2000 - Mixing of Injected, Connate and Aquifer Brines in Water Flooding and Its Relevance to Oilfield Scaling

Ž .Journal of Petroleum Science and Engineering 27 2000 85–106www.elsevier.nlrlocaterjpetscieng

Mixing of injected, connate and aquifer brines in waterfloodingand its relevance to oilfield scaling

K.S. Sorbie), E.J. MackayDepartment of Petroleum Engineering, Heriot-Watt UniÕersity, Research Park, Riccarton, Edinburgh EH14 4AS, UK

Received 22 July 1999; accepted 14 March 2000

Abstract

Waterflooding is one of the most common methods of oil recovery although it does lead to certain production problemsafter water breakthrough, e.g. corrosion, scaling, etc. The issue of concern in this paper is mineral scale formation by brine

Ž .mixing as occurs in barium sulphate barite, BaSO scaling. Barite formation in the production well and tubulars occurs in4Ž . Ž Ž .. Ž .many oilfields when sulphate-rich injection water IW often seawater SW mixes with barium-rich formation water FW

close to or in the wellbore. However, when a brine is injected into the reservoir, it may mix to some extent with theŽ .formation or connate brine deep within the system. Such in situ mixing of barium-rich and sulphate-rich brines would

certainly result in barite deposition deep within the reservoir due to the low solubility and rapid kinetics of this precipitationprocess. Conversely, in order to estimate how much of this type of in situ precipitation might occur in reservoirs, we must beable to model the appropriate displacement processes incorporating the correct level of dispersive brine mixing in thereservoir formation. In this paper, all of the principal mechanisms of brine mixing in waterflood displacements are

Ž . Ž .considered and modelled. Mixing between the IW, the oil leg connate water CW and the aquifer water AQW is analysedŽ . Ž .starting from a one-dimensional 1D frontal displacement, extended Buckley–Leverett BL analysis. This particular

mechanism occurs in all other types of displacement and reservoir mixing process including those in both heterogeneouslayered systems and in areal flooding situations. Of vital importance to brine mixing is the level of reservoir sandbodydispersivity, and field values of this quantity are estimated. Results from the numerical modelling of oil displacement andIWrFW mixing are presented to illustrate various points which arise in the discussion. These calculations show that quite

Ž .complex patterns of mixing of connate, aquifer and injection brines can occur in relatively simple two-dimensional 2Dsystems. The significance of in situ brine mixing to barite scaling is discussed in some detail. q 2000 Elsevier Science B.V.All rights reserved.

Keywords: brine mixing; reservoir dispersivity; barite scale; barium sulphate; reservoir mixing; dispersive mixing

) Corresponding author. Tel.: q44-131-451-3139; fax: q44-131-451-3127.

Ž .E-mail address: [email protected] K.S. Sorbie .

1. Introduction

Waterflooding is one of the main oil recoveryprocesses being applied in many reservoirs aroundthe world. In the North Sea, and in many other

Ž .offshore developments, seawater SW is the mainŽ .injection water IW . As well as displacing oil, this

0920-4105r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0920-4105 00 00050-4

( )K.S. Sorbie, E.J. MackayrJournal of Petroleum Science and Engineering 27 2000 85–10686

IW will also displace and mix with either the con-Ž .nate water CW in the mobile oil zone of the

Ž .reservoir or with the aquifer water AQW , depend-ing on where the injection well is completed. TheCW, often referred to as the ‘‘irreducible water’’,

Ž .occurs at a saturation denoted by S or S ; thewc wir

CW and AQW may be referred to collectively as theŽ . Ž .formation waters FW . All of these waters brines

will generally have different ionic compositions andthere may be several chemical consequences of theirmixing both directly in the reservoir and aquifer andalso as they are co-produced at the producer afterwater breakthrough. For example, the composition ofthe IW — which will be out of equilibrium with thereservoir rock substrates — may cause mineral dis-solution, ion exchange or other clayrfluid interac-tions to occur. Also, the mixing of incompatiblewaters — for example, SW containing SO2y with4

FW rich in Ba2q — may cause mineral scale precipi-Ž .tation of BaSO , in this case both within the forma-4

tion and also on co-production at the wellbore.A combined reservoir simulation and chemical

precipitation model for in situ brine mixing in theŽ .context of oilfield scale barite formation was pro-

Ž .posed some time ago by Bertero et al. 1988 . This

paper focused principally on the development of themodel and few results were presented on examplesof brine mixing. More recently, there has been arevival of interest in the issue of IW and CW mixingstimulated in the UK by the following two different— but related — activities.

Ž .i Firstly, field observations have been made byseveral companies that barium sulphate scale may bedropping out deep in the reservoir, implying ‘‘in-timate’’ IWrCW mixing. For example, in theChevron operated Alba field, the barium levels at theproducers consistently fall below those expected ei-ther of the SWrFW mixing line or of the aquiferrFW

Ž .mixing line as shown in Fig. 1 White et al., 1999 .The apparent barium loss implies dropout either deepin the reservoir or in the near-well region. However,if the latter explanation were correct, then significantproduction loss would be expected and this is not thecase in the field, possibly due to the very highpermeability of the Alba reservoir sands, k;1–4Darcy.

Ž . Ž .ii Secondly, Coleman 1999 has establishedthat the composition of produced brines may initially

Ž .be connate formation or aquifer brines rather thanbeing IW as shown in Fig. 2. Coleman has made

Ž .Fig. 1. Barium development for all wells in the Alba Field from White et al., 1999 .

( )K.S. Sorbie, E.J. MackayrJournal of Petroleum Science and Engineering 27 2000 85–106 87

Ž . Ž . ŽFig. 2. Results from Coleman 1999 showing the early breakthrough of in situ brines CW and AQW in an oil reservoir from Coleman,.1999 .

observations which are ‘‘expected’’; e.g. early CWproduction. Other more intriguing observations onthe compositional trajectories of produced brinespoint to the fact that oil leg FW and AQW composi-tions are usually different and, indeed, that composi-tional variation may occur in different parts of afield. It is expected that the produced brine composi-tion should vary depending on whether injection wasinto the oil leg or aquifer.

