A Mechanistic Pore-Scale Analysis of the Low-Salinity Effect in Heterogeneously Wetted Porous Media
AA Mechanistic Pore-Scale Analysis of the Low-Salinity Effectin Heterogeneously Wetted Porous Media
Michael G. Watson ∗ School of Mathematics and Statistics,University of Sydney, NSW 2006, Australia
Steven R. McDougall
Institute of Petroleum Engineering,Heriot-Watt University, Edinburgh, EH14 4AS, Scotland (Dated: August 9, 2019) a r X i v : . [ phy s i c s . g e o - ph ] A ug bstract Over the last two decades, the enhanced oil recovery technique of low-salinity (LS) waterfloodinghas been a topic of substantial interest in the petroleum industry. Many studies have shown that LSbrine injection can increase oil production relative to conventional high-salinity (HS) brine injection,but contradictory results have also been reported and a complete understanding of the underlyingmechanisms remains elusive. We have recently developed an innovative, steady-state pore networkmodel to simulate oil recovery by LS brine injection in uniformly wetted pore structures (Watson etal., Transp. Porous Med. , 201–223, 2017), and we extend this approach here to investigate themechanisms of the low-salinity effect (LSE) in heterogeneously wetted media. We couple a model ofcapillary force-driven fluid displacement to a novel tracer algorithm and track the evolving salinityfront in the pore network as oil and HS brine are displaced by injected LS brine. The wettabilityof the pore structure is modified in regions where the water salinity falls below a critical threshold,and simulations show that this can have significant consequences for oil recovery.For networks that contain spanning clusters of both water-wet and oil-wet (OW) pores prior toflooding, our results demonstrate that the OW pores contain the only viable source of incrementaloil recovery by LS brine injection. Moreover, we show that a LS-induced increase in microscopicsweep efficiency in the OW pore fraction is a necessary, but not sufficient, condition to guaranteeadditional oil production. Simulations further suggest that the fraction of OW pores in the net-work, the average network connectivity and the initial HS brine saturation are key factors thatcan determine the extent of any improvement in oil recovery in heterogeneously wetted networksfollowing LS brine injection. This study clearly highlights the fact that the mechanisms of the LSEcan be markedly different in uniformly wetted and non-uniformly wetted porous media.
I. INTRODUCTION
As the global demand for energy continues to grow, new and innovative enhanced oilrecovery (EOR) methods are required to maximise oil production from hydrocarbon reser-voirs. One method that has attracted increasing interest over the last decade is low-salinity(LS) waterflooding, which has been shown to be capable of improving oil recovery at the ∗ Corresponding author: [email protected] consequence of LS brine injection is a shift inthe rock wettability, and that this shift is most likely towards increased water-wetness ordecreased oil-wetness [30, 31]. Direct evidence for wettability alteration has recently beenprovided by Khishvand et al. [32], who used micro-CT imaging to visualise a reduction inoil-water contact angles from an average of 115 ◦ to an average of 89 ◦ inside miniature coresamples following LS brine injection. This change in wettability led to an overall increasein oil recovery, which was explained by increased water invasion of small- and medium-sizedpores due to their reduced capillary entry pressure thresholds. Note that here, and in allthat follows, the oil-water contact angles in individual pores have been measured throughthe aqueous phase. By convention, we use the term water-wet (WW) to refer to pores thathave equilibrium contact angles less than 90 ◦ and OW to refer to pores that have equilibriumcontact angles greater than 90 ◦ . The strength of the wetting preference in a given pore isdetermined by the proximity of its contact angle to 90 ◦ (e.g. a strongly water-wet pore hascontact angle close to 0 ◦ and a weakly water-wet pore has contact angle close to 90 ◦ ).Pore network models have been widely applied in the oil industry to study pore-scalefluid displacement phenomena in a variety of contexts (see Blunt [33], Joekar-Niasar andHassanizadeh [34] and Blunt [35] for excellent reviews of this work). Following on from thetheoretical approach developed by Sorbie and Collins [36], we have recently developed steady-state and unsteady-state pore network modelling approaches to investigate the pore-scaledisplacement behaviour that arises from LS-induced wettability modification in uniformlywetted networks [37, 38]. In these studies, we have used tracer algorithms to track theevolution of water salinity when LS brine is injected to displace oil from in silico networksthat also contain HS brine. Oil-water contact angles of individual pores were modified whenlocal brine salinity fell below a critical threshold, and the results have indicated that thiscan have a dramatic impact on both the sequence of pore-filling and the overall volume ofoil recovered from the network.The fluid displacement methodologies in steady-state and unsteady-state pore networkmodels are distinctly different, and this has important implications for the spatio-temporaltracking of water salinity in LS brine injection simulations. In steady-state models (suchas Watson et al. [37] and the current study), viscous forces are assumed to be negligibleand the displacement of oil from the network is driven purely by capillary forces between4he oil and the injected brine. This approach is most applicable to far-field regions of theoil reservoir (i.e. the bulk of the reservoir away from injection and producing wells). Thecapillary pressure P c (which we define as the pressure difference between the oil and waterphases, P c = P o − P w ) is reduced in a stepwise manner and oil is displaced from several poresat each stage of the simulation. Saturation changes in the network in steady-state modelsare therefore distinct and governed solely by the capillary entry thresholds of the pores.Hence, in steady-state LS waterflooding models, the evolution of brine salinity cannot betracked explicitly and must be estimated a posteriori . In unsteady-state models (such asBoujelben et al. [38]), the displacement of oil from the network is driven by both viscous andcapillary forces (note, however, that the underlying methodology is equally applicable at lowwater injection rates where viscous forces may be negligible). For unsteady-state models, afixed rate of water injection is typically assigned at the network inlet. Individual oil-watermenisci are then explicitly tracked as they move through each pore, and the simulationproceeds by completely filling one single pore with water at each timestep. Unsteady-statemodels are therefore much more computationally expensive than steady-state models andthis is prohibitively restrictive for 3D simulation.Much like the experimental coreflooding studies discussed above, our recently publishedsteady-state and unsteady-state LS waterflooding models have demonstrated a range ofoutcomes that include positive, neutral and negative effects of LS brine injection. Boujelben et al. [38] performed unsteady-state LS brine injection simulations on uniformly wetted2D pore networks and showed that the outcome of dynamic contact angle reduction byLS brine can be influenced by parameters such as the water injection rate and the oil-water viscosity ratio. Results showed that dynamically reducing capillary forces in thenetwork can lead to a marked change in the displacement regime (e.g. from capillary fingeringto stable displacement, or from capillary fingering to viscous fingering) and that this canultimately result in either a positive or negative LSE. Boujelben et al. [38] also found that theparticular combination of initial and LS-modified wettability (e.g. strongly OW to weaklyOW, strongly OW to neutral wet) can be an important factor. This finding is consistentwith the results of Watson et al. [37], who performed steady-state simulations of LS brineinjection in uniformly wetted 3D pore networks. Watson et al. [37] explored a range ofpossibilities for the combination of initial and LS-modified wettability, including scenarioswhere LS brine caused contact angles to increase — indeed, the phenomenon of increased oil-5etness following LS brine injection has been reported on several occasions [20, 21, 39, 40]and our results indicated that this wettability shift can lead to increased oil production,particularly if LS brine causes contact angles to change from WW to OW. Watson et al. [37]also demonstrated that the pore size distribution (PSD) of the network can be a key factorin determining the extent of any LSE. In particular, for networks with pore radii uniformlydistributed between R min and R max , we showed that the potential for LS brine injectionto improve oil recovery by reducing pore contact angles increases as the ratio R max /R min decreases.In more general terms, the key insight from the study of Watson et al. [37] was the identi-fication of two distinct pore-scale effects that can be used to explain the outcome of dynamicLS-induced contact angle modification in a particular scenario. First, the “pore sequenceeffect” , which is characterised by an overall change in the size distribution of pores displaced by injected LS brine, and second, the “microscopic sweep efficiency effect” , which is charac-terised by an overall change in the total fraction of pores displaced by LS brine. Importantly,we showed that these two effects can act independently or synergistically depending on thechoice of parameters, and also that both can have positive and negative consequences foroverall oil recovery following LS brine injection.The studies by Watson et al. [37] and Boujelben et al. [38] have provided many newinsights into the pore-scale displacement phenomena that can accompany dynamic wetta-bility modification and demonstrate the potential of pore network modelling as a powerfuland inexpensive tool to help interpret the results of experimental LS coreflooding studies.However, the results from these studies were obtained by considering only in silico networksthat initially contain either 100% WW or 100% OW pores. Such idealised distributions ofwettability are unlikely to hold for many of the rock samples utilised in experimental studies,where a combination of heterogeneous mineralogy and core preparation techniques (e.g. ag-ing) will lead to a mixture of wettability conditions within the core. Therefore, in this paper,we extend the steady-state modelling approach of Watson et al. [37] to investigate the LSEin networks with non-uniform distributions of initial wettability. Our results suggest thatthe conditions required to achieve a positive LSE in non-uniformly wetted networks maybe much more stringent than for uniformly wetted networks, and, as will be seen, dependon parameters that we have not reported to date. We will begin in the next section byrestating our modelling methodology in full, and we will then present a series of simulation6esults and parameter sensitivities in Section III. We will provide an in-depth discussion ofthe implications of our results in Section IV, before concluding with a broad summary ofthe outcomes of the study in Section V. II. MODEL METHODOLOGYA. Pore Network Initialisation
We stress from the outset that our goal in this paper is not to make quantitative predic-tions, but rather to use pore network modelling techniques to gain new, qualitative insightsinto the mechanisms of the LSE in networks of non-uniform wettability. Therefore, as inour previous studies of LS brine injection in uniformly wetted networks [37, 38], we chooseto represent the pore space as a simple 3D network of interconnected capillary elements —more complex network architectures with irregular distributions of connectivity and poregeometry could be considered, but we maintain an idealised approach at this stage to betterhighlight the underlying pore-scale behaviours and to simplify the analysis of our results.Hence, we assume that, in general, the properties of the individual pores in our networkscan be described using the scaling laws proposed by McDougall and Sorbie [41]. In thistheory, a capillary entry radius R can be assigned to each pore, and the pore volume V andpore conductance g can then be described by the proportionalities V ( R ) ∝ R ν (0 (cid:54) ν (cid:54) g ( R ) ∝ R µ (2 (cid:54) µ (cid:54) ν = 2 and µ = 4), but we will later reduce the value of ν togive a more representative distribution of individual pore volumes across the network. Notethat the assumption of cylindrical pore elements precludes the possibility of explicit film andcorner flow modelling in our networks. As discussed extensively in both Watson et al. [37]and Boujelben et al. [38], wettability alteration by LS brine transported in the corners ofangular pores could have additional implications for oil recovery. We will focus here simplyon bulk transport mechanisms but note that we are currently exploring the consequences ofLS brine transport in films and will report this work in a future publication.For each simulation, we generate a 3D pore space by defining the network dimensions(i.e. the number of nodes in each direction), the average pore length L , and the networkco-ordination number ¯ Z , which determines the average number of capillary elements (pores)7 (cid:3040)(cid:3028)(cid:3051) (cid:1844) (cid:3040)(cid:3036)(cid:3041) WWOW (a) (cid:1844) (cid:3040)(cid:3028)(cid:3051) (cid:1844) (cid:3040)(cid:3036)(cid:3041)
