Optimal slug size for enhanced recovery by low-salinity waterflooding due to fines migration

Journal of Petroleum Science and Engineering 177 (2019) 766–785 Contents lists available at ScienceDirect Journal of Petroleum Science and Engineeri...

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Journal of Petroleum Science and Engineering 177 (2019) 766–785

Contents lists available at ScienceDirect

Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol

Optimal slug size for enhanced recovery by low-salinity waterﬂooding due to ﬁnes migration

T

L. Chequera, K. Al-Shuailia, L. Genoletb, A. Behrb, P. Kowollikb, A. Zeinijahromia, P. Bedrikovetskya,∗ a b

Australian School of Petroleum, The University of Adelaide, Adelaide, SA, 5005, Australia Wintershall Holding GmbH, EOT/R, Friedrich-Ebert Straße 160, 34119, Kassel, Germany

A R T I C LE I N FO

A B S T R A C T

Keywords: Low-salinity waterﬂooding Slug injection Fines-assisted waterﬂood Fines migration Sweep eﬃciency Mobility-control EOR

The objective of this study is modelling of low-salinity water (LSW) slug injection followed by continuous highsalinity chase drive. We discuss the case of ﬁnes-assisted low-salinity waterﬂood, where low salt concentration of the injected water results in mobilisation and migration of the in-situ reservoir ﬁnes and the consequent formation damage. This diverges the injected water ﬂux into low-permeability zones and enhances sweep eﬃciency. We extend the mathematical model for two-phase ﬂow with varying salinity and ﬁnes migration for the case of over and undersaturated ﬁnes, where the detaching drag torque, exerting the ﬁne particle, exceeds or is below the attaching electrostatic torque, respectively. The laboratory study has been performed to determine how movable clays and the induced permeability damage is distributed between the layers. We distinguish between two competitive physics eﬀects of “ﬁnes-assisted low-salinity”: (i) plugging of the high-permeability layer that results in the ﬂux diversion into unswept zones yielding enhanced oil recovery, (ii) induced formation damage behind the waterfront in low-permeable zones that reduces the recovery. These opposite eﬀects indicate the existence of an optimal low-salinity slug size that results in maximum recovery. The modelling shows that for all simulated cases there does exist an optimal low-salinity slug size, which is the main result of the work. It was found that the volume of optimal low-salinity slug has an order of magnitude of the pore volume of the high-permeability layer.

1. Introduction Low salinity waterﬂooding (LSW) is currently one of the most promising methods of enhanced oil recovery (EOR) (Bartels et al., 2019; Lake et al., 2014; Muggeridge et al., 2014). Successful results have been shown in numerous laboratory studies and several ﬁeld pilots (Morrow and Buckley, 2011; Sheng, 2013, 2014), where a signiﬁcant decrease in residual oil saturation sor and relative permeability for water krw was observed. The above eﬀects are explained by decrease of contact angle and interfacial tension with consequent wettability alteration, osmosis with mobilisation of the fraction of residual oil, ionic exchange between ﬂuids and rock detaching some oil, oil-water emulsiﬁcation, chemical reactions between the injected water and rock, ﬁnes migration, and some others (Aghaeifar et al., 2015; Aksulu et al., 2012; Al-Ibadi et al., 2018; Al-Saedi et al., 2018; Farajzadeh et al., 2015, 2017; Mahani et al., 2015a, b, 2017; RezaeiDoust et al., 2009; Sheng, 2010; Torrijos et al., 2016).

∗

Fines migration is the most controversial topic in LSW. Intensive discussions on the role of ﬁnes migration in LSW started in late 90th (Sarkar and Sharma, 1990; Scheuerman and Bergersen, 1990; Tang and Morrow, 1999) and are on-going (Al-Yaseri et al., 2016; Bartels et al., 2019; Schembre and Kovscek, 2005; Song and Kovscek, 2016). Some low salinity core ﬂood studies have reported the release of signiﬁcant amounts of ﬁnes (Bernard, 1967; Morrow et al., 1998; Pu et al., 2010; Tang and Morrow, 1999), while others have reported no evidence of ﬁnes migration (Jerauld et al., 2008; Lager et al., 2008; Rivet et al., 2010) even though additional oil was recovered. Sarkar and Sharma (1990), Sharma and Filoco (2000), and Al-Sarihi et al. (2018) observed that sor decreased along with permeability damage during LSW injection and consequent ﬁnes migration. Moreover, they proposed including the formation damage factor into the capillary number to describe the enhanced recovery eﬀects by desaturation curves (Hwang and Sharma, 2018; Mendez, 1999). Hussain et al. (2013) and Zeinijahromi et al. (2016) observed not

Corresponding author. E-mail address: [email protected] (P. Bedrikovetsky).

https://doi.org/10.1016/j.petrol.2019.02.079 Received 10 November 2018; Received in revised form 22 February 2019; Accepted 24 February 2019 Available online 02 March 2019 0920-4105/ © 2019 Elsevier B.V. All rights reserved.

Journal of Petroleum Science and Engineering 177 (2019) 766–785

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permeable layers, yielding higher permeability damage due to LSW injection. Flux diversion from high-permeability layers to low-permeability layers, resulting in sweep eﬃciency increase, occurs if the formation damage increase with permeability decrease is relatively low. In this case, the main formation damage is induced in highly permeable layer swept by the injected water (Fig. 2). For the opposite case of high permeability damage increase with decrease of permeability, LSW can cause the opposite ﬂux diversion, i.e. from the low-permeable to highly permeable layers. So, if the overall induced hydraulic resistance in lowpermeable layers is signiﬁcantly higher than in the swept area, it can have higher eﬀect on injected water diversion than that by the fast water breakthrough in the swept area and the consequently induced formation damage (Zeinijahromi and Bedrikovetsky, 2014). The corresponding reservoir characteristic indicating whether LSW yields higher or lower recovery, if compared with HSW, would be the formation damage factor versus permeability of the corresponding layer. However, those distributions for natural reservoirs are unavailable. Presently so-called multicomponent polymer-ﬂood model is used for reservoir simulation of LSW (Johansen and Winther, 1989). The wettability and sor-reduction are modelled by the ionic-composition dependent relative permeability and capillary pressure (Al-Shalabi and Sepehrnoori, 2016; Al-Shalabi et al., 2016; Dang et al., 2016; Khorsandi et al., 2017; Qiao et al., 2016). The ﬁnes-assisted LSW is implemented into the two-phase multicomponent model by introduction of the suspended and attached ﬁnes, along with the concentration of ions present in the aqueous phase and attached to clays (Zeinijahromi et al., 2013, 2016). The model assumes the “saturated” state of the reservoir ﬁnes, where the attached particle remains in the state of mechanical equilibrium during the salinity decrease. The ﬁnes detachment is modelled by the weakening of the electrostatic particle-rock attraction and perturbation of the mechanical equilibrium upon the arrival of water with a diﬀerent composition. Yet, water is injected with a velocity that exceeds or is below the velocity threshold that mobilises the attached ﬁnes under the unperturbed reservoir conditions. Those cases correspond to so-called over and undersaturated ﬁnes, respectively. Basic equations for single-phase migration of over and undersaturated ﬁnes and their validation by comparison with the experimental results are presented by Chequer and Bedrikovetsky (2019). Either of the two cases occurs during any water injection. However, a mathematical model for LSW accounting for two-phase transport of over and undersaturated reservoir ﬁnes is not available. Usually, available volumes of LSW are limited, which requires slug injection with high salinity (HS) water chase drive. The slug version of LSW injection has been investigated using the reservoir simulation (Jerauld et al., 2008; Attar and Muggeridge, 2016, 2018). The studies show strong eﬀect of dispersivity on slug propagation and incremental oil. Jerauld et al. (2008) shows that small slugs can loose integrity destroying the EOR beneﬁts, and suggested slug size over 0.4 PV. The systematic study undertaken by Attar and Muggeridge (2018) suggests that 0.6 PV of LSW slug is necessary to reach 0.95 of the recovery that would be achieved by continuous LSW. For smaller slugs, dispersivity yields the salinity increase in the slug, resulting in signiﬁcant recovery decrease if compared with continuous LSW ﬂood. However, the eﬀects of slug size on ﬁnes-assisted LSW ﬂood have not been investigated. In the current paper, we develop a novel mathematical model for two-phase ﬂow with ﬁnes migration accounting for over and undersaturated states of the attached ﬁne particles. The model is applied for modelling of LSW slug injection with HS chase drive in two-layer cake reservoir. A laboratory study has been carried out to determine how clays and the induced permeability damage are distributed between two layers with contrast permeability; a sample curve of induced formation damage versus permeability is obtained. Two competitive physics eﬀects of ﬁnes-assisted LSW have been distinguished: highpermeability zone plugging with the ﬂux diversion into unswept zones, and induced formation damage behind the waterfront in low-permeable zones. This indicates the existence of optimal slug size. Indeed, for all

Fig. 1. Schematic for ﬁnes detachment and permeability damage: a) detachment of ﬁnes in the body of a pore and their size exclusion in the pore throat; b) torque balance as a condition of mechanical equilibrium of ﬁne particle on the rock surface.

only decrease of relative permeability for water during salinity decrease, but also water relative permeability decrease with water saturation increase at its high values, i.e. where s approaches to 1-sor. Here and further in the text, unless speciﬁed, “saturation” means water saturation s. This eﬀect of non-monotonic krw-curves was explained by ﬁnes lifting from continuously increasing water-rock contact area during the saturation increase. The term “ﬁnes-assisted low-salinity waterﬂood” corresponds to reduced permeability for water due to straining of the mobilised reservoir ﬁnes that could result in sweep enhancement (Zeinijahromi et al., 2011) and residual oil reduction (AlSarihi et al., 2018). The main physics mechanism of ﬁnes migration is the permeability decrease, which is explained by ﬁnes straining in thin pore throats and consequent ﬂow tortuosity increase (Muecke, 1979). Fig. 1a shows the ﬁne particle capture in this pore throat, pore plugging and the consequent tortuosity enhancement of the ﬂow trajectory. During LSW, this decelerates water in the swept zones and diverts it into the unswept reservoir areas. It yields the delay in water breakthrough and sweep enhancement. Fig. 2 shows a signiﬁcant propagation of water in the high-permeability layer, straining by the mobilised ﬁnes and induced formation damage, causing the injected water diversion into the lowpermeable layer. The eﬀects of prolongation of the waterless oil production and sweep enhancement during so-called ﬁnes-assisted LSW, if compared with high salinity waterﬂooding (HSW), are clearly illustrated by the Dietz's model of the displacement from layer-cake reservoirs (Zeinijahromi et al., 2011). Despite the presence of sor weakens the permeability-damage eﬀect of ﬁnes migration (Sarkar and Sharma, 1990), the Dietz's-based modelling exhibits signiﬁcant sweep enhancement during LSW. Usually low-permeability layers contain more clays than highly