To address the above field observations, we mustconsider all of the possible displacement and mixingmechanisms which may occur between an IW, the

Žoil leg CW if regional compositional variation is. Žobserved, then CW , CW , etc. and the AQW in1 2

principle, there may also be more than one —.AQW , AQW , etc. . This paper discusses all of1 2

these mixing mechanisms starting from a one-dimen-Ž .sional 1D frontal displacement, extended

Ž .Buckley–Leverett BL analysis. This mechanism isimportant in all other types of displacement andreservoir mixing process including those in bothheterogeneous layered systems and in areal floodingsituations. The issues of IWrCW brine mixing andreservoir dispersivity within a given sandbody areconsidered and field values of this quantity are esti-mated. Finally, we consider the numerical modellingof oil displacement and IWrCW mixing and recom-mendations are made on this in the light of thedevelopments in this paper. Some examples of nu-

merical modelling of IWrCW mixing are presentedto illustrate a number of points in the discussion.

2. Reservoir displacement processes and watermixing mechanisms

IW can displace in-situ brines — either CW orAQW — in a number of different situations. How-ever, underlying all of these is the basic frontaldisplacement mechanism, which is fully describedbelow.

We consider injectedrin-situ brine mixing in avariety of situations which closely parallel howreservoir engineers often think of the oil recoverymechanisma, as follows:

Ž .i Linear frontal displacement and brine mixing inŽa‘‘1D’’ waterflooding microscopic displacement

.efficiency ;Ž .ii Displacement and mixing in heterogeneousŽ . Ža .layered cross-sections vertical sweep ;Ž .iii Areal displacement and mixing in the water-flooding of a horizontal sandbody or single layerŽa .areal sweep ;Ž .iv Combinations of all of the above — as wouldoccur in a real heterogeneous three-dimensionalŽ .3D reservoir.

Despite the complexities of the above cases, theycan all be understood in terms of combinations of


simple frontal displacement theory. This will bediscussed below, although the mathematical detailsare relegated to Appendix A. The gradually morecomplex cases will then be built up using schematicfigures which develop the various scenarios. It will

Ž .be seen that intimate i.e. 100% IWrCW mixingcannot occur due to the nature of the frontal dis-

Žplacement mechanism. However, at the IWrCW or.IWrAQW interface, significant levels of mixing

Ž .can occur due to local rock permeability k —Žheterogeneity. Thus, in a precipitating system such

2q 2y. Žas Ba rSO , some reservoir deposition of4.BaSO may occur. This is consistent with the earlier4

Ž .results of Bertero et al. 1988 who showed that veryŽsmall amounts of in situ barite deposition only

.;0.16% PV maximum were expected in a 100%brine homogeneous aquifer displacement calculation.

In this work, we will assume that the transportedspecies do not interact with the matrix either chemi-cally, by adsorption or by ion exchange. Such pro-cesses may occur for certain species, such as Ca2q,Ba2q and this may change the relative velocities of

Ž .theses species relative to local inert brine velocityŽ .which may enhance or indeed retard mixing. Fresh

brinerclay interactions may also occur to change theionic composition of the IW and biological activity,involving SO2y ion for example, may also lead to4

compositional changes in the reservoir. For example,significant changes in the chemistry of producedwater associated with the interaction of injected CO2

with reservoir mineralogy have been reportedŽ .Bowker and Shuler, 1991 . These various phenom-ena will not be considered here since, in large scalemixed IWrCWrAQW waterflood displacements,they are probably second order effects superimposedon the principal reservoir displacementrmixing

Žmechanisms. No temperature effects e.g. reservoir.cooling, barite solubility etc. are modelled in this

work.

3. Linear displacements and IWrrrrrCW brine mix-ing in waterflooding

3.1. Oilrwater displacement and CW banking

Since this process is of central importance to allthe other displacementrmixing processes in reser-

voirs, we will consider this in some detail. Firstconsider a simple 1D waterflood in a linear homoge-

Ž .neous constant k and f sandbody, e.g. a linearchannel between injector and producer as shown inFig. 3, where all terms are defined. The viscousdominated displacement of oil by IW is well under-stood for this system and is described by BL theoryŽ .Buckley and Leverett, 1942; Dake, 1978 . However,we wish to investigate what happens to the IW andCW; particularly, we want to establish whether these

Ž .mix intimately i.e. 100% IWrCW mixing orwhether frontal displacement occurs. BL theoryshows that there are various regions of single andtwo-phase flow as shown in Fig. 4, where the issueof IWrCW mixing is also indicated. In fact, anextension of BL theory to deal with the fate of theIW and CW is presented in Appendix A where

Žmathematical details are given following Pope, 1980;.Lake, 1989 . The essential result is summarised in

Fig. 5 which shows that, at time t, CW ‘‘banking’’occurs behind the BL waterroil shock-front. Thereis, in this ideal case, a sharp front between the IW at

Ž .CW at xsx at time t . Hence, when water isbŽ .produced i.e. when x sL , then the first water willf

be 100% CW which will continue until all the CWis produced and the rear of the CW bank reaches the

Ž .outlet well x sL . At this point, the producedb

water will change immediately to 100% IW as shownin Fig. 6 where the watercut development and thenature of the produced water are shown. At break-through time, t , the waterfront reaches the outlet1

Fig. 3. Idealised linear 1D channel of homogeneous sand or rockin which an IW displaced oil and CW from initial conditions;

Ž . Ž .S s 1yS ; S sS . Volume of CWs D xD yD zfS .0 wc w wc wc


Fig. 4. Schematic of a displacement along a simple 1D sandbodyŽor channel. IWs injection water; CWsconnate water formation

.brine . The regions of oil and water flow are well established fromBL theory but the questions arise concerning the nature of themixing zone between the IW and the CW as shown.

Ž .end of the 1D system x sL and the watercutf

jumps immediately to the fractional flow of water atŽ .S sS , i.e. to watercuts f S . This is entirelyw wf w wf

CW as shown and the watercut rises until all the CWŽ .is produced at times t see Fig. 6 and, from thenb

onwards, IW is produced. All of these quantities canbe calculated analytically from a quite simple exten-

Ž .sion of BL theory Pope, 1980 ; see Appendix A. Infact, this has been observed experimentally in water-floods in cores where the IW and CW have been

Žlabelled as shown in the effluents of Fig. 7 after.Sorbie and Walker, 1988 . In this figure, the injected

Žbrine was labelled with radioactive chlorine-36 i.e.36 .Na Cl which is a b-emitter and can be assayed

very accurately using a scintillation counter. Notethat, in Fig. 7, the first five samples after break-

Ž .Fig. 5. Snapshot of the water saturation profile, S x,t , at time tw

showing: CW banking; the sharp front between the IW and CW atxs x ; the single- and two-phase flow regions which develop.b

Fig. 6. Watercut development in a 1D BL type displacement asŽshown in Fig. 5, indicating regions of CW and IW production no

.in situ brine mixing .

through contain no IW which breaks through verysharply in a single sample which is ;96% IW.