OWWW (b) (cid:1844) (cid:3040)(cid:3028)(cid:3051) (cid:1844) (cid:3040)(cid:3036)(cid:3041)
WWOW (c)
FIG. 1. Schematic plots of the WW and OW pore size distributions in (a) FW, (b) MWL and (c)MWS systems. connected to each junction (node). Individual pores are assigned both a radius R from achosen pore size distribution (PSD) and a contact angle θ that defines the initial wettabilityof the pore. Of course, in a real porous medium, we might expect a certain level of intra-porewettability variation due to localised heterogeneity in rock mineralogy and pore geometry.However, wettability variation at the scale of a single pore would be difficult to implementin any meaningful way and, moreover, this type of data is rarely measured experimentally.We therefore assume that θ represents an effective contact angle that captures some spatialaverage of the wetting properties in each pore. In this paper, we are only concerned withnetworks that initially contain both WW and OW pores, so we introduce the parameter α to represent the fraction of pores that are initially assigned OW contact angles. If WW andOW contact angles are assigned to pores at random (i.e. with no correlation between poresize and wettability), the network is termed fractionally-wet (FW). However, in addition toFW networks, there also exist two well-recognised wettability classes that display a distinctcorrelation between pore size and pore wettability. These are the mixed-wet large (MWL)class, where only the largest pores are OW, and the mixed-wet small (MWS) class, whereonly the smallest pores are OW (Figure 1). See Skauge et al. [42] for an interesting discussionregarding these three wettability classes in the context of related experimental work. Here,we will mostly use networks of the MWL class, but, where possible, we will also interpretthe implications for similar FW and MWS networks. B. Steady-State Oil Displacement
The steady-state approach to waterflood simulation is well-established and has been re-ported previously [43, 44]. Briefly, water is introduced at the inlet face of an oil-bearing8etwork and a stepwise reduction of the P c allows water invasion of oil-filled pores with se-quentially decreasing capillary entry pressures. As we are neglecting film flow at this stage,water ingress can only occur via piston-like displacement of pores with a bulk water connec-tion to the network inlet — this takes place at capillary entry pressure 2 σ cos ( θ ) /R , where σ denotes the oil-water interfacial tension. In a heterogeneously-wet system, a representativeHS flood proceeds as follows: (i) the flood begins at a high, positive P c and water first in-vades the smallest WW pores that are connected to the inlet; (ii) the displacement continuesat positive (but decreasing) P c , filling successively larger WW pores until all displaceableWW pores have been invaded; (iii) the water then fills the largest OW pores that are con-nected either to the inlet or to bulk water, and proceeds to displace oil from progressivelysmaller OW pores as the P c becomes increasingly negative. Oil that becomes isolated fromthe network outlet by invading water is trapped and remains immobile. Note that we do notconsider snap-off of oil-filled pores due to notional water film accumulation in WW pores,nor do we allow oil to drain from the network through OW pathways. Simulations proceedaccording to the above rules until no further oil can be displaced from the network.Note that, in this stepwise oil displacement procedure, the simulated rate of P c decline canbe regarded as an implicit measure of the physical waterflood injection rate. Under low waterinjection rate conditions (small P c steps), displacement of oil by water will occur pore-by-pore. Conversely, at higher injection rates (large P c steps), many capillary entry thresholdswill be satisfied at the same time, and water will displace oil from many pores simultaneously.Hence, different injection rates ( P c step sizes) result in different pore displacement patterns. C. Tracer Injection Algorithm
The steady-state LS waterflooding model uses an innovative approach to track the evo-lution of water salinity during oil displacement from the pore network. In this section, weprovide a complete description of our salinity tracking methodology, followed by a detailedphysical interpretation of our modelling assumptions.9 . Methodology
Following Watson et al. [37], we track the dispersion of HS and LS brines in the networkby coupling the steady-state oil displacement model to an unsteady-state tracer injectionalgorithm. We assume that HS brine (tracer concentration of zero) and LS brine (tracerconcentration of one) are miscible and that they instantaneously mix in pore elements. Thenormalised brine salinity in a general water-filled pore is therefore given by 1 – ( tracerconcentration ). During periods of HS brine injection, there is no need to track the watersalinity in the network and simulations proceed as detailed in Section II B. However, whenLS brine is injected into a network that contains HS brine, the tracer algorithm is appliedafter each saturation change to track the dispersion of the newly-introduced fluid. In theabsence of explicit, dynamic information about the LS brine influx, we effectively “rewind”time to before the most recent saturation change and use the tracer approach to estimatenew brine salinities in the water-filled pores. Contact angles of oil-filled pores are thenupdated according to their local brine salinity (see Section II D), and this localised wettabilitymodification can alter the sequence of pore-filling at subsequent displacement steps.In this paper, we consider two different modes of LS waterflooding: (1) a secondary modewhere LS brine is injected into a network that contains an initial (spatially-distributed) HSbrine saturation S wi ; and (2) an early tertiary mode where LS brine injection commencesat water breakthrough following a period of HS brine injection ( S wi = 0). Our simulationstherefore involve both pre-breakthrough and post-breakthrough LS brine injection, and weuse different methods to update the salinities of water-filled pores in each of these twoscenarios. Prior to water breakthrough, the volumes of HS and LS brine that occupy eachwater cluster are known exactly and we estimate the water salinity in each pore using acluster-scale averaging algorithm. After breakthrough, however, the resident HS and LSbrines disperse due to flow and estimation of new water salinities requires a more detailedcalculation. See Figure 2 for some images of the salinity evolution during a typical simulationas oil and HS brine are displaced from a pore network by LS brine injection.In the case of pre-breakthrough LS brine injection, the tracer concentrations in water-filled pores are updated after each saturation change according to the following steps:(a) Identify each unique inlet-connected water cluster.10 IG. 2. Snapshots of an evolving salinity front during a typical model simulation where injectedLS brine (salinity = 0) displaces oil from a pore network that contains HS brine (salinity = 1).Colours of individual pores correspond to normalised salinity values as follows: red [0, 0.25); pink [0.25, 0.5); white [0.5, 0.75); light blue [0.75, 1); dark blue (b) For each cluster, determine the total cluster volume V clus , the total mass of tracernewly added to the cluster M new (i.e. pores just displaced by injected LS brine) andthe previous mass of tracer in the cluster M clus (i.e. pores displaced by HS or LS brineat earlier times).(c) Assign to each pore in the cluster the averaged tracer concentration C clus =( M new + M old ) /V clus .For post-breakthrough LS brine injection, we use a two-stage process to calculate the newtracer concentrations associated with the latest saturation change in the network. First, weassign initial tracer concentrations to all newly displaced pores by estimating the salinity ofthe water driven into them by the invading LS brine. We then simulate a period of tracerflow in the spanning water cluster, where the total injection time accounts for the fact thatthe volume of brine injected to achieve the latest saturation change will typically be largerthan the volume of oil produced (i.e. some of the injected LS brine will simply bypass theoil and flow out of the network). Note that we assume a Poiseuille flow law in each poreelement for all tracer flow calculations.The salinity assignment procedure at each Pc step for post-breakthrough LS brine injec-tion can be summarised as follows:(1) For an arbitrary pressure drop between the inlet and outlet faces of the network,calculate the global pressure solution in the spanning water cluster and determine the11ndividual pore flow rates. Rescale this solution to achieve the desired fixed injectionrate Q , and determine the associated pressure drop ∆ P new .(2) Calculate the volume of oil ∆ V just displaced from the network.(3) Estimate the time T ∆ V required to achieve this oil displacement, according to theequation: T ∆ V = ∆ VQ (cid:20) ∆ P old ∆ P old − ∆ P new (cid:21) , (1)where ∆ P old denotes the pressure drop from the previous displacement step. If T ∆ V exceeds the time T P V required to inject one whole pore volume of fluid, set T ∆ V = T P V .(4) Traverse flow pathways through the spanning water clusters and assign to all newlydisplaced and newly connected pores the flow-weighted average tracer concentrationof their upstream neighbours.(5) Follow steps (a)–(c) described earlier for all non-spanning inlet-connected water clus-ters and for all “dead-end” water clusters (i.e. pores that branch from a flowing watercluster but do not yet support flow themselves). For dead-end clusters, the sourceconcentration for new mass M new is the flow-weighted average tracer concentration inthe upstream neighbours of the node from which the cluster branches.(6) Calculate the total mass of tracer M added to pores in steps (4) and (5), and calculatethe time T required for this period of pre-assignment according to T = M /Q . Simu-late dynamic tracer injection in the flowing water-filled pores for the remaining periodof time T ∗ = T ∆ V − T (see the Appendix for a fuller description of the methodology).Equation (1) is a crucial component of the tracer assignment procedure, and its formathas been chosen to reflect qualitative observations from experimental coreflooding studies.The term ∆ V /Q quantifies the minimum length of time T min required for the most recentnetwork saturation change, while the viscous pressure ratio term provides a modulatingfactor that determines the extent to which this minimum time is lengthened by water egressfrom the network. If the reduction in the pressure drop between successive P c steps is large(e.g. when a percolating water cluster becomes more connected), we assume that minimalwater will be lost to the outlet during the oil displacement ( T ∆ V not much larger than T min ). However, if only a small reduction in the pressure drop can be achieved (e.g. when12he water phase has a well-established spanning cluster), we assume that the process of oildisplacement will be highly inefficient ( T ∆ V significantly larger than T min ). Note that thechoice of water injection rate Q provides a physical time scaling for the tracer injection.We briefly comment on the limiting criterion in step (3) above and emphasise that, inpractice, the condition T ∆ V > T P V is very rarely satisfied. Moreover, when this conditionis satisfied, it is always towards the end of a simulation when little untrapped oil remains,and the global pressure drop in the water phase becomes insensitive to further small satu-ration changes in the network. The inclusion of the limiting criterion therefore has no realconsequences for the simulation results, and we include it in the model simply to preventexcessively long run times in the circumstances just described.