Fig. 2. Enhanced sweep eﬃciency in heterogeneous reservoirs with ﬁnes-induced formation damage. 767

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Fig. 3. Maximum retention curve with over and undersaturated states of ﬁnes in porous media: a) trajectories Io→A→B→J for initial oversaturated ﬁnes system, and Iu→B→J for undersaturated initial ﬁnes system during salinity change from γI to γJ; b) maximum retention functions for constant velocity and saturation, and initial attached confor high-permeability centrations I2 (k2 = 500 mD) and I1 low-permeability (k1 = 10 mD) layers. Index 1 corresponds to low-permeability layer and index 2 to highly permeable layer.

simulated cases there does exist an optimal slug size. It was found that the optimal slug size has the same order of magnitude as the volume of the highly permeable layer. The simulation shows that ﬁnes undersaturation yields some recovery factor (RF) reduction due to delay in ﬁnes release during LSW propagation in the high-permeability layer. Oversaturation of ﬁnes in low-permeability layer highly increases RF due to the lateral low-permeability barrier created by LSW slug drift along the interlayer boundary, isolating layers and yielding the increased sweep. The simulation also show that the size of the optimal slug is not aﬀected by over and undersaturation of ﬁnes concentration, despite the recovery factor does. The structure of the paper is as follows. First, we describe the background physics eﬀects of LSW on rock and ﬂuid properties (section 2). In section 3 we describe the laboratory study of ﬁnes-induced formation-damage in rocks with diﬀerent permeability. Afterwards, we develop the mathematical model for two-phase ﬂow accounting for over and undersaturation of the reservoir ﬁnes and deﬁne the laboratory-based reservoir model (section 4). This follows by the analyses of the modelling data, determination of the optimal slug size and sensitivity analysis (section 5). Discussion of the model limitations and main conclusions ﬁnalize the paper.

decreases with decrease of brine salinity (Derjaguin et al., 1975; Derjaguin and Landau, 1941; Elimelech et al., 2013; Gregory, 1981; Israelachvili, 2011), while drag force monotonically increases with ﬂow velocity (Bedrikovetsky et al., 2011, 2012; O'Neill, 1968). The particle detachment and mobilisation occurs if the detaching torque from drag force exceeds the attaching torque from electrostatic force, i.e. τ > 0. Therefore, as it follows from Eq. (1), particles are mobilised at high ﬂow velocity or high water saturation, since the water velocity increases as saturation increases, or at low salinity. For a single-phase ﬂow, applying the Eq. (1) to any attached ﬁne determines whether the particle remains attached to the rock, or is removed under a given velocity U and salinity γ. The total of attached particles per unit of the rock volume is called “attached particle concentration, σa”. Under the conditions of mechanical equilibrium, attached particle concentration is a function of U and γ. This function is called the “maximum retention function, σcr(γ,U)” and expresses the rock capacity to hold ﬁnes at given velocity and salinity (Bedrikovetsky et al., 2011, 2012)

σa = σcr (γ , U )

For a known porosity, either Darcy velocity U or interstitial velocity u can be used as the second variable in expression (2). Eq. (2) is a consequent of the mechanical equilibrium condition (Eq. (1)) and assumes that the torque balance is fulﬁlled for all particles attached to the rock surface. Since reduction of salt concentration weakens the electrostatic force Fe, the maximum retention function monotonically decreases as salinity decreases (Fig. 3a). Torque balance given by Eq. (1) is applied to mono-size ﬁne particles attached to cylindrical pores of the bundle of parallel capillary using Poiseuille formula for ﬂow in tubes (Bedrikovetsky et al., 2011; Yuan and Shapiro, 2011). This yields the expression for maximum retention function, which is derived in Appendix A. Experimental veriﬁcation of the concept of maximum retention function (Eq. (2)) derived from the mechanical equilibrium, given by Eq. (1), can be found from Zeinijahromi et al. (2011, 2012, 2013), and Bedrikovetsky et al. (2012). Fig. 3b shows maximum retention curves versus salinity for two rocks with low permeability and porosity (k1 = 10 mD, ϕ1 = 0.18) and high permeability and porosity (k2 = 500 mD, ϕ2 = 0.25), as calculated using formulae (A.15). The lower is the permeability and the smaller are the pores’ radius the lower is the maximum concentration of the particles that can remain on the rock surface, i.e. σcr. The maximum concentration of attached particles σa for diﬀerent salinities γ (on the plane (γ,σa)) for a given constant velocity U is shown in Fig. 3a. Each point on the maximum retention curve corresponds to equality of the attaching and detaching torque, given by Eq. (1):

2. Background theory: physics of detachment for oversaturated and undersaturated ﬁnes Following papers by Yuan and Shapiro (2011), Bedrikovetsky et al. (2011, 2012), Zeinijahromi et al. (2011, 2013, 2016), Yuan and Moghanloo (2018a, b), Yuan and Wang (2018), and Chequer and Bedrikovetsky (2019), in this section we brieﬂy present the physics phenomena of low-salinity ﬁnes-assisted waterﬂooding, and introduce over and undersaturated ﬁnes. This background is used further to formulate the laboratory procedures (Section 3) and the mathematical model (Section 4). Fig. 1a shows movable ﬁne particles (kaolinite, illite, or chlorite) on the rock surface. Initially, the ﬁne particles are attached to the rock by electrostatic forces that are stronger than the detaching viscous drag force from the ﬂowing ﬂuid. The particle zoom on Fig. 1b shows the force balance on a particle and the detachment moment when the particle rotates around the rock surface asperity or the attached neighbour particle. The condition of mechanical equilibrium of a particle on the rock surface is an equality of attaching and detaching torques from electrostatic and viscous forces (Bedrikovetsky et al., 2011; 2012):

τ = Fd (u, rs ) l − Fe (γ , rs ) = 0,

l = l d ln ,

u= Uϕ

(2)

(1)

• For points above the curve, the detaching torque exceeds the attaching torque, i.e. τ > 0; hence particle detachment will occur. • For points below the curve, the detaching torque is lower than the

Here Fe and Fd are electrostatic and drag forces, respectively, ln and ld are the corresponding lever arms, l is the dimensionless lever arm ratio, τ is the resultant torque, u is the interstitial velocity, U is the volumetric ﬂow rate per unit of cross-section (Darcy's velocity), γ is the water salinity, ϕ is the porosity, and rs is the particle radius. According to DLVO theory, the attaching electrostatic force Fe

attaching torque where τ < 0; hence no ﬁnes mobilisation will occur.

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the permeability damage is determined by the product βσs, which is called the formation damage factor. In large-scale approximation, where all mobilised particles are captured by the rock during their migration, the mass balance of strained, suspended and attached particles yields:

The solid black line in Fig. 3a corresponds to the maximum retention function (equilibrium condition) of a given reservoir. γI is the initial salinity of the reservoir water that determines the maximum concentration of the ﬁnes particles that can stay attached to the rock surface. This corresponds to point A with ordinate σcr(γI).

• If the initial concentration of the reservoir in-situ ﬁnes σ

σaI − σcr (γ , U ), σaI > σcr (γ , U ) σs = ⎧ ⎨ 0, σaI < σcr (γ , U ) ⎩

is great than the maximum value of σcr(γI), then this is referred to as oversaturated, i.e. point Io. If the initial concentration of the reservoir in-situ ﬁnes σaI is less than the maximum value of σcr(γI), then this is referred to as undersaturated, i.e. point Iu. aI

•

3. Laboratory study In this section, we present the laboratory study to determine formation damage distribution over the layers with diﬀerent permeability. We describe the properties of rock and water (section 3.1), the experimental set-up used for the laboratory study (section 3.2), followed by detailed description of the methodology (section 3.3). The experimental determination of the dependency of the formation damage on permeability is also presented (section 3.4). The obtained “permeability dependency” of the “ﬁnes-induced permeability damage” is later used for setting the reservoir model for simulation of low-salinity slug injection and its optimisation.

The oversaturated and undersaturated cases correspond to τ > 0 and τ < 0, respectively. Let us discuss the oversaturated and undersaturated curves in more details. Consider maximum velocity of natural subterranean ﬂow during the overall geological history (before any injection or production) via a given reservoir point

σaI = σcr (Ucr , γI )

(3)

where σaI is the initial attached particle concentration, Ucr is called the critical velocity and γI is the initial water salinity. If the ﬂow velocity during injection or production exceeds the critical velocity, the initial state of the local ﬁnes system is oversaturated. Therefore, after the commencement of injection and production, a ﬂow with velocity U is created and all the ﬁne particles in the rock with initial ﬁnes concentration that corresponds to point Io will be mobilised (Fig. 3a). Under the incompressible ﬂow, the velocity is instantly established throughout the reservoir. Hence, the attached concentration instantly drops until the critical value σcr(γI) and the diﬀerence σaIσcr(γI) is mobilised at the beginning of injection. For an undersaturated state of the local ﬁnes system, the ﬂow velocity during injection or production is lower than the critical velocity and the initial attached point Iu is located below the curve. Hence, the ﬁnes remain immobile at ﬂow velocity U and until salinity drops up to the critical value γcr (Khilar and Fogler, 1998), (Fig. 3a):

σaI = σcr (γcr , U ),

γcr < γI

3.1. Rocks and ﬂuids Six Berea and one Bentheimer sandstone core plugs were used in this study. The Berea plugs were drilled from the same outcrop block using a water-cooled saw and dried prior to the tests. The XRD analysis of the core plugs showed that the Berea and Bentheimer samples contain 5.7% and 3.3% of clays (kaolinite and illite). The properties of six Berea sandstone cores and one Bentheimer core are shown in Table 1. The brines were prepared by dissolving the desired amount of NaCl salt in Milli-Q water (ultrapure deionized water, ﬁltered with a 0.22 μm ﬁlter). The sodium chloride solutions with 35000 ppm of NaCl are used as the high salinity water (HSW). 3.2. Laboratory set-up

(4)

Fig. 4a shows the photo of the set-up. The schematic of the experimental set-up with the main components is presented in Fig. 4b. The set-up consists of a coreholder, pressure transducers, data acquisition system (PC), injection pump and a particle counter. The pump injects water with a ﬁxed salinity γJ at a constant rate while the pressure transducers monitor the inlet and outlet pressures. Samples of the produced ﬂuid are collected continually to measures ﬁne particle concentrations of the eﬄuent samples using a particle counter. A more detailed description of the set-up is presented by Russell et al. (2017) and Chequer et al. (2018).

