- Analytical-Numerical Method in Waterflooding. .

HUBERT J. MORE~SEYTOUXJUNIOR MEMBER AIME I CHEVRON RESEARCH CO.

LA HABR& CAUF.

—.. ——.—..—--*qti., ... . ______

Predictions

ABSTRACT.

Methods of predicting the influence of patterngeometry and mobility ratio on water floodingrecovery predictions are discussed. Two methodsof calculation are used separately o+concrmently.

The analytical method yields exact solutions ina convenient fo~ for a unit rnobilit y rat io piston-likedisplacement, A/ewtypical pressure distributions,sweep efficiencies and oil recoveries aie presented/or various patterns. For non.unit inobility ratio,one may resort toanurnerical method,, such us thatof Sheldon and Daugherty. 1I‘: Because the domainsof applicability y of the analytical and nurhericaltechniques over!ap, the exact solutions provideestimates of the ewors in the nurnericul procedures.

The advantages of the analytical andnume?icalmethods can be comb fried. To develop a numericaltechnique as independent oj geometry as possible,the physical space is transformed into a standardrectangle, The entire ef feet of geometry is renderedtbrotigb one term, the ‘‘scale- fact~”, derived hornmappin~ relation% The scale factor can be calcu-.lated from the exact unit-mobility ratio solution forthe particular pattern of interest. By this meansrecovery performances for arbitrary mobility ratiocan be obtained for many patterns,, A sample ofresults obtained in t ha’s manner is presented,

INTRODUCTION

Pattern geometry and mobility ratio are two majorfactors in making a waterflood recovery prediction.Beca~se assisted recovery has become increasinglyimportant to the OH industry, pe ttern configurationand mobiIity ratio also assume a greater significancein the asses srneim of the economic value of recoveryprojects. The influence of pattern geometry andmobility raeio in shaping a recovery curve and onthe. other quentitie”s--ef interest to the reservoirengineer is the main subject of this paper. Mucheffort has already been spent on estimating quanti-tatively the influence of either pattern or mobilityratio or both on oil recovery. The literature reports

—

Orighal mwwwcipt reaeived h’ society of Petrolewm Wiin-soffice JUIY 21, 1964. ~evi-ed mmuSo~lPt of ~~ 985 receivedAuz, 3, 196S, Paper presented at SPE AWUIO1Fall Meeting heldin Hmmton,Oot. 11-14, 1964.

SEPTEMBER, 1965

many investigations of this nature. 3-g_H*wever,many results or methods of recovery predictionpresented in the literature cannot be consideredfully satisfactory. Even for unit mobility ratio andpiston-like displacement, where analytical solutions”are avaifable, the literature shows discrepancies.For non-unit mobility ratio, the divergence in theresults ie extreme. For infinite mobility ratio in arepeated five-~pot, depending on the investigator,the sweep efficiency ranges from O per cent to 60per cent. With reepect to the influence of patternon recovery, only the repeated five - spot hasreceived much attention. Other confined patternsand pilot configurations have received very Iittleattention.

Two calculation methods are presented in thispaper, either separately or concurrently: theanalytical “method of potential theory and thenu.rnericel method of finite-difference approximation.The analytical method is mor& restricted in scopethan the finite-difference method, but it has thedefinite advantage of providing exact solutionswithin ite range of applicability. If a unit-mobilityratio piston- like displacement is asaumed, theanalytical approach is poseible. A few typicalresults ate reported in this paper; the detaileddescription of the general method and of a great

,’

variety of results will be the subject of otherarticles. For non-unity mobility ratio, we must resortto a numerical scheme. The Immerical technique isthat which wae described by Sheldon andDougherty. 112 It is not limited to piston - likedisplacement. However, mainly single interfaceresults will be presented here. Because therespective domains of applicability of the analyticaland the numezical method overlap, uaefu 1 compari-sons’ of exact and numerical solutions can be made .for a variety of patterns.

The advantages of the analytical and numericalapproaches can be combined. The reason for thesuccess of this analytical-numerical approach canbe summarized in the following two points:

1. The numerical solution for arbitrary mobilityratio can be programmed. most efficiently when. the. . .physical space in which the displacement actualIytakes pla-ce is transformed @to a standard shape;and

2. Thie can be ddne with remarkabl~ simplicitywhenever an analytical expression for the pressure

847

discribumni is known for the unit mobility ratiodisplacement proceaa.

Aa a side-line {t is shown by a mathematicalproof that under certain conditions an approximatesolution for waterflooding predictions can lead toerroneous results. The approximate method Is onethat aima at predicating non -unit mobility-ratioperformances from a unit mobility-ratio flow net.When applied to piston-llke displacement, it leadsto the conclusion that sweep efficiency ia independ-ent of mobHity ratio. This contradicts well-knownexperimental data.6

ANALYTICAL SOLUTIONS

Both pattern geometry and mobillty ratio influencethe results of a waterflooding prediction. As usualwith a function of two variables, the relativedependence of a function on one of the two variablesia most easily demonstrated by maintaining theother con stant. To obtain an analytical solution,the mobi~!tv tatio was held constant at tbe value of

.

.one, If in addition, it is asaumed that the diaplace-menc is piston-like, that gravity and capillary effectsare either negligible or nonexistent, then the flowpattern of the two fluids ia the same as that of asingle fIuid. Consequently, the pressure at anypoint of the pattern does not change with time aslong as a constant flow rate is maintained. Oncethis pressure distribution is known, it is possibleto calculate the quantities of interest to thereservoir engineer—sweep efficiency, oil recoveryand water-oil tatio vs pore volume injected.

