In-Situ Gel Calculations in ComplexReservoir Systems Using a NewChemical Flood SimulatorT. Soott, SPE, U.K. Atomic Energy Authority, MrinfrithL.J. Roberts, U.K. Atomic Energy Authority, WinfrithS.R. Sharpe, U.K. Atomic Energy Authority, WinfrithP.J. Ctifford, U.K. Atomic Energy Authority, WinfrithK.S. Sorbic, SPE, U.K. Atomic Energy Authority, Winfrith

SP /$4?34/

%mma~. This paper presents results for a series of calculations on the deep emplacement of a polymer gel in a stratifiedreservoir nmdel. These calculations were performed with a new chemical flocding simulator that has the facility to describegeneralized chemical reactiom between components. This code is described, and prelimimry calculations on an in-situ geltreatment in a large model reservoir are presented. We find that to obtain significant amounts of incremental oil while avoidingvery large pressure buildup, the polymer geI system must have the correct mmnbimtion of long gelation times and goodpermeability-reducing properties in the high-permeability streak (residual resistance factor FR -10 to 40 in the cases studied here).No commercial polymerl cmsslinker systems are currently available that have the very long gel times required to obtain deepemplacement in a large reservoir system,

IntroductionDuring recent years several chemical systems have been suggestedfor blocking off ve~-hi h-permeability channels in heterogeneous

3petroleum reservoirs. 1. These systems involve the injection ofpolymer [either polyacrylamide. (PAM) or xantban polymer] anda crosshnking-ion or redox system to form a suitable gel, Reacramsmay be mixed jnst before i~jection imo the reservoir, or lhey maybe injected m alternating slugs of the two materials.

To date, no catcukationshave appred in the Iiteramre that atmmptto qatify the flow patterns .md oil recoveries expected when apolymer gel system is injected into a heterogeneous reservoir. Inthis paper, such calculations are presented, assuming a simple mcdelof the gel kinetics and bebavior in the porous medium. These cal-culatiom have been carried ow witi a new chemicat tlcd simulator,Simulator for Chemical Oil Recove~ and Polymer Injection(SCORP1O),developed at Winfriti. SCORP1Ois a genemd-purpxe,mukiphase, mukicomponem chemical-flood simulator fhat may beapplied to polymer, surfacta.nt, or caustic flodlng o either the fieldor laboratory scale. This paper includes a description of the simwlaters underlying mathematical formulation, amplifying those fea-tures that cleat with in-situ gelation. Results from in-siru gelcalculations are discussed in some detail, particularly the effectsof gel formation rate and pore blocking on oil recovery efficiency.

Simulation ModelSCORPIOis based on the metbcd of tinife differences and is desigmdto be flexible enough to handle a wide range of petroleum engi-neering problems. Up to 10 chemical componems may be imluded,and these can be distributed among up to three liquid phases (aque-ous, oleic, and micellar). Flow may be treated as either compres.sible or incompressible, making due allowance for rockcompressibility. Two new features for a chemical flood simulatorare included in SCOSPIO(1) a generalized mcdel of chemical reac.tion that defines the rates and stoichiometry for any series of cou-pled chemical reactioms between components witMn a given phaseand (2) a heat-bal?mce equation that allows calculation of tempera-ture fronts and contours within a reservoir caused by injection ofcool water into a hot reservoir. Heat flow across reservoir bound-aries is treated with aquifer-type models.

The calculated temperature may feed back onto such physimtproperties as fluid viscosi~ and reaction rates. Some calculationsthat use facilities of this type have been presented previously in

C.aPY,@,987S.acie!y0?Petro[eumEngineers6?4

a study of the effects of temperature on the chemical degradationof polymer i a stratified reservoir system. 3 This module is usedto describe the chemical reactions tiat fake place i gel fomtions.In addition, modules are available to model the influence of mm.pled adsorption of compocrds and the effects of frontal spreadinglhrough velocity dispersion and molemlar diftisiom The cede canaccommodate several different rock types for relative permeabili-ty. adsorption, polymer residual resistance, and capillary pressure,in addhion to allowimg for regional variations in rock permeabil-i~. ~j~iOn ~d pr~OctiOn wells ~ ~ O~~t~ under ei~er PICS-sure or rate constraints and may be completed in any number oflayers of the reservoir.

The compositional fornwlation used in SCORP1Ois m extemioof that introduced by Acs et al.4 and subsequently used by~~er~5.6 i heir de development work. This approach has b~na&pted to chemical flooding and has proved a good foundation onwhich to establish the framework of the sinmlator.

Simulator Equations. fn thk section, the differential equations thatgovern fluid flow in rhe reservoir are presented, together with astatement of the physical significance of the various terms represem-ing important phenomena. The corresponding difference equationsand tie solution strategy used to solve for the primary variablesof the system are described later.

If we assume that there are N, chemical components, includingwater and oil, then for each Component i, conservation of massyields the following differential equation i standa.td oil mser-mirnotation

a(dfi;)++~scj%+qi=, . .

. &(1)

where

trii=zpmsac; +QRr;l+. . . . . . . . . . . . . . . . . . . . . . . . ...(2)a

WE ReservoirEneineerine. No.emba 1987

Mass- rather than volume-based equations are used because fhe mainrelationships used when describing displacement processes arc basedon the mass conservation of each component. The quantity fi;, re-ferred to as the mass density for Component i, is the mass of Com-ponent i per imit volume of fluid. In general, this includes theadsorbed and mobile components, as indicated in Eq. 2.

The miss concentration of Component i in ,Phase a is denotedby C: and is equal to the mdss of Component i per unit mass of,Phase w In the present study, the phase will be either aqueous oroleic. The term OR represents the density of the rock, includingits pore space, whereas pa denotes the density of Phase ct. Con-tributions from molecular, diffusion and velocity dispersion ire in-corporated in the, term D:. Gel formation is handled through thereaction term, R; , which accounts for the formation of Compo-

nent i in Phase a; this is represented as the mass of Componenti per unit volume of Phase c-iper unit time. The well terms are rep-resented by qi, which is the mass of Component i being inject-edlproduced per unit volume per unit time.

Along with Eq. 1,which is used to evaluate the change in massof Component i, the model also requires the derivation of a pres-sure equation. This is usually obtained by summing Eq. 1 over alli ad using constraint relations on the C;. In the present forrmda-tion, however, a differen: approacti is adopted.