During a gradual salinity decrease from initial water salinity γI to injected water salinity γJ, the oversaturated particles move along the path Io → A → J, where the jump Io → A occurs at the beginning of injection (t = 0), and continuous path A → B → J correspond to gradual salinity decrease. Undersaturated particles perform the movement along the path Iu → B → J. Here the segment Iu → B corresponds to particle equilibrium on the rock for γ > γcr, i.e. until salinity reaches the critical value. Eqs. (3) and (4) along with scenarios Iu → B → J and Io → A → J have been validated by detailed comparison between the data of laboratory and mathematical modelling (Chequer and Bedrikovetsky, 2019). The number of degrees of freedom for the measured breakthrough curve and pressure drop across the core exceeds the number of tuned model parameters, so the close agreement allows claiming the model validity. The mobilised ﬁne particles migrate with the ﬂow until they are strained in thin pore throats, yielding a signiﬁcant permeability decline. This permeability decline is expressed as a linearly increasing function for normalised reciprocal to permeability versus strained concentration, which corresponds to zero- and ﬁrst-order terms of Tailor series (Pang and Sharma, 1997)

k (0) k (σs ) = 1 + βσs

(6)

3.3. Methodology Before placing the core in the core-holder, all cores were dried in the oven under a constant temperature (60 °C) for the period of 24 h. The dry cores were weighed and then saturated by HSW under vacuum. The Table 1 Rock properties.

(5)

Here k is the permeability, σs is the strained concentration, and β is socalled formation damage coeﬃcient. Initial permeability without accounting for ﬁnes migration corresponds to σs = 0. According to Eq. (5), 769

Rock type

Diameter (mm)

Length (mm)

Porosity (%)

Original permeability k0 (mD)

Bentheimer Berea Berea Berea Berea Berea Berea

38.02 36.04 36.14 38.10 37.67 38.11 38.10

48.40 52.26 52.71 49.69 51.08 47.97 49.38

23 17 18 20 21 20 19

1398 35 50 71 87 84 108

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Fig. 4. Laboratory set-up for ﬁnes migration in porous media: a) photo of setup; b) schematic of laboratory setup: 1- injection ﬂuid, 2 - pump, 3, 9,11 - pressure transmitters, 4, 8, 20 - control valves, 5 - core-holder, 6 - viton sleeve, 7 - core sample, 10 – overburden pressure generator, 12–15 - diﬀerential pressure transmitters, 16–19 - 3 port switching valves, 21 - sample collector, 22 - sampling tube, 23 - particle counter/sizer, 24 - data acquisition module, 25 - signal converter, and 26 - PCbased data acquisition system.

compared with high permeable samples. Hence, the concentration of mobilised ﬁnes and consequent formation damage is higher in low permeable cores if compared with high permeable ones. The continuous curve in Fig. 5 was obtained by least square method treatment of the points presented. This curve is used later in the paper for simulation of LSW in two-layer cake reservoir to determine the permeability reduction of each layer after the invasion of low-salinity water.

saturated cores were weighed again and the porosity was determined by weighting method. The saturated cores were then installed in the core-holder and injection of HSW was carried out in order to calculate the initial permeability k0. Then HSW injection was switched to fresh water and continued until permeability stabilisation. The pressure drop across the core and produced ﬁnes at the core eﬄuent were measured throughout the tests. The tests have been performed under the room temperature. Flow velocities varied from 8.1 × 10−6 to 1.5 × 10−5 m/s. The overburden pressure was 1000 psi.

4. Mathematical model for ﬁnes-assisted LSW In this section, we extend the basic system of equations that governs the displacement of oil by low-salinity slug with ﬁnes migration for the cases of over and undersaturated ﬁnes. It includes reservoir simulation for LSW injection with oversaturated and undersaturated ﬁnes migration scenarios.

3.4. Experimental results During the coreﬂoods with low-salinity water, we calculated the permeabilities before and after the ﬂoods. The formation damage factor βσs = β[σcr(γI)-σcr(γJ)], induced by ﬁnes mobilisation, migration and size exclusion is determined using Eq. (5). Fig. 5 shows the dependency of induced formation damage (formation damage factor βσs) versus initial permeability for the seven cores ﬂood test performed in this study (red points). Other data are taken from works by Jones (1964), Mungan (1965), Sarkar and Sharma (1990), Oliveira et al. (2014), and Torkzaban et al. (2015). These tests have been performed using the methodology and under the conditions, which are similar to our tests. The general tendency exhibited by Fig. 5 is: decreasing of formation damage factor βσs with the increase of the initial permeability. This phenomenon can be explained by the initial clay content of the samples. Usually, the low permeable samples contain higher clay content if

4.1. Basic equations for ﬁnes-assisted LSW The assumptions of the governing system, formulated in Appendix C are immiscible incompressible ﬂow of oil and water with varying salinity and negligible suspended particle concentration. Concentrations of detached and strained ﬁnes are low, so the detachment and straining do not aﬀect the volumetric balance of both phases. The ratio between the lateral dispersive and advective ﬂuxes is equal to the ratio between the dispersivity coeﬃcient and the reservoir length. In large-scale approximation, this ratio has order of magnitude 10−3-10−1 (Barenblatt et al., 1989; Lake et al., 2014). Therefore, the lateral dispersivity is neglected. We also neglect vertical dispersivity across the layers. For two-phase ﬂow, the attached ﬁnes are mobilised only from the rock surface that is accessible to injected LSW. It is assumed that oil cannot mobilise attached ﬁnes. So, the attached ﬁnes can be removed only from the fraction of the rock surface accessible to water, i.e. AW(s,γ) (Fig. 6). Increasing water saturation increases the fraction of the rock surface accessible to injected water and consequently increases the number of attached in-situ reservoir ﬁnes exposed to the injected water. In addition, the contact angle decreases as salinity γ decreases; hence, the wetted area increases with salinity decrease (Zeinijahromi et al., 2013; Borazjani et al., 2017). To study the eﬀects of low-salinity-induced ﬁnes migration under no wettability change, we assume that relative permeability and capillary pressure are independent of salinity. Torque balance given by Eq. (1) is valid in water-ﬁlled pores, where the interstitial velocity is equal to uW/ϕs (Yuan and Shapiro, 2011). Therefore, for two-phase ﬂow, Eq. (2) for maximum retention function takes the following form (Borazjani et al., 2017)

Fig. 5. Initial permeability dependency for permeability damage induced by ﬁnes migration. 770

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be reached during a simultaneous decrease in salinity and increase in saturation, where the maximum retention function decreases. In the case where point Iu is located below point J in Fig. 3a, i.e.

U ,γ⎞ σaI < σcr ⎛ − 1 sor J ⎠ ⎝ ⎜

(

)

(7)

where σcr(γ,u) is the maximum retention function for single-phase ﬂow, uw is the water velocity, and s is water saturation. Consider three cases of initial reservoir ﬁnes concentration, Fig. 3a: i Oversaturated reservoir: initial reservoir ﬁnes concentration (point Io) is located above the maximum retention function curve (“saturated” point A), i.e. initial reservoir ﬁnes concentration is higher than the maximum concentration of ﬁnes that can stay attached to the rock at reservoir brine salinity γI and injected brine salinity γJ. ii Undersaturated reservoir: initial reservoir ﬁnes concentration (point Iu) is located below maximum retention function curve and above point J (point J corresponds to the injected brine salinity γJ), i.e. initial reservoir ﬁnes concentration is lower than the maximum concentration of ﬁnes that can stay attached to the rock at reservoir brine salinity γI and is higher than the maximum concentration of ﬁnes that can stay attached to the rock at injected brine salinity γJ. iii Point Iu is located below point J. Initial reservoir ﬁnes concentration is lower than the maximum concentration of ﬁnes that can stay attached to the rock at injected brine salinity γJ.

krw (s, γ , σs ) =

⎧ ⎪ ⎨⎡σ − σ cr ⎪⎣ aI ⎩

The mathematical model formulated in the previous section is applied for two-layer-cake reservoir. Permeability, porosity, and thickness of each layer are constant, their values are presented in Table 2. Index 1 corresponds to low-permeability lower layer and index 2 to highly permeable upper layer. Water is injected with constant ﬂowrate into the injection well (left-hand side of the Fig. 2) and produced from the production well (right-hand side of the Fig. 2). The abandonment moment corresponds to water-cut equal 0.95.

(8)

σaI < σcr

0,

(

uW , s

)

γ ⎤ AW (s ), σaI > ⎦

( (

uW , s

u σcr W , s

) γ)

(11)

4.2. Input data for the reservoir model

For undersaturated case, where point Iu is located above point J, no particles are mobilised and strained until the maximum retention function reached the value of the initial attached concentration σcr(γcr)

σs =

krw (s, γ , 0) 1 + βσs

In two-phase multicomponent LSW models, nominator of the ratio (Eq. (11)) corresponds to salinity-eﬀects on relative phase permeability for water, i.e. decrease of sor and krwor (Al-Shalabi et al., 2016; Dang et al., 2016; Farajzadeh et al., 2009, 2012; Khorsandi et al., 2017; Qiao et al., 2016). Denominator corresponds to permeability decrease due to straining and size exclusion of the migrating ﬁnes in thin pore throats. The system of equation (C.10-C.14) is solved by the polymer option in reservoir simulator CMG STARS. The details of the equivalence of ﬁnes-assisted LSW system with polymer ﬂooding system can be found in Zeinijahromi et al. (2013).