The total volume of CW that is present in a verysimple 1D system as shown in Fig. 3 is given by;volume CWsD xD yD zfS . This indicates that,wc

with direct injection into the oil leg, there will be a

ŽFig. 7. Comparison of the watercut and the traced Chlorine-36.labelled IW where the CW is not labelled. Note that the first

Žwater produced is quite clearly the CW replotted from Sorbie and. )Walker, 1988 . Note: ‘‘Pore volumes’’ refers to the total PV of a

Ž .layered pack used in this flood see reference for details .


relatively small and complete production of the CWand a fairly rapid changeover from CW™ IW andhence a very short period of co-production of CWand IW. If this were a scaling system, then we would

Ž .expect very little or a short-lived problem. Broadly,this conclusion has some truth in it although thesituation is rather more complex if other mecha-

Ž .nisms are present which they often are .

3.2. DispersiÕe mixing at the IWrCW front

The first level of complexity that may occur isthat the sharp front between the IW and CW, asshown in Fig. 8, may be spread due to some level ofpermeability heterogeneity within the ‘‘linear’’ sand-body. This is shown in Fig. 8 where the IWrCWfront at xsx is somewhat spread or dispersed andb

Ž .a mixing zone of length, L t , develops with time.m

The level of dispersion is related to level of perme-ability heterogeneity with greater spreadingrmixingoccurring in more heterogeneous sandbodies.

An estimate of the mixing zone length can bemade from the dimensionless 1D Convection–Dis-

Ž .persion CD equation for single phase tracer dis-placement, as follows:

EC 1 E2 C ECs y 1Ž .2ET N EXEXPe

Ž . Ž . Ž .where C X,T sc x,t rC where c x,t is the0

tracer concentration and C s injected concentration;0

XsxrL where Lssystem length; Ts tVrL is theŽ .dimensionless time and N s VLrD , is the PecletPe

number where D is the dispersion coefficient and VŽ .is the fluid velocity, VsQr Af . N is a dimen-Pe

Fig. 8. Linear waterflood as shown in Fig. 5 showing the develop-ment of an IWrCW mixing zone of length, L .m

sionless measure of mixing or dispersion where a‘‘high’’ N gives a very sharp displacement frontPe

and a ‘‘low’’ N gives a more spread out front. ThePe

dimensionless concentration, C, is normalised in therange 0FCF1, where Cs1 may represent 100%

Ž .IW and Cs0 is100% CW. Eq. 1 has a simpleanalytical solution from which a dimensionless mix-ing zone length, D X, can be calculated as the differ-ence between the IW normalised concentrations of

Ž .0.9 and 0.1, as follows Lake, 1989 :

D XsX yX 2Ž .Cs0.9 Cs0.1

giving:

TD Xs3.625 . 3Ž .(NPe

Ž .Since Eq. 3 is dimensionless and Ts1 is theŽ .time for the injected front of IW to go from the

Ž . Ž .inlet at Xs0 to the outlet at Xs1 , then:

3.625D X s 4Ž .bt N( Pe

where D X is the fractional length of the 1D systembt

taken up by the mixing zone; e.g. for N s200,Pe

then D X f0.26 of the length of the system. If webt

only had a CW saturation of 0.2 PV initially, thenthis would all be quite well mixed with the IW bythe time the IW front reached the producer along theID sandbody. Note that if N s1000 for the samePe

system, D X s0.1 and about half of the CW wouldbt

have mixed with the IW.Hence, the critical issue is: what is a realistic

value of N in a single sandbody or layer? It is notPe

straightforward to estimate N as we will showPeŽbelow. The dispersion, D, in a porous media Per-

.kins and Johnston, 1963 is given by:

Dsa V 5Ž .L

Ž .where a , the dispersivity dimensionss length , isL

a measure of the amount of mixing which occurs.Substituting for this quantity in the equation forPeclet number gives:

ÕL ÕL LN s s s . 6Ž .Pe D a Õ aL L


Thus from an estimate of L and a , we can calcu-L

late N and hence the likely mixing zone length,Pe

D X . For example, in a tracer flood in a 1-mbtŽClashach sandstone core, a f0.004 m Sorbie,L

.1991, p. 218 . Thus, if our 1D reservoir layer was ashomogeneous as a Clashach core, but was say 100 m

Žlong, then N f25,000 and D X f0.02 from Eq.Pe btŽ ..4 and the mixing zone would be very small.

ŽObservations taking dispersivities measured di-.rectly in the laboratory and estimated in the field,

indicate that a increases with system length asLŽ .shown in Fig. 9 see Arya et al., 1988; Lake, 1989 .

However, this is a little misleading since some of theŽ .results in Arya et al. 1988 are for Õery heteroge-

neous systems including layering — which are notin the dispersive flow regime appropriate for a singlesandbody. Indeed, if Arya et al.’s results are taken atface value, then for a system of Ls1000 m, areasonable range for a would be: 10 mFa F100L L

Ž .m see Fig. 9 . Even the lower value for a wouldL

give N s100 which would give, D X f0.32 andPe bt

hence, full IWrCW mixing in virtually all cases. IfŽ 2q.this were true, then mobile barium ions Ba

would virtually neÕer be seen at producer wells andthey very frequently are if this species is present inthe CW. Therefore, a more reasonable estimate ofa may be found from Arya et al.’s shorter length-L

Ž .scale data 10 mFLF100 m which are far more

Ž .Fig. 9. Laboratory and field levels of dispersivity, a after Arya et al., 1988 .L


likely values of this quantity within a single sand-body. At these length scales, the range of a valuesL

Ž .is Fig. 9 :

10y2 m Lf10 m, N ;105 FaŽ .Pe L

F2 m at the lower end Lf100 m, N ;500Ž . Ž .Pe

In a system of Ls1000 m, these would lead to arange of mixing zones with D X f0.01 to f0.15bt

which would lead to anything from virtually noŽIWrCW mixing to quite significant but not quite

. Ž .complete mixing at S f0.2 . This range agreeswcŽ .with the value quoted by Stalkup 1998 in the

context of gasroil mixing in enriched gas injection;Ž .at the reservoir scale ;300 m a dispersivity of

Ž .a f0.3 m ;1 ft is quoted.L

The issue of what effective dispersivity should betaken in the field is still open and some estimate ofthe range of values which this quantity can have maybe obtainable by analysis of field produced brinedata. However, in a given reservoir, it is vitally

important to establish the magnitude of a if we areL

to accurately determine the degree of in situ brineŽmixing and hence level of in-reservoir scale

.dropout . It will be shown below that this also hasimportant consequences for the numerical modellingof displacement and IWrCW brine mixing in thereservoir. In particular, the level of physical mixingin the reservoir has a strong influence on the numberof grid blocks required to model such processesaccurately.

