1

0 01

( ) ( )

−−− −

−− −

⎡ ⎤⎢ ⎥

Δ = +⎢ ⎥⎢ ⎥⎣ ⎦

∫ ∫ ∫

wi

wi or

wi

wi or

s ss s

rwor rowiD D D D

w o s ss s

k kp t U k k Z dZ k Z dZ dxμ μ

(C-2)

Now perform the integration for time before breakthrough, tD<1/Dwi, accounting for dependency of the solution (B-3) on the self-similar variable, ξ:

( ) ( )( )( ) ( )

( )

11

0

1

1

, , , ,

orD

wi

st wior D

D Dor wis

DD D

D

Df z dzD tdP t t

s s z s

xs s x t f d f dst

−⎛ ⎞−⎜ ⎟′′⎜ ⎟Δ = = − +

Λ Λ − Λ Λ⎜ ⎟⎜ ⎟⎝ ⎠

′ ′′= = = = =

∫ ∫ξ

ξ ξ ξ ξ ξ (C-3)

Introducing constant G:

12 SPE 139239

( )( )

1

1

or

wi

sw or wi

o rwor rowis

Drowi

f z dzD DGk z k

P Gtk

− ′′= − −

Λ

Δ = +

∫μμ

(C-4)

and substituting the expression (A-1) for dimensionless time tD into (C-3):

( )dU tU L

dt=φ

(C-5)

yields an ordinary differential equation for rate U(t):

( ) 1 DD

o rowi

dtk p t Gt LL k dt

⎛ ⎞Δ = +⎜ ⎟

⎝ ⎠φ

μ (C-6)

The solution is obtained by separation of variables:

( )22

0

2 2 t

D Drowi o

M kt t p z dzGk G L

+ = Δ∫μ φ (C-7)

The explicit expression for accumulated injected volume is obtained from the quadratic equation:

( )( )22

0

1 2 1t

Drowi o rowi

kt p z dzGk G L Gk

= − ± Δ +∫μ φ (C-8)

Expressing pressure drop and integrating in xD results in the expression for the pressure drop across the reservoir after breakthrough, tD>1/Dwi:

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( )( )

1 11 1

0 0 0 0

1

1

11 , ,

; , , ,1 1

orD D

or

D

or or

Dt tD

D D DD D D D

t sor or D

D D D Dor or DD s

P P P dx d d ds P dx t tx x s x s s s s

f z dzD d D xP t t x t f d f dss s s z t

ξ ξ ξ

ξ ξ ξ ξ ξ−

⎛ ⎞⎜ ⎟⎛ ⎞∂ ∂ ∂

= −Λ − = Δ = − = = = + =⎜ ⎟ ⎜ ⎟∂ ∂ Λ ∂ Λ Λ Λ Λ⎝ ⎠ ⎜ ⎟⎝ ⎠

⎛ ⎞⎛ ⎞′′⎜ ⎟

′ ′′Δ = + = + = = = =⎜ ⎟⎜ ⎟ ⎜ ⎟Λ − Λ Λ − Λ⎜ ⎟ ⎝ ⎠⎝ ⎠

Δ

∫ ∫ ∫ ∫ ∫

∫ ∫

( )( )( )( )

1

1,1

or

D

sor

Dor s t

f z dzDP ts z

−⎛ ⎞′′⎜ ⎟= −⎜ ⎟Λ − Λ⎝ ⎠

∫

(C-9)

This results in the following ordinary differential equation for (C-9) rate U(t)

( ) ( )( )( )

1

21,

or

D

sw orD

Do o rwor s t

f z dzDdtk p t tL dt k z

−⎛ ⎞′′⎜ ⎟Δ = −⎜ ⎟Λ⎝ ⎠

∫μ

μ φ μ (C-10)

Solving by separation of variables:

SPE 139239 13

( ) ( )( )( )

12 2

21,2

orD

wi Dwi D

sttD Dwi w or

D Do o rwort t s t

f z dzk t t Dp y dy t dtL k z

μμ φ μ

−⎛ ⎞′′⎛ ⎞⎛ ⎞−⎜ ⎟Δ = −⎜ ⎟⎜ ⎟ ⎜ ⎟Λ⎝ ⎠⎝ ⎠ ⎝ ⎠

∫ ∫ ∫ (C-11)

yields the dependency t=t(tD) allowing for calculating the overall velocity U(t).

14 SPE 139239

Fig. 1- permeability change with salinity variation

Fig. 2- Straining of detached particles in a single pore

SPE 139239 15

Fig. 3- Salinity dependency of retained concentration

Fig. 4- waterflooding in layer cake reservoir and induced fines migration

16 SPE 139239

Fig. 5- Comparison between fractional flow curves with and without induced fines migration

Fig. 6- Effects of induced fines migration on water cut versus time for different heterogeneity (μo=10 CP)

SPE 139239 17

Fig. 7- Effects of induced fines migration on recovery factor versus time for different heterogeneity (μo=10 CP)

Fig. 8- Effects of induced fines migration on PVI versus time for different heterogeneity (μo=10 CP)

18 SPE 139239

Fig. 9 - Effects of induced fines migration water cut versus time for different oil viscosities (CV=0.6)

Fig. 10- Effects of induced fines migration on recovery factor versus time for different oil viscosities (CV=0.6)

SPE 139239 19

Fig. 11- Effects of induced fines migration on PVI versus time for different oil viscosities (CV=0.6)

Table. 1-Indicators of the recovery efficiency for normal and fines-migration-induced waterflood in reservoirs with different heterogeneity

CV BT time, days Water cut after 6 months Cumulative injection after 6 months, PV

Normal WF Low salinity WF Normal WF Low salinity WF Normal WF Low salinity WF

0.3 26 80 0.7 0.5 0.5 0.35 0.6 14 68 0.77 0.56 0.57 0.34 1.5 10 57 0.84 0.65 0.64 0.3

Table 2.-Comparison between the normal and fines-migration-induced waterflood in reservoirs with different oil viscosities

Oil Viscosity, cp

BT time, days Water cut after 6 months Cumulative injection after 6 months, PV

Normal WF Low salinity WF Normal WF Low salinity WF Normal WF Low salinity WF

5 25 151 0.56 0.04 0.36 0.26 10 14 68 0.77 0.56 0.57 0.34 50 5 12 0.95 0.9 2.3 1.1