The arrows on Fig. 3a show the path for ﬁnes detachment at the three scenarios. At the reservoir scale, the distance that mobilised particles can travel before being captured by the rock (free-run of particles) is signiﬁcantly lower than the reservoir size, so all detached particles are instantly strained (size excluded) in thin pore throats, i.e. instant capture of σaI - σcr(γJ) for the oversaturated state Io:

u σs = ⎡σaI − σcr ⎛ W , γ ⎞ ⎤ AW (s ) ⎝ s ⎠⎦ ⎣

(10)

no ﬁnes are lifted (mobilised) during LSW injection. It corresponds to a single-phase ﬂow where point γJ lays above point Iu in Fig. 3a. In this case, the process corresponds to two-phase ﬂow with varying salinity without ﬁnes migration, modelled by Eq. (C.10-C.14) with σs = 0. To summarise, the excess of ﬁnes in the oversaturated case is released and strained instantly, adding to initial conditions for strained particles and yielding the decreasing relative permeability for water. The LSW injection in undersaturated case is (i) equivalent to two-phase ﬂow without ﬁnes for the period γI →γcr that correspond to Eqs. (C.10C.14) with σs = 0; (ii) after the maximum retention function reaches the initial attached concentration, the ﬂow equation (C.10-C.14) correspond to the saturated caseγcr→γJ. All particles that are removed by the aqueous phase are suspended and travel with the aqueous phase until being strained and decrease water relative permeability. We assume that phase permeability for water decreases following Eq. (5) (Hussain et al., 2013; Zeinijahromi et al., 2016):

Fig. 6. Fines detachment by the injected water from rock surface fraction Aw accessible to water; ﬁnes attached to the surface Ao remain immobile.

σa = σcr γ , uW ϕs AW (s, γ )

⎟

Table 2 The rock properties, grid dimensions and Corey parameters used in layers 1 and 2.

γ

Layer 1 (bottom)

(9)

Rock properties Permeability 10 mD Porosity 0.18 Thickness 8m Corey Parameters swi 0.33 sor 0.23 krowi 0.7 krwor 0.06 nw 2.7 no 2.7 Leverett's function threshold values J(sor) 2.1 × 10−4 J(swi) 1.6 × 10−2

The exact solution for one-dimensional (1D) LSW shows that the salinity front lags behind the waterfront (Borazjani et al., 2017; Jerauld et al., 2008). Therefore, during LSW injection, initially water saturation increases from the initial value of swi with initial formation water salinity γI after the arrival of the waterfront. The salinity starts decreasing with a delay after the arrival of the salinity front. Since LSW is injected, the water salinity decrease corresponds to an increase in saturation and, consequently, in the increase of interstitial water velocity in Eq. (7), so the maximum retention value can drop until σaI before the salinity front arrival. If at the salinity front arrival σaI is still below the maximum retention function, the equality (Eq. (7)) can 771

Layer 2 (top)

500 mD 0.25 2m 0.3 0.2 0.8 0.05 3 3

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the wetting phase (water), and is given in the following parametric form

Table 3 The ﬂuid properties used in the simulation study. Property Oil viscosity (cp) Water viscosity (cp) Oil density (kg/m3) Water density (kg/m3) Initial water salinity (ppm) Injected water salinity (ppm)

Value 4 0.5 850 1000 35000 0

∫ y2 f (y) dy ⎡⎢∫ y2 f (y) dy⎤⎥ 0 r

=

−1

∞

r

s (r ) =

∫ 0

⎣ ⎡ yf (y ) dy ⎢ ⎣

∞

∫ 0

, Aw (r )

⎦

0

−1

⎤ yf (y ) dy⎥ ⎦

(12)

where f(r) is pore size distribution. Here, a log-normal pore-size distribution for the parallel tubes is assumed: mean pore radii are 7.07 and 1.17 μm for high- and low-permeability layers, respectively; the coefﬁcient of variance is equal to 1.4 for both layers. The details of calculations for maximum retention function in single-phase ﬂow are presented in Appendix A, and their extension to two-phase ﬂow is shown in Appendix B. The electrostatic constants presented correspond to kaolinite ﬁnes and silica (sandstone) rock. Fig. 3b shows the maximum retention curves for low-permeability and high-permeability layers. The initial clay content in the low permeability layer is higher than that in the highly-permeable layer; it corresponds to points I1 and I2, respectively. So, initially the high-permeability layer is undersaturated by ﬁnes, and the low-permeability layer is oversaturated.

Following two-phase multicomponent modelling of LSW, the Corey functions are used to model relative permeabilities and capillary pressure (Dang et al., 2016; Al-Shalabi et al., 2016; Qiao et al., 2016; Afekare and Radonjic, 2017; Khorsandi et al., 2017; Rabinovich, 2017). Table 2 presents the Corey parameters for both layers. The Leverett's function is the same in both layers; Table 2 shows the threshold value J (sor) and maximum value J(swi). The eﬀects of ﬁnes-induced formation damage for water relative permeability are given by Eq. (11). Yet, Eq. (11) shows the residual oil saturation is independent of the retained concentration of the strained ﬁnes σs. The distance between rows of injectors and producers L = 50 m. Injection velocity is 3.5 × 10−6 m/s. Table 3 shows the injected and formation water salinities; oil and water density and viscosity are independent of salinity. The area of the rock exposed to water Aw(s) is calculated for a bundle of parallel tubes, assuming that all the small pores are ﬁlled by

5. Analysis of numerical results Following the mapping from low-salinity waterﬂood equations to

Fig. 7. Saturation and salinity proﬁles during injection of LSW slug into two-layer-cake reservoir: a), b), c), and d) – salinity proﬁles at the moments tD = 0.3, 0.5, 0.7, and 1.0 PVI, respectively; e), f), g), and h) – saturation proﬁles at the moments tD = 0.3, 0.5, 0.7, and 1.0 PVI. 772

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water forms an oil-water bank in both layers, yielding delay in salinity front if compared with the water front, which is clearly exhibited in Fig. 7a and e, 7b and 7f, 7c and 7g, respectively. The salinity proﬁles show that HSW drive sweeps the slug from highly-permeable layer due to high velocity, while the slug sweep occurs in low-permeable layer signiﬁcantly slower. After the breakthrough, the slug moves inside the low-permeable zone (Fig. 7b, c, and 7d). Deep bed ﬁltration of lifted and mobilised ﬁnes inside the slug decreases the permeability in low-permeable zone. Therefore, LSW slug injection yields the appearance of low-permeability barrier near to the interlayer interface. Fig. 8a shows signiﬁcant decrease in water phase permeability near to the interface between the layers. Here the curves 1, 2, 3, and 4 correspond to reservoir cross section xD = 0.2 at the same four moments as in Fig. 7. However, the eﬀect of low-permeable lateral barrier is partly masked by the monotonic saturation increase as depth increases, resulting in some non-monotonicity of the permeability proﬁles for water. Fig. 8b shows the proﬁles of formation damage factor, exhibiting high values of formation damage induced in the barrier.

the polymer-ﬂooding model, derived by Zeinijahromi et al. (2013), and accounting for over and undersaturated ﬁnes (Eqs. (3) and (4)), we use the polymer ﬂood model for reservoir simulation of LSW injection with induced ﬁnes migration. In this section, we analyse the eﬀects of over and undersaturation, mobility ratio, heterogeneity and thickness of high-permeability layer on RF and optimal LSW slug size. Several scenarios for LSW injection are simulated where a slug of LSW is injected to the reservoir followed by continuous injection of HSW. 5.1. Existence of optimal slug size for low-salinity ﬁnes-assisted waterﬂooding Fig. 2 shows the eﬀects of ﬁnes migration on LSW in a two-layer cake reservoir. The injected low-salinity water moves preferably into the high-permeability layer yielding ﬁnes detachment and consequent permeability decline. This decelerates the water encroachment in the high-permeability layer and redirects it to the unswept zones of the lowpermeability layer. This waterﬂux redistribution prolongs the water breakthrough time and increases the reservoir sweep. Consider three injection scenarios related to diﬀerent slug sizes: (i) continuous injection of LSW, (ii) injection of LSW slug that is signiﬁcantly smaller than the pore volume of the highly-permeable layer and (iii) injection of LSW slug that is signiﬁcantly larger than the pore volume of the highly-permeable layer. First we consider continuous injection of LSW. Under high permeability contrast between the layers (k2 > > k1), breakthrough in highpermeability layer occurs when water in the low-permeable layer is still near to the inlet (Figs. 7, 10 and 11). High-permeability layer is homogeneous, so water-cut in high-permeable layer reaches maximum value soon after the breakthrough. Therefore, no incremental oil is produced and no formation damage is induced in highly permeable layer after the breakthrough. Yet, continuous injection of LSW yields permanent induction of permeability damage behind the displacement front in low-permeable layer, resulting in the decrease of fraction of the injected water that enters this layer. Switching to HSW after sweep of the highly-permeable layer avoids increase of formation damage in lowpermeable layer, so injection of LSW slug increases sweep eﬃciency of two-layer-cake reservoir if compared with continuous LSW injection. Consider injection of LSW slug that is signiﬁcantly smaller than the pore volume of the highly-permeable layer. Small LSW slug mixes with the high-salinity formation water and water from the chase drive, so the dispersion causes signiﬁcant salinity increase, and only a small fraction of highly-permeable layer is damaged (Jerauld et al., 2008). The recovery is just slightly higher than that for HSW ﬂood; all recovery curves in Fig. 9 intersect at the point of zero-slug, i.e. HSW ﬂooding. In this case, increasing the size of LSW slug yields the recovery enhancement. Consider LSW slug signiﬁcantly larger than the pore volume of the highly-permeable layer. Similar to continuous injection of LSW, in this case, LSW does not enhance sweep in the homogeneous low-permeable layer after the oil has already been displaced from the highly-permeable layer. Large LSW slug only induces the permeability damage, which diverts water ﬂux into the already swept high-permeability layer. Therefore, the size of LSW slug must be decreased. Hence, the above speculations of the three scenarios indicate the existence of an optimal slug size.