3.3. Consequences of brine mixing within the reser-Õoir

We now consider the consequences for a scalingsystem where in situ mixing occurs, e.g. Ba2qrSO2y.4

Clearly, if complete mixing occurred, then all the2q ŽBa would usually precipitate since real systems

2y.usually have a large excess of SO . This would4

give a non-scaling zone between the IW and CWthat would grow gradually — depending on the

Žw 2yx w 2qx.Fig. 10. Schematic profiles of scaling ion concentrations SO and Ba along a simple 1D system where a mixing zone between the4

IW and CW develops. This results in a zone depleted of barium ions but not of sulphate ions.


dispersivity, a . Generally, the local quantity ofL

BaSO that would deposit would be insignificant and4

it would not accumulate locally. For example, if all2q Ž .the Ba in a 2000-ppm solution Õery severe

deposited locally in the pore space it would cause aporosity change of -1r2000 with virtually no per-meability change. That is, scale deposition deep inthe reservoir should not lead to any significant levels

Žof formation damage or productivity decline cf..Bertero et al., 1988 . Thus, the reservoir mixing

mechanism in Fig. 10 — which probably occurs tosome extent in most reservoirs — helps to alleviatethe situation as far as scaling problems at producersare concerned by:

Ž .i leading to ‘‘harmless’’ scale deposition deepwithin the reservoir; andŽ .ii causing a non-scaling ‘‘spacer slug’’ to de-

'Ž .velop and grow with time, L ; t further sepa-m

rating CW and IW as they arrive at the producer.

However, if the IWrCW mixing levels are lowŽ .as in low-heterogeneity sandbodies then most ofthe Ba2q in the CW will reach the producer. In asimple 1D linear sandbody, this would be of nosignificance since we would firstly see the Ba2q inthe CW and then the SO2y at a later time in the IW.4

There would be no near-well or in-wellbore scalingin such a case. Although real reservoirs are not 1D,this frontal 1D displacement mechanism is Õery im-portant since it is this mechanism that delivers the

2q ŽBa in a banked form to the producer where IW or. Ž .AQW from other mechanisms see below can mix

with it and hence cause scaling problems. The subtlebut important role of the frontal displacement andbanking mechanism is essential to understanding insitu brine mixing in reservoirs.

This point is central to explaining why we eÕersee BaSO problems at producers; if 100% reservoir4

mixing of IWrCW occurred, then we would neÕersee such problems. Unfortunately, we do because theBa2q is banked in the CW and ‘‘delivered’’ to theproducer. How depleted the Ba2q becomes before itreaches the wellbore depends on how much actuallydropped out in the reservoir. An important corollaryto this is as follows: if any unprecipitated bariumoccurs at the producer in a given layer, then it isalmost certainly at the original barium concentration,

w 2qxBa , as in the CW. If it appears to be lower thanthis level, then it is almost certainly because ofdilution in another brine stream from another layerŽ .possibly IW or from the aquifer. This is the casebecause of the banking and the level of mixing at therear of the CW slug, as shown in Fig. 10. It is clearfrom this figure that the Ba2q is precipitated gradu-ally from behind. If the SO2y reached the front of4

the Ba2q slug, then all the Ba2q would be missingrather than just some of it. Hence, if some mobile

2q Ž . Ž .Ba appears at the well, then one or more layer sw 2qxmust be producing at the full Ba of the CW.

( )4. Two-dimensional 2D vertical and areal dis-placements and water mixing

4.1. Displacements in heterogeneous Õertical cross-sections

The linear displacement mechanism describedabove will again play a key role in vertical heteroge-

Ž .neous cross-sections and in all other scenarios . Thisis illustrated when we apply these earlier ideas to theschematic five-layer cross-section in Fig. 11 fromwhich we note the following points:

Ø layers 1, 3 and 5 are producing only oil;Ø layer 4 has broken through early and is producing

ŽIW the CW ‘‘bank’’ went through some time.previously ;

Ø layer 2 is currently producing its CW bank.

Fig. 11. Schematic of the displacement processes that may occurin a heterogeneous vertical cross-section of a reservoir. See textfor discussion.


Fig. 12. Schematic of the displacement processes in a heteroge-neous vertical cross-section of a reservoir with an aquifer. Notethat simultaneous production of oil, AQW, CW and IW is possi-ble.

The problem here comes from the mixing of IWŽ . Ž .layer 4 and CW layer 2 which occurs at andabove the layer 2 perforations. No scaling problemsare yet present below layer 2. Also, no deep reser-voir scaling occurs due to the IW and CW flowing in

Ž .adjacent vertical layers 1, 3 and 5 Fig. 11 sincethese layers are flowing in parallel and there is no

Ždirect displacement Vs0 in the displacement direc-.tion , and hence no significant mixing. If there were

a little diffusive mixing this would give a small —but insignificant — depleted zone of Ba2q betweenlayers.

Following the current problem shown in Fig. 11Ž .layer 2 CWrlayer 4 IW mixing , a layer 1 CWbank will also appear later giving similar problems.The heterogeneity between layers staggers the arrivaltimes of the CW banks hence spreading the problemout in time. The problem with vertical heterogeneitymay be complicated further by the presence of an

Ž .aquifer which may indeed frequently does have adifferent composition from either the CW or the IWŽ .Coleman, 1999 . This is shown schematically inFig. 12 where the co-mingling of the AQW makes itquite possible for a well to be producing IW, CWand AQW with the resulting scaling problems.

Thus, the vertically heterogeneous case is just acombination of different 1D displacement processeswhich cause CW banking and delivery of CW and

Ž .IW and possible AQW to the producer at differenttimes.

4.2. Displacements in areal horizontal sand layers

The situation in a 2D areal sand is very similar tothat in the vertical 2D cross-sectional case. The areal

Žcase is a sequence of 1D displacements which again.bank CW which are ‘‘staggered’’ in their arrival

times by the different length streamlines as shown inFig. 13. Note that the ‘‘fastest’’ streamline between

Ž .injector and producer streamline 1; Fig. 13 causesearliest water breakthrough — first of CW and laterof IW. As IW is produced by streamline 1, the CW

Ž .along streamlines 2 and 3 Fig. 13 break through.This will cause a scaling problem by IWrCW mix-ing which is again stretched out in time by thearrival of later streamlines.