2. Physical Interpretation
Based on the tracer injection methodology described above, it should be clear that theoutcome of a given LS waterflooding simulation will intrinsically depend on the number of P c steps that are taken to displace the resident oil. This is a deliberate feature of the modeland reflects the fact that the P c step size is correlated to the physical waterflood injectionrate. In any given simulation, oil is displaced from the network by discretely reducing the P c from some initial (maximum) value to some final (minimum) value. If the P c is reducedover many small steps (mimicking a low water injection rate), the tracer injection algorithmwould predict an inefficient displacement of the oil after water breakthrough (i.e. an injectedwater volume much larger than the produced oil volume over the course of a simulation).On the other hand, if the P c is reduced over few large steps (mimicking a higher waterinjection rate), the algorithm would predict an efficient displacement of the oil after waterbreakthrough (i.e. an injected water volume only slightly larger than the produced oil volumeover the course of a simulation).For all simulations performed in this study, the P c has been stepped from its maximumvalue to its minimum value in no more than 100 steps (mirroring an intermediate waterinjection rate and providing a sensible base case scenario). As a simulation proceeds, eachnew P c is chosen to ensure that the entry pressures of approximately 1% of the oil-filledpores in the network are newly satisfied. If more than 1% of the oil-filled pores have becomenewly accessible due to LS-induced contact angle modification at the previous simulation13tep, the P c is maintained at its existing level for the subsequent simulation step. For 100steps, we find that our simulations require around 2–4 PVs of water injection to remove allof the displaceable oil from the network. A total injection volume of 2–4 PVs is broadlyconsistent with the volume of water required to displace the resident oil from core samplesin experimental studies.The tracer injection algorithm detailed in Section II C 1 combines both advection mod-elling and a cluster-scale averaging procedure to track the evolution of brine salinity inthe aqueous phase. A physical and methodological justification for this hybrid approach isgiven below, where we interpret the implications of our assumptions in the context of bothtertiary and secondary LS brine injection. In tertiary LS simulations, we initially considercases where S wi = 0 and mixing of HS and LS brines occurs only after injected HS brine hasbroken through. At breakthrough, the HS brine in the network will be mostly distributedin one or two spanning clusters and several large (inlet-connected) non-spanning clusters.In the spanning water cluster(s), and any others that subsequently become spanning, tracerevolution is determined by means of the global flow solution. Here, we consider only ad-vective mixing of the HS and LS brines, but this is reasonable because the frontal advancerate in a typical coreflood (1–5 m/day) should limit the capacity for diffusive mixing in thetortuous pore space. In the non-spanning water clusters that grow (or emerge from theinlet) by LS brine injection, we assume perfect (instantaneous) mixing of the HS and LSbrines. While the assumption of perfect mixing (rather than advective mixing) of HS andLS brines in non-spanning water clusters seems restrictive, note that, in any non-spanningwater cluster that contains mostly HS brine or mostly LS brine, perfect mixing should makelittle difference to the salinity in most pores in the cluster. If a non-spanning water clusterinstead contains similar volumes of HS and LS brine, the perfect mixing assumption is lessvalid and (relative to advective mixing) one would anticipate a slight reduction in contactangle modification events proximal to the network inlet. Accurate tracking of LS ingress innon-spanning water clusters requires a more involved unsteady-state modelling approach,such as that reported by Boujelben et al. [38] for uniformly wetted systems.In secondary LS simulations, we include cases where LS brine invades a network that isinitially populated by randomly distributed clusters of HS brine. We again assume perfectmixing of HS and LS brines in non-spanning water clusters. Therefore, as clusters of LSbrine emerge from the network inlet, they immediately assimilate with the connate HS brine14lusters that they contact. In reality, this assimilation process would be a local phenomenon,whereby each HS brine cluster that is contacted by LS brine is diluted locally by convectivemixing with its neighbouring LS brine-filled pores. However, as LS brine clusters developwithin the tortuous pore space, connate HS brine clusters are contacted around the entireperimeter of each LS brine cluster (this can be envisaged as a series of fingered LS brinestructures that are “studded” with small HS brine clusters). Consequently, the assumptionthat HS brine clusters undergo perfect mixing with their host LS brine cluster(s) at each P c step is not as restrictive as it may first appear.We further remark that while we have neglected to explicitly consider mechanisms ofsalinity dispersion, we surmise that dispersive effects would play a secondary role comparedto the first-order impacts of network topology and wettability distribution during advectivewater transport. Phenomena such as Taylor dispersion and diffusion within dead-end loopscould be included in a more refined future model, and these mechanisms may well provideadditional insights into the complex LS waterflooding process. D. Coupling Tracer Concentration to Contact Angle
After appropriate tracer concentrations have been assigned to water-filled pores, we usethe updated salinity map in the network to locally modify the contact angles of remainingoil-filled pores. In the absence of definitive data on the relationship between brine salinityand the pore-level wettability change within rock structures, we adopt a modest wettabilitymodification methodology that reflects key qualitative observations from LS corefloodingexperiments. Results from laboratory studies suggest that the LSE will be negligible forinjected brine salinities above approximately 5000 ppm [29], but may become increasinglyefficacious at lower salinity [9, 14]. An expression that captures increasing contact anglemodification with increasing tracer concentration therefore seems reasonable, and to simplifythe analysis of our results we propose the following Heaviside model to describe the dynamicmodification of contact angles in the network: θ = θ HS , C N < C ∗ θ HS − ∆ θ, C N (cid:62) C ∗ . (2)Here, θ HS denotes the unmodified contact angle of an oil-filled pore, ∆ θ quantifies the extentof LS-induced contact angle modification, C N denotes the maximum tracer concentration15n all neighbouring water-filled pores and C ∗ denotes the critical tracer concentration re-quired for contact angle modification. Equation (2) effectively assumes that when brineof sufficiently low salinity invades a pore, the brine will contact the oil/rock interface ofneighbouring oil-filled pores and locally alter the surface wettability at the neighbouringpore entrances. If the capillary entry pressure of any neighbouring pore is then satisfied,LS brine will enter the pore and there is an implicit assumption that the wettability of theentire host pore surface will be progressively altered as the brine invades. We assume thatthe modification of wettability in each pore arises from an appropriate chemical reaction(e.g. multi-component ion exchange or electrical double layer formation). However, as thisreaction operates on a much faster timescale than the time between successive P c steps, wedo not explicitly model the associated chemical mechanisms. III. RESULTS
As discussed in Section I, we have previously used the above modelling approach to in-vestigate pore-scale mechanisms that potentially underlie the LSE (or lack of) in networksof uniform initial wettability (i.e. networks that consist of either 100% WW pores or 100%OW pores prior to LS flooding; Watson et al. [37]). The results of this earlier study, whichexplored the implications of dynamic wettability alteration by LS brine injection, revealeda suite of potential modifications to pore filling sequences that can offer some insight intothe apparent inconsistencies that have been reported in the experimental LS waterfloodingliterature. To obtain a more complete understanding of the pore-scale displacement phe-nomena associated with LS brine injection, the current study investigates the consequencesof dynamic, LS-induced wettability alteration in networks of non-uniform initial wettability.Note that by non-uniform we specifically refer to networks that initially possess spanningclusters of both WW and OW pores . Percolation theory therefore stipulates that we require: D ¯ Z ( D − (cid:54) α (cid:54) − D ¯ Z ( D − , (3)where D denotes network dimensionality. Figure 3 shows a plot of the parameter combina-tions (cid:0) α, ¯ Z (cid:1) that satisfy this condition for regular 3D pore networks: note that, for networksthat contain both WW and OW pores but only one spanning wettability cluster, our modelproduces results that can be interpreted from the uniform wettability study performed in16
456 0 0.25 0.5 0.75 1 Z (cid:302) WW poresspanning OW poresspanningWW and OWpores spanning
FIG. 3. Plot that shows how the number of spanning wettability clusters in a regular 3D porenetwork is determined by the fraction of OW pores in the network ( α ) and the average networkco-ordination number ( ¯ Z ). In the left-hand region of the plot only WW pores span the network, inthe right-hand region of the plot only OW pores span the network, and in the central region bothWW and OW pores span the network. The curves that delineate these regions are derived usingresults from percolation theory (see equation (3) in the main text). Watson et al. [37].Although there is a commonly held view in the experimental literature that LS water-flooding can lead to increased water-wetness in COBR systems, a LS-induced increase inoil-wetness has also been reported on several occasions [20, 40]. In our previous study withuniformly-wetted networks, these contrasting reports motivated us to explore a range ofpossible scenarios by assuming that LS brine could lead to wettability modification towardseither increased water-wetness or increased oil-wetness. However, given the additional com-plexity introduced by using networks with non-uniform initial wettability, in the currentstudy we shall focus predominantly on cases where LS injection causes initially WW poresto become more WW and initially OW pores to become less OW. The implications for otherpossible scenarios, including those where LS brine causes pores to change their wetting class(i.e. OW to WW or vice versa), will be summarised in Section IV. For the various scenarios,we will also consider how the LSE is likely to vary according to the wettability classificationof the underlying pore network (i.e. FW, MWL or MWS).In Watson et al. [37], we adopted a LS injection methodology that lay somewhere be-tween secondary and tertiary in that we captured elements of both approaches by assumingthe injection of HS brine until water breakthrough and the injection of LS brine thereafter.17hile we return to this protocol for the initial part of the current study, our findings suggestthat a more conventional (and experimentally-relevant) secondary LS injection methodologyis required to provide additional important insights into the mechanisms of the LSE in net-works of non-uniform wettability. Hence, the final results section is dedicated to secondarysimulations, with LS brine introduced into a HS-bearing network at the very outset of thewaterflood. Regardless of the particular scheduling of brine injection, for each parametercombination investigated in this study, two separate simulations are performed: one pro-ducing a standard steady-state displacement with HS brine injection only, and the otherinvolving LS brine injection with associated wettability modification. By generating bothsets of results, the impact of any alteration to pore filling sequence by LS flooding can bereadily assessed.