5.3. Eﬀects of ﬁnes over and undersaturation Figs. 9–12 present the comparative study of oversaturated and undersaturated initial ﬁnes concentration on LSW slug performance. We discuss ﬁve cases of waterﬂooding in a two-layer-cake reservoir: a) HSW without ﬁnes mobilisation; b) LSW with initial “saturated” ﬁnes concentration in both layers; c) LSW with “undersaturated” initial ﬁnes concentration in high-permeability layer and “saturated” ﬁnes in lowpermeability layer; d) LSW with “saturated” initial ﬁnes concentration in high-permeability layer and oversaturated ﬁnes in low-permeability layer; e) LSW with “undersaturated” initial ﬁnes concentration in highpermeability layer and “oversaturated” ﬁnes in low-permeability layer. Fig. 9 shows the ultimate recovery factor vs slug size for the ﬁve abovementioned cases. Fig. 10 presents ﬁeld water saturation for ﬁve cases at tD = 0.5 pore volume injected (PVI), while Fig. 11 presents ﬁeld water saturation at the abandonment time. Fig. 12 shows the normalised pressure drop between injectors and producers for ﬁve scenarios. One can see that in all scenarios, injection of LWS slug has enhanced the RF. The low RF obtained during continuous injection of HSW (the zero-slug size point in Fig. 9) is due to the fast water channelling through high permeable layer. The premature water breakthrough in highly permeable upper layer has left large upswept areas in the lower low permeable layer after 0.5 PVI (Fig. 10a) and also at the abandonment time (Fig. 11a). Consider the case where both layers are at saturated initial ﬁne concentration (Fig. 2). Here the advancing LSW plugs the high-permeability layer and diverts the water ﬂux into the low-permeability layer that signiﬁcantly enhances the sweep at tD = 0.5 PVI (Fig. 10b) and also after the abandonment (Fig. 11b). Since small volume of LSW quickly dissolves in HS connate and chase-drive waters, the RF increases by increasing the volume of the LSW slug from zero (Fig. 9). The maximum RF is obtained at LSW slug size of 0.3 PVI. Injection of larger slug sizes induces formation damage in the low-permeable layer also that results in some additional re-direction of injected water into the high-permeability layer which decreases the sweep in the low-permeability layer. Hence, the RF decreases for large volume LSW slugs. The pore volume of the highpermeability layer is 0.25 of the overall pore volume that is close to the optimum volume of LSW slug size (0.3 PVI). The same speculations on the existence of the optimal slug size are valid for other LSW cases presented in Fig. 10c, d, and 10e and Fig. 11c, d, and 11e. All RF curves in Fig. 9 demonstrate a maximum point. Figs. 10c and 11c show the case where the initial ﬁnes concentration is undersaturated in the high-permeability layer. Usually, clay concentration in highly permeable rocks is low; hence the movable clay concentration is also low, which justiﬁes the assumption that the ﬁnes

5.2. LSW slug dynamics in two-layer-cake reservoir Salinity and water saturation proﬁles during LSW slug injection into the reservoir, where the initial reservoir ﬁnes are saturated, are presented in Fig. 7. The model assumes that equality of salinities in injected and connate water due to small-scale diﬀusion is established instantly, yielding the piston-like displacement of HS connate water by LS injected water under no lateral dispersivity. The displaced connate 773

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Fig. 8. Low-permeability barrier created near to interface between the layers during LSW slug injection: a) phase permeability for water at instances tD = 0.3, 0.5. 0.7, and 1.0 (curves 1, 2, 3, and 4, respectively) at the reservoir cross section xD = 0.2; b) the corresponding proﬁles for formation damage factor βσs.

Fig. 9. Ultimate recovery factor for LSW for initial concentration of ﬁnes: saturated in both layers (blue curve), undersaturated in high-permeability layer and saturated in the layer with low permeability (yellow curve), saturated in high-permeability layer and oversaturated in the layer with low permeability (orange curve), undersaturated in high-permeability layer and oversaturated in the layer with low permeability (grey curve). (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the Web version of this article.)

774

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Fig. 10. Eﬀects of over and undersaturated ﬁnes on oil displacement for ﬁnesassisted low-salinity waterﬂooding in a two-layer cake reservoir (500mD and 10mD) at tD = 0.5 PVI: a) continuous injection of HSW without ﬁnes detachment; b) LSW for ﬁnes saturated in both layers; c) undersaturated in highpermeability layer and saturated in the layer with low permeability; d) saturated in high-permeability layer and oversaturated in the layer with low permeability; e) undersaturated in high-permeability layer and oversaturated in the layer with low permeability.

Fig. 11. Eﬀects of over and undersaturation of initial reservoir ﬁnes on ﬁnal oil recovery for ﬁnes-assisted low-salinity waterﬂooding in a two-layer cake reservoir (500mD and 10mD) at the abandonment time: a) continuous HSW without ﬁnes detachment; b) LSW for ﬁnes saturated in both layers; c) undersaturated in high-permeability layer and saturated in the layer with low permeability; d) saturated in high-permeability layer and oversaturated in the layer with low permeability; e) undersaturated in high-permeability layer and oversaturated in the layer with low permeability.

become mobile at salinity lower than the initial formation-water salinity, γcr < γI (Russell et al., 2017). The ﬁnes at each point of the reservoir are mobilised only after the concentration front γ = γcr passes through that point, so high-permeability layer plugging occurs with some delay if compared with the case of saturated ﬁnes. Unswept areas in Figs. 10c and 11c are slightly larger than those in Figs. 10b and 11b. The corresponding yellow RF curve in Fig. 9 is located below the blue curve. Figs. 10d and 11d correspond to the case of saturated initial ﬁnes concentration in the high-permeability layer and oversaturated initial ﬁnes concentration in the low-permeability layer. Usually, the concentration of clay in low-permeability layers is high; it justiﬁes the assumption that LSW ﬂow mobilises some ﬁnes at the instance of its arrival, when salinity start increasing from the value γI. After the arrival of LSW front, residual oil and LSW are established in the high-permeability layer, while connate HSW remains “below” in the low-permeability layer. The saturation diﬀerence causes some imbibition of LSW into the low-permeable layer. It causes salinity decrease below the interface along with ﬁnes migration, straining and permeability decline. The above occurrences create a low-permeability barrier (interface barrier) between the two layers. In the oversaturated case, the amount of strained ﬁnes is higher than in the saturated case. Fig. 10d and e

shows almost piston-like displacement in the low-permeability layer, while Fig. 10a, b, and 10c exhibit wide zones where saturation changes from swi to 1-sor. It results in a higher sweep at abandonment (Fig. 11d). The corresponding orange curve in Fig. 9 lays above blue and yellow curves. The assumption of undersaturated initial ﬁnes concentration in high-permeability layer yields some delay in its plugging by LSW. Sweep in Figs. 10e and 11e are slightly lower than that in Figs. 10d and 11d. Grey curve in Fig. 9 is located slightly below the orange curve. The displacements in Fig. 11a, b, 11c, 11d, and 11e stop at the abandonment corresponding to water cut value of 0.95; the corresponding abandonment times are 2.13, 1.08, 1.11, 1.96, and 2.14 PVI. For all the above cases, the ultimate recovery tends to that for HSW ﬂood, where the slug size tends to zero. Four curves in Fig. 9 have the joint point (0, 0.518). Small slugs are quickly spread by the lateral dispersion, so small low salinity slug can result only in insigniﬁcant additional recovery (Jerauld et al., 2008). Pressure drop curves in Fig. 12 are normalised by dividing by the maximum pressure drop at the end of LSW for the oversaturated case. The normalised pressure drop curves are located in the same order as the curves in Fig. 9. Therefore, the higher is the induced formation damage the higher is the incremental oil recovery due to ﬁnes-assisted LSW under the slug option. 775

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Fig. 12. Pressure drop across the reservoir for continuous HSW without ﬁnes detachment (green curve), LSW for initial ﬁnes concentration: saturated in both layers (blue curve), undersaturated in highpermeability layer and saturated in the layer with low permeability (yellow curve), saturated in highpermeability layer and oversaturated in the layer with low permeability (orange curve), and undersaturated in high-permeability layer and oversaturated in the layer with low permeability (grey curve). (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the Web version of this article.)

results show that the higher is the heterogeneity the lower is the recovery for continuous HSW injection (points of zero-slug size) and also for LSW (three curves). For the saturated cases, where the initial attached concentration is equal to maximum retention function in both layers, the role of the interface barrier is less than that in the oversaturated case. The eﬀects of permeability decline in the low-permeable layer and ﬂux diversion into high-permeability layer are dominant for large slug sizes, where RF can be lower for HSW. The lower is the ultimate recovery, the higher is the incremental recovery by LSW injection due to the ﬂux diversion.

The optimal slug size in Fig. 9 is also almost the same for four LSW cases and has the same order of magnitude as the volume of the highly permeable layer (0.25 PVI). 5.4. Eﬀects of mobility ratio The eﬀects of mobility ratio on oil recovery and optimum slug size for ﬁnes-assisted low-salinity waterﬂooding are shown in Figs. 13 and 14. For all cases injection of a LSW slug has enhanced recovery if compared with continuous injection of HSW (LSW slug size = 0). The results show that the higher is the oil viscosity the larger is the optimal slug (Fig. 13). Fig. 14a–c shows ﬁeld's water saturation at tD = 0.5 during HSW. The higher is the oil viscosity the lower is the sweep and the recovery. Fig. 14d–f presents LSW for the saturated case and optimal slug size. Signiﬁcantly better sweep is reached at LSW if compared with continuous injection of HSW. The higher is the oil viscosity the lower is the recovery, but for LSW this eﬀect is less pronounced.