The pressure eqwdion IS obtained by introducing the conceptsof fluid and effective PV. PV is taken to be a function of pressurealone, whereas the timctional dependence of the fluid volume, V,

fis smmizrized as V= Yf(p, T, ~), where ~ = {ml, .P%Pressure is referred to by p and dependence on temperamre ~asbeen included explicitly in T; mj represents the mass of Compcrnent i. In tie applications to be described in this paper, however,it is noi necessary to include explicit dependence on temperaturebecause it is reasonably assumed in chemical flood processes that.variation in temperature takes place slowly within the reservoir.Of course, this would not necessarily be so in the presence of avapor or, gas phase. Thus differentiating Vf witi, respect totime gives

avf, afi ap avf ami~=---z+ ~= . . ........ . .,...........(3)

The partial volume WITI8V/am; represents the increase in thefluid volume caused by the addition of a unit mass of Componenti. As Dodge23 showed, these partial volumes are related to the totalvolume by

~f= ~ !vm~a,ni;. . (4)

Use of the product mle for differentiation shows that

Substimting this into Eq. 3 and adopting the relation in Eq. 4 andthe definition of mass density, fit =nzJVfi shows that

a~ avf ap Vf

[avfa(dfii)

~.+ ~ap at + i. hi al

( )1v:?at VfAgain the product rule for differentiation can be applied to give

SPE Resemoir Enginemig, November 1987

TABLE 1RESERVOIR ANO FLUIOPROPERTIESAND COMPUTATIONALMESH SPECIFICATION

FOR TWO-LAYERSTRATIFIEDSYSTEM

Resewoir PropertiesResewoir dimensions

length, x, ft 3,000width, y, ft 1,000thickness, z, ft 100

Mesh definitionNX 20NY 1NZ 6Ax, fl 150Ay, ft 1,000Az,ft,for four low-permeability layers 20

for two low-permeability layers 10Horizontal permeability, k~ (100:1 permeability ratio)

Low-permeability regiox k. =ky = 100 mdHigh-permeability region: k. =ky = 10,000 md

Vertical permeability, kvRatio (kv/kH) is varied in each region

between 0.1 and i O4.Porosity, .$

$=0.25 (high permeability)c$=0,20 (low permeability)

Injection rate34,500 ftslll =6,140 BID.Corresponds to 1 PV in 5 years and is maintained for allcases (see text). Wells are located at each end of thereservoir (NX= 1 and NX= 20) and are completed in alllayers except during gel placement.

Rock density, pRRequired to calculate adsohtion tevelsp.= 129,8 lbm/fts =2.08 g/cm$

Fluid Properties .ViscosiPj data, cp

Water viscosity, p ~ 0.5Oil viscosity, p. 1.0

Fluid densities, p, IbmIftsWater (polymer solution or gel) density, Q. 62.4Oil density, p. 50.0

Relative permeabilities, km, km(See Fig. 1 for these curves)The end points are the same in both region%SW.= 0.25 SW=0.22

Compressibility dataAll fluids are incompressible in these calculations

OiI in place, bblTotal OOIP=4.725x107 fts=8.415x106Target (movable~ OOIP=3.339XI07 ft =5.947 X1O3

which, when substituted into the previous Sic, leads to the tiredform of the pressure equation used in SCORPIO

.+ avf ap av, a(+mi) a+ ap-+~;=~;. . . . . . . . . . . . . ...(5)v,ap at , at

Physically, Eq. 5 can be interpreted by saying that the rate of changein volume o? fluid Eresent caused by pressure changes Plus the volu-metric rate of fluid influx equals the rate of change in PV.

The derivative .3(.#@/dt is eliminated according to Eq. 1. Thegoverning equations of the system are therefore Eqs. 1 and 5, whichmust be solved for the N,+ 1 primary variables of pressure andoverall component mass densities. Art additional equation is alsoneeded to calculate the temperature as a primary variable. SCOR-P1Oincludes such an equation, based on heat balance, but becausetemperamre dependence is not included in our current study, weomit further discussion of temperature and its calculation. Somedetails on this point are included in other work. 3

if the fliids and rock are taken to be incompressible, then onlythe summation term in Eq, 5 is nonzero.

Auxiliary relations that are required to solve Eqs. 1 and 5 in;elude the usual saturation constraint:

Xse=l, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . ...(6).

,wherc the saturation, Se, is defined in terms of the volume, Ye,occupied by Phase a and the total fluid volume, Vfi such that

sa=ve/vp .......................... . . . . . . . . . . ...(7)

An additional equation is provided through the capillary relation,

Pc=pap, . . . . . . . . . . . . . . . . . . . . . . . . . ,.., . . . . . . . ....(8)

where Pc denotes capillary pressure between the phase. pressurepa and the pressure obtained by solving Eq. 5.

The soltio of Eqs. 1 and 5 by use of the constmits given LiyEqs. 6 through 8 is described in the following sections.

Difference Equations. The first step is to apply block itegmtionto Eq, 1 to reduce the system of component differential equationsto discrete form, This implies the costmction of gridblcck volumesaround fixed gridblock centers with the subsequent ilerivation ofa tiite-difference equation for each block, Application of thk ap-proach to obtain the difference equations is particularly significantin reservoir simulation because the interface and bmmdmy condi-tions may be handled more readily.

Thus, suppose a typical gridblock face is dmtotcd by At, wherein dmee dimensions f= 1, . ..6. ad that Pdenotes tic center pointof this block, which has a vohime V?,. The distance between thisblock center and the neighboring block that shares the face At isspecified m ~

AX# =(Axp+Azp,)/2,

with Art and AxV at the edge lengths i that direction of the neigh-boring block and the block under mnsideratio, respectively. Thefinal form oFEq. 1 for the gridblock with center f, ad mmpoentmass tnl,p = Vrcptfzi is then taken as

+AtQi,tZ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

Thcsou~e term Qifor Blockt, icldes wcllad chemical rac-tion terms,

Qi=$~saRi+gi, . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(10).

and the coet%cient .@ is a compact notation for

A4&=(km/#JEJ& . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(11)

The time increment, At, is given as the difference hctween the be-ginning, tfl, andthcend, tn+l, of the timestep (Ar=#+l-tn),with a similar meaning for AIrq.

636

Quantities with a bar, such as phase density, denote evsfuationat block interfaces using weighted arithmetic averages such that

~t+ ,h=(Ax@tc +Ax@P)/(Axg+Ax/,).

Themass concentmuio, C~, a?dmobility quotient, k,=lw=, inEq.1I are calculated at block interfaces by use of either one-or two-pointupstteamin~. Rock permeability iscalculated by harmonicaveraging as follows:

k8, =ktktr(A.q+Axt, )/(Axttk2+Axtkt).

Note that the source term Qi,f, is indep&dent of the neighboringblocks.

For the pressure equation, the corresponding discredzation is ob-tained inaslightly different manner. As already stated, the fluidand PVs in a block are assumed to be equaf at the end of the .dmestep, At. AcseraL4showed thstasmall discrepancy betweenfluid and PVs measured at the beginning of the time intervat willarise because the pressures cslctdatcd at the previous time level arenot exactly correct, which leads to tie fluid in that block not fillingthe effective PV completely.

Therefore, starting with theassumptiontbat ingener-al

vf(J+AP, T+ AT, ~+b+)=VP(P+Ap)~t time fI+l wimp, T, md ,7? the primary variables at time t,linearization leads. to the required pressure equation:

()v,Vp a v Ap avf Ami avp~++x=.At ap At i ami At ap At. . . . . . . . . . . . . . . (12)

As explained previously, temperahye dependence hss been assumednegligible in the derivation of this equation. Such an assumptionwould not hold for thermal rccove~ processes. The term iirparen-theses is the volumetric error term it arises omfyio the case of com-pressible flow. Indeed, for incompressible flow. only tie terminvolving a summation over i remains. For either type of flow,Ami/Af is replaced for computational purposes by the expression; R. 9.l..