A. Post-Breakthrough Low Salinity Waterflooding
For consistency with the approach of Watson et al. [37], we begin our investigation ofheterogeneously-wetted networks by assuming that HS brine is injected until water break-through and LS brine is injected thereafter. Recall that we use a tracer algorithm to simulatemixing of this LS water with pre-existing HS water in the network. This leads to a spatio-temporal evolution of salinity within the aqueous phase and contact angle modification inneighbouring oil-bearing pores that satisfy the condition given in equation (2). Where ap-plicable, we have used base case parameters that are identical to those used in Watson et al. [37]. Thus, unless otherwise stated, the following properties have been utilised in thissection:– 30 nodes ×
25 nodes ×
25 nodes network (with an average pore length of 333 µ m);– uniform PSD with minimum radius R min = 1 µ m and maximum radius R max = 50 µ m(note that we deliberately choose a uniform distribution for ease of analysis);– pores are perfectly cylindrical: volume scaling V ( R ) ∝ R and conductance scaling g ( R ) ∝ R ;– oil-wet pore fraction α = 0 . Z = 5;18 initial water saturation S wi = 0;– critical tracer concentration for localised wettability change C ∗ = 0 . θ = 20 ◦ (note that this value is consistent with theaverage 16 ◦ shift reported by Khishvand et al. [32]).We begin by performing waterflood simulations on a MWL network that comprises50% moderately WW pores and 50% strongly OW pores — the initial contact anglesfor WW and OW pores are assumed to be θ HS,W W = 60 ◦ and θ HS,OW = 140 ◦ , respec-tively. Dynamic wettability modification by LS injection, which is achieved when the lo-cal tracer concentration exceeds the critical value C ∗ , therefore leads to pores of eitherstronger water-wetness ( θ LS,W W = θ HS,W W − ∆ θ = 40 ◦ ) or more moderate oil-wetness( θ LS,OW = θ HS,OW − ∆ θ = 120 ◦ ). Note that, while it would be entirely feasible to considera range of different values for θ HS,W W , θ HS,OW and ∆ θ across the pore network, this wouldonly serve to obfuscate the behaviours that we are seeking to understand. The P c curvesobtained from the HS and LS simulations are presented in Figure 4 (recall that HS brineis injected until breakthrough in the latter case, so the two displacements are identical upto that point). From these curves, a number of interesting observations can immediatelybe made, including: (i) during the respective LS imbibition and drainage cycles, the ma-jority of the oil is recovered at a higher P c than previously possible; (ii) the shape of theLS P c curve during the imbibition leg of the waterflood (oil displacement from WW pores)is quite different to that seen with HS injection and features a period of recovery at fixed P c ; (iii) the HS and LS P c curves during drainage (oil displacement from OW pores) appearto differ only by a uniform shift in the P c ; and (iv) LS brine injection fails to improve theoverall oil recovery at the end-point P c . To fully understand the mechanisms that underliethese results, a thorough investigation of the respective HS and LS pore filling sequences isrequired.Fluid occupancies in pores of different sizes at various time points during the HS and LSsimulations are presented in Figure 5, where pores are characterised as being either water-filled (blue), oil-filled with unmodified contact angle (dark green), oil-filled with modifiedcontact angle (light green) or oil-filled and trapped (red). Recall that in a MWL system, thesmallest pores are deemed to be WW (since R min = 1 µ m and R max = 50 µ m, the switch from19 P c (Pa) Water Saturation LSHS
FIG. 4. Simulated P c curves for HS brine injection (solid line) and post-breakthrough LS brineinjection (dashed line) into a MWL network with 50% WW pores ( θ HS,W W = 60 ◦ ) and 50% OWpores ( θ HS,OW = 140 ◦ ). WW to OW occurs at a pore radius of 25 . µ m). During HS water imbibition, WW poreswill generally be filled in order from smallest to largest according to the displacement rulesdefined in Section II B; however, in the case where LS injection leads to dynamic contactangle modification, this pore filling sequence can be drastically altered by the associatedchanges in capillary entry pressures. This phenomenon can be clearly seen by comparingFigures 5a and 5b, which show fluid occupancies shortly after breakthrough in the respectiveHS and LS simulations. While in the HS case only pores of radius up to approximately17 µ m have been displaced, LS injection has recovered oil from WW pores of all sizes. Thisphenomenon, whereby the accessible WW pore radius is increased by reducing pore contactangles, is familiar from our previous work [37], and has been shown to be governed by thefollowing equation: R LS,W W max = cos ( θ LS,W W )cos ( θ HS,W W ) R HS,W W max , (4)where 0 (cid:54) θ LS,W W < θ
HS,W W < π/
2. For a fixed P c value, R LS,W W max and R HS,W W max denote the radii of the largest WW pores that can be displaced in the presence and absenceof contact angle modification, respectively. The ratio of the cosines is approximately 1.53in this case, so R LS,W W max ≈ µ m and it becomes clear why WW pores of all possiblesizes have been displaced. This simple analysis reveals the reason why much of the oilcan be displaced at a higher P c than previously possible, and also explains the period ofrecovery at fixed P c that is observed during LS brine imbibition. Since all WW pores thatundergo contact angle modification immediately satisfy their capillary entry condition, the20radual ingress of LS water after breakthrough leads to self-reinforcement of the imbibitiondisplacement. Hence, without the requirement for any further reduction in P c , a sizeablefraction of modified WW pores of all sizes is invaded (as well as a fraction of unmodifiedWW pores with radii up to 17 µ m). Note that while some OW pores will also have theircontact angles reduced during the imbibition cycle, they remain inaccessible to the invadingbrine until the P c becomes sufficiently negative.Following this period of equilibration in the LS simulation, further reduction in the P c is required to displace the remaining WW pores. A plot of the pore size fluid occupancytowards the end of the imbibition cycle is given in Figure 5d, with Figure 5c showing the HSsimulation results at an equivalent stage. These figures indicate that, despite the significantlydifferent pore filling sequences up to this point, the simulations have converged towards thesame outcome — that is, the majority of the WW pores in the network have been displaced.This highlights the important point that in heterogeneously-wetted networks of this type— where LS injection does not alter the overall proportion of WW and OW pores — thepotential for additional oil recovery from the WW pores is likely to be very limited. Hence,any significant increment in oil production following LS injection will have to be obtainedfrom the OW pores. Given that displacement of oil from OW pores by HS brine injectiongenerally occurs in a volumetrically favourable largest to smallest sequence, this implies thatany possible improvement in oil recovery by LS-induced wettability modification could onlybe achieved by increasing the fraction of OW pores that are displaced. Such an outcomewould require a reduction in the topological trapping of oil within the OW pores of thenetwork by a mechanism that we have termed the “microscopic sweep efficiency effect” [37].Taking these factors into consideration, the question that remains is: why was the micro-scopic sweep efficiency of the OW pores not improved in this case? The answer lies in closerinspection of Figure 5d, which reveals widespread contact angle modification in the OWpores prior to the commencement of drainage in the LS simulation. Although these poreshave become less strongly OW than those in the HS simulation (and are therefore displacedat a less negative P c ), in both the HS and LS drainage cycles the water is effectively invadinga uniformly-wetted OW cluster and, consequently, the pore filling sequences are more-or-lessidentical (Figures 5e and 5f). Hence, an increase in microscopic sweep efficiency requiresthe capillary entry pressures of a sizeable fraction of the OW pores to be increased during the drainage process. As remarked previously, this could potentially lead to immediate dis-21 No. ofPores R (µm)
Oil (Trap)Oil (HS)Oil (LS)Water (a)
No. ofPores R (µm)
Oil (Trap)Oil (HS)Oil (LS)Water (b)
No. ofPores R (µm)
Oil (Trap)Oil (HS)Oil (LS)Water (c)
No. ofPores R (µm)
Oil (Trap)Oil (HS)Oil (LS)Water (d)
No. ofPores R (µm)
Oil (Trap)Oil (HS)Oil (LS)Water (e)
No. ofPores R (µm)
Oil (Trap)Oil (HS)Oil (LS)Water (f)
FIG. 5. Pore size fluid occupancy plots that correspond to the (a, c, e) HS brine injection and (b,d, f) LS brine injection P c curves presented in Figure 4. Plots show the fluid occupancy of poresarranged by capillary entry radius (a, b) at the first displacement step after water breakthrough,(c, d) at the time at which approximately 45% of the pores contain water, and (e, f) at the timeat which approximately 55% of the pores contain water. The occupancies of pores are denoted bycolours as follows: blue water; dark green oil (unmodified contact angle); light green oil (modifiedcontact angle); red oil (trapped). placement of these pores and a concurrent reduction in topological oil trapping as the LSfront progresses across the network.In our previous study of LS injection in uniformly wetted networks, we reported casessimilar to that described above where, once again, no additional oil was recovered followingdynamic contact angle reduction. In those cases, it was observed that incremental oil could R max /R min ), increasing the extent of contact angle modification and/or shifting the initialnetwork wettability towards a more neutral state. However, given the sequencing of pore-level displacements outlined above for the failure of LS brine injection to improve oil recovery,it is clear that none of these parameter modifications would significantly alter the outcomesreported for this particular scenario. Furthermore, it should be emphasised that the aboveresults are not unique to the choice of a MWL system, and equivalent simulations for FWand MWS networks also demonstrate little or no additional oil recovery (results not shown).This is despite a significant increase in the HS brine saturation in the network in both theFW and MWS simulations prior to LS brine injection (7.0% for FW and 12.8% for MWSvs. 