In the most general case, the 1D banking of CWprobably always occurs but the arrival times of IWand CW at the producer can be staggered by bothvertical heterogeneity and areal spreading of thestreamlines. Areal heterogeneity may also have arole and indeed may result in some degree ofIWrCW mixing which may also cause scale dropoutin the reservoir.

Ž .Fig. 13. Schematic of an areal 2D waterflood showing theŽ .flooding pattern A and the displacements along the associated

Ž .streamlines B where the banking of the CW is shown. It is thevariation of areal velocities along the streamlines that causes thespreading of arrival times of the CW and IW at the producer.


5. Numerical modelling of displacement and brinemixing in reservoirs

5.1. Numerical simulation and the use of numericaldispersion to simulate physical mixing in the trans-port equations

Numerical simulation is used to model oil dis-placement by IW and the associated brine mixing inreservoirs. The equations governing the transport of

Ždifferent components in the aqueous phase different.IW, CW and AQW compositions as well as the

normal multi-phase flow equations must be solvednumerically. A number of commercially availablereservoir simulators can carry out this task, e.g.

Ž .ECLIPSE 100 98A and ECLIPSE 200 1998 ,Ž .STARS version 98 1998 , VIP Reservoir Simulator

Ž . Ž .version 3.3 1996 , SCORPIO Scott et al., 1987 ,Ž .UTCHEM 1999 , etc. However, all of these codes

are based on finite difference discretisation of thepressure and transport equations and this introduces

Ž .an artificial diffusion or mixing at any transportedŽfronts such as the oilrwater front or at the IWrCW. Žinterface due to numerical dispersion, D Peace-num.man, 1978 . The quantity D is essentially a mix-num

ing term very similar to the actual physical disper-sion term D in N , but it arises as a numerical errorPe

in the finite difference approximation of the flowequations. However, since this numerical error isdispersive in nature, it may be possible to actuallyuse numerical dispersion to represent the actual lev-els of mixing seen in the reservoir. Indeed, this has

Žbeen done in many previous studies e.g. Sorbie et.al., 1992; Stalkup, 1998 . To do this requires that we

estimate the level of numerical dispersivity, a ,Lnum

that a given finite difference scheme gives and thenmatch this — by appropriate choice of numbers ofgrid blocks, N — to the physical levels of disper-grid

sivity, a , that we think is correct for the reservoir.L

It is well known that the level of numerical disper-sion for a one-point upstream discretisation of thesingle phase convection equation with grid block

Ž .size, d x, and time step, dt, is Lantz, 1970 :

d x V dtD s y V 7Ž .num 2 2

Ž . Ž .Fig. 14. Water saturation A and IW B profiles, CWrIW splitŽ . Ž .at 300 days C and water mix in production watercut D for

1000=0.3048 m cells.


which implies that the numerical dispersivity, aLnum

is given by:

d x V dta s y . 8Ž .Lnum 2 2

For a sufficiently small time step where the level ofnumerical dispersion is due entirely to the spatialterm then:

d xa s . 9Ž .Lnum 2

Hence, the level of dispersivity inherent in thenumerical scheme is approximately half of the gridblock size. Considering the values which we dis-

Ž .cussed in Section 3.2 i.e. a f0.3 m implies thatL

the grid block size must be d xf0.6 m in order toŽobtain the correct levels of reservoir mixing cf.

.Stalkup, 1998 . If the injector and producer are;300 m apart, then N f500 in order to obtaingrid

the correct reservoir mixing levels by using numeri-cal dispersion. This is easy in a 1D model but it maybe impossible computationally in a full 3D reservoir

Ž .3 Žmodel where up to 500 grid blocks i.e. 125.million may be required. However, in a layered 3D

Žmodel, we may only require a finer areal grid 500=. Ž .500 and possibly fewer vertical grid blocks ;10

if there is little crossflow — a calculation with 2.5million blocks is difficult but possible. It may also bepossible to perform some numerical scoping studieson a coarse grid to find out where the IWrCW mainmixing paths are and then to carry out a local grid

Ž .refinement LGR in these regions of the model tocapture the local brine mixing with the appropriatelevels of dispersion.

5.2. Examples of 1D and 2D oil displacement andbrine mixing calculations

In order to demonstrate the effects discussed inthis paper, we present some illustrative examples of1D and 2D oil displacements and brine mixing calcu-lations. Figs. 14 and 15 show the results from 1D

Ž . Ž .Fig. 15. Water saturation A and IW B profiles, CWrIW splitŽ . Ž .at 300 days C and water mix in production watercut D for

50=6.096 m cells.


displacement calculations in a 1000-ft-long ‘‘linear’’reservoir where the grid block sizes are, d xs0.3048

Ž . Ž . Žm 1 ft and 6.096 m 20 ft , respectively i.e..a f0.15 and 3 m . The water saturation profilesLnum

at various times are shown in Figs. 14A and 15A.ŽThe clear development of the BL shock front Ap-

.pendix A can be seen in Fig. 14A but this is moredispersed in Fig. 15A due to the larger grid blocksize. The corresponding IW concentrations withinthe water phase are shown in Figs. 14B and 15Bwhere they clearly lag behind the water saturation

Žfronts leading to banking of CW as predicted Ap-.pendix A . Again, a much sharper IW front is seen

for the fine grid calculation in Fig. 14B. This isshown more clearly in Figs. 14C and 15C where theproportions of IW and CW are shaded within the

Ž .water phase at a single time ts300 days and thelengths of the mixing zones in these two figuresshould be noted. The 0.3-m grid block calculationsin Fig. 14C leads to a sharp front with a short mixingzone. The 3-m grid blocks used in the calculation in

ŽFig. 15C lead to the development of a numerically.induced mixing zone of )60 m. This is approxi-

mately 0.2 of the well to well reservoir distance butis rather less than the D X;0.36 estimated by Eq.Ž . Ž .4 . However, Eq. 4 is for fully miscible singlephase tracer mixing and the effect of the two-phaseflow is to somewhat shorten this numerical mixingzone. The main point to note in Fig. 15C is that thereis predicted to be very extensive IWrCW mixing inthe reservoir if we use a grid block size, d xs3 m.Figs. 14D and 15D show the numerically predictedwatercut development with time at the producer andthe shaded fractions of this which are CW and IWŽ .cf. Fig. 6 for the ideal — no mixing — case . Arelatively short transition time is seen from CW,which is produced first, to IW in the fine grid caseŽ .Fig. 14D , whereas a lengthy period of co-produc-tion of CW and IW is seen in the coarser grid modelŽ .Fig. 15D .