The pressure distribution can be obtained in aclosed form (in other words not in the form of aninfinite series) for a variety of patterns within the3 following categories: (1) isolated ‘pattern (e.g.isolated five-spot~ (2) doubly periodic pattern (e.g.repeated five-spot); and (3) singly periodic pattern.

The isolated pattern corresponds to a smallnu~ber of weIls in a large reservoir (mathematicallyof infinite extent). The doubly-periodic pattern (Fig.1) consists of a large number of rows of injectionand production wells reguIarly spread over a largefield (mathematically an infinite array). lo Thep&tern is constituted of an infinity of basic cellssuch as ABCD. Any cell can be superimposed onany other ceII by an’ integral number of translationsin the direction i31, or 6!2. Flow patterns ereidentical in each basic cell. A singly-periodicpattern corresponds to the case when one of thevectors al or 22 becomes infinite. The basic cellbecomes a basic strip.

It is convenient to calculate first the dimension-less potential distribution @ The pressure is thencalculated from the formula:

P=- 88.74 ~Q+c

---- ----- . --- .-. *-.. —.- :- . .—.- - (1) -

where p is pressure (psi), p viscosity (poise), Qflow rate (barrel/day), k permeability (darcies), hthickness of reservoir (feet), and the constantdepends on the boundary conditions. The potential

S4a---- . . . . .. -”.-...-. .-—. . . . . . . . . . .- .-. . . . . . . . . .

itself is obtained as the real part of a complexanalytic function, which is known as the complexpotential, w(z). The imaginary part Wof the complexpotential w(z) is known as the stream function. Inthe case of isolated and periodic patterns theconstruction of the complex potential is a relativelysimple task. Sample reauIts for a few patterns arebriefly presented for a homogeneous isotropicporous medium, Because of space limitation, ,allthe quantities of interest could not be listed for ‘allgeometries. To give an idea of the range of geometriesthat were investigated, each derivation is associatedwith a different geometry.

ISOLATED (N + 1)-SPOT

Typical of the isolated configuration is the (N+l)-spot, which corresponds to the case of N wellsregularly distributed on a circle with one wel 1 atthe center. Either all the fluid injected is producedor a fraction of it moves away from the pa:ter!l(mathematically is lost at infinity). Both cases areinvestigated.

NO LOSS OF FLUID AT INFINITY

h this instance, the expression for the complexpotential is

()~Nu(z) = A h —.. . , (2)

211N 1.8N

- The breakthrough sweep efficiency is obtainedby the same method described in a later paragraphon the staggered line-drive:

(E)=N . . . . . . . . .~b

. (3)N+ 2

. . .The” area of the circle is arbitrarily chosen as theunit of pore voIume occupied by the mobile fractionof the fluids:

~OMY PERIODICWELLARRAY

x I INJECTION WELL 0, PRODUCTION *LL

FIG. 1- DOUBLY PERIODIC WELL ARRAY,

90 CIETY OF PETROLEUM ENGINEERs JO URINAL

r‘-””-.. ... ----- .. . . . . .. . .. . . .

,. ,’-

-’”-”I

where cn(z,k) is the elliptic cosine lo c11 of modulusk. and K(k) is the comcdete elliptic integral of the

From the knowledge of the stream function Y andfrom considerations of relative areal spread of thedifferent streamtubes, the parametric equations ofnormalized oil recovery (E;) and water-oil ratio(Fwo) vs normalized pore volumes injected ( v~l )can be obtained. For an isolated three-spot (N = 2)

f~rst kind. Calculation; of the p&ential ~ieldI

4Y(X8y)=fi ha ( )l-m2z on2y ,

k2m2x + ktzmzzpthey are very simple:

V* , 1=—p% 1 -I- C08 4’Ry

+ tan (2v Y)

. . . . . . . . . . . . ..oo. e . . (6) I

where CFZXstands for cti(x,k), ctry for cn(y,k’), andk,’is the complementary modulus (k’z + kz = 1). Thevalue of the modulus k is determined from therelation

,,E; == (1-4Y) V*. +p%

F4Y

tie” ~”””””xt(k) d—=-- $. .”.... ..* (7)

2K(k) a. . . . . . . (4)

where V is the stream function parameter, varyingbetween O and % The corresponding curves areshown on Fig. 2.

LOSS OF FLUID AT INFINITY

Usually in a pilot flood only a fraction of thevolume of f[uid displaced by injection is produced.The capture factor @depends on the relative valuesof injection, production and reservoir pressure.

.Once @ has been calculated, it is possible topredict the performances of the ‘pilot flood (withinthe Iimirarions of tbe assumptions). Curves ofbreakthrough oil recovery vs capture factor and ofoil recovery vs pore volumes injected for samplevalues of /3 are shown respectively on Figs. 3 and4 for an inverted, isolated five-spot (injection atthe center of the ,parteri). The capture factor cancake values greater than unity if primary recoveryis still important.

For the repeated five-spot, which is a special caseof the staggered line-drive (d/a = I/2), Eq. 6 sim-plifies to

because k = k’. Th t term in k is omitted because

,.UNIT MOBILITY RATIO

,6-

5~

4 -“

.3–

.2-

.1— — ANALYTICAL S4LUTION

o0 .10 .20 .!0 40 .50 Eo 10 ,80 .w

ti

STAGGERED LINE-DRIVE

The staggered line-drive (Fig, 5) is typical of thedoubly-periodic pattern. Detailed analysis showsthat the complex potential is

CfiPTURE 6P.CTOR

— BREAKTHROUGH RECOVERY VS CAPTURE~AG6~OR, INVERTED ISOLATED FIVE - SPOT WITH

LEAK AT -... . .