1.1% for MWL). This suggests that by injecting LS brine only after HS brine has brokenthrough, the HS brine in the network has limited capacity to suppress subsequent LS-inducedwettability modification.One method that could be used to improve the overall oil recovery by LS brine injectionin the above cases would be to introduce the LS brine at a later stage of the flood (i.e. during the drainage phase). This approach would prevent the widespread modification ofcontact angles in the OW pores at an early stage of the displacement, and potentially allowfor additional oil displacement by improving the microscopic sweep efficiency in the OWpores. We have performed simulations that support this theory (results not shown), but,in practice, the window of time available for such an approach to be successful may benarrow and, moreover, difficult to predict in an experimental setting. Indeed, other thaninvestigations of tertiary LS brine injection following HS brine injection to irreducible oilsaturation, we are not aware of any existing experimental studies that have attempted suchan approach.In Watson et al. [37], our in silico LS waterflooding approach demonstrated clear potentialfor a strong LSE in uniformly wetted pore networks. In contrast, the above investigationsuggests that the potential for a LSE in non-uniformly wetted networks may be relativelylimited. It is tempting to conclude that uniformly wetted networks may generally be bettercandidates for LS injection than non-uniformly wetted networks, but such a conclusion doesnot appear to be supported by the available experimental evidence. The coreflooding studyof Ashraf et al. [14], for example, demonstrated significantly elevated oil recovery followingsecondary LS injection across a range of different initial core wettabilities. Furthermore, in a23omprehensive review of the experimental LS coreflooding literature, Hamon [45] noted thatsecondary LS injection produces more oil than secondary HS injection in the overwhelmingmajority of published studies. This evidence suggests that improved oil recovery by LSbrine injection should be possible across the full spectrum of initial wettability states andindicates that our in silico approach needs to be adapted to simulate a truly secondary LSinjection protocol. These modifications are discussed next.
B. Secondary Low Salinity Waterflooding
In the simulations performed in Section III A, we assumed zero initial HS brine saturation( S wi = 0) in our networks and injected HS brine until water breakthrough. This establisheda HS brine saturation in the network prior to the injection of LS brine. For secondary LSinjection, however, HS brine is already present as connate water before the oil displacementcommences. To create an initial HS brine saturation in our networks, we perform a steady-state primary drainage simulation that involves the injection of oil into a 100% WW and100% water-filled network until the desired S wi is achieved. Note that this process follows thedrainage methodology detailed in Section II B, except that oil is now the invading phase.The initial HS brine will therefore be distributed uniformly throughout the network andreside predominantly in the smallest pores. We assign a non-uniform wettability state tothe network only after the final S wi has been established (note that the pores that containinitial HS water are assumed to remain WW, even if we impose a FW or MWS wettabilityclass).Results from our earlier simulations of post-breakthrough LS injection suggest that incre-mental oil production will only be observed from non-uniformly wetted networks if contactangle modification can be sufficiently delayed to allow for a more efficient displacement ofthe OW pores. One candidate for the source of such a delay in secondary LS injectionsimulations could be the resident HS brine that will mix with the injected LS brine andinhibit localised changes in wettability. For this mechanism to be effective, it is importantthat the network is initialised with a realistic initial water saturation S wi . To achieve this inour simplified networks, we narrow the width of the PSD and modify the volumetric scalingproperties of the pores (through the volumetric exponent ν ). Additionally, we reduce theco-ordination number of the network ¯ Z to better reflect measured rock data — this change24lso means that a significant fraction of pores can be filled with initial HS brine withoutapproaching the percolation threshold of the network too closely (for 3D cubic systems, thepercolation threshold is given by
32 ¯ Z ). If the fraction of initial HS water-filled pores wereto be close to the percolation threshold, then breakthrough would occur shortly after thecommencement of LS injection and the simulation protocol would, in effect, be the same asthat used in Section III A. We therefore make the following specific amendments to our basecase parameter values in order to better mimic a more realistic pore system without havingto incorporate additional layers of complexity associated with irregular network topologiesand pore geometries at this stage (all other parameter values remain unchanged):– uniform PSD with minimum radius R min = 1 µ m and maximum radius R max = 16 µ m;– we assume V ( R ) ∝ R . to reflect a more physical scaling relationship for pore satu-ration values (note that we adopt a constant pre-factor to keep the total pore volumeof the network consistent with the case where ν = 2);– ¯ Z = 3 . Z = 5);– S wi = 0 .
12 (corresponding to HS brine in approximately 20% of the pores).Note that, in comparison to the networks used in Section III A, all of the above modificationsbring the features of the derived in silico pore networks closer to those associated with realrock structures utilised in experimental coreflooding studies. We reiterate that we are notstriving for quantitative prediction at this stage of the modelling: we are simply aimingto provide some insight into the underlying displacement sequences operating during LSfloods. More intricate pore systems could be examined but inclusion at this juncture wouldcomplicate the interpretation of results without adding significant benefit.For consistency with the simulations in Section III A, we focus on networks with theMWL wettability class (although note, again, that this is of little consequence to the quali-tative outcomes of the study). The base case simulation again begins with 50% WW pores( θ HS,W W = 60 ◦ ) and 50% OW pores ( θ HS,OW = 140 ◦ ), and ingress of LS brine again causesdynamic, localised reduction of contact angles (∆ θ = 20 ◦ ). The P c curves from the HS andsecondary LS brine injection simulations are presented in Figure 6, and it is immediatelyapparent that the qualitative features of the results are quite different to those observed25 P c (Pa) Water Saturation LSHS
FIG. 6. Simulated P c curves for HS brine injection (solid line) and secondary LS brine injection(dashed line) into a MWL network with 50% WW pores ( θ HS,W W = 60 ◦ ) and 50% OW pores( θ HS,OW = 140 ◦ ). previously (c.f. Figure 4). In contrast to the case where LS was introduced after break-through, here we see a limited response to LS injection throughout the imbibition phaseand, moreover, an overall increase of approximately 5% in the total volume of oil produced.This is a surprising result — our earlier (post-breakthrough LS) simulations suggested that delayed LS brine injection would be necessary to improve oil recovery but here (in secondaryLS mode) additional oil has been obtained by injecting LS brine at an earlier stage.An examination of the pore filling sequences during the HS and LS displacements againallows us to understand the mechanisms responsible for this result. Pore size fluid occupan-cies at various stages of the HS and LS simulations are presented in Figure 7. The plots inFigure 7a (HS) and Figure 7b (LS) are taken at an early stage of the respective displacementsand highlight the fact that secondary LS brine injection causes the pore filling sequencesto diverge from the very outset. At this stage, some of the largest
WW pores (which areproximal to the network inlet) have already been invaded by LS due to the reduced connec-tivity of the pore space. Recall, however, that the pores displaced over the entire imbibitioncycle should be effectively identical for both HS and LS brine injection. Figure 7c (HS)and Figure 7d (LS), which are plotted at the end of the respective imbibition cycles, provethis to be the case. The more important issue at this stage is the extent of contact anglemodification in the OW pores, and a comparison of Figures 5d and 7d reveals that, in thecase of secondary injection, significantly more OW pores remain unmodified by LS brineas drainage begins. At this stage of the secondary LS injection simulation, breakthrough26
No. ofPores R (µm)
Oil (Trap)Oil (HS)Oil (LS)Water (a)
No. ofPores R (µm)
Oil (Trap)Oil (HS)Oil (LS)Water (b)
No. ofPores R (µm)
Oil (Trap)Oil (HS)Oil (LS)Water (c)
No. ofPores R (µm)
Oil (Trap)Oil (HS)Oil (LS)Water (d)
No. ofPores R (µm)
Oil (Trap)Oil (HS)Oil (LS)Water (e)
No. ofPores R (µm)
Oil (Trap)Oil (HS)Oil (LS)Water (f)
FIG. 7. Pore size fluid occupancy plots that correspond to the (a, c, e) HS brine injection and (b,d, f) LS brine injection P c curves presented in Figure 6. Plots show the fluid occupancy of poresarranged by capillary entry radius (a, b) at the first displacement step, (c, d) at the end of theimbibition cycle, and (e, f) at the time at which approximately 50% of the pores contain water.The occupancies of pores are denoted by colours as follows: blue water; dark green oil (unmodifiedcontact angle); light green oil (modified contact angle); red oil (trapped). has already occurred and a reasonable amount of LS water (0.33 pore volumes) has beeninjected into the system. Hence, the lack of contact angle modification can only be explainedby the ongoing influence of the connate HS brine in the system coupled with the reducedconnectivity.As highlighted previously, a delay in LS-induced contact angle modification in the OWpores of a non-uniformly wetted network can be beneficial for oil recovery because it al-27 Water-FilledPores R ( (cid:541) m) LSHS (a)
Water-FilledPores Nodes from Inlet Face
LSHS (b)