The partial derivatives in Eq. 12are treated in various ways., Fornonzero but tixed compressibility, we use. the relations

avp =Cpvp

ap

for PV and

sp. .~fvfap

for fluid volume, where the totaf fluid compressibility, c , is taken1to be a weighted sum by saturation of the individual p ase com-

pressibilities,

The fluid volume, Vfi is evahtated as the snm of the individualfluid-phase volumes, Va, such that

v,= xv. ,.

where Vu= (mass in Phase rY)/Pe.In the present work, we assume tiat atl components apart from

oil partition wholly into the aqncous phase. Oil partitions totally

SPE ReservoirE@min& November 1987

into the olcic uhase. Thus the Dartkdvolume term for a typical wid-block ii com@d is

.

avf I

ami pm

where mi is wholly partitioned into Phase a.As discussed by Watts, 6 the discrete form of the pressure equa-

tion, Eq. 12, presents an attempt to find the pressure that causesthe fluid volume in a block to fill the PV of that block exactly. Forinstance,, if there is too much fluid in the block, the equation shouldforce the pressure to increase, thereby compressing the fluid in ad-dition to forcing some of it to leave the block. If just the right pres-sure is found, the fluid remaining in the block at the end of thetimestep should precisely fill the PV of the gridblock.

fn summary, the key equations are therefore Eqs. 9 and 12. Thesolution stratefg for these equations is dkcussed next.

Method of Solution. The methcd used to solve &e fundamentalsystem of equatiom for the primary variables is based on the familhrimplicit-pressure, explicit-saturation (3MPES) approach, 8 exceptthat here the component masses are evaluated explicitly. The cd:culation of the phase saturations is then performed with the expticiftydetermined component masses.

Thus we solve Eq. 12 implicitly for the oil-phase pressure. Otherphase pressures are obtsined through the capillary relation, Eq. 8,where the basic fMPES assure tion that capilkwy pressure is con-

~stsnt over a timestep is used. For each Component i, Eq. 9 issolved to obtain the change in mass, Ami, explicitly for each grid-block. Given the mass of Component i in each gridblock for alli, the mass in each phase is evaluated, somedmcs referred to asthe flash calculation. If the fluids are compressible, tie new phasedensity is calculated (it is currently assumed that density is a func-tion of pressure alone). Oti&iwise rhe overall calculation goesstraight to the evaluation of the volume, V., occupied by Phasea. It is then a simple matter to obtain the phase saturations at time~n+I from Eq,~.To adv~ce to the next timestep, the sam~ti0ris3component masses, and phase pressures arc used to evaluate thevarious coefficients appearing in Eqs. 9 and 12. The timestep isupdated and the process of solution repeated, starting with Eq. 12.

In our model; wells are treated explicitl~ fhey maybe complet-ed in ay number of gridblock or layers of the reservoir. The con-trols can be either pressure or.rate consrmjnts and may be switchedfrom one to the other under prescribed conditions. Rates can bedefined in terms of mass or volume.

The discrete-pressure equation (Eq. 12) gives rise to 2 symmet-ric banded coefficient matrix that could possess up to seven bands,imldig the diagonal. SCORPIOincorporates direct, and iterativealgorithms to solve the system of linear equations. The direcfsolv-ers include the multifrontal method of Duff and Reid, 9 which isparticukly suited to solving sparse mattii systems whose sparsitypattern is unchanged from one timestep to the next. Tbe iterativeoptions include a preconditioned conjugate gradient method thatstems from the work of Meijerink and van der Vorst0 and Ker.~kw, t i We have found that thk latter solution technique is ex-tremety fart and efticient for selected premnditionings and is suifablefor relatively low numbers of gtidblocks rather than restricted tothe cbamcteristically highir number of gridblock USWJIYassociatedwith iterative solvers.

The SCORPIOcode is based on a modular structure so that themajority of modules essentially supply the various terms neededto construct the coefflci&rs in the difference equations. Once tbcsecoefficients have been provided, the main solver module will evalu-ate the primay variables, i.e., pressure and the N, componentmasses. .Othermodules will then determine from these primary vari-ables the seconda~ variables, such as component concentrationsin each phase present (the flash c~culation), phase densities (forcompressible flow), viscosities, phase vol~es and saturations, ad-sorption and reaction terms, relative permeabilities and capillarypressures as functions of satmadon, diffusion coe.fflcients, etc. Theprogram has been written to take as much advantage of the vector-p~cessing capabilities of tie Cray computer as possible. Wifh someminor modifications, however, it easily could be run on anothermachine.

SPE ReservoirEngineering,November 1987

Reaction and Adsorption Modules in SCORPIOTo perform calculations in polymer gel SYStetIIS,it is necessary tohave some generalized representation of the reaction terms RL inEq. 1. These tetis may be very simple and describe a single-stepA +B+ C reaction, or they may be more complex and describe amukistep process in gel formation. It is also necessary to describethe main physical chzacteristics of the reactants (polymer and cross-linker) and gel prcducts. The main properties that must be modeledare (1) the effects of the polymer and gel on the mobile aqueous-phase viscosi~; (2) the adsorptiqtl retention of polYmer and PIO-duced gel; and (3) any corresponding reduction in permeabiliv that,e,i,mnce fader,, ~R~results from the adso tionlretcntmn of material, i.e., residual

The first of these relations concerning the effective viscosity ofthe polymer gel systems is specified by the user tiom appropriateexperiniental data.

Reaction Module. The SCORPIOreaction package calculates theproduction (or degradation) rate of Component i in any giVCnPhase& This rate is allowed to depend on the concentration of every com-ponent in that phase, including itself, and on temperamre. Both mo-bile and adsorbed components (except water and oil) may beproduced or degraded by chemical reaction: The total mass changeof Component i in the gridblock is then calculated.

The reaction package has two main options for calculating ratesof change of Component i in Phase a caused by reactiotidegi-ada-tion, i.e., (a C!#Jt)rxn. The first of these uses tabular data enteredby the user to calculate this quantity, and the second uses an em-pirical formula.

The total mass rate of change of Component i caused by reac-tion, including both mobile and adsorbed material, is given by

where the adsorbid concentration per unit rock mass in confact withPhase a, ri,a,is assumed to be.identical for each Phase a, but witha different reaction rate for adsorbed material in contact with eachphase. The quantity fa denotes the fractioq of rock surfacein con-tact with Phase e and is a fimcticm of salutation and nettability.Note that f= is not necessarily equal to the safurafion, .%. For ex-ample, in a water-wet rock, the fraction of rock surface in contactwith the aqueous phase may be close to unity.

The empirical equation that may be used to catcutate the reaction-mte term is constmcted on the assumption that the rate of produc-tion of a component from a given reaction in Phase a is propor-tionstoaprcduct of powers of the concentrations of all componen~associated wifii that reatiiow i.e.,

a.;(:)= UO(CL), .(cJJj

(C3awexp(-cw+ I/23,

where the quantities a.. .aN+, are constants for each Componenti, and N denotes the total number of mobile and adsorbed concen-trations contributing to the reaction. In some simple reactions, theexponents a, aN are integers, but in general they are nonnega-tive real numbers. The tem~erature dependence of the reaction meis assumed to follow the Arrhenius law through the final term inthe equation. The quantity aN+ I.represents EJR, where Ea is theactivation energy of the reaction and R is fbe gas constant:

The quantities of each component produced by reaction over atimestep are recorded and incorporated into the overall material-baknce calculation in the simulator.

Simplified Gef Kinetics. Several smdies have appeared in the liter-ature on the kinetics of gcl formation. 1318This may be a multistcpkinetic process if complex mixtures of polymer and re@x systemsare used. For example, we previously studied modeling thePAM/Cr207~- Ithiourea system in experimental floods. 19

637

I mY.