IAPERFORMANCE CURVES FOR AN ISOLATED A

1.$- 3 3POT

SINOLE INTERFACE[.( ‘uNIT MOBILITY RATIO

- 1

%s-

1,0- -5=

J -

.8 -

.4- -2

— ANALYTICAL SOLUTION

.‘i- 0 u A tvuMERICAL SOLUTIONS -1

0

NVVERTEO ISOLATED ~ srOT* 5.0 - 01L RECOVERY ierws PORE - vOLUMES INJECTEO~ p.1.50> D cCAPTURE FACTOR

‘g 4,0 —, VOLUME OF PRODUCEQ FLW?

n VOLUME OF INJECTED FLulD

UNIT MOBILITY RATIO

j 3.0:

Ez p.lao

A 2.0-&~ “

8/3to.70

ml .0-

~.

“DD 1,0

.-a ,.-

NORMBLIZEO PORT. VOLuMfs INJEcTCoNORMALIZED PORE vOLUMES INJECTEO

FIG. 4 — INVERTED ISOLATED FIVE -SPOT, O~LRECOVERY VS PORE-VOLUMES INJECTED.

FIG, 2—PERFO&NCE CURVES FOR AN ISOLATi$DTHREE-SPdT.

. . ... .. . .. . ... . .. . ... . .. .,..-?.

-... . . .. .- -.

the potential is defined only to an arbitraryconstant, The breakthrough streamline by reason ofsymmetry is the diagonal line y = x and the break-through time (~b) is:

= 4n

I

J&=v2, . . . . ..(9)

2+R2 .u

Conseqtmntly the breakthrough sweep efficiencyfor a repeated five-spotlz is

~A)b = ~,=4x(1::,40?)2

= 71,78%

1 . . . .(10)

The result is strikingly simpIe and is exact.Breakthrough sweep efficiencies for the direct andthe staggered Iinti-drive, 13 for a flank flood and forother p?ttems can also be evaluated by this method.

The breakthrough sweep efficiency is interesting,but the complete recovery curve is more valuable.This recovery curve can be obtained’ from theknowIedge of the location of the water-ii interfaceas follows.

The location of the water-oil interface and therate at which this interface moves can be found inevery steamtube of the flow pattern by considers tionof mass conservation. Along the streamline W theelementary arc length is da, and dA is the steamtubeelementary cross-section. Then if we depart fromthe assumption of piston-like displacement-for amoment, expression of the conservation of massyields

$dA ~ +f’(s)du~= o,. .(11)ati aO

where qi is porosi~, S water saturation, / waterflow fraction and du is die volumetric flow rate inthe steamtube. If we now consider the streamlinesW and an orthogonal set of lines, the @lines, as acoordinate system, Eq. 11 transforms to

$’ = + J(Q, Y)q f’(S)~ = O~ ‘(12)at a+

-where- q-.=-&-n

is the flow rate per .unit. streamsube... -

and /(@t W)is the Jacobian of the transformation. 14< From Eq. 12 the frontal advance kquation, i.e., the

equation for the veIocity of a front of constantsaturation is

2s0

& ,= ‘1

dt TJ(t##Y)q f’(S)’ ● “ ● ● (13)

In the event the steamtube is straight, then @ =x,J = 1 and Eq. 13reduces to the well-known i3uckley -Leverett equation’. 15 Returning to the assumptionof piston-like unit mobility ratio displacement, Eq.13 becomes, after integration

I~_ d@ ...,..... (14)~ - J(@a Y)

For the repeated five-spot,

. . . . . . . . . . . . . . . . . . , (15)

so that after explicit integration and inversion theequation of the water-oil interface at a given timet becomes

co8h(8v@) - co B(81r Y)

8722 {$ . K(C08411y)a

COS411y}..C.O.S..... (16)

The breakthrough time of a particular streamline Vis obtained from Eq. 16 for the value O = m:

=l?w$x( e”),.,,,, ,(17)

At time of breakthrough of streamline W, theinstantaneous producing water-oil ratio is (1 -8Y)/By; therefore, the equation of FWO vs V~j curve isobtained:

~;{ .( ‘“o)= ,46LS{ ‘0 F1 + Fuo

‘} . (18)

Oil recovery and Fwo vs V~i curves for a repe~tedfive-spot are shown on Fig. 6.

REPEATED SEVEN-SPOT

The complex potential can be obtained for therepeated seven-spot and other patterns by super-position of simpler patterns. For the repeatedseven-spot, the complex potential is expressible interms of the elliptic cosine of modulus k = sin (15°)

-..

‘-

(1-cn2a). M.(z L .5.. &n...{? l+eflz g..-.... . .–.- .. . .... . ... . _.

and calculations yield for breakthrough arealefficiency

() T2‘A ~= i?K(16°) K’(l S”)

=74,37 %””’ ””’” ”(2°)

Analytical solutions have been obtained for awide variety of patterns. These sohttions yieldsimple exact results.

NUMERICAL SOLUTION

As mentioned earlier, the analytical method,however advantageous within its domairr ofapplicability, has serious drawbacks. It isdifficult to obtain anaIytic soh.trions when themobility ratio is not one. Therefore, at present, wemust resort to a finite-difference numerical tech-nique. The folIowing mathematical analysis andthe description of the numerical procedures carriedout in a corresponding computer program areessentially a summary of Sheldon 1 and Dou.gherty ’s2approach. However, in their work the results ofBuckley and Leverett, obtained for a linear system,were extended to two dim&nsions without proof.The more rigorous approach briefly outIined hereshows that this extension is valid.