FIG. 8. Plots of (a) the size distribution and (b) the spatial distribution of water-filled pores atresidual oil saturation for simulations of HS brine injection (solid line) and secondary LS brineinjection (dashed line) into a MWL network with 50% WW pores ( θ HS,W W = 60 ◦ ) and 50% OWpores ( θ HS,OW = 140 ◦ ). lows a more efficient displacement of the remaining oil to be achieved. The mechanism ofimproved microscopic sweep efficiency can be seen in Figure 7f, where the drainage cyclein the LS simulation proceeds by filling successively smaller pores with modified contactangles (unmodified pores initially remain inaccessible). Gradual reduction of the P c allowsthis to continue and, as more and more pores are contacted by LS, any modified pore withradius larger than the smallest accessible pore will automatically satisfy its capillary entrythreshold and be immediately invaded. In volumetric terms, this sequence of pore fillingis less favourable than the equivalent HS simulation (Figure 7e); however, the LS-inducedphenomenon of immediate invasion increases the total fraction of pores displaced (62.2% forLS vs. 58.1% for HS) and ultimately increases the overall oil recovery. Figure 8 displaysthe final distributions of water-filled pores for the HS and LS simulations organised by bothpore entry radius (Figure 8a) and by spatial position in the network (Figure 8b). Figure 8ashows that, despite increased trapping of oil pores with radii in the range 14–16 µ m, the LSsimulation has improved overall oil recovery by significantly increasing the displacement ofoil-filled pores with radii in the range 10–14 µ m. Figure 8b, meanwhile, confirms that thisincrease in microscopic sweep efficiency is not a localised effect. Compared to the injectionof HS brine, LS brine injection has increased the displacement of oil-filled pores across theentire width of the network. 28ur results in Section III A suggested that a significant improvement in oil recoveryin non-uniformly wetted networks would require an explicit delay in the injection of LSbrine, but the above simulation indicates that significant incremental oil can be producedeven if LS brine is injected in secondary mode. This may appear to be something of acontradiction but recall that we have made quite different assumptions in these two cases.The distribution of HS brine in the network prior to LS injection is one aspect that haschanged considerably. For post-breakthrough LS injection, the injected HS brine spansthe network and is largely distributed in inlet-connected clusters, whereas for secondaryLS injection the connate HS brine is both non-spanning and widely dispersed throughoutthe pore structure. The secondary LS base case simulation suggests that the influence ofthe connate HS brine plays a critical role in the success of secondary LS injection, and weinvestigate this below by performing sensitivity simulations on the initial HS brine saturation S wi .We have studied the impact of both a decrease ( S wi = 6%; ∼
10% of available pores) andan increase ( S wi = 18%; ∼
30% of available pores) in the initial HS water saturation. P c curves from the HS and LS simulations are compared to those from the base case simulationin Figure 9. We find that a reduction in S wi has a negative impact on the efficacy of oilrecovery by LS brine injection. The profile of the P c curve during the imbibition phaseindicates an accelerated rate of contact angle modification within the network comparedto the LS base case simulation. This limits the scope for an efficient displacement of theOW pores, and ultimately results in a smaller increment in the total volume of oil produced(1.9% vs. 4.9% for the base case LS simulation). An increase in S wi , on the other hand, couldbe expected to significantly delay modification of contact angles by the injected LS brineand ultimately lead to greater incremental oil production than the base case LS simulation.However, this does not prove to be the case and, in fact, the results from the two simulationsare remarkably similar. This appears to be related to the fact that an increase in S wi leads toearlier water breakthrough (after 0.030 pore volumes water injection rather than 0.065 porevolumes water injection in the base case LS simulation): the elevated ingress of LS brineafter breakthrough accelerates the removal of HS brine from the network and leads to a rateof LS-induced contact angle modification that is very similar to the base case simulation.This simulation with S wi = 18% is particularly interesting because it demonstrates thatLS brine injection can displace significant incremental oil from non-uniformly wetted net-29 P c (Pa) Water Saturation LS (Swi = 6%)HS (Swi = 6%)LS (Swi = 12%)HS (Swi = 12%)LS (Swi = 18%)HS (Swi = 18%)
FIG. 9. Simulated P c curves for HS brine injection (solid lines) and secondary LS brine injection(dashed lines) into a MWL network initialised with a range of S wi values. Results for the basecase parameter value ( S wi = 12%; blue lines) are compared to results for both a smaller ( S wi =6%; green lines) and a larger ( S wi = 18%; red lines) value of S wi . The network contains 50% WWpores ( θ HS,W W = 60 ◦ ) and 50% OW pores ( θ HS,OW = 140 ◦ ). works, even when the LS brine is injected close to the point of water breakthrough. Ourearlier simulations in Section III A had suggested that this phenomenon was unlikely, so itappears that the changes we have made — other than modifying the water injection protocol— may also be influential in the positive LSE that we have observed in the above secondaryLS injection simulations. It seems unlikely that narrowing the width of the PSD wouldproduce such dramatic consequences for the efficacy of LS injection. Indeed, according toresults from Watson et al. [37], the potential for an efficient displacement of the OW poresshould increase as the ratio of the largest OW pore radius to the smallest OW pore radiusis reduced. For a change in the PSD from U (1 ,
50) to U (1 ,
16) in a MWL network with α = 0 .
5, this ratio falls only slightly from 1.96 (i.e. 50 µ m / 25 . µ m) to 1.88 (i.e. 16 µ m /8 . µ m). Therefore, we focus instead on the connectivity of the pore network and performa sensitivity simulation on the co-ordination number ¯ Z . Figure 10 compares the base caseHS and LS P c curves with those from a simulation with increased network connectivity( ¯ Z = 4 . Z = 4 . Z = 3 . Z = 3 . Z = 4 . P c (Pa) Water Saturation LS (Z = 3.5)HS (Z = 3.5)LS (Z = 4.5)HS (Z = 4.5)
FIG. 10. Simulated P c curves for HS brine injection (solid lines) and secondary LS brine injection(dashed lines) into MWL networks with different ¯ Z values. Results for the base case parametervalue ( ¯ Z = 3 .
5; green lines) are compared to results for ¯ Z = 4 . θ HS,W W = 60 ◦ ) and 50% OW pores ( θ HS,OW = 140 ◦ ). to a combination of delayed water breakthrough and increased residence time of HS wateras it navigates circuitous pathways through the pore structure towards the outlet. Networkconnectivity can therefore have a crucial influence on the LSE in non-uniformly wetted net-works, and the lack of incremental oil recovery in our post-breakthrough LS brine injectionsimulations can be largely explained by the well-connected network that was utilised for thestudy. IV. DISCUSSION
The mechanisms and efficacy of LS waterflooding have been topics of much recent debatein the oil industry. While the LSE is yet to be fully understood, a consensus has emergedthat injected LS brine operates by altering the wettability of its surrounding rock structure.We have recently developed novel pore-scale models to investigate the dynamic LSE inuniformly wetted pore networks [37, 38], and we have extended our methodology here tostudy networks of non-uniform initial wettability. Our results indicate that the conditionsrequired for a positive LSE in non-uniformly wetted networks are typically more stringentthan those for uniformly wetted networks and, moreover, we have identified several potentialreasons for the inconsistent performance that is observed in experimental LS waterfloodingstudies.Following Watson et al. [37], we have simulated LS waterflooding by coupling a steady-31tate oil displacement model to a tracer algorithm that estimates the spatio-temporal evo-lution of water salinity as HS and LS brines mix in the in silico pore networks. We havechosen not to consider explicit chemical reactions in the COBR system and we have insteadcaptured the apparent consequences of LS brine injection by assuming that brine salinitiesbeneath a critical threshold induce localised wettability modification. Changing the contactangles of individual pores in this way influences the sequence of pore filling by the invadingLS brine and, under certain circumstances, increases the total volume of oil produced duringthe displacement.While the steady-state LS waterflooding model that we have reported here is less sophisti-cated than the unsteady-state model of Boujelben et al. [38], the steady-state approach doeshave certain benefits. In particular, the steady-state model can simulate the displacementof fluids from large 3D pore networks at substantially lower computational cost than theunsteady-state model — this is significant because a meaningful investigation of the LSEin regular non-uniformly wetted pore networks can only be achieved using 3D simulations.For a regular pore network to contain spanning clusters of both WW and OW pores, thepercolation threshold of the network must be (cid:54) (cid:62) α = 0 . Z = 4)and the WW and OW pores are randomly distributed (irregular, poorly-connected 2D net-works would be even more compromised). Moreover, were this restrictive case to be used toinvestigate LS waterflooding, model simulations would provide little or no insight into themechanisms of the LSE. Indeed, for any LS waterflooding simulation where individual poresmaintain their initial wetting class (i.e. WW or OW), oil from most of the 50% WW poreswould be displaced during the imbibition cycle. Then, at the end of the imbibition cycle,both the fraction of water-filled pores and the fraction of oil-filled pores would be close tothe network percolation threshold (50%), meaning that most of the remaining oil would beisolated from the network outlet and trapped inside the OW pores. Hence, only oil from theWW pores could ever be displaced and there would be no scope for LS brine to improve oilrecovery by increasing microscopic sweep efficiency in the OW pore fraction.For consistency with our earlier work in uniformly wetted media, we began this study bysimulating an early tertiary LS waterflooding protocol where LS brine was introduced into32he network after breakthrough of injected HS brine. We initialised a network with an equalbalance of WW (60 ◦ ) and OW (140 ◦ ) pores and assumed a contact angle reduction of 20 ◦ inall oil-filled pores that were contacted by water of sufficient freshness (note the implicationthat no OW pores could become WW over the course of the simulation). The assumed extentof wettability modification by LS brine is consistent with the recent experimental findings ofKhishvand et al. [32], who observed an average LS-induced contact angle reduction of around16 ◦ in miniature sandstone core samples. While this simulation