The consequences of the level of mixing in theŽ .coarser d xs3 m model for a sulphate-rich IW

displacing a barium containing CW would be exten-sive in situ precipitation of barium sulphate. If thetrue level of sandbody dispersivity in this 1D dis-placement is of order ;0.15–0.3 m, then clearly thelevel of mixing is vastly overestimated. In fact, a3-m grid block in a reservoir simulation model would

be considered to be ‘‘very fine’’. Therefore, it maybe very difficult or practically impossible to performsuitably fine grid calculations in full 3D reservoirmodels and we will return to this point at the end ofthis section.

2D cross-sectional displacement calculations havebeen performed for the model detailed in Table 1.Cross-sectional models of this type are quite typicalof those used in standard reservoir engineering calcu-lation for layered heterogeneous formations. In thesesystems, early water breakthrough frequently occursand then subsequent water fronts breakthrough invarious layers. The calculations presented here differin two important respects from those which are

Ž .routinely performed, as follows: a a very fine gridof 1000 blocks is taken in the x-direction to give a

Ž .sharp mixing zone with a s0.15 m ; withinLnumŽ . Žeach layer; b the tracer transport option in.ECLIPSE, 200 is used to label the IW, the CW and

Ž .the AQW. A fairly fine vertical grid d zs0.3 m isused to resolve vertical flows in the layered modelbut this is less important in the mixing calculationsthan taking a fine x-grid. In these cross-sectionalcalculations, the injector and producer wells are onlycompleted aboÕe the aquifer. Results are shown attimes 100, 400 and 700 days in Figs. 16A–L, whichshow the time evolution within the system of:

Ž .Ø water saturation, S Figs. 16A–C ;wŽ .Ø injected water, IW Figs. 16D–F ;Ž .Ø connate water, CW Figs. 16G–I ;Ž .Ø aquifer water, AQW Figs. 16J–L .

The main points to note from these results are asŽ .follows: a AQW is produced very early by coning,

Ž .see Fig. 16J; b within the oil bearing layers, theCW banking by IW can be seen and, in the aquifer,

ŽIW displaces AQW although the injector is com-pleted only above the aquifer, injection readily entersthe aquifer mainly due to the higher mobility of the

. Ž .aqueous phase ; c IW breakthrough is seen in layer4 at ;400 days — see Fig. 16B — where some

Ž .banked CW can also be seen; d later in the flood at700 days, layer 2 is about to break through, firstlywith CW and then with IW.

The results in Fig. 16 show that, at differenttimes, the produced water may contain varying mix-tures of IW, CW and AQW. This is shown more


Table 1Geometric, rock and fluid properties of 2D simulation of a waterflood in a multi-layer reservoir

clearly in Fig. 17 which shows the watercut develop-ment at the producer over time indicating the propor-tions of IW, CW and AQW in the produced water.Essentially, these results confirm the qualitative de-scriptions discussed in Section 4.

5.3. Consequences of reserÕoir mixing on baritescaling

To demonstrate the impact of CW and IW mixingin the wellbore, a calculation was performed of the


Ž .Fig. 16. Water saturation, and distribution of IW, CW and AQW at 100, 400 and 700 days for 1000=20 0.3048=0.3048=0.3048 mgrid cells in six-layer system. Average frontal advance rate is 0.3048 mrday.

scaling potential along the production well in this 2Dmodel. An in-house scaling tendency prediction codeŽ .SCALEUP, 1993; Yuan et al., 1994 , was used tocalculate precipitation and supersaturation tendenciesas a function of IWrFW mix. An example showingthe mass of barite and the corresponding supersatura-

Fig. 17. Mix of waters in produced watercut in six-layer 2Dmodel.

tion vs. SW fraction is shown in Fig. 18, where theIW is SW and the FW is a typical Forties type brine.Water production history along the 4-m completedinterval of the well in the 2D model described aboveis shown in Fig. 19. It is seen that the majority of theproduced water cones up from the aquifer and entersthe well over the bottom 0.6 m. IW breaks throughin the bottom of the well after 400 days, and pro-gresses up the well, so that by 900 days, the wholewell is producing IW as shown in Fig. 20. However,the IW fraction decreases in the middle section of

Ž .the well 1.5–2.5 m after 900 days, so that althoughthe top and bottom sections are producing 100% IWafter 1300 days, the middle section is producing only20% IW. This does not necessarily mean that themiddle section is the most prone to scaling, as Fig.21 demonstrates; this figure shows the barite dropoutprofile along the well in kgrmrday. The IWrFWfraction at every point in Fig. 20 is used, together


Fig. 18. BaSO scaling tendency prediction showing the mass of4

precipitate and the supersaturation for Forties type water at 1.7447 Ž .=10 Pa and 403.15 K 1308C .

with the precipitation tendency predicted byŽ .SCALEUP Fig. 18 , to calculate the maximum pos-

Fig. 19. Combined water production profile along well in 2Dmodel as a function of time. Note that most water is producedfrom the bottom of the well.

Fig. 20. SW fraction of total produced water along the well vs.time. Note that the complex behaviour as SW breakthrough at thebottom of the well leads to production along the entire well

Ž .length, but subsequently reduces in the middle section 5–8 ft .

sible dropout. This is then multiplied by the waterŽ .flow rates Fig. 19 to give the possible mass dropout

per unit length of well per day. Because the majorityof water production comes from the bottom of thewell, the highest dropout will occur when the bottomof the well is producing 5–40% IW. However, itshould be noted that the values calculated represent amaximum possible dropout, and the actual precipita-tion will also be determined by the local supersatura-tion, which is shown along the well as a function oftime in Fig. 22.

It is clear from these results that a relativelysimple 2D layered system, such as the one modelled

Fig. 21. Maximum possible BaSO dropout based on calculation4

of precipitation as a function of SW production. Note that theperiod of greatest scale dropout is immediately after SW break-through at the bottom of the well.


Fig. 22. Supersaturation profile along the well vs. time calculatedfrom SW fraction in produced water.

here, can produce complex flow patterns that willlead to scale formation. It therefore follows that it

may be very difficult to predict scale dropout in real3D reservoir systems. This is especially true whenthe luxury of very fine scale models, such as the oneused for these sample calculations, is not an option.