MATHEMATICAL ANALYSIS

‘ For each fluid i the law of mass conservationrequires that the following equation be satisfied:

STAGGERED[IRE-DRIVE

tv

BASIC CELL FOR cn (z,k)

4K ‘ +-

C@lPLEX POTENTIAL

w(z).= $m In [m(z+ K,k)]

-. —.P67ENTiAL

.- —. -— .

FIG. 5 — STAmERED LINE’DRIVE.

where S{ is the saturation of fluid i and % thi totalvolumetric flow-rate vector (the fluids are assumedincompressible, gravity and explicit capillaryeffects are neglected, the medium is assumedhomogeneous). The fundamental information to bedrawn from Eq. 21 is that the rate of advance of agiven saturation surface in any direction is propor-tional to the total flow-rate component in thatdirection, and that the constant of proportionalityis /’(S)/+. This resuit allows generalization of theBuckIey-Leverett method to two-dimensional apace.

Eq. 21 can be rewritten in terms of pressure. If,moreover, the smooth saturation profi Ie behind theBuckIey-Leverett front is approximated by a seriesof steps, between two successive steps the satura-tion is constant and in this region (Rk ) Eq. 21simplifies to LapJace’s equation:

a2pk a2pk.v~~=~ +W=O, ””” (22)

where pk is the pressure in region (Rk).In addition, the following conditions, which express

“continuity of mass flux and of pressure at thesaturation-step surface, must be satisfied:

Pk ~Pk+l, .,. ,.. ,.. .(23)

where didn denotea differentiation along the localnormaI to the interface k and Mk is the total mobilityratio at the interface k. The total mobility, isdefined as the sum of the nobilities of. the fluidsin a region.

The riormal rvelocity component of the interfaceis proportional to the toral flow rate in thatdirection “and proportional to the slope of the secantapproximation to the fractiona I flow curve. Thus,

‘.= (’$)k=(i%+’)‘n.

‘-(=)’~‘“’*(24)where Ak is the toral mobiIity of the fluids inregion h. The coefficient of d#k/& in Eq. 24 isindependent of pressure, pattern geometry orlocation of the interface. It characterizes thedisplacement process. Eq. 24 can be written moregenerally:

Dk will be a different expression depending on the=recovery -’process 2.. (wate~ flooding, .~nriched gas. _

drive, carbonated water flood),

NUMERICAL PROCEDURES

: On the basis of the above equations, a computer

2s1

.. . .. .....,—,

.,...:.. --.w

.—

program was developed by Sheldon and Dougherty.Eqs. 22, 23 and 25 comprise what we refer to as

a “two-dimensional fluid-fluid interface problem. Ineach subregion of constant aaturation, Laplace’sequation ia replaced by a fini m-difference equation,

factor”. The point of the matter is that the scalefactora can be calculated from the unit-mobilityratio analytical solutions for the geometries ofinterest and then applied co the numerical method ‘for. non-unit mobifity ratio.

s taadard away- from interfaces or boundaries, specialin their vicinity, The first step in the calculationis to determine the coefficients for the pressureequations. Then the system is solved by the methodof successive -over-relaxation. Next the interfacevelocities are calculated. J3ecauGe the pressuredistribution changes with time, a pressure ‘solutionat a given time cannot be used to describe themotion of an interface over a long increment oftime. The time-step si~e is determined in such away that the maximum distance travelled by anyinterface over this period At will be a fraction ofthe mesh size. Then the new position of a set ofpoints on the interface is calculated. Volumes ofinjection and withdrawal, and incremental’ volumeswept are then computed. This concludes one cycleof operation. The cycle is repeated until thequantity of pore volume injected or the water-oilratio exceeds a specified vaIue. ,

COMPARISON WITH EXACT SOLUTIONS -

Numerical procedures are subject to errors. Onthe other hand, though limited to unit mobilityratio, exact solutions are avaiIabIe for a variety ofgeometries. An estimate of the error in the numericalsolution is possible by comparison with the exac’tsolutions. Figs. 2 and 6 are typical of the goodagreement even for such a sensitive quantity aswater-oil tatio.

ANALYTICAL-NUMERICAL SOLUTION

So far che analytical smd numerical methods were,considered independently, bur the advantages ofthe analytical and numerical approaches can becombined. In the numerical solution, the space inwhich the displacement takea place is transformedinto a standard rectangular form. The entire effectof the original geometry for the transformed prob Iemis expressed through a single term, the ‘‘scaIe

COORDINATE TRANSFORMATION

~~With a general curvilinear system 1Aof coordinate J(I$ v), di~ (VP) = O takes the form

+(<g)++pg)=o................... . (26)

where eqdq and e~df are the elemental Iengthsalong the lines < = constant and q = constant. Ifthe & q system of coordinates corresponds to aconformal mapping of the Cartesian system, thene(=eq = e and Eq. 26 becomes:

$+$= O,. . . . . . . . . . (27)

P

I.e., Laplace’s equati~n is invariant under theconformal transformation.. The interest of thistransformation is double. First, for a welI systemsuch as is depicted in Fig. 7, the transformationgreatly expands the critical region in the neighbor-hood of the wells, thereby facilitating more accuratenumerical computations. Second, the boundaryconditions (constant pressure or constant flowrate) and Eqs, 22 and 23 are invariant under therranaformation. However, I?q. Z5 becomes

;kI+&8pk ... . . . .k --%’

. (28)n

where e is the scale factor for the coordinatetransformation and ~$ is the interface velocity inthe transformed space. The scale factor is a pointfunction obtained from the transformation equationequation: < = ~ i- iq = <(z), via the expression

13 . PERFoRMANcE CURVES FOR A REPEATEDE? 1?-

5 SPOT

~.~

1?09 ,, SINGLE INTERFACE

UNIT MOSILITV RATIO~ lb -, 100

9 .Q:

:, 10

~, -0

~<

$1 M =

A ,5-g

:, 0 ●

$’w

~: o— AnalytiCal SOLUTION 29

d, o NuMER(CAL SOLUTION. .... . . .

b~– ‘-5!0 1$ ?0 ?$ 30 it 4/

NORMkLIZEO PORE VOLUMES tNJtCTED

FIG, 6—PERFORMANCE CURVES FOR A REPEATEDFIVE-SPOT.

lY /

:.