5.6. Eﬀects of pore volume of high permeability layer Fig. 16a, b, 16c, and 16d show that the thicker is the high-permeability layer, the lower is the relative ﬂux fraction via the low-permeability layer, and the higher is residual oil in the low-permeability layer. The same tendency is observed for LSW injection (Fig. 16e, f, 16g, and 16h), despite the “ﬂux diversion (due to high-permeability layer plugging) partly compensates the negative eﬀect of a “thick highlypermeable layer”. Fig. 17 shows the optimal slug sizes for diﬀerent fractions of pore volume of the highly permeable layer. One can see that the optimal

5.5. Eﬀects of heterogeneity Fig. 15 shows RF versus LSW slug size for three reservoirs with a permeability ratio of 500:10, 750:10, and 1000:10. The simulation

Fig. 13. Eﬀects of viscosity ratio on oil recovery and on the optimum slug size for ﬁnes-assisted low-salinity waterﬂooding. 776

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Fig. 14. Eﬀects of viscosity ratio on oil recovery for optimum slug size at tD = 0.5 PVI: a) HSW with oil viscosity μo = 4 cp; b) HSW with oil viscosity μo = 10 cp; c) HSW with oil viscosity μo = 20 cp; d) LSW at optimum slug size with oil viscosity μo = 4 cp; e) LSW at optimum slug size with oil viscosity μo = 10 cp; f) LSW at optimum slug size with oil viscosity μo = 20 cp.

realistic input into the reservoir model, we carried out the laboratory tests on LSW measuring permeability decline in cores with diﬀerent permeability. The obtained data along with the results from 5 other published data allowed proposing an averaged curve for induced formation damage versus permeability. This curve can be used in simulation of LSW injection in reservoir for the development of the static reservoir model.

LSW slug has almost the same volume as the high permeable layer implying that: to obtain the maximum RF, switch from LSW to HSW must occur after water breakthroughs in the high-permeability layer; otherwise some LSW water may enter the low-permeability layer and damage it. 6. Discussions 6.1. Fines-migration-induced formation-damage distributed over layers

6.2. Isolation of layers during LSW injection

The results of reservoir simulation show that the RF is very sensitive to the distribution of induced formation damage over layers with different permeability. To the best of our knowledge, this information about natural reservoirs is not available. Therefore, in order to create a

Low sweep in heterogeneous layer-cake reservoirs is due to preferential oil displacement in high-permeability layers. The streamlines are directed from low-permeability to highly permeable layers, i.e. the injected water moves across the interlayer boundaries. It yields to

Fig. 15. Eﬀects of heterogeneity on oil recovery and on optimum slug size for ﬁnes-assisted low-salinity waterﬂooding in two layer cake reservoir. 777

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Fig. 16. Eﬀects pore volume fraction of high permeability layer on optimum slug size for ﬁnes-assisted low-salinity waterﬂooding: a, b, c, and d) HSW with fraction of highly permeable layer equal to 0.25, 0.37, 0.67, and 0.84 PV, respectively; e, f, g, and h) LSW at optimum slug size with fraction of highly permeable layer equal to 0.25, 0.37, 0.67, and 0.84 PV, respectively.

7d). LSW slug moves below the interface and causes ﬁnes lifting and permeability decrease. The permeability damage in the barrier across the interface is signiﬁcantly higher for the case of oversaturation in lowpermeability layer than in the saturated case. It explains wide zones where saturation changes from swi to 1-sor during the saturated case (Fig. 10b) or undersaturated case in high-permeability layer (Fig. 10c)

signiﬁcant volume of by-passed oil in unswept zones (Fig. 2). Existence of low-permeability shale that separates the two reservoir layers, decrease the inter-layer ﬂuxes and results in sweep eﬃciency increase, i.e. isolation of layers from each other yields an incremental recovery. Early LSW breakthrough through high-permeability layer creates an interface between LSW under sor and oil with swi of HSW (Fig. 7b, c, and

Fig. 17. Eﬀects of the pore volume fraction for high permeability layer on optimum slug size for ﬁnes-assisted low-salinity waterﬂooding. 778

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decrease in oil recovery. Low-salinity water breakthroughs in highpermeability layer while the water front in low-permeable layer is still near to injector (Fig. 7a). Permeability damage in highly permeable layer decreases water ﬂux via this layer and redirects the injected water into low-permeable layer yielding increase of water front velocity. This results in sweep eﬃciency increase. Yet, if movable ﬁnes concentration in low-permeable layer is signiﬁcantly higher than that in high-permeability layer, and the induced formation damage in low-permeable layer is signiﬁcantly higher than that in high-permeability layer, the water ﬂux is re-directed into high-permeability layer, decelerating water front in low-permeability layer and decreasing sweep eﬃciency. The EOR-favourable case corresponds to low decrease in the “damage vs permeability” curve in Fig. 5, while the recovery decrease occurs in the case of its steep decline. The above speculations lead to some screening criterion for continuous and slug-like LSW ﬂoods, relating the reservoir heterogeneity and the LSW-induced permeability damage distribution. This is the topic of the forthcoming work. The advantage of slug if compared with continuous LSW injection is the decrease of unfavourable water diversion from low-permeability into highly permeable layer due to formation damage, induced in the unswept area. The existence of an optimum slug size is important for planning and design of LSW slug injection followed by HSW chase drive. However, the conclusions about optimal slug size are restricted to two-layer-cake reservoirs discussed. More general conclusions require two- and threedimensional modelling in multi-layer reservoirs.

and more piston-like displacement in the low-permeability zone in two cases of oversaturation in low-permeability layer (Fig. 10d and e). Initial ﬁnes oversaturation in low-permeability layer yields higher formation damage behind the salinity front if compared with the saturated case, yielding some additional ﬂux diversion into the highpermeability layer. So, if compared with the saturated case, the recovery decreases. However, oversaturation creates a low-permeability barrier between the layers, resulting in recovery increase. The simulation shows that oversaturation in low-permeable layer causes recovery increase, so the eﬀect of layer separation dominates over the ﬂux diversion eﬀect. The similar eﬀect of separation of the adjacent layers with contrast permeabilities have been reported by Sorbie and Mackay (2000) due to chemical reaction between the incompatible formation and injected waters in the neighbourhood of the inter-layer boundary. Precipitation of the solid product of the chemical reaction yields signiﬁcant permeability reduction across the inter-layer boundary. 6.3. Mobility control EOR Eﬀects of ﬁnes migration make LSW injection a mobility-control EOR method. Therefore, the higher recovery usually corresponds to larger damaged unswept area and higher pressure drop across the reservoir (Fig. 12). The eﬀect of pressure drop increase must be accounted for fulﬁlling the restrictions on fracturing, i.e. injection pressure must be kept below the fracturing pressure. Wettability alteration during LSW ﬂood yields the sor reduction, so the method is analogous to chemical EOR (Mahani et al., 2015b; Qiao et al., 2016; RezaeiDoust et al., 2009; Rivet et al., 2010). Fines-assisted waterﬂooding decreases the mobility of the injected water and increases the sweep, so the method is similar to mobility-control EOR. Depending on mineral compositions of rock, water and oil, either of EOR mechanisms or neither of them can occur (Morrow and Buckley, 2011; Pu et al., 2010; Zeinijahromi et al., 2011, 2013, 2016). In this study, to investigate the eﬀects of ﬁnes migration during LSW separate from the wettability alteration eﬀects, we assume that relative permeability is independent of salinity but is highly aﬀected by the concentration of strained ﬁne particles, following Eq. (11).

6.6. Necessary extension to the mathematical model The mathematical model given by Eqs. (C.10-C.14) is obtained by merging two-phase multicomponent model for low-salinity waterﬂood (Al-Shalabi et al., 2016; Dang et al., 2016; Khorsandi et al., 2017; Qiao et al., 2016) and ﬁnes-migration permeability-damage eﬀects (Al-Sarihi et al., 2018; Al-Yaseri et al., 2016). In the present paper, we implemented over and undersaturated ﬁnes (Eq. (4), Fig. 3) into this model. However, the model does not account for several signiﬁcant physics eﬀects. Osmosis through residual oil ﬁlms is one of them (Sandengen et al., 2016). The multicomponent ionic exchange between clays and aqueous phase is not accounted for in the polymer ﬂood model (D.3), used in the present study. Accounting for ion exchange in ﬁnes-assisted LSW ﬂood can be achieved by using LSW option in CMG STARS. In large scale approximation, we neglected lateral dispersion (Barenblatt et al., 1989; Lake et al., 2014). However, vertical dispersion can highly increase mix-up of the slug in two layer and signiﬁcantly aﬀect the recovery (Jerauld et al., 2008; Brodie and Jerauld, 2014; Attar and Muggeridge, 2016, 2018). Besides, lateral dispersion can highly aﬀect the dynamics of small LSW slugs. Basic equation (C.10-C.14) assume ﬁnes migration within the aqueous phase, while some ﬁnes can be transported by water-oil menisci (Huang et al., 2018; Shapiro, 2016, 2018). The velocity of oil-water separating surface can signiﬁcantly diﬀer from either oil or water phases, so the particle capture rate and induced permeability damage can be signiﬁcantly diﬀerent. Another important extension of the model includes ﬁnes mobilisation by both moving phases, i.e. by the viscous force in the oil phase and by the capillary oil-water-particle force, like it occurs in foams (Farajzadeh et al., 2009, 2012, 2015). In the present work, we have used Corey parameters for relative permeability and capillary pressure for both core- and reservoir scales (Rabinovich, 2017, 2018). However, salinity variation is an important feature of the model. Therefore, basic up-scaled equations for twophase multicomponent ﬂow must be developed to describe salinityvariation triggered ﬁnes migration (Afekare and Radonjic, 2017; Rabinovich et al., 2015, 2016). The model assumes instant ﬁnes detachment during salinity