Thus, the main frontal displacement mechanismand the water mixing mechanisms in a verticalcross-sectional model have been confirmed numeri-cally, and the impact on potential scale dropout hasbeen clearly demonstrated. No areal calculations arepresented here since they would simply confirm thequalitative description given in Section 5. It is clearfrom these calculations and from our analysis ofdispersion and reservoir mixing that very fine simu-lation grids are required to avoid spurious numericalmixing of IW, CW and AQW. These grids are muchfiner than those usually employed in numerical reser-voir simulation. This is clearly illustrated by theresults in Fig. 23 which shows the level of the IW

Ž .Fig. 23. 3D reservoir model showing distribution of oil grey — around middle and toe sections of horizontal producer Y1 and waterŽ . Ž .black , with contours showing IW fractions from 10% to 90%. Note the unrealistic spread in the IWrFW mixing zone )165 m .


and CW mixing zone in the down dip aquifer of aŽ .crestally placed reservoir red is oil saturation . The

Žgrid blocks are of areal size, d xsd ys30 m aLnum.f60 m . This leads to mixing zones in the size

Ž .range 150–300 m depending on flow directionsbetween the injector and producer, which is a crestalhorizontal well in this case. If the true levels ofdispersivity within the reservoir layers are of order,a f0.3m, then this grid is completely inappropriateL

for brine mixing calculations in reservoir displace-ments.

6. Discussion and conclusions

In this paper, we have presented a survey of themixing mechanisms between IW and in situ watersŽ .CW and AQW . The consequences of this for scal-

Žing systems which arise by in situ brine mixing such.as barium sulphate has been considered. Frontal

Ž .displacements with CW or AQW banking occur inall cases and the degree to which there is mixing atthe IWrCW front depends on the level of dispersiv-

Ž .ity within the sandbody or layer . Some estimates ofsandbody dispersivity in reservoirs have been dis-cussed and the importance of this parameter has beendemonstrated in numerical calculations. The furthereffect of vertical heterogeneity and areal flow pathsis to spread the arrival times of the various brinefronts — banked CW, IW and AQW — at theproducer wells. Hence, co-production of brine mix-tures is the expected case and is not unusual andthis, in turn, may lead to mineral scale depositiondepending on the precise brine compositions in-volved. Again, numerical calculations have been pre-sented in a 2D heterogeneous layered cross-sectionalmodel illustrating these conclusions. Indeed, thesecalculations show that quite complex patterns ofCW, AQW and IW production can occur in rela-tively simple 2D systems.

The main conclusions from this work are as fol-lows.

Ž .i The simple 1D frontal displacement of CW byIW leads to the banking of the CW. Thus, in such adisplacement the first produced water is CW fol-lowed later by IW.

Ž .ii A ‘‘mixing zone’’ at the IWrCW front maydevelop during linear displacement as a result of the

sandbody heterogeneity. This can lead to some scal-ing ion dropout thus forming a spacer zone depletedin one of the ions, usually Ba2q. This mixing processwill never produce enough scale deep in the reser-voir to significantly affect the local porosity or per-meability and will hence not adversely affect reser-voir productivity. Only this 1D frontal displacementmixing mechanism offers the potential for scalingion dropout in the reservoir.

Ž .iii Within a given 1D sandbody, the mixingzone is characterised by a dispersivity, a , whichL' .leads to a t growth of the length, L , of this zonem'Ž .i.e. L ; t . The more heterogeneous the sand-m

body, the larger is a and hence the more mixingLŽ .that occurs and vice versa . To accurately determine

Žthe degree of in situ brine mixing and hence scale.dropout, if appropriate in a displacement process in

a given reservoir, it is very important to establish themagnitude of a .L

Ž .iv Despite the mixing at the rear of the CWbank, it will often not catch up with the front of theCW zone. Thus, a zone that is producing any scaling

Ž 2q.ion such as Ba will probably be producing it atfull CW concentration. If the scaling ion concentra-tion is found in the wellhead brine to be below thislevel, then either mineral scale dropout is occurringdownhole or other zones are producing brines that

Ž .are diluting the barium producing FW CW or AQW .Ž .v The role of this 1D CW banking mechanism is

very important in all reservoir mixing mechanismssince it works in conjunction with them in both

Ž .vertically heterogeneous layered and areally exten-sive systems.

Ž .vi In vertically heterogeneous systems, the bank-ing mechanism ensures that CW reaches the pro-ducer. However, the layer to layer permeability het-erogeneity ensures that the CW and IW can arrivefrom different layers at the same time. Thus, simulta-neous arrival may be considerably staggered in timegiving a scaling problem at the producer for anextended period. AQW may also play a role in thisprocess and, in the most general case, a producermay produce IW, CW and AQW at the same time.

Ž .vii The situation of staggered arrival of IW andbanked CW in a 2D areal system is very similar tothe vertically layered case. However, the mechanismof delay in a 2D areal system is due to the differentlengths of areal streamlines. This causes the spread-


Ž .ing of the IW and CW and oil arrival times at theproducer.

Ž .viii In a complex 3D heterogeneous reservoir,all of these mechanisms may operate together whereagain:

Ž .a the 1D banking ensures that CW arrives at theproducer; andŽ .b the vertical and areal ‘‘delay mechanisms’’

Ž .spread out the IW and CW and possible AQWarrival times at the producer.

Ž .ix Both 1D and 2D numerical simulations havebeen performed which confirm the linear and hetero-

Ž .geneous cross-sectional IWrCW water mixingmechanism discussed in this paper.

Ž .x If within sandbody heterogeneity leads tolevels of physical dispersivity of order ;0.3 m, thismay be modelled using numerical dispersion. How-

Ž .ever, the grid sizes d x and d y in particular mustŽ .also be of this order a fd xr2 . This impliesLnum

that very fine grids are required to model in situIWrCW mixing processes accurately. The finenessof the required grid may mean that it is impracticalto carry out such simulations for many field cases.

NomenclatureŽ .A System 1D cross-sectional area

Ž .AQW Aquifer water brineBL Buckley–Leverett analysis for 1D dis-

placementsŽ .CW Connate water brine

Ž .c x,t Concentration of ‘‘tracer’’ as function ofx and t

Ž .C X,T Dimensionless concentration of tracer;Ž . Ž .C X,T sc x,t rc0

c Injected tracer concentration0Ž 2 .D Dispersion coefficient units m rs ; where

Dsa VL

D Numerical dispersion arising for the finitenum

difference scheme used; where D snum

a VLnumŽ . Žwf S Fractional flow curve; f s 1r 1 qw w w

Ž .Ž .xk rk m rm — all quantities de-ro rw w o

fined belowŽ .FW Formation water brine ; may refer to CW

or AQW

Ž .IW Injected water brinek Permeabilityk , k Water and oil relative permeabilities; theserw ro

are functions of Sw

L System lengthLGR Local grid refinement

Ž .L t Length of the mixing zone as a function ofm 'time; L ; t for dispersive flowmŽ .N Peclet number dimensionless ; where NPe Pe