-EEEw$..PROOUCINOWtLL

‘c?%c

&

PRODUC{NO WELL

(: v (

s T ~,

6 ~ (, (z (, (a x

t

c INJECUOU WELL PFiOOUCIUO WELL

a. . . . .. A —---- . . . . . . . . .

INKCTION WELL ~

FIG. 7—INVERTED ISOLATED FIVE-SPOT COORDIN! “ATE TRANSFORMATION. ,

iiOCIETY ok” PE’I’ROLE1!M tiXCINEEl?s JO II RNAL

—.. .— .- —....- .— ..——,-. ———,.--. .-— -..

If the scale factor is known for a variety of geometry,performances of a particular well pattern areobtained by changing a single subroutine in thecomputer program with an otherwise unchangedmain program.

SCALE FACTORS

The complex potential was derived earlier for avariety of geometries. The equipotential andstreamlines corresponding to this complex potentialcan be chosen as the system of coordinates, andwith this system of coordinates the physical apaceis mapped onto a basic rectangle. Therefrom,

e2 =.‘(B) 2,

4m2J(Q8Y) =1+1

. (30)f z)

because the complex potenriaI is always of theform

u(z) = & in {f(z)} . . . .(31)

The function /(z) can be constructed ‘generally forall isolated and periodic patterns. A few exampleswere given in the section “Analytical Solutions ”.”Following arc a few typical expressions for thesquare of the scale” factor.

‘Regular Isolated (N + I)-Spot

y“

@2=e {li2 + lVe-2g

&N*, ,. {3*)

+ Ze-g Cosrl}

Repeated Five-Spot

.( )1.

4s?2= cosh4g + cos4n “2 . . (33)2

Staggered L,ine-Drive

~2 = [2( CoSh2~ _ S~y12Q)

~(k4e2E+k1be ‘2g+2kzkt2sZn2n)l

. . . . . . . . . . . . . . . . . . . . (34)

It must be noted that the scale factors irivolveonly elementary functions., and are thus simplerthan the pressure distributions which for doublyperiodic patterns involve elliptic functions. Knowingthe scale factor:, we can now proceed to uae thecomputer to obtain the numerical sohttion for arbi-trary geometry and mobility ratio.

SINGLE INTERFACE FOR ALL M_-.

E-ven though the- “assumption of” piston - l’ike-displacement may seem restrictive, curves ofwhich Figs. 8, 9, 10 and 11 are examples can beused by engineers for quick estimates. In Fig. 10

SEPTEMBER, ‘196s

.. . .. . .“- ----- —. .. . . . . . .. . . . ... .. . .. . . . . ... . . .. . . . . . . . . . . . . . . .

. . —. . . . .

Qpoinjectivity is defined as —, where Q ~a thekhAb

injection flow rate, F. the viscosity of oil, &the formation permeability, b formation thicknessand hp pressure drop between injector andproducer. In addition to direct use, these curvescan tie incorporate into standard predictioncalculation methods. For example: (1) the Dykstta-Parsons 16 method, which assumes one-dimensionalflow in layers of different permeabilities, can beextended to two dimensions; and (z) the Hurstsmethod, which ass”umes unit mobility ratio, arepeated five-spot pattern and a single layer ofaverage permeability, can be extended to non-unitmobility ratio and a number of weIl geometries.

Note that the curves of water-oil ratio on Fig. 9intersect one another. This behavior is alwaysobserved for, confined patterns, never for unconfinedpatterns. It can be proved that for confined patternsthe FWO curves must intersect. Let (Fwo)k, ( Fwo)ibe the water-oil ratio for mobility ratios M&, Mjrespectively. The total amount of oil that can berecovered after an urfinite time is the original ,finite quantity of oil irt place, which is arbitrarilychosen as unity. Thus, ‘

Vc=’z2=z=: 01RSC7 LINE C-””’-

(d/b.l/2)

IC1 ! .9 13 1? 2.1 M 2.9 !!. if 4,1 4,5 L’

kORhfhLIZED PORE VOLUMES INJECTED

FIG. S — INFLUENCE OF MOBILITY RATIO ON OILRECOVERY.

DIRECT LINE DRIVE (d/o* 1/2)

SINGLE INTERFACE

4

. 1

,1 .3 i ,1 .9 1.1 13 1.5 1.1 1,9 ?1 ?$ ?5NORMALIZED PORE VOLUME S INJECTED :

FIG, 9 — INFLUENCE OF. MOBKLITYRATIO ON PRO-, DUCING WATER OIL RATIO.

J2!ss

. . .. .. .. .. . . ..... .... . ... -~... . .-.....-”... . ...,”...... ......

. —..

1

“and, taking the difference,

. . . . . . . . . . . . . . . . . . . . . (3$)

13q. 36 cannot be satisfied unless the quantity

(F-wo)j- (Fwo)k changes sign, i.e., un&s,, the--F ~. curves intersect. Thus, an FWOcurve, for agiven Mk will intersect all the other Fwo cdr~escorresponding to a different mobility ratio MjThis result brings additional evidence that. th Lnumerical solution is correct.