6.4. Analogy between a polymer ﬂooding and ﬁnes-assisted LSW injection The mathematical model for ﬁnes-assisted LSW ﬂood is a particular case for polymer ﬂood model, where the eﬀect of viscosity enhancement is not accounted for (Appendix D). Resistance factor of adsorbed polymer is a sweep enhancement mechanism for continuous polymer injection; residual resistance factor enhances sweep during polymer slug injection (Lake et al., 2014; Sheng, 2010, 2014). Due to the same reason, lifted and migrating ﬁnes yield the sweep increase. The factor of higher polymer adsorption and larger formation damage in low-permeability layer can yield some sweep decrease. However, the eﬀect is completely overwhelmed by the viscosity enhancement in the polymer case. Therefore, the larger is the polymer slug the higher is the sweep, i.e. there is no optimal polymer slug size. The same corresponds to slug injection of LSW under the wettability alteration and no ﬁnes migration – the recovery at continuous LSW injection is above than that with slug injection (Jerauld et al., 2008; Brodie and Jerauld, 2014; Attar and Muggeridge, 2016, 2018). The competing eﬀect of higher damage of low permeable layer by LSW is more pronounced than for polymer; water viscosity remains the same for low-concentration aqueous ﬁnes suspensions. It explains the existence of an optimal slug size for LSW injection. 6.5. Optimum slug size for LSW The permeability decrease in two layers due to induced ﬁnes migration yields two competitive factors, which can yield increase or 779

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release in this layer and decrease in sweep eﬃciency. 6 There are two competing factors during injection of LSW slug that aﬀect the recovery factor during LSW slug injection: (i) induced formation damage in swept high-permeability layer by LSW with consequent ﬂux diversion into unswept low permeable layer, (ii) the induced formation damage in the swept part of low permeable layer. For the simulated cases of two-layer cake reservoir, there always does exist an optimal size of LSW slug with HSW chase drive. 7 Optimal slug size strongly depends on reservoir heterogeneity, relative volumes of both layers, and mobility ratio. Initial over and undersaturation concentration of ﬁnes almost has no eﬀect on the optimal slug size, but aﬀect the RF. 8 Early LSW breakthrough in the high-permeability layer followed by LSW imbibition into the low-permeability layer and counter-current salt diﬀusion yield partial isolation of the layers. This restricts interface injected water ﬂow from the low-permeability to high-permeability layer and increases sweep eﬃciency.

decrease or velocity increase, while the delay in wettability alteration and ﬁnes mobilisation is signiﬁcant at the reservoir scale (Mahani et al., 2015a, 2015b, 2017). 7. Conclusions Laboratory determination of ﬁnes-induced formation damage in rocks with diﬀerent permeability and reservoir simulation of LSW slug injection accounting for under and oversaturated ﬁnes allows concluding as follows: 1 The extension of maximum retention function for two-phase ﬂow includes multiplication of the single-phase maximum retention function by the contact water-rock area, and substitution of the single-phase velocity by the phase velocity for water. 2 The model for ﬁnes migration in reservoirs with undersaturated initial ﬁnes concentration is equivalent to the reservoirs with saturated initial ﬁnes with the maximum retention function that is equal to the initial attached concentration for salinities above the critical salinity. 3 The model for ﬁnes migration in reservoirs with oversaturated initial ﬁnes is equivalent to the reservoirs with saturated initial ﬁnes, where the initial attached concentration is equal to critical attached concentration, and the diﬀerence is equal to the initial strained concentration. 4 Injection of LSW slug results in creation of a low-permeability barrier near to layers interface due to mobilisation and straining of the reservoir ﬁnes inside the slug, which yields RF increase. 5 Undersaturated ﬁnes in high permeability layer yields delay in ﬁnes

Acknowledgements The term “ﬁnes-assisted low-salinity waterﬂood” has been suggested by Prof. Cor van Kruijsdijk (Delft University, Shell Research) during fruitful discussions with Drs. R. Farajzadeh and H. Mahani. The authors are grateful to Dr Yong Wang and Dr Pengxiang Diwu (China University of Petroleum) for helping in numerical coding. L. Chequer acknowledges CNPq for ﬁnancial support. Many thanks are due to David H. Levin (Murphy, NC, USA), who provided professional Englishlanguage editing of this article.

Appendix A. Calculation of the maximum retention function Following Bedrikovetsky et al. (2011) and Zeinijahromi et al. (2013), in this Appendix we present the technique generating maximum retention function for a single-phase ﬂow. Calculations for gravitational and lifting forces indicate that their order of magnitude is between 10−14 and 10−13, while drag and electrostatic forces range between 10−11 and 10−8 (Bedrikovetsky et al., 2012; Kalantariasl and Bedrikovetsky, 2013). Therefore, gravitational and lifting forces are disregarded and only drag and electrostatic forces are considered relevant to determine conditions for particle detachment. This allows for simpliﬁed formulae for maximum retention function if compared with the expressions presented by Zeinijahromi et al. (2013). We assume ﬁnes lifting in situ the aqueous phase, neglecting ﬁne particle mobilisation by the moving oil. The condition of mechanical equilibrium of the ﬁnes particle on the top of either pore wall or internal cake built by other ﬁnes is the equality of the torque balances of electrostatic attraction and detaching drag forces, given by Eq. (1). The lever arm for normal forces is determined by mutual particle-rock deformation under the action of electrostatic attraction. The lever arm is equal to the size of contact ﬁne-grain surface and is calculated from Hertz theory (Derjaguin et al., 1975; Schechter, 1992)

ln3 =

Fe rs , 4K

4

K≡ 3

(

1 − ν12 E1

+

1 − ν22 E2

)

(A.1)

Here K is the composite Young's modulus, ν is the Poison's ratio and E is the Young's modulus. Indexes 1 and 2 correspond to matters of particles and rock, respectively. Once the normal lever arm is calculated, the drag lever arm can be calculated from the geometrical relationship between the particle radius and the normal lever arm. The drag lever arm for solid particles is assumed to be equal to the particle radius rs, because the contact area size is signiﬁcantly smaller than rs. Let us calculate the maximum retention function for a bundle of parallel capillary distributed by radius with density distribution function f(rp). Fig. 1b shows the forces exerting a single ﬁne particle on the surface of an internal ﬁlter cake or of a pore. The drag force is given by the expression derived from the analytical solution of the Navier-Stokes equations for ﬂow around a ﬁnite size particle ﬁxed on the plane (Altmann and Ripperger, 1997; O'Neill, 1968).

Fd =

u=

ωπμrs2 u (rp) rp

(A.2)

U ϕ

(A.3)

where u(rp) is the average interstitial velocity in a pore, rp is the pore radius, μ is the ﬂuid viscosity, and the dimensionless drag constant ω is an empirical coeﬃcient varying in the range 10–60. For a spherical particle on the plane surface, ω = 10.2 (O'Neill, 1968). More precise formulae than (A.2) for particles with sizes comparable with the pore radii, can be obtained by numerical solutions of Navier-Stokes equations using computational ﬂuid dynamics software packages. The total electrostatic force is the derivative of the potential energy 780

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Fe = −

∂V ∂h

(A.4)

where the total energy V is the sum of the London-van-der-Waals potential (VLVW), Electrical Double Layer potential (VEDL) and Born Repulsive potential (VBR), given by the extended DLVO (Derjagin-Landau-Verwey-Overbeek) theory (Derjaguin and Landau, 1941; Elimelech et al., 2013; Gregory, 1981; Israelachvili, 2011; Khilar and Fogler, 1998)

VLVW = −

A132 rs ⎡ 5.32h ⎛ λw ⎞⎤ 1− ln 1 + ⎥ ⎢ 6h ⎣ λw 5.32h ⎠ ⎦ ⎝

128πrs n∞ kB T ψs ψg e−κh κ2

VEDL =

;

(A.5)

zeζs ⎞ , ψs = tanh ⎛ 4 ⎝ kB T ⎠ ⎜

⎟

6

VBR =

A132 ⎛ σLJ ⎞ ⎡ 8 + Z 6 − Z⎤ ; + 7 7560 ⎝ rs ⎠ ⎢ (2 Z ) Z7 ⎥ + ⎣ ⎦ ⎜

⎟

Z=

zeζg ⎞ ψg = tanh ⎜⎛ ⎟ 4 ⎝ kB T ⎠

(A.6)

h rs

(A.7) (A.8)

V = VLVW + VEDL + VBR −21

J for kaolinite and quartz interaction, λw = 100 nm is the characteristic wavelength of interaction, Here the Hamaker constant A132 is 9.54 × 10 h is the surface-to-surface separation distance, kB = 1.381 × 10−23 J/K is Boltzmann constant, T = 298 K is temperature, κ is the inverse Debye length, n∞ is the bulk density of ions, ψs and ψg is the reduced zeta potential for particle and grain, respectively, z is the valence of the cation in solution, e = 1.602 × 10−19 C is the elementary electric charge, ζs and ζg are the zeta potentials for particle and grain, respectively, and σLJ = 0.5 nm is the atomic collision diameter adopted from Elimelech et al. (2013). Substitution of the expression for drag force (A.2) into the torque balance (Eq. (1)) yields

Fe =

ωπμrs2 u l rp − hc

(A.9)

where hc is the cake thickness and Fe is constant for ﬁxed particle size and salinity. The expression for cake thickness hc(rp) follows from Eq. (A.9):

h (rp) = rp −

ωπμrs2 u l Fe

(A.10)

The porosity in the parallel bundle of capillary is ∞

∫ rp2f (rp) drp

ϕ = πn

(A.11)

0

where n is the capillary density in the unitary rock cross-section and f(rp) is the pore size distribution. Permeability in the parallel bundle of capillary follows from the Poiseuille formula

k=

πn 8

∞

∫ rp4f (rp) drp

(A.12)

0

Consider a log-normal distribution of pore radii f(rp), which is determined by average radius < rp > and covariance coeﬃcient Cv. It makes second and fourth radii moments < rp2 > and < rp4 > Cv-dependent. If permeability and porosity and known, the values of < rp > and n are determined by Cv only. The concentration of attached particles is ∞

σcr (U , γ ) = n (1 − ϕc )

∫ [rp2 − (rp − hc (rp))2] f (rp) drp

(A.13)

0

where ϕc is the porosity of the cake of attached particles. Substitution of cake thickness from Eq. (A.10) into expression (A.13) results in ∞

σcr (U , γ ) = n (1 − ϕc )

2

ωπμrs2 u ⎞ ⎤ l ⎟ ⎥ f (rp) drp ⎝ Fe ⎠⎦

∫ ⎡⎢〈rp2〉 − ⎛ ⎜

0

⎣

(A.14)