ultimately produced thesame volume of oil as the equivalent case with only HS brine injection, the results did providevaluable insight into the necessary conditions for improved recovery by LS brine injectionin non-uniformly wetted networks of this type. The simulation demonstrated that anyaction of the LS brine on the oil-filled WW pores is essentially superfluous, since WW poreswill always be displaced by imbibition regardless of their contact angle or the compositionof the injected brine. Correspondingly, the OW pores contain the only viable source ofadditional oil and, due to the volumetrically-favourable nature of the standard OW porefilling sequence, any increase in the overall oil recovery by LS brine injection can only beachieved by improving the microscopic sweep efficiency in the OW pores.The above requirement for improved microscopic sweep efficiency in the OW pores isparticularly stringent, and it has several implications for oil recovery by LS brine injectionin non-uniformly wetted networks. Firstly, increased microscopic sweep efficiency is a phe-nomenon that is associated only with dynamic, LS-induced contact angle reduction [37].Hence, if LS brine injection were to lead to an increase in oil-wetness — a phenomenonthat has been reported on several occasions [21, 39] — there would be no potential for anassociated improvement in oil recovery in a non-uniformly wetted network. Note that werefer here only to cases where the modified WW pores remain WW; cases where contactangles cross 90 ◦ are less clear-cut and will be discussed in detail below. Secondly, increasedmicroscopic sweep efficiency in the OW pores requires a sizeable fraction of OW contactangles to be reduced during the drainage cycle . Hence, parameters that contribute to thedelay of widespread contact angle reduction by injected LS brine in non-uniformly wettednetworks are crucial to the production of incremental oil. Note that we did not encounterthis type of restriction in our earlier studies of LS brine injection in uniformly wetted net-works, where parameters such as the PSD, the initial network wettability and the extent ofcontact angle modification were identified as the most critical parameters in determining the33agnitude of any positive LSE. For non-uniformly wetted networks, these parameters canstill have an important influence on the extent of any positive LSE. However, it is only thePSD and the contact angle properties of the OW pore fraction that are significant in thisregard, and even then only under the condition that wettability modification by injected LSbrine can be sufficiently delayed.The observation that the potential for improved oil recovery by LS injection requiresa stricter set of conditions for non-uniformly wetted networks than for uniformly wettednetworks prompted us to adopt a more experimentally-relevant LS brine injection protocoland to focus on pore networks with more physically realistic features. Consequently, inSection III B, we adjusted the properties of the pore space (i.e. PSD, ¯ Z , ν ) to ensure thata realistic HS S wi could be initialised in the network and we simulated secondary LS brineinjection rather than a post-breakthrough LS brine injection approach. These changes to ourmethodology and parameters allowed us to perform a more thorough investigation of the LSEin non-uniformly wetted pore networks. For parameter values that were otherwise identicalto the earlier post-breakthrough LS brine injection simulation, we found that secondary LSbrine injection led to a marked increase in oil recovery compared to HS brine injection alone.Although LS brine was injected earlier , the modification of contact angles in many OW poreswas delayed and a more efficient displacement of the OW pore space was ultimately achieved.This result highlights that the parameters that we modified — most notably S wi and ¯ Z —are crucial factors that can control the extent of any observed LSE in non-uniformly wettednetworks.We explored the impact of HS S wi on the efficacy of secondary LS brine injection byperforming sensitivity simulations for S wi values that were both smaller and larger thanthe base case value. For the non-uniformly wetted network that we considered, our resultsindicated an interesting nonlinear relationship between S wi and the volume of incrementaloil recovered by LS brine injection. When the initial fraction of HS brine-filled pores was farfrom the network percolation threshold, the additional oil recovered by LS brine injectionincreased with increasing S wi . This result reflects the fact that increasing the connate HSbrine saturation increases the delay in LS-induced wettability modification and thereforeimproves the microscopic sweep efficiency during the drainage cycle. Interestingly, the trendof increasing oil recovery by secondary LS brine injection with increasing HS S wi is con-sistent with experimental results reported by Zhang and Morrow [18] for waterflooding of34erea sandstone cores with permeabilities of 500 mD and 1100 mD. As more pores in ournetworks were filled with initial water and the fraction of HS brine-filled pores approachedthe percolation threshold, the trend for increasing incremental recovery with increasing S wi saturated and our results showed a marginal decline in the efficacy of LS brine injection.This result appears to reflect the fact that early water breakthrough — and the associatedegress of HS brine from the network — can accelerate the localised modification of wettabil-ity by injected LS brine and consequently reduce the potential for an efficient displacementof the OW pore fraction.We also investigated the impact of the network co-ordination number ¯ Z on the efficacy ofsecondary LS brine injection, and our simulations showed an increase in the relative volumeof incremental oil recovered by LS brine injection as ¯ Z was reduced. For the non-uniformlywetted networks that we considered here, this result reflects the fact that a reduction in thepore space connectivity increases the residence time of connate HS brine in the network anddelays the widespread modification of wettability until the drainage stage of the waterflood.The trend of increasing efficacy of secondary LS brine injection with decreasing ¯ Z is notnecessarily restricted to non-uniformly wetted networks, however, as unpublished simulationresults indicate a similar result for uniformly wetted pore networks. This is because oiltrapping is prevalent for poorly connected networks of all initial wettability states and,hence, the potential for LS brine injection to improve microscopic sweep efficiency increasesas ¯ Z is reduced. It would be interesting to test this theory against experimental corefloodingobservations, but the average pore space connectivity is a parameter that is not typicallyconsidered in laboratory studies. The permeability of the core is generally reported, butneither the network connectivity nor the network PSD can be explicitly determined fromthis measurement. Our simulation results indicate that both ¯ Z and the distribution of poresizes can influence the LSE by distinct pore-scale mechanisms and an ideal experimentalanalysis should therefore aim to control for these factors independently.A potentially important parameter that we have yet to explicitly consider in this study isthe fraction α of OW pores in our in silico networks. Given that any incremental oil recoveredby LS brine injection into non-uniformly wetted networks should be mostly recovered fromthe OW pore fraction, the value of α seems likely to be a key factor in determining theextent of any LSE (note that we refer only to values of α that give spanning clusters ofboth OW and WW pores; c.f. Figure 3). Based on the results reported in Section III, our35imulations would predict that the potential for improved oil recovery by LS brine injectionshould increase with increasing α (i.e. for networks with larger fractions of OW pores).For networks with large α values, the imbibition phase is shorter, and more wettabilitymodification should therefore take place during the drainage phase. This should enhancethe prospect of improved microscopic sweep efficiency in the OW pores and could potentiallylead to an overall increase in oil recovery. Networks with small α values, on the other hand,would have less potential for improved microscopic sweep efficiency because a larger fractionof OW pores would be likely to undergo wettability modification before the drainage phaseis reached. Note that, for these theoretical considerations, we are again assuming thatthe LS-induced contact angle reduction does not cause the OW pores to become WW.The results presented in Section III were all generated using networks with identical OWand WW pore fractions. Therefore, to test the theory outlined above, we have performedadditional simulations (not shown) for networks with different α values (note that thesenetworks still have spanning OW and WW clusters, and that all other parameters remain attheir Section III B base case values). For α = 0 . α = 0 . α = 0 . α , which is entirely consistent with ourtheoretical underpinnings.We shall now briefly consider cases where LS brine injection causes the contact anglesof the WW or OW pores in a non-uniformly wetted network to cross 90 ◦ (such that thenetwork becomes uniformly wetted over the course of the displacement). We emphasisefrom the outset of this section that, while we do not report any explicit numerical results,the theoretical arguments that we present are all based on observations from an extensivemodel sensitivity analysis. Model simulations indicate that, when pores can dynamicallychange their wetting class , the sequence of pore filling (and extent of any LSE) in a givensimulation is strongly dependent upon the particular distribution of initial and modifiedpore contact angles (amongst other factors). Therefore, to present results from just one ortwo simulations would be misleading about the wider qualitative trends that we observe,while to present results from a full suite of simulations would be disproportionate. For36he scenarios that we have discussed to this point, the wettability class of the network(MWS, FW or MWL) has had no real consequence for the qualitative outcome of LS brineinjection. The situation is rather different, however, when LS brine causes a subset of poresto change their wetting preference. To fully appreciate the mechanisms by which LS brineinjection may or may not improve oil recovery under this assumption, it is instructive tofirst consider the qualitative outcome of standard HS brine injection in networks of eachwettability class. Assuming, for illustrative purposes, that α = 0 .