Fig. lRelative-permeability cutves for the two rock typesin the model reservoir.

A simplified reaction scheme for gel formation maybe modeledby assuming that the reaction is of the form polymer (C I)+cross-linker (C2)+gel (C3), where the Cis refer to themass concen-trations in the aqueous phase. It is assumed that the gel formationis first order in each of the reactions, i.e., second order overall.Defining C! o and C2 as the initial concentrations of polymer andcrosslinker, respectively, the reaction rates may be specified througha single rate constant, K, where

1

-( )dC, = KC1C2,

Cl dt

and

1 ()dC3 =KC1C2.(Cl +C2) dtThis simple system may be integrated analytically to give the gelconcentration as a fynction of time r

Kc,c~(c,+c2)fq =l+ KC, C2t

Adsorption Modufe. The central problem in a generalized adsotp-tionlphase package inamukiphase, multicomponent simulatorsas follows: given thetotal mass densify fii inagridblock forallcomponents (i= 1, 2:. .N,), howmethese componenf.sdistributedbetween the Phases a(cz=aqueous, oleic, micellar) andthe rocksurface? A more restricted module is currently used witlin SCOR-!+o, butthii is being generalized, fnthepresent work, theadsorp-tion level of Component ion the rock surface maybe specified as

638

a general function of the aqueous-phase concentrations of all com-ponents and tempaamrc. This is perfectly adequate to describe pcdy-mer and gel adsorption onto the rock nmtrix.

The mass density of adsorbed Component i, ?iq, is given as (seeEq.2)

lfii=pwswcL+Pxri(c;..n/+3

wherepw, SW,and Ci aretheaqueous-phase densiW,saWmtiOn,andconcerdration of Componenti, respectively, and C,q istbe en-

tire concentwion vector in the aqueous phase. The adsorptionproblem in. this case involves solv@the fOllOw&&,setOf nOn-

Iinears imultaneouse quations for C~a!d. hence fYG. T) fOragiven mass densi~ vector, r%

,Z=O,.S,,Z.+PRF(ZW,7)1+.

When component adsorption depends on only its own cqcentm-tion (in the aqueous phase), it is possible to use a fast interpolationmmhodtos cdvet hisproblem. Iftheadscyptioni sothermdepcndson two or more components, however, an iterative Newton-.Rapbfmn20scheme is used. If the maximum number of componenbinvolved in coupled adsorption is N., then asmal syst?m OfN~nordinear equations must besolvcdto find adsorbed and mobilelevels of components This is a fairly straight fonvard process re-quiring the solution of an No xNa, matrix equation at ~ch iteration.

Many materials that adsorb onto reservoir rock do so !r.rcversi-bly; i.e., they desorb very slowly compared with other time scalesinthesystcm. Irreversible adsorption istreated intbemodnlebytracking the amounts of each component adsorbed in each gridblock.It is assumed that after some maximum irreversible level of ad-sorption is reached, further adsorption iseitber reversible or doesnot occur.

Residual Resistance Factors. Permeability reduction (pore block-ing) caused by adsorbed polymer or gel is treated through residwd*&tmce faflors, FR.12 These defmethedrOp inthe mObility OfPhase a, h=(usually tbeaqueotis phise), caustiby adso~tionasfollows

X. =k,,&aFf@j]These resistance factors are specified,through tabular input and maybe functions of all the zdsorbed species.

Computational Results.TheRcsemoir Model. A simple two-dimensional layered reser-voirsystcm was used in the calculations p:esentedin this paper.This model is adequate here because it shows all the main featuresof themechanism.of gel emplacement ina stratified systcrn anditallows ustoinvestigate several sensitivities, including thee ffect.sof vertical crossflow (kV/kH), gel reaction rate, and injection strat-egy, inastr~ght fOrward way.

Table 1 gives the reservoir and fluid characteristics used in thecdcukuions.rhere semoiris 3,000ft[9l5mllong~d LWfC[30III] thick and bas two distinct permeability regionsa low-permeability region of 80-ft [24-m] total Wlckness and a high-permeability streak of 20-ft [6-m] totat thickness. In most calcula-tions, che reservoir is represented by 20 gridblock in the x dlrec-tion and by sixv.mtical gridblock: four inthe low-permeabilityrcgionand twointie high-permeablli~ region. Calculations onafiner mesh show that all the mtin conclusions maybe reached withthis grid. The horizontal permeability contrast is taken to be 100:1,with the higher-petmwability streal at thebottom to reduce the help-ful effects of gmvity in the oil recovery mechanism. This high per-meability conhast is adopted so that very early water breakthmugbwill be observed, and in addition, a gel treatment would be expect-ed to pefform substantially better than conventional polymer flood-ing. The relative permeabilbies for the two reservoir regions areshown in Fig. l. Alinear saturation dependence isusedin tiehigh-permeabili~ streak. This and the fact tludtthe mobility ratio in thksystem is very favorable (0.67) produce piston-like oildisplace-ment. Thus polymer will not contribute to the oil recovery mecha-

SPE Resenmir Engineering,November 1987

.,

TABLE 2PROPERTIES OF MODEL BIOPOLYMER AND SYNTHETICPOLYMER SYSTEMS USED IN THE CALCULATIONS PRESENTEO

PropertyInjection concentrationViscosity

Adsorption/retentionHigh permea~lity.LOW permeability

Residual resistance factors,FR

High permeabifkyLow permeability

Polymer degradation

Injection strategy

W8S of polymer injected

Amount of oil recovered, perunit mass of polymer

biopolymer(Model Xanthan)

1,000 ppm4 cp at injection conceritcation

(No adsorption)

1,01.0

Stable

Synthetic Polymer(Model PAM)

1,500 ppm2 CP at injection concentration

0.28 x 104 (160 lbm/acre-ft)0.44x 104 (250 lbm/acre-ft)

1.s2.5

Stable

Q=24,500 ft31D(6,140 BID)throughout. Polymer injected for150 days after 450 days of waterlood (i.e., an 6.2% PV slugof polymer)

32.3 x 104 Ibm 48.4x 104 Ibm(147 tonnes) (220 tonnes)

306 Ibm oilllbm polymer (1 .0S 212 Ibm oill!bm polymer (0.75bb[llbm) bblflbm)

CommentConcentrations by weightFor viscositylconcentiation

curves see Fig. 2Given in fractional

concentration (Ibm polymerlIbm rock, assumingPR = 121.7 lbml~ts in thehigh-permeability streak ad129.8 lbmlft3 in the 10wer-permea~lity streak), astripping irreversibleisotherm is assumed.

FB is quoted for the syntheticpolymer at the mwimumadsorption level (see above]it is.taken to be a hearfunction of adsorbedconcentration

No polynier degradation isassumed in thesecalculations (see Ref. 3)

Because of the low visco&ieSand FRs for thesepolymers, large pressurebuildup was not found forthese injection rates

Results at 1,500 days

nism by significantly impmving microscopic displacement efficiencyin this region. Hence, any additional oil will arise through fluiddiversion and crossflow, which are the important mechanisms inthese systems. 3,21

Onc of the qmin objectives is to investigate plugging by a suitz-ble polymer gel in-depth. That our reservoir model has a largevolume and well spacing is ve~ important because the requirementof long gel times is implied. We intend to define some of the keypttrametcrs that are required for a successful application of thischemical system. We stress that we are not studying small near-well gel treatments,

Water amf ?olymer Ffooding Resufts. In both the water and poly-mer floods, water is injected into the whole formation at a rate of34,500 fc3/D (6,140 B/D [980 m3/d]). This corresponds to the in-jection of 1 reservoir PV in 5 years. It is desirable to maintain thisrate, although high pressure buildup may arise in some cases. Insuch cases, we define maximum acceptable differences between thepressures of the injectorlpmducer well pair. .