Because actual displacement of oil by watereven tuaII y deviates appreciably from a piston-likeone as the mobility ,becomes more tinfavorable,estimates of oil recovery on the basis of singleinterface curves will be in error when the mobilityratio exceeds a value of the order of five to seven.The breakthrough displacement efficiency (ED )1may be known from a core flood or from a correlationof (ED); vs M. (ED); is defined by the relation

10.00

I.N

,(

.0

———- -.

)IRECT LINE DRIVE[d/a: 1/2)

/’--+”””c1 ----- -.

4

+M= I

Me+----,... .

“ ASYMPTOTE ‘--

n- ----- -M,= 2

NORMALIZED PORE VOLUMES INJECTEO .’

where ~w is the average water saturation in thecore at breakthrough, S~c connate water saturationand So, residual oil saturation.

The normalized recovery at breakthrough in apattern is for all practical purposes the product:

(displacement efficiency) x (sweep efficiency,. for M .= MBL) .

,.MB~ is the mobility ratio across the Buckley -Leverktt front. Again, this value of. hlBL can beobtained from permeability data or from a correlation .o~”hfB’L,VS M,MBL is usually much smaller than M,For M = 8, on the average, (ED);= 0.63 and MBL=1.50. From Fig. 11, (ED); = 0.67 for M = 1.50. Thisgives a breakthrough recovery of the order of 0.42,After breakthrough a good estimate of the shape ofthe recovery curve is obtained by selecting thecurve corresponding to the value of M equal toMBL. Drawing a line parallel to the curve corre-sponding to M = 1.50 (Fig, 11) from the breakthroughpoint (V~i = 0.42,’ oil recovered = 0.42) will yielda more reaIistic estimate of the recovery curve dran,the straight use of the singIe interface recoverycurve for M = 8.

BREAKTHROUGH SWEEP EFFICIENCYFOR UNFAVORABLE M

From Figs; 8 or 11, curves of breakthrough sweepefficiency vs mobility ratio can be deduced.Because sweep efficiency depends on mobilityratio and pattern geometry, we can hope to relarethe breakthrough sweep efficiency in a given’pattern for a given M to the value of M and to thevalue of ’breakthrough sweep efficiency in the samepattern, for M= I, (.f?A)~.For the repeated five-spotand three cases of the direct line-drive (d/a = 1/2,1 and 2) a simple relationship fitted reasonablywelI the calculated curves of area 1 efficiency forl+i~l:

I SINGLE INTERFACE

ii “t E-===——

-~-” -”,”-,- --.” .- ..- — I ..—...

.2 4 .6 ,6 I.L II !.6 1,6 1.8 u. 2.? M MNORMALIZED PORE vOLUMES INJECTEO

FIG. 11 _ INFLUENCE OF MOBILITY RATIO ON OILRECOVERY.

sOCIETY 6iF PETl\66LuIllt ENGISEEEt!i J(’JliftNAL -

.-. . .-..—..-. ..—.-. .—. — ..-. .— . .. .. .. . ... .-- ——.-. .-.—-—-- .-- —-—-

() LL1M ‘A b‘A ~ =(1+ $/ . . . . (38)

2

MULTIPLE INTERFACE SOLUTIONS FOR ALL M

Whenever the curves of re]ative m?rmeabilitv for. .oil and water are known, a Buckley - Leverettcalculation for the pattern can be performed.Results for a repeated five-spot geometry are ‘shownon Fig, 12. ”Relative-permeability curves used forthe calculations sre those of Fig. 13, The samepermeability curves were used to obtain theperformances of a direct line drive (d/a = 1/2)shown on Fig. 14. A comparison of experimental vscalculated normalized oil recoveries using thepermeability data and experimental results reportedby Douglas 17 is shown on Fig. 15.

L/‘0?$66101?1$ 1618’20ZZ

I24 26&ORMALIZCO PORE VOLUMCS IwEC?CO

FIG. 12 — tNFLUENCE OF MOBILITY RATIO ON OIL.-RECOVERY.

ANALYTICAL TEST OF ANAPPROXIMATE SOLUTION

The approximation consists of predicting oilrecovery in a pattern for arbitrary mobility ratiofrom the unit mobility-ratio flow net. For non.unitmobiIity ratio, the performances of a pattern willremodified in two ways because (I) the streamlines -are continually changing and (2) the resistivities . ...in the invaded and the non-invaded zonks are —.

different. If the shift in streamline is assumed (O ‘–-

be’ a minor effect, predictions can be made byperforming a BuckIey-Leverett calculation in thepotential fIow streamtubes. This now means@ppIY:ng Eq. 13 with a flow rate per unit steamtubeq, “which changes with time as the flood proceeds.

The value of the procedure can be tested bycomparison with experimental results 3 using actualpermeability data. It csn also be tested on itsability to accurately represent limiting cases forwhich reliable solutions are known. In particular,the application of the method for a piston - iikedisplacement throws some light on the nature of

10DIRECT LINE DRIVE ,ti,.*”

-/

j MULTIPLE INTERFACE /*’”’ I

- ““’.””---’- ““’”-”””r”-’””’- ““”b ! :0 Is 20 26

/--

RATIO

0,08 - ● .●

● :● *,

●●

●

1----,4 IAAA

AA

AA

S“w NORh@EO PORE VOLUMES INJECTED.. .

-. .-

FIG, 13 — RELATIVR PERMEABILITY CURVES FOR F~G. 15 —WATER AND OIL..