Ž .s VL rDN Number of grid block used in the 1Dgrid

finite difference modelPV System pore volumeQ, Q Volumetric injection rate — specificallyw

for waterŽ .S x,t Water saturation profile along the systemw

Ž .0FxFL at time tS , S CW saturation; Irreducible water satura -wc wir

tionS Residual oil saturationor

S Water saturation at BL shock frontwf

S Water saturation at the IWrCW interfacewbŽ .in 1D linear displacement no mixing

t Breakthrough time of the BL shock frontbŽproducing only CW in the no mixing

.caset Breakthrough time of the IWrCW frontb

T Dimensionless time; Ts tVrL — time toinject 1 PV

V Total Darcy velocity V sQrAT TŽ .V Miscible fluid brine total velocity Vs

Ž .Qr Af

Ž .V S Velocity of water at saturation level, S ,w w wŽ .where, V sV d f rdSw T w w

x, t Space, timeX Dimensionless length variable; XsxrLx Distance to the IWrCW front in the nob

Ž .mixing case at a given time, tŽx Distance to the BL shock front at a givenf

.time, tŽ .a Dispersivity units, mL

a Dispersivity due to the numerical finiteLnumŽ .difference scheme units, m

D x, D y, D z System size in each directiond x, dt Grid block size in x-direction and time

step size in finite difference schemeD X Dimensionless mixing front length; D Xs

Ž .3.6256 TrNPe


D X Dimensionless mixing front length at breakbtŽ .through; D Xs3.6256 1rNPe

Ž .f Porosity fractionm , m Water and oil viscositiesw o

Acknowledgements

The authors would like to thank a number ofcolleagues for very helpful discussions over the yearson the issue of mixing of injected and in-situ reser-voir brines, including: Richard White, Jess Brookley,Alden Carpenter, Bill Milliken and Pat Shuler ofChevron; Max Coleman of Reading U.; Gordon Gra-ham of Heriot-Watt U.; Gerald Hamon of Elf; MylesJordan of Nalco-Exxon; Rex Wat of Statoil; MingDong Yuan of Baker-Petrolite. Schlumberger Geo-Quest are thanked for providing free access to theirECLIPSE suite of reservoir simulation tools.

Appendix A. Extended BL theory for connatebrine displacement

This explanation of IW and CW displacement is asimple case of the more general treatment of PopeŽ .1980 . Conventional BL theory for 1D linear water-flooding is based on the fractional flow formulation

Ž .Fig. A1. The fractional flow curve, f s , showing the BL-Welgew wŽ .construction at the shock front saturation, S . See S x,t atwf w

time t in Fig. A2.

Ž .Fig. A2. The instantaneous water saturation profile S x,t alongwŽ .system on length, L, at times t before water breakthrough .

Žof the 1D conservation equation Buckley and Lev-.erett, 1942 :

ES Ewf sy V A1Ž .wž /Et Ex

Ž .all terms explained in the Nomenclature . Using thefact that V sV f gives:w T w

ES Ewf syV f . A2Ž .T wž /Et Ex

Ž .Since the fractional flow, f S , is a function of Sw w w

only then the chain rule may be applied to obtain:

ES d f ESw w wf syV . A3Ž .Tž / ž /ž /Et dS Exw

Thus, the velocity of water saturation level, S , orwŽ .V S is identified as:w w

d fwV S sV A4Ž . Ž .w w T ž /dSw

Ž .and hence Eq. A3 above may be viewed as anormal convection equation for the saturation asfollows:

ES ESw wf syV S . A5Ž . Ž .w wž / ž /Et Ex

Thus, we may use the usual BL-Welge construction,as shown in Fig. A1, to find the shock front height,S , and position, x , at a given time, t. The instan-wf f

Ž .taneous saturation profile at time t, S x,t is shownw

in Fig. A2.Ž . ŽThe area under the S x,t profile AqBqC inw

.Fig. A2 represents the sum of both the volume ofŽ .injected brine AqB and the volume of CW up to

Ž .the shock front C . However, although Fig. A2


Ž .Fig. A3. Water saturation profile, S x,t , at times t showingw

clearly the ‘‘banking’’ of the CW by the IW. The rear of the CWwater bank has travelled distance, x , and the correspondingb

saturation is, S .wb

Ž .shows the total injected plus CW AqBqC , itgives no indication of whether this is a completemixture of IW and CW or whether there is someother distribution. Since the IW and CW may bevery different, it is very important to determine this‘‘fluid configuration’’. For example, the IW and CWmay have radically different salinities, they may

Ž 2q 2y.contain separate scaling ions e.g. Ba and SO ,4

etc.In fact, since the IW miscibly displaces the CW, it

must simply ‘‘bank’’ this CW as shown in Fig. A3.ŽBy simple material balance arguments, Area D2q

.D3 in Fig. A3 must be equal to Area E1, since theCW in Area E1 has been swept into the CW bank.Note that the total IW area at time t in Fig. A3 is

Ž .given by Area E1qE2qE3 . However, simpleBL theory tells us that:

E1qE2qE3sV t . A6Ž .T

From Fig. A3, it is evident that these areas can becalculated as follows:

x S qx S ySŽ .b wc b wb wcŽ . Ž .E1 E2

d fŽ .1yS worqV t 1yS dS sV t .Ž . HT or w Tž /dSS wwb

Ž .E3

A7Ž .Ž . Ž . w Ž .The integral in Eq. A7 E3 is simply f 1ySw or

Ž .x Ž . Ž .y f S and since f 1yS s1, Eq. A7 be-w wb w or

comes:

x S qx S yx S qV tyV t f S sV t .Ž .b wc b wb b wc T T w wb T

A8Ž .

Ž .Fig. A4. Fractional flow curve, f S , showing the Welge-typew wŽ .construct for the rear of the CW bank line b .

Ž .Cancelling terms in Eq. A8 and simplifying leadsto:

x S sV t f S A9Ž . Ž .b wb T w wb

and expressing this in Welge tangent form leads to:

f S y0Ž .w wbx sV t . A10Ž .b T S y0wb

Ž .The term in square brackets in Eq. A10 above isclearly identified as being the appropriate Welge

Ž .construct for the quantity d f rdS which isw w S sSw wb

Ž .shown in Fig. A4 line b . Thus, from the appropri-ate fractional flow curve, we can find values of Swf

Ž .and S either by graphical construct as in Fig. A4wb

or, more commonly, by numerical solution by com-puter.

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2000 - Mixing of Injected, Connate and Aquifer Brines in Water Flooding and Its Relevance to Oilfield Scaling

Documents

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