.The properties of the two polymer typs used in the calculationsarc pm.scnrcdin Table 2. We assume that .fheinjection water is quitesaline and hence that PAM is a poorer viscosilier than xanthan.However, this is compensated for by the pore-blocking propertiesof PAM. Fig. 2 shows the viscosilylconccntration curves assumedfor the two polymers.

Polymer flooding calculations were perfommd for both xamhanand PAM and compared with the corresponding waterflood calcu-lations over a range of vertical-to-horizonti permeability ratios from10-1 to 104. fn all cases, a polymer slug was injected over 150days after 450 days of previous watertloochg, when the water cuthad risen to 95 %, At tie injection rate used, this led to an 8.2%PV slug of polymer. The cumulative oil recovery profiles with timeare shown in Fig. 3 for kvlkH =0.01 for a waterflood, a xanfhanflood, and a PAM flood. Corresponding water-cut profiles areshown in Fig. 4. The effect of kvlkH on oil remve~ at 1,5CQdaysis summarized in Fig. 5. The following is noted.

8PE Reservoir Engineerig3November 1987

Fig. 2Concentration dependence of viscosity for the modelxanthan and polyacrylamlde polymers.

.,,.,,,,4 .,,, 5,,, --- .,. ,,0

~

u

:

: ,,

j. ------------------

: o

: ,6

m,. ,,,,,,0

,,_

i9. 3Cumulative oil recoveiy profiles for .waterfl.aod and]ast%case polymer floods. ~.

m===--__,7yL-z-= =-==

; 4.

v

/..m,.m,,,, m ,,,m -... ,,,,,. ,,, ,,,,,,,,00

1., For the polymer properties assumed in this work (fable 2,Fig. 2), both xznthan and PAM give very simifar amounts of oil,although they operate through slightly different mechanisms.

2. Both the waterflood and polymer tlocds recover more of thetarget oil. for the higher values of kvlk~.

3. Polymer flocding gives a larger incrementzi oil recove~ cont-pared with the waterflood at higher values of kv/kH because of in-creased oil crossflow from the low- into the higher-permeabilityzone during polymer flooding, as well as fluid diversion. 3.21

4. At the highest crossflow level investigated (~v/kH =0. 1), thepolymer produces anadditiotxd 9% of the,target oil, i.e., a 30%improvement over waterflood.

Analysis of these results has led us to choose the kv/kH =0;01case as our base case. This gives an improvement of about19%over the corresponding waterflood (i.e., a further 6% of the brgetoil), which, given that the simulation model tends to be rather op-timistic, in these systems, is sufficiently poor tojustify a geltreatment.

G&I Properties. In the simple kinetic model described above, itis assumed that the polymer and cmsslinker are in stoichiometricratios, whicli in this case are taken such that l,OWlppm by weightof polymer reacts with 20 ppm of crosslinker to give 1,020 ppmof gel. Therefore, the time taken for conversion of half of the1,000-ppm-polymer/20-ppm-cmsslinker mixture to gel is given by

1 _sxlo7f,A.

KC, C20 K

Thus, a 10-&y half-conversion time corresponds. to K=5 x.106day - 1; in our applications, werequire t,A to be about 100 daysand hence KG 5 X 10s day 1. The time profile of gel conversionfor a range of K values is shown in Fig, 6. Here, we are primarilyinterested in the longer gel times, which allow deep penetrationof the polymer into the large reservoir model.

The physical propetiies of the resulting gel in tbe porous medi-um must be de,tined to,describe bow the gel actually behaves inthe reservoir system. In practice, polymerl crossligkef fluids ap-pear to operate by first aggregating into small (pregel) clusters ofpolymer molecules and, in some cases,. subseqtkntly forming fulltl@e-dimensional structures. Even in the pregel stage, the ag-gregates will be retained and will reduce the effective permeabilityof the aqueous phase. It is this pore-blocking mechanism that isinco~orated into the model gel considered in.thk work., Neitherthe polymer nor the crosslinker is adsorbed onto the reck, and oidythe gel is assumed to be adsorkdretained. It is known that poly-mers and some crosslinkers, such as A13+ and Cr3 +, ~e adsorbedon reservoir reck. Ordy prelimikuy results ate ptesented here, tiow-ever, and further work is in progress to study the effects of ad-sorption in more detail.

The properties of the model gel are given in Table 3. Two casesare defined showing higher and lower levels of residud resistancefactor, FR. fn each of these cases, the FR in the low-permeabiiigJregion is taken m be four times that in the high-penneabitity w.@.Whife this factbr is somewhat arbkmy, it reflectt the fact that block-ing is moresevere if geI enters or is formed in the low-permeabilitystrata and follows earlier reported practice. 7.22However, the ab-solute value of the FR in the high-permeability region has beenfound m be a much more important parameter. As expected, the

i9. 5Effect Of vertical-to-horizontal permeability ratio on111recovety (after 1,500 days) for v@terfloods and polymerIoods.

,.! ---------------------------------

-------- ------------!:7, , !, .0.,,.,

,---, ,., , !, @A,,.e

g., - ,

,.,. ,!,,.,,,

8 ,5 ,, f r. ,s . ,,. ,.7,,

t, i

, !, . .! s !,, ,,,,,

,,, ,mAs,

640 SPE Reservoir Engineering, November 1987

.,,

TABLE 3PROPERTIES OF THE GEL USED IN THE CALCULATIONS PRESENTED

Property Values CommentInjection concentrations Polymer, Co= 1,000 ppm These are fixed for all calculations

.Cmssfinker, Co= 20 ppmRate constants K(d~y -l) t,fi (day) A range of rate constants is Studfee

20 te is the time for conversion of half80 62.5 the original material to gel (C and25 20 Co as above)80 62.525 20so 62.5

Polymer viscosity (C)=0.5 +1.75X103 C, This is a convenient analytical form at+1.75 XI06 c; C, = 1,000 ppm, the polymer

viscosity is 4 cpAdsorption/resistance ,, 1. Gel is assumed to adsorb

factors Polymer crosslinker does not adsorb irreversibly with a strippingisotherm

Gel Residual 2, Given in tractional concentration.adsorption resistance

Permeability.(Ibm polymerl[bm rock assuming

(1,2) factor (3) PR = 121.7 Ibmift in the high-

Case 1 High 5.65S x 105 (300 lbm/acr6-ft) 40permeability streak and 129.8 lbm/ft

LOW 8.844x 105 (500 lbm/acre-ft) 80 in the lower-permeability streak)

Case 2 High 5.658 x 105 (300 Ibmlacre-ft) 10LowS. Twocases with. higher and lower

8.644 x10-5 (5001 bm/acre-ft) 20 levels of pore blocking by gel arestudi@FR isquoted at themaximum adsorption level

levels of F;for the gel case are hi~her than those for the ordinarypolymer flood (see Table2) presented previously.