Substitution of Eq. (A.11) into (A.14) yields the expression for the critical retention function: 2

ωπμrs2 U ⎞ ⎞ ⎛ϕ l⎟ ⎟ σcr (U , γ ) = (1 − ϕc ) ⎜ − n ⎜⎛ π ⎝ Fe (γ , rs ) ⎠ ⎠ ⎝

(A.15)

Appendix B. Maximum retention function for two-phase ﬂow We assume that ﬁnes are detached in aqueous phase only, neglecting ﬁnes detachment by the oil. So, interstitial velocity u in Eq. (2) for maximum retention function is substituted by the interstitial velocity of water (Yuan and Shapiro, 2011)

u=

U ϕ

→

uW ϕs

(B.1)

Also, ﬁnes are detached from the rock surface fraction accessible to water AW(s) (Fig. 6). The ﬁnes are not detached from the rock surface fraction 781

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1- AW(s), which is accessible to oil. Therefore, maximum retention function for two-phase ﬂow has the form given by Eq. (7) where σcr(γ,U) is the maximum retention function for a single-phase ﬂow. Appendix C. Governing equations for two-phase ﬂow with ﬁnes migration Following Zeinijahromi et al. (2011) and Bedrikovetsky (2013) here we describe equations for two-phase ﬂow in porous media with salinity alteration, ﬁnes detachment and migration. The equivalence between the models for ﬁnes-assisted low-salinity waterﬂood and polymer ﬂooding allows using the polymer ﬂood option of reservoir simulation of the ﬁnes-assisted low-salinity waterﬂooding. First, we discuss the case where all ﬁnes are in the saturated state, i.e. Eq. (2) holds. In large-scale approximation, the capillary pressure, diﬀusion and non-equilibrium eﬀects are neglected (Lake et al., 2014). The volumetric balance for an incompressible ﬂux of carrier water and oil is

→ ∇U = 0

(C.1)

The volumetric balance for incompressible water is

→ ∂s ϕ + U ∇f (s, γ , σs ) = 0 ∂t

(C.2) −1

kro (s, γ ) μ w (1 + βσs ) ⎤ f (s, γ , σs ) = ⎡1 + ⎥ ⎢ krw (s, γ ) μo ⎦ ⎣

(C.3)

where ϕ is the porosity, s is the water saturation, t is time, f is the fractional ﬂow of water, β is the formation damage coeﬃcient, kro and krw is the oil and water relative permeability, respectively, and μo and μw are the oil and water viscosity, respectively. The mass balance of salt in the aqueous phase neglects diﬀusive ﬂux:

ϕ

→→ ∂ (sγ ) + U ∇ (γ ) = 0 ∂t

(C.4)

The modiﬁed Darcy's law for two-phase ﬂow accounts for permeability damage:

→ k (s, γ ) k (s, γ ) ⎤ → U = −k ⎡ rw ∇p + ro ⎢ μ (1 + βσs ) μo ⎥ w ⎦ ⎣

(C.5)

where p is the pressure and k is the absolute permeability. In a ﬁxed reservoir point at low saturation, water velocity is low, drag force exerting from water on the attached ﬁne is low, and ﬁnes remain in an undersaturated state

σaI < σcr (γ , u),

u=

Uf (s, γ ) ϕs

(C.6)

After the arrival of salinity front at the reservoir point, salinity decreases, electrostatic force decreases, and ﬁnes are immobile until the inequality (C.6) turns into equality. Until it happens, i.e. when inequality (C.6) fulﬁls, strained concentration remains zero and σs = 0 in the system (8, C.1, C.2, C.4, C.5). The initial conditions for LSW in a virgin reservoir are

t = 0:

s = s wi, γ = γI , σs = 0

(C.7)

If the initial ﬁnes in the reservoir are in the oversaturated case, i.e.

σaI > σcr (γ , u),

u=

Uf (s, γ ) ϕs

(C.8)

initial strained concentration is equal to the diﬀerence between the initial attached concentration σaI and the maximum retention value, i.e. the excess is instantly strained by the pores. Let us introduce the following dimensionless parameters x

xD → L , tD → u→

u u0

=

qt , ϕLWH

uWH , q

p→

σs → k0 p u0 μo L

σs , σaI

u0 =

=

k 0 pWH , qμo L

q , WH

γ→

γI − γ γI − γJ

(C.9)

where x is the spatial coordinate, xD is the dimensionless spatial coordinate, tD is the dimensionless time, L is the reservoir length, q is the volumetric ﬂow rate, W is the reservoir width, H is the reservoir thickness, u0 is the initial overall ﬂow velocity, and k0 is the initial absolute permeability. The system of equation (8, C.1, C.2, C.4, C.5) in dimensionless forms transforms is the following system

∇ (u) = 0

(C.10)

∂s + u∇f (s, σs ) = 0 ∂tD

(C.11)

∂ (sγ ) + u∇ (γf ) = 0 ∂tD

(C.12)

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u σs = ⎡1 − σcr ⎛ W , γ ⎞ ⎤ AW (s ) ⎝ s ⎠⎦ ⎣

(C.13)

krw (s ) μo + kro (s ) ⎤ ∇p u = −⎡ ⎥ ⎢ μ (1 + β (σaI − σcr (γ ))) ⎦ ⎣ w

(C.14)

The system of three equations (C.11), (C.12), and (C.13) determines three unknowns s, γ, and σs. Then pressure p is found from Eq. (C.14). The equivalence between the polymer ﬂooding and ﬁnes-assisted low-salinity waterﬂooding models used for reservoir simulation of LSW by polymer option is explained in the next Appendix. Appendix D. Equivalence between the polymer ﬂood and ﬁnes-assisted low-salinity waterﬂooding models The “dictionary” below translates the ﬁnes-assisted low-salinity waterﬂooding model (C.11-C-14) into the polymer ﬂood model (Zeinijahromi et al., 2013):

cp =

cp 0 (γI − γ ) (γI − γJ )

,

ca =

δcp 0σs (γI − γJ )

,

Rk (ca) = 1 + βσs

(D.1)

where cp and ca are concentrations of dissolved and adsorbed polymer, and δ < < 1 is a small parameter. The initial condition for salt concentration is γ = γI, where polymer concentration in the aqueous phase is zero, cp = 0. The inlet boundary condition for salt concentration γ = γJ corresponds to polymer concentration in aqueous phase equal to cp = cp0. The polymer adsorption is expressed through the maximum retention function. For over and undersaturated states, the adsorbed (strained) concentration is given by Eqs. (8) and (9), respectively. Expressing the resistance factor as a function of the formation damage variables yields

1 + (RRF − 1)

ca = 1 + βσs ca (cp 0)

(D.2)

Substituting Eq. (D.1, D.2) into the system for ﬁnes-assisted low-salinity waterﬂooding (C.11-C.14) yields ∂s ∂tD

+ u∇f (s, cp, ca) = 0

∂ (scp + ca) ∂tD

+ u∇ [cp f (s, cp, ca)] = 0 k

(s ) μ

u = −⎡ μ (rwc ) R (oc ) + kro (s ) ⎤ ∇p ⎣ w p k a ⎦

(D.3)

the equations for continuity of water, of polymer with vanishing adsorption, and modiﬁed Darcy's law for two-phase ﬂow with formation damage by adsorbed polymer (Lake et al., 2014). The equations for the total two-phase ﬂux for polymer ﬂooding (D.3) and ﬁnes-assisted low-salinity waterﬂooding (C.11-C.14) are equivalent.

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t: Time, s tD: Dimensionless time T: Temperature, K u: Interstitial Velocity, m.s−1 U: Darcy's velocity, m.s−1 V: Total potential energy, J VBR: Born Repulsive Potential, J VEDL: Electrical Double Layer Potential, J VLVW: London-van-der-Waals Potential, J W: Reservoir width, m x: Spatial coordinate, m xD: Dimensionless spatial coordinate z: Valence of the cation in the solution

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Nomenclature

Greek letters

Latin letters

β: Formation damage coeﬃcient βσs: Formation damage factor γ: Salinity, mol/L ζg: Zeta potential for grain, V ζs: Zeta potential for particle, V κ: Inverse Debye length, m−1 λw: Characteristic wavelength of interaction, m μ: Dynamic viscosity, cp ν: Poison's ratio σa: Volumetric concentration of attached particles σaI: Initial attached particle concentration σcr: Maximum retention function σLJ: Atomic collision diameter, m σs: Volumetric concentration of strained particles τ: Resultant torque, N ϕ: Porosity ϕc: Cake porosity ψg: Reduced zeta potential of grain ψs: Reduced zeta potential of particle ω: Dimensionless drag coeﬃcient

A132: Hamaker constant, J Aw: Fraction of the overall rock-liquid surface accessible to water ca: Polymer adsorption concentration cp: Polymer concentration in the aqueous phase cp0: Polymer concentration in the injected water Cv: Covariance coeﬃcient e: Elementary electric charge, C E: Young's modulus, Pa f: Fractional ﬂow of water F: Force, N h: Surface-to-surface separation distance, m hc: Cake thickness, m H: Reservoir thickness, m k: Permeability, mD kB: Boltzmann constant, J/K krowi: Oil relative permeability at the initial water saturation krwor: Water relative permeability at the oil residual K: Composite Young's modulus, Pa l: Lever arm ratio ld: Drag lever arm, m ln: Normal lever arm, m L: Reservoir length, m n: Pore density in a cross section, m−2 n∞: Bulk density of ions p: Pressure, Pa q: Volumetric ﬂow rate, m3s−1 rp: Pore radius, m rs: Particle radius, m Rk: Resistance factor RRF: Maximum value of the resistance factor s: Water saturation sor: Oil residual saturation swi: Initial water saturation

Super/subscripts 0: Initial, for permeability and velocity cr: Critical, for salinity and velocity d: Drag, for force e: Electrostatic, for force I: Initial, salinity J: Injected, for salinity o: Oil, for viscosity ro: Oil relative, for permeability rw: Water relative, for permeability w: Water, for viscosity and velocity

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Optimal slug size for enhanced recovery by low-salinity waterflooding due to fines migration

Optimal slug size for enhanced recovery by low-salinity waterflooding due to fines migration

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