OIL RECOVERED VS PORE VOLUMES

IINJECTED. ‘

. ...... . . . . .-. .-— ..— . . . .. . .- -. — .- . —

the approximation. If a piston-like displacement isassumed, with a pressure drop maintained constant,the value of q is obtained from considerations ofthe total resistance of the steamtube in which thefrootha sreache dtheequipotential line@:

A@q=

(M-l- 1)4 + (-) A@’

. . . . . . . . ...* $* *...’ . .[39)

where A@is the potential increase from injectionto production well. Substituting g into Eq. 13 forthe particular case of a repeated five-spot, andintegrating yields the equation of the front at atime t in the ~, Ysystemof coordinates:

The equation of the front when water first breaksthrough is

aosh($wo) - 1 =8n2 {IIcoah(8w4) - C08(8ny1

().

2 M-1’.x[oos(41r YJI -; m

G(Qa Y)a

Cros(411Y) }””””””””””” ““(42)

A acudy of G(@,Y) indicates that-its limit whenAQ=M (i.e., rw= O) is zero, and that, even forextremely large values of rw, its contribution tothe argument of the elliptic sine is negligible evenfor infinite mobili~ ratio. In other. words, break-through sweep effi~iency for any M is practicallythe same as for unitmobility ratio. This holds alsofor post-breakthrough areal coverage. However,after breakthrough the water-oil ratio is differentfrom that of unit mobility ratio, because the flowrate in the steamtube having broken through is hi,whereas for! the other streamtubea. stiIl produe!ng..

efficiency on mobility ratio for piston-like displace-ment, in contradiction of experimental evidence.After breakthrough the prediction of the water-oilratio may be somewhat in error, but it does notconflict with experimental evidence. ..The resultscalculated by this method are shown on Fig, 16,These results differ considerably from those ofFig, 11, ‘calculated by the numerical techniquedescribed earlier.

When the displacement is ‘no longer piston-like,i.e., when a relative permeability curve is used tocalculate /’(S), this approximate method is likelyto yield improved resuita because the mobiIity ratioat a Buckley -Leverett front is often not in excessof two to three. .-

SUMMARY AND CONCLUSIONS

For unit mobiIity ratio, single-interface solutionscan be obtained exady in a convenient form. Thesesolutions are valuable per se, but the method toobtain them also yields the means to caIculate thescale factor of a coordinate transformation used inobtainitlg the numerical soIution for non-unit mobilityratio. This greatly expands the potential of thenumerical technique.

By the combined analytical-numerical approach,performances of a variety of patterns can beobtained. They can be obtained for a displacementthat is assumed piston-like (single interface) or foran actual displacement in a given porous mediumwhenever permeability data are available,

Single-interface curves of oil recovery vs porevolume injected for a given value of M can beused directly or with other. methods, for exampleas areal ‘coverage vs area processed in the Hurstmethod, thereby improving the method withoutadditional ‘effort.

Under certain conditions, the analytical methodof solution for unit mobility ratio allows an estimateof the error in the approximate method which usesthe unit-mobility ratio flow net in calculating non-unit mobility ratio displacement. This method willnot represent very accurately a displacement process

r I

‘gIo - REPEATED 5 SPOTSINGLE INTERFACE $’ .? f -? .$6 $:

i .9 .

? .8 -

e* ,1~

~ .6 -

gi -.

Y8 ,4 -s

RECOVERY CURVES AS CALCULATEDWITH THE ASSUmptiOn OF NO STREAM

: .5 - FLOW CHANGE w ITII M

.2 1.2 .4 .6 .8 . ..10. .12 l.!- !.! L!. .2) .22. . L- 1.$ : – , .

NORM~L12E0 PORE VOLUWS INJECIED

FIG. 16 _ RECOVEFiY CURVES AS CALCULATEDWITH THE ASSUMPTION ,OF NO STREAM FLOW

CHANGE WITH M..-

SOUIETY OF PETROLEOM EN211NEERS JO U22NAL

.- .- —

with a non - unit mobility ratio that is almostpiston-like.

a .

b=

Cn (Z,k) =

d=

du =

dV;i =

e .

W

~=.

k.=

k’ =

n=

NOMENCLATURE

distance between adjacent injectionwells

subscript for breakthrough

elliptic cosine bf argument z andmodulus k

distance between two adjacent rowsof injection and production wells

volumetric flow rate in an elernentatysteamtube

incremental normalized injectionvolume

scale factor for coordinate transfor-mation

fractional flow of waterfractional flow values on the two

sides of a front

fractional flow of fluid ireservoir thickness

formation permeability, or moduIusof Jacobian elliptic functions

complementary modulus

normal direction to an interface

p = pressure

pk = pressure in region kq =‘flow rate

r = radial distance

sn (z, k) = elliptic sine of argument z andmodulus k

t = time ,

= breakthrough time~’= breakthrough timefor streamline Y

-+U = total volumetric flow-rate vector

Uti = normal component of 2

vn’ = velocity normal to interfacew(z) = complex potential

x, y = coordinates in the physical space

z = complex n“u’mberDk = interface velocity constant

(EA )& = breakthrough areal efficiency

(EA)~ = breakthrough areal efficiency formobility ratio M

(EA )1 = breakthrough areal efficiency forunit mobility ratio”

(ED ); = normalized breakthrough displace-ment efficiency ~

E; = normalized recovery efficiency or oil“-rkcove-iy”in “units of ‘norm-alized

pore volume

F We, = instantaneous producing water oil. . ratio ‘

SEPTEMBER, 1965

..-- . ... . . . .. . . . ... ....-, ,.. . . . . .. . .... . . . . .. ... . . . . .