Remtfts of Gel Calculations. When a gelling system is injectedinto a reservoir at a constant rate, there is inevitably a pressurebuildup. To put these pressures in context, consider two limitingcases in our model reservoir system (1) when all injected waterenters the flooded-out 10-darcy streak, at the injection rate consid-ered here (34,500 ft3/D [980 m3/d]), then, if 100% water flowis assumed, AP across the reservoir is -140 psi [-965 @a], and(2) when all injected fluid is diverted at the same rate into the 100-mdIow-psnmeability tigion as a result of the total blocking of the streakby a ve~ efficient gel treatment; then, if 100% oil flow is assumed,the.correspottding AP across the reservoir is -2,300 psi [- 15.9MPa].

For simplicity in our calculations, we maintain a constant injec-tion rate at above, but we specify a maximum acceptable pressuredrop across the system of 2,300 psi [15.9 MPa]. If the pressurerises shove this value in a given gel flood, we assume that the floodis not feasible. This pressure drop does not include the pressuredrop between the wellbore and the adjacent formation. This de-pends very much on the local conditions, such as the well skin fac-tor and well fracturing, and therefore it not included. Only relativelylow-viscosity fluids ~e being injected into tbe fimnation (maxi-mum polymer viscosi~ is 4 cp [4 mpa. s]), however, and unlessthere is considerable reduction in pe~eability close to the well,injectivity should not be impaired vew significantly.

In all cases,, the 8el injection strategy is similar to that for thepolymer cases discussed previously, The gel systems are injected,however, into only tie high-permeability streak for 150 days after

m/~ /JOO30

.jY//~.

- xANTHANCUMULATIVE(NOCROS5L1NKER1 / A>,x- ...-

/-wATERFLOOD CUWLATIVE ~//-e ..4-:-

_-.--=- V -----cm 1 XANIHkN SLUG

WAmnnooo&10* LONG1,,, 104 ,~, SHORT t,,,

R,lE CONSTANT. K

Fig. 7Effect of reaction rate and residual resistance factor on cumulative oil recovety (af.ter 1,500 days) and injectivity for pol ymerlcrossfinker injection.

8PE Rse;oir Engineering, November1937

,0

0

CASE I GEL CALCULATIONS---- XANTNAN SLUG WA: ERFLOOO

---------------------

__- - -J---

/ POLYHERORPOLYHER)CROSSLINKERIN JEC,, ON

Fig. 8Cumulative oil rec6ve!y profiles for waterflood, polymer flood,. and Case 1 gel calcuIations.

lg. 9Waler-cut development for waterflood, polymer floodmd Case 1 gel calculations.

450 days of water flooding. This avoids excessive gel blocking iiithe low-permeability region close to the injection well. Watertlo@i-ing is then resumed over all layers for the remainder of the flood.

[n the first series of calculations, with the data in Table 3, theeffects of both residual resistance factors, FR, generated and diegel reaction rate, K, were investigated. The effects of FRand Kon oil recovery (after 1,500 days) and maximum pressure buildupacross the resctvoir are summarized in Fig. 7. The maximum pres-sure value cccurs at the end of polymerlcrosslinker injection. Fkstconsider the higher blocking situation (Case 1, Table 3), whereatmaximum gel adsorption, the FR is 40 and 80 in the higlier andlower strata, respectively. A range of reaction rats with2.5x103 SKS8X 104 day-l is considered for Case 1; the max-imum reaction at which tie AP across the reservoir remains ~c-ceptablc is K=8 x 104 day 1. ~is is equivalent to t ,Aof 625 days,which is above rhe mrredy available practical gel times by a factorof about 20. For boih this case and that where K=2.5 x 104day -1, it is noted that increnmmal oil recoveries are considerablyabove those for either the waterflood or the polymer floods; thesetwo cases recover a futtber 17 and 27 % of the farget movable oil(quoted at 1,500 days), r+spcctively, compared with the polymerflood.

fA2..-

lle recovery profiles and water cuts for tlese cases are alsoshown in Figs. 8 and 9, where they are compared with the water-flood and polymer flood. Note that the gel applications show ap-proximately the sanie onset time for increased oil production rate(or wider mt drop). The better gel application lowers water cutsin the simple kesemoir model fmm -95% to as low as 32%, how-ever, compared with a lowest water cut of - 72% in tic polymerflood. Because of the permeability-reducing action of the gel, theeffect oit water cut is more persistent. Final water cuts in tie twocases shown in Fig. 9 ~e -85 and 80% compared with a worsepostflood water cut of - 97% in the case of the mo~$lV-cOntrOlpolymer. Contours of water saturation, residual resistance factors,and pressure Sre shown i Figs. 10, 11, and 12, respectively, ata rangeof times during the tlood for the Case 1 example withK= 8 x 104 day-1. The resistance factors in Fig. 11 are a meas-ure of where the gel is actually located. The coriespondmg watersaturations in Fig. 10 show the progress of the aqueous phase dur-ing and after the gel injection. Fig. IOa shows that very little waterhas entered the low-permeability region after 450 days. After thepolymerlcrosslinker reactants have been injected (at 600 days), ticwater invasion of the Iow-petineability strata is much improved,as shown in Fig. 10b through d..

It is inmuctive to examine the pressure contoup at different stagesof the flood shown in Fig., 12. Before polymerlcrosslinker injec-~on, pressures aie low (s 150 psi [s 1034 kPa]) and somewhathigher in the.streak than in the low-permeability region. The pres-sure reaches a maximum of.- 2,100 psi [- 14.5 MPa] after 600days at the end of chemical injection. At this time and over the next300 days, there is a lower-pressure region just in front of the gelslug that causes some crossflow of oil from the low- to the high-permeabiliw zones. The effat of this can be seen in Fig. 10b andc, where water saturations in tbe streak clearly drop a little. Thesecrossflow mechanisms are discussed elsewhere. 21 Long afief the.resumption of normat waterflooding, the pressure tield is as showni,nFig. 12d. The pressure across the system has:dmpped to -1,400.psi [-9653 !&a], and pressures in the high- and low-permeabilitystreaks have approximately equalized. tn this case, the effect ofthe treatment is clearly very long-lived because gel degradation hasnot been included in. our calculations. 3

If gel blocklng, FR, is reduced by a factor of four in both thehigh- and low-permeability regions (Case 2), then increased oilrecoveries are obtained only at higher gel reaction ratei, as shownin Fig. 7. For a value of K-4.5 x105 day-i (ZIL-111 days),however, the pressure constraint across the reservoir is again just

SPE Reservoir Engineering, November 1987

mm sANwrm6 .

a !Ca: am3 a 3m a 4m:. asm

a 6ma 7m

: am

t = 450 days (start of p.al~erlczoss linker injection) i: c-ml1. cm

t . 600 days (end of polymer/cro.s linker injection)

t = 900 days

, = 1500 day,

i9. 10W4W SatUmtiOn COntOUW during polymer gelation treatment (Case 1; K= a x 104Iay-i).

wm=hiw itsm=imum acceptable value. The oil recove~ forthis case is between the Case 1 values for K=2.5x 104 and 8 x 104day-l.

We not; that the FRvalues used in these two cases for the high-permeabilhy streak (40 for Case 1 and10 for Case 2) are relative-ly low. It has been found that if high gel blocking is assumed, thenthe penetration depth is much redu;cd and pressures become uriac-ceptably high, Gel adsorption levels are also quite low in this study(see Table 3) for the same rea.mu i.e., at high adsorption levels,the gel system cannot penetrate deeply into fhe streak. Higher block-ing Td adsorption levels may be acceptable for smaller systems;this is currently under study.