5, Figures 11a to 11c showschematic diagrams of the groups of pores that we would expect to be displaced by HS brineinjection for MWS, FW and MWL networks, respectively. The oil-filled pores that would betrapped during the flood are shown in black and, in each case, represent the smallest
OWpores in the network (i.e. those with the most negative capillary entry pressures).Consider first the scenario where LS brine injection dynamically reduces pore contactangles and causes the initially OW pores in the network to become WW (as has beendirectly observed in the experimental study of Khishvand et al. [32]). For these initiallyOW pores, the largest-to-smallest drainage filling sequence would no longer occur, and thepores would instead be imbibed in a smallest-to-largest filling sequence. Consequently, the largest of these oil-filled pores (rather than the smallest) may ultimately become trapped.Hence, it appears that there would be no explicit volumetric benefit to this process and anyimprovement in oil recovery by LS brine injection would likely require an overall increasein microscopic sweep efficiency. High microscopic sweep efficiency could be guaranteed inthis case by delaying the injection of LS brine until the drainage phase has begun (i.e.once P c < P c (and the P c remained positive thereafter due to subsequent contact angle modification),the chances of an improvement in microscopic sweep efficiency would be strongly dependentupon the features of the pore space.To improve the microscopic sweep efficiency by LS brine injection in this case, it isessential that a subset of the newly WW pores can realise capillary entry pressures that arelarger than the prevailing P c and thereby undergo immediate imbibition. It is complex topredict the frequency with which this may occur for a given in silico network (depending asit does upon the timing of the changes in wetting preference, the PSD of the network and37 (cid:3040)(cid:3028)(cid:3051) (cid:1844) (cid:3040)(cid:3036)(cid:3041) WWOW (a) (cid:1844) (cid:3040)(cid:3028)(cid:3051) (cid:1844) (cid:3040)(cid:3036)(cid:3041)
WWOW (b) (cid:1844) (cid:3040)(cid:3028)(cid:3051) (cid:1844) (cid:3040)(cid:3036)(cid:3041)
WW OW (c) (cid:1844) (cid:3040)(cid:3028)(cid:3051) (cid:1844) (cid:3040)(cid:3036)(cid:3041)
WWWW* (d) (cid:1844) (cid:3040)(cid:3028)(cid:3051) (cid:1844) (cid:3040)(cid:3036)(cid:3041)
WWWW* (e) (cid:1844) (cid:3040)(cid:3028)(cid:3051) (cid:1844) (cid:3040)(cid:3036)(cid:3041)
WW WW* (f) (cid:1844) (cid:3040)(cid:3028)(cid:3051) (cid:1844) (cid:3040)(cid:3036)(cid:3041)
OW WW OW* (g) (cid:1844) (cid:3040)(cid:3028)(cid:3051) (cid:1844) (cid:3040)(cid:3036)(cid:3041)
OW OW*WW (h) (cid:1844) (cid:3040)(cid:3028)(cid:3051) (cid:1844) (cid:3040)(cid:3036)(cid:3041)
OWWW OW* (i)
FIG. 11. Schematic plots showing idealised depictions of potential outcomes from representativesimulations of (a-c) HS brine injection, (d-f) LS brine injection where OW pores can become weaklyWW (here denoted WW ∗ ) and (g-i) LS brine injection where WW pores can become weakly OW(here denoted OW ∗ ). Plots are based on in silico observations of pore-filling sequences for (a,d, g) MWS, (b, e, h) FW and (c, f, i) MWL networks initialised with 50% WW pores and 50%OW pores. The blacked-out area on each plot denotes the subset of oil-filled pores that could beexpected to be trapped in each case. See the main text for a detailed explanation of the featuresof the various plots. the instantaneous distributions of the initially WW and newly WW pore contact angles),but it is possible to broadly categorise the different wettability classes in terms of theirlikely potential for improved microscopic sweep efficiency by LS brine injection. In a MWLnetwork, for example, any newly WW pore would be larger than all initially WW pores.Therefore, to undergo immediate imbibition, a newly WW pore would have to become moreWW than at least some of the initially WW pores. In contrast, for a MWS network, any38ewly WW pore would be smaller than all initially WW pores. Therefore, the immediateimbibition of a newly WW pore in a MWS network could potentially occur even if it was lessWW than some of the initially WW pores. This condition seems far less stringent than thecorresponding condition for MWL networks, and we therefore envisage that MWS networkswould have greater potential for improved microscopic sweep efficiency than MWL networksin this case. Since FW networks initially contain WW pores and OW pores of all sizes,we also envisage that the potential for improved microscopic sweep efficiency by LS brineinjection in FW networks would lie somewhere between that for MWS and MWL networksin this case. Based on these theoretical considerations, Figures 11d to 11f show schematicdiagrams of hypothetical outcomes from LS brine injection into idealised MWS, FW andMWL networks (note that WW ∗ denotes newly WW pores). The MWL diagram assumes noincrease in microscopic sweep efficiency, the FW diagram assumes a small increase, and theMWS case assumes a slightly larger increase. Note that the trapped oil pores all have largerradii than those in the equivalent HS cases (Figures 11a to 11c). Hence, in this scenario(where some OW pores can become WW following exposure to LS brine), even an increasein microscopic sweep efficiency would provide no guarantee of an overall improvement in oilproduction.Finally, we consider the scenario where the contact angles in a non-uniformly wettednetwork are dynamically increased by invading LS brine and the initially WW pores changetheir wetting preference to become OW. Note that LS brine injection cannot improve mi-croscopic sweep efficiency in this case (increased oil trapping is, in fact, more likely) andany improvement in oil recovery requires a pore filling sequence that is more volumetricallyfavourable than the sequence for a standard waterflood (c.f. the “pore sequence effect” ).An immediate consequence of this condition is that MWS networks — the most favourablewettability class in the above scenario — have no potential for improved oil recovery wheninjected LS brine causes WW pores to become OW. As shown by Figure 11a, a standardwaterflood in a MWS network will tend to trap only the very smallest oil-filled pores andthis outcome cannot therefore be bettered by simply altering the pore filling sequence. Ingeneral, for the case that LS brine injection causes WW pores to switch their wetting pref-erence, production of incremental oil requires some of the larger initially OW pores (whichwould otherwise be trapped) to be recovered at the expense of some of the smaller newlyOW pores (which would otherwise be displaced by imbibition). The greater the number39f pores for which this phenomenon occurs, the greater the potential volumetric benefit tooverall oil production. Hence, the optimum approach in this scenario would be to introduceLS brine into the network at the earliest possible stage to maximise the number of smallWW pores that become OW.As for the above scenario of OW pores becoming WW, the efficacy of an approach whereLS brine injection causes WW pores to become OW is difficult to predict a priori . It ispossible to assert, however, that the potential for the displacement of large, initially OWpores at the expense of small, newly OW pores should be greater in MWL networks than insimilar FW networks. Unlike for a FW network, in a MWL network the initially OW poreswill all be larger than any newly OW pore. Given that the OW pores in a network willtend to fill in a largest to smallest sequence, the MWL wettability class therefore appearsto maximise the chances for oil trapping in small, newly OW pores. Figures 11g to 11i showschematic diagrams of hypothetical outcomes from LS brine injection in networks of eachwettability class, where the newly OW pores are denoted OW ∗ (for illustrative purposes,we assume that 50% of the initial WW pores become OW during the imbibition phase).Based on the above theoretical considerations, the MWS case exhibits no overall change tothe filling sequence (Figure 11g), the FW case exhibits a small change (Figure 11h) and theMWL case demonstrates a larger change (Figure 11i). Figures 11h and 11i both suggest thatthe oil recoveries for LS brine injection would exceed those obtained from equivalent HS brinedisplacements (Figures 11b and 11c, respectively). Note, however, that these plots do notaccount for the increase in topological oil trapping that is typically observed in simulationswhere injected LS brine dynamically increases contact angles. Hence, the likely outcome forLS brine injection in a given network is not clear-cut, and both increases and decreases inoverall oil recovery could again be possible in this scenario. V. CONCLUSIONS
It is important to emphasise that we have taken a relatively modest approach to ourinvestigation of the LSE in this study. We have utilised a steady-state fluid displacementmodel in which corner flow is neglected, and where the evolution of brine salinity in the porespace is estimated using a dynamic tracer injection algorithm. The model also neglects anyexplicit pore-scale chemistry and assumes that LS brine injection leads to a direct alteration40f local wettability. However, this pragmatic approach, combined with a focus on idealised in silico pore structures, has made it feasible to identify several factors that may be criticalto the LSE in non-uniformly wetted networks. In line with experimental observations, oursimulations and analysis demonstrate that LS brine injection can have a positive, negativeor neutral effect on overall oil recovery.For non-uniformly wetted networks in which LS brine injection cannot alter the actualwetting class of individual pores (only the strength of the wetting preference), our resultshave indicated that a dynamic, LS-induced reduction in contact angles can significantly in-crease the production of oil. We have shown that the OW pores provide the only viablesource of incremental oil in non-uniformly wetted networks and, consequently, that oil re-covery by LS brine injection is optimised when wettability modification occurs primarilyduring the drainage cycle. Simulations demonstrate that the fraction of OW pores in thenetwork α , the average network connectivity ¯ Z , and the initial HS water saturation S wi canall strongly influence the degree to which LS-induced wettability modification takes placeduring drainage. Hence, we conclude that all three of these parameters can play a criticalrole in determining the extent of any positive LSE in non-uniformly wetted porous media.The results reported in this study demonstrate that the factors that promote incremen-tal oil production by LS brine injection can be distinctly different for uniformly wettedand non-uniformly wetted pore networks. These findings, which can help to clarify theprevailing uncertainty in the experimental literature, further underline the importance ofidentifying the initial and final wettability states when assessing coreflood performance inLS waterflooding studies. Appendix: Dynamic salinity updates in flowing water-filled pores
For each post-breakthrough P c reduction that permits the ingress of LS brine, a periodof convective tracer transport is simulated to determine the new salinity distribution in theflowing water. Tracer of concentration one (i.e. salinity = 0) is introduced at the networkinlet and, at each timestep ∆ t , tracer concentrations C old of flowing water-filled pores areupdated according to their elemental flow rates q and volumes V as follows:(i) calculate the mass of tracer that leaves each pore, M out = q · ∆ t · C old ;41ii) calculate the total flow q node and total mass of tracer M node that enters each node bysumming the q and M out values from immediate upstream pores;(iii) assume perfect mixing in each node and calculate the mass of tracer that enters eachimmediate downstream pore, M in = ( q/q node ) · M node ;(iv) calculate the new tracer concentration in each pore, C new = C old + ( M in − M out ) /V .At each timestep of the above procedure, the mass of tracer that leaves a pore cannot exceedthe initial mass of tracer in the pore (i.e. M out (cid:54) C old · V ). The timestep length is thereforegiven by ∆ t = min ( V /q ), where the minimum is taken over all flowing water-filled pores.
ACKNOWLEDGMENTS
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