— . . . . .. . ..

j (@,w =

K, K(k),K(t$’) =

K’ =

M=

MB= =

Q=Re( ) =

Rk G

‘s=Si =

sor =

~$=VP =

V;i =P“

(?,(F’==

c.=

Jac#i~~)of the transformation (@,

complete elliptic integral of the firstkind

complementary complete eHiptic in-tegral .

mobility ratio (> 1 when unfavorable)mobility of water at residual oil

saturation=mobility of oil at connate water

saturation

sum of nobilities of water and oil atthe BuckleY-Leveretc front saturation

mobility of oil at connate watersaturation

mobility ratio at interface knumber of wells regularly distributed

on a circle in an (N + 1)-spot

total flow rate

Real part of ( )region bounded by interfaces k and

k+l

water saturation

saturation of fluid iresidual oil saturation

Sw- Swcnormalized saturation =

. 1- So,-swcconnate water saturation

average saturation at breakthroughsaturations on the two sides of the

frOnt ;

pore volume .

normalized pore volume .Y~ S“(1 -s- Sw,)Vp

nor~~lized pore volume injected

capture factor ..=

angle, angle in degreescomple”xnumber in transformed space

=g+r’q ~

A,hi = mobility, of fluid z’Ah = total mobility in region k

/4 Pw, /.10 = viscosity, of watet, of oil

~, q = system of coordinate in transformedspace

pi = density of fluid i= arc length

~ . porosity

@ = potential “,

@i, Q?p= @w = potential at the injection well, at.theproductiifn~ell.. . - . . :-. .

W = stream function:-“- al, W2 = periods of a doubly periodic function .

,.,./

9s7

-“/’

... ----- .. ... ..:. :.”

,., ,. ‘.

.. .. ..

. . .—.-—. . .—. _. _--.,-----

I

L

2,

3.

4.

.5,

6,

A@= potential increase from injection toproduction well

REFERENCES

Sheldon, J. W. kd Dougherty, 1%L.: “A Numerl?alMethod for Computing the Dyna~cal Behavior ofFluid-Fluid Interfaces tn Permesble Media”, Sot.Fe;, fhrg,Jour( June, 1964) 158,Dougherty, E, ‘L. and Sheldon, J. W,: ‘Whe Use ofFluid-Fluld Interface to Predict the Behavior of OilRecovecy Process’~ Soc. Pet. En& Jww.(June,1964)171.Higgins, R, V. and Leighton, A., J.: $’Computer %e-diction of Water Drive of Oil and Ges MixturemThrough Irregularly Bounded Porous Medis-fireePhaae Flow”, Jour. Pet. Tech. (Se@., 1960!O@M~@ket, M.: ~~~e F1OW of EIOfIIOgCWMrOUS Fluids

Through Porous Media?’, J.”W.”Edwards, Ann Arbor,Mich. ( 1946),Hurst, W.: “Determination of Performance Curve inFive-Spot Water Flood”, Pet, Engi (1953) Vol. 25,No. 4, 538.

Dyes, A. B., Csudle, B. H. and Erickson, R. A.:tioil pro~uction After B*eakthrough-As kfiuenced

by Mobility Ratio’; ‘f’fr?ns. , AIME(1954) VO1, 201, 81.

Prats, M., Matthewa, C. 8., Jewett, R, L. and Bsker,J, D,: $~Prediction of Injection Rate and ProductionHistory for Multi-Fluid Five-Spot Floode’ ~, Ttans.,AIME (1959) Vol. 216, 98.

$.

9,

10*

11,

12,

13,

14:

1s.

16.

17.

Hauber, W. C.: 4~Prediction of Waterflood Pe?fonesncefor Arbitrary -Well Patterns and Mobility RatioaJ~,&?WCPet. Tech (h., 1964) 95.

Jacqusrd, P,: t#CaI=u~ Num&iques de D6p1acement6de Fronta~\ 6tb WorldPetroleum Congress, Frankfurt/Msin (1963),

Whittaker, E. T, and Wateon, G, N.: A Course Oj

Modem Atrdysis, Cambridge University Press, 4thEdition (Reprinted 1962) 429-462,

hfllne,l%ome~, L. M.:~“Jacobien Elliptic FunctionTtiblea”, Dover (KWO),

Prats, M,, Strickier, W, R and Matthews, C. S,: %ingle-FluId Five-Spot Floods in Pipping ReservoIra?J,?kW2S,, AIME (19S5) Vol. 204, 171,Prats, ”M.: ~f~e Breakthrou@ Sweep Efficiency Ofthe Staggered Line Drive }~, Jour. Pet. Tech. (Dec.,1956) Vol. VIII,, 67.

KeUog, O. D.: Fourrdationa of Potential Theory,Dover (1953) 17S-183.

Buckley, S, E, and Leverett, M, C,: $~Mechsniam ofFluid Displacement in Sands”, T~dns., AIME (1942)vol. 146, 107,

Dykstra, H, and Psraons, R. L.: 4(Oil RecoveryPredicUon, by Water Flood”, Secondary Recovery ojOil in #be United States, 2nd Edition, API (1950),

Douglas, Jim, Jr., Peaceman, D. W, and Rachford,H, H,: ‘~A Method for Calculating Multi-dimensionalImmiaclble Displacement”, Trans., AIME (1956) Vol.216, 297.

***

—.,

... ---- —.. . ,. -.... .. ..:— ._ ..- —.. . . . . . .-. —.-----

{

SOCIETY OF PET NO Ll!ki M NxGIXEEUS JOtlaN.AL

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