.%unmsry of Results. A number of points emer~e from thesepreliminary studies of gel formation in a layered system.

1. To obtain deep penemationof large stmthied systems, gels withve~ long setting times are required if pressure problems are to beavoided. No systems with such long gel times are currently known.

2. Both the incremental oil recove~ and pressure buildup incr.$aseas the pore blocking, FR, and reaction rate increase over the range

SPE ReservoirE@meting, November 1987

. .

of reaction rates studkd. For higher-blocking gels, however, thereaction rate must be sufflcicnOy low toavoid high pressure build-up across the reservoir.

3. It is clear from our calculations that the gel properties havenot been optimized. For example, in the reaction whereK=8.OX 104 day -1 in Case 1, Fig. 6 shows that the conversionof reactants is less than 62% after 1,000 days of reaction. Clearly,kjection cancenuntions and slug size of reac+ts could be oprimizgxtin terms of oil recovery for 8iven F.q values and rate behavior.

4, If the technical problems of designing a very lonXtime-settingpolymer)croislinker system can be overcome, then this approachmay have considerable potential in large reservoir systems wherestr@4ng is known to occur. The size of the reservoir studied inthis work imposes severeconstraints on the rate constant in the gelsystem. If these constraints are relaxed somewhat, so that requitedpenetration depths are about 2Cilrather than 1,000 II [60 rather than300 m] and setting times, t,h, are 30 to 60 days rather than 100m 1,000 days, then the time-setting properties of a suitable gel aremore technically feasible. Future work will examine the size seal-ing of systems in which gel applications may be performed.

643

ESISIM FACIM vNIEs

t = 450 days (,*care of polymerlcrms linker i.jec. ion) 1:

c = 600 days (end of polyme.1.ross linker ipjectiom)

r = 900 days

t = 1500 devs

z Om4.Om&c@& 003,korm12.6X14.mlKm.,anm20.m

Fig. 11-Residual resistance factors for polymer gel treatment showing deep penetration 0material. (Case 1; K=8x104 day-).

Concluding Remark? and Future WorkA number of simplified gel calculations have been carried out witha new chemical flood simulator, SCORP1O.This is the first time thatcalculations of this type have been presented in the Iiteramre. Thesimulator has performed very satisfactorily in these calculations.Taken as a whole, SCORP1Orepresents the cmrent state of chemicalflooding technology; it is flexible and has a wide range of versatil-ity. Care must be taken, however, with the well models whm bigh-viscosity fluids or permeability reduction of the formation are in-volved. Further work is in progress to incorporate h more implicittreatment of wells in tbe SCORP1Ocode.

Akhough our calculations are quite prelimina~, some interest-ing coriclusions on gel systems have been ~achd. The results inthis paper apply to large resefvoir systems coritaining very severehigh-permeability streaks in which we have attempted to specifycondkions for deep gel emplacement. Our most important conclu-sion is that, to obtain significant amounts of incremental oil whileavoiding excessive pressure buildup; the gel system must have a.acceptable combination of long gelation times and good

644

permeability-reducing properties in the high-penneabiliv streak(FR-10 to 20 in the cases stadied).

Areas of work that require further study include the following.1. More refined models of the gel kinetics that allow for the mul-

tistep nature of tie coagulation-type reaction andlor describe thekinetics of a time-delay redox system to produce tie cmsslinker.

2. The effects of interactions, such as those betieen themckmatrix and the crosslinker-e. g., when Cr3 + isthe cr:sslinkingion, it is known to adsorb onto the rock.

3. The effects of reservoir sizs in this work a large reservoirsystem was use~ that requires very long gel times hcycmd ,tioseof currently available polymer-gel technology, but in sma!ler sYs-tems, gel times on the order of 20 to 30 days may have considera-ble beneficial effect:

4. Beiter model descriptions of the physical bebav~or of tie gelin the porous medkun (in this work, the gel was adsorbedlrefainedand permea.biliy reduction was described through residual resistancefactors, FR).

5: A wider range of sensitivities than has been, covered in thispapertie effects of gelation rate, initial rw.cmnt concentmtions,

SPE Reservou Engineering,November 1987

c = 450 days (start of polymerlcro~s linker injection)

t = 600 days.(end of p.lpcrlcmss linker injection)

I t = 1500 days

PRESSURE VALUES:

I 2100..001700.00

: !300.004 900.005 500,00~. 200.007 150.00

100.00; 50.00

10 0.00

i9. 12Resem~ir pressure distribution through polymer gelat@ treatments (case I;(=8x104 day- ). ~~

slug size, permeability coritrast, and blocking characteristics need f. = frmtibn of Phase a in contact witi adsorbedto be examined. ,-

Nomenclatureaj = coefficients in general chemical reaction relation4P. = area of interface between Gridblock t and!,

fi2 [mz]C,fR total fluid compressibility, psi-1 [kpa -I]CP = pore compressibili~, psi -1 [kPa -1]c = compressibility of Phase a, pi-1 &Pa-] .$ = adsorbed component comqitratio vectorCL .= mass concent;tion of Component i in Phase a Ci = initiallinjection concetmwion of Component i

D = depth below line of zero gravitational potential, .ft [m]

D~ =, physical diffision of Component i in P@se, a,c t12/D [mZ/d]

~a = activation energy, cal/g mol. [kJ/kMOll

SPE R.?semoirEngineering,November 1987-.

rnataiaI~ = residual resiswnce factor for Phase a

g = gr2vit2ti0nal constantk = rock permeability, md

kra = relative permeabili~ for Phase o!kH = horizonml rock permeability, mdkv = vertical rock permeabili~, mdK = reaction rate constant for crosslinhg reaction,

&y-l

~ = vector of component masses, lbm Kg]mi = mass of Component i, Ibm [kg]

Aq = mass increment for Component i=mp+l rff,lbm [kg]

~i = mass densily of Component i, lbm/ft3 [kglm3 ].ML = coefficient =(k,a/pJp@C~

N = total number of adsorbed and mobile reactingcomponems

645-- . . . .

AJa = mtmkr of components involved in co~piedadso~tion

N, = total number of components, including waterpa pressure of Phase a, psi [kpa]AP = pressure increment over tinre=pn+ 1Pn, psi [kpa]Fe = capillary pressure, psi [kPa]

e = pressure drop across reservoir, psi [kPa]9! = SOUrCe/SiItk tCrm for Component i, lbtn/ti3 -D

[kg/m3. d]Q = overall source fsink term for Component i idding

chemical generationldcgradation, lbm/ft3 -D[kg/m3 d]

R = gas constant, cal/gmol-K [MJ/kmoI K]R~ = reaction rite of Component i in Phase & lbrnlt13-D

[kg/m3 d]S= = satrxztion of Phase a, fraction

f = time, daysAt = length of timestep =t +1 -f, daysts = half-conversion time of second-order crosslinking

reaction, daysT = temperature, F ~C]

Vf = total volume occupied by all fluids, ft3 [m3]VP = volume of Gridblock PI, R3 [m3]VP = effective PV, ft3 [m3]

Vpb = block PVVm = volume occupied by Phase a, ft3 [m3].Ax = incremental disfance, ft [m]Ti = adsorption of Component i

Ii,a = adsorption of Component i on rock surface incontact with Phase a

@= typical quantity to be evaluated at block interfaceskm = mobility of Phase a, l/cp [l/Pa.s]I% = viscosity of Phase a, cp [Pa. S]PW= density .of Phase a, lbm/t13 [kg/m3]PR = density of rock, including pore space, Ibmlft3

[kg/m3]+ = pOrOsity

Subscriptsi,j = component label

P = gridblock connected to Gridblock ?,f = current gridblock

rxn = term associated with chemical reacti.mw = water ;.a = phase label (aqueous or oleic)

4. Acs, G., Doleschall, S., and Farkas, E.; General Purpose Compmi-tiond Model, SPEI (Aug. 1985) 543-53.

5, Kendall, R.P, a cd.: .$Developmentof a MultipleApplicationReser-voir Simulator farsUse cma Vector Cmnpww,,, paper SPE 11483mesentcd at the 1983 SPE Middle East Oil Tectuical Conference,Bahr.uh, March 14-17.

6. Watts, 1.W.: ..A CompositionalFormulation of the Fmss.re and Sm-mticm Equation,,, SPERE (May 1986) 243-52.

7. Sorbic, K. S., Robem, L, J., and Foulser, R. W, S,:.. Polymer Flood-ing Calculations for HighlyStratified Brent Sands in the North Sea, Pmc,, Secmd European Symposium on EOR, Paris (Nov. 19g2)175-90.

8. Aziz, K. and Settari, A.: P.Iroltwm Re.?en$ofr Simudarfon, AppliedScience Publishers Ltd., London (1979) 135-38.

9. Duff, 1.S. and Reid, J.f$.: .The MukifrontalSoludon of IndefiniteSparse SymmetricLinear Systems,, Tntns., AWL for CmnPutigMachinery, Mafh. Software (19g3).9, 3~-25.

10. Meijerink,J.A. andvan&r Vorst,H.A.: A. IterativeSol.tim Meth&for Limar Systemsof Which the Cmtlicimt Matrix is s.SymmetricM-Matrix,,. Mtifh. COW.. (1.977) 31, 148-62.

11. Kershaw, D. S.: .Thc [ncomplefe Cholesky-Conjugate Gradient Methodfor the Iterative Solution of Systems of Linear Equations, J. Commit.Phys. (1978) 26,43-65.

12. Jennings, R.R., Rogers, J.H., and West, T.J.: .$Fa.!ors Iutl.emigMobilityCmtrol by PolymerSolutions,,>JPT (March 1971)391-401;Trans., AIME, 251.

13. Dovm, H.T. and Hutchim, R.D,: kDevelopmmtof a New Aluni-mm/PolymerGelSystemforPermeabilityAdjwrnem,, SSPERE (May19S7) 177-83.

t4. Pmdhomme,R,K. et al.: .RheologicalMonitoringof the Formationof PolyactylamidelCr3+Gels, SPEJ (Oct. 1983)8W-08.

15. Aslam, S., Vosso.gbi, S., and Wrllhite,G.P.: Viscom.trtc Messmeme.t of ChromiumlU-PolyacrylarnideGelsbyWeissenbergRheo-goniom.ter,,, paper SPE 12639 presented at tie 1984 SPE/GGEEnhanced Oil, Recovery Symposium, Tulsa, April [5- IS

16. Pmdhomme,R.K. andUtd, J.T.: Xinetiis of Polymer/Metal-IonGe-lation, paperSPE 12640presentedat the 1984SPEiD.OEEnhancedOil RecoverySymposium,Tulsa, April 15-18.

17. Southard,M,Z,, Green, D,W., ad Willhke, G.P,: Kineticsof theChromiumVtlThioureaReactionin fhePresenceof Polyacrylamidc,DaDerSPE 127[5mese.ted at the 1984SPEIDOEEnhawedOil Recov-~ti Symposium,?.1s., April 15-18,

18. Huang,C., Green,D.W..,ind Willhhe, G.P.: An ExperimentalStudyof theIn-sit Gelationof Chromium(+ 3)/PolyamylamidePolymeri.Pomw Mti,,, , SPERE (No. 1986) 583-92.

19. Sortie, K.S., Roberts, L,J., and Clifford, P.J.: ..Cdculado.$ m theBehavi.w of Tirne-SctdngpolymerGelsi PorousMedia,,, pqer 62.presentedat the 1985 AIChE NatL Spring Meeting, Houston, March24-2X.

20. ~cr~d, C.F.: Aj@ed Numericcd Analysis, se&mdedhion, Addison-Wesley PublishingCo. Inc., Reading, MA (1980) 15-20.

21. Clifford,P.J. and Sortie, K. S.: .Polymer Flooding in Stratifkd Sys-tems Recnvery Mechanisms andthe Effert of Chemical Degradation,

m EOR, Tmmibeim (Oct. 1984).22. Vela, S., Peaceman, D, W., and Sm.ik, E.I.: .GEvalwuionof Polymer

Superscriptn = timestep index

Corwersion facfms ceded m achieve mmistency are implicit ithe equations.

Acknowledgments -We thzmkK.L. Rolf for her assistance with the compwing and R.I.Hawes for his helpful comments on the manuscript. This work wastimded by the U.K. Dept. of Energy.

Reterences1, Balycky, J.P., Maii, B.B., md Milmz, G.: CAStudyof the Applica-

tion ofPolymmicGelsin Porom Media,,, paperSPE 10620presentedat the 1982SPEItl, SyrnpmsiwncmO,ltieldmd GeothermalChemis-try, DaUaS, Jan. 25-27.

2. Navradl, M., Sovak,M,, and Mitchell,M,S.: Dive~ng AgentsforSweepknprovemmts i ,FloodingOperationsLaboratoryStudies,,,paperSPE10621presented at the 1982SPE 1.0. Symposiumon Oil-field and GeothermalChemimy, Dallas, Jam 25-27,

3. Clifford,P.J, md Sorbic, K.S.; $TheEffects of ChemicalDegrads-tio on polymer Ftooding,,3paper SPE 13586presmted at the 1985SPE ImL Syrnposiun m Oilfieldand Gco!herm?JChemistry,Phoe-.nix, April 9-11.

dation,,, paperSPE5102 presented at the 1974 SPE Amu.1 Meet;ng,Hcmstm. Ott. 6-9.

23. Dodge,B.F.: ChemicalEngineeringT&nnodynamc.tM.Graw-.HillBook Co., New York City (1944) 104-07.

S1Metric Conversion Factorsacre-ft x 1.233489 E+03 = m3

bbl x 1.589873 E-01 = m3Cp x 1.0* E03 = Pasft x 3,048* E01 = m

ft3 X 2.g31 685 E02 = m3lbm X 4.535924 E01 = kg

lbm/ft3 X 1.601846 E+OI = kg/m3tonne X 1.0* . E+OO = Mg

.C..vers:.. fm.r !. e=+ SPERE

Odgin~SPEnmwScdptreceivedforreviewSepI.22,1985.PaperacceptedforPublic..Ho.June 11,1986,Revised rnanus.xlplmcelwd Sepl, ?7, 1986,PaPer(SPE 14234)firstPresentedat the 1985SW AnnualTechnic.)Conference..d Exhwtim held,. Lasvwa,,Sept. ,985.

646 SPE Rese&ok En8i.neetig, November 1987