352+

Mathematical

jxz y

Simulation of Polymer Flooding inComplex

P. L, BONDOR

G. J. HIRASAKI

JUNIOR MEMBERS AIME

M. J. THAM

MEMBER AIME

ABSTRACT

Simulation of polymer flooding in many complexreservoirs has requirements that preclude the useof either three-phase stream tube or two-phasefinite-difference simulators. The development of apolymer flooding model used in a three-phase,four-component, compressible, finite-differencereservoir simulator that allows the simulation of avariety oi complex situations is discussed.

Tbe polymer model represents tbe polymersolution as a fourth component that is included inthe aqueous phase and is fully miscible with it.Adsorption of polymer is represented, as is both(1) the resulting permeability reduction of theaqueous phase and (2) the resulting lag of thepolymer injection front and generation of a strippedwater bank. Tbe effects of fingering between thewater and polymer are taken into account using anemp in”cal “mixing parameter” model.

~,~g ?.esz!~ing simgjg~g? ~~ Cgpgbje o~ rne&@

reservoirs with nonuniform dip, multiple zones,desaturated zones, gravity segregation, andirregular well spacing and reservoir shape.

Two examples are presented. The first illustratesthe polymer flooding of a multizone dipping reservoirwith a desaturated zone due to gravity drainage.The second illustrates the flooding of a reservoirwith a gas cap and an oil rim with polymer injectionnear the oil-water contact. In this example, theeffects of nonuniform dip, irregular well spacingand field shape, and gravity segregation of the floware all taken into account. The two examplespresented illustrate the versatility of tbe simulatorand its applicability to a wide range of problems.

INTRODUCTION

The design of a polymer flood for a complex

reservoir requires a model that represents thereservoir features that have a significant ef feet onthe performance of the flood. These features may

Original manuscript received in Society of Petroleum Engineersoffice July 28, 1971. Revised manuscript received April 5, 1972.Paper (SPE 3524) was presented at the SPE 46th Annual FallMeeting, held in New Orleans, Oct. 3-6, 1971. ~ Copyright 1972American Institute of Mining, Metallurgical, and PetroleumEngineers, Inc.

1Ref~~~~ceS given at end of PaPer.

OCTOBER, ,972 Jpzr

Reservoirs

SHELL DEVELOPMENT CO.

HOUSTON, TEX.

%include the presence of a gas cap or a desaturated

zone due to gravity drainage in a dipping formation,the presence of an aquifer, irregular well spacingand reservoir boundaries, multiple zones, reservoirheterogeneities, and a well performance that is

limited by state proration, injectivity, andproductivity. These reservoir features are being

represented by most compressible, three-phase,

three-dimensional simulators. However, to modelpolymer flood projects, it is necessary to include a

conservation equation for the polymer, and to

represent the adsorption of polymer, the reduction

of the rock permeability to the aqueous phase after

contact with the polymer, rhe dispersion of the

polymer slug, and the non-Newtonian flow behaviorof the polymer solution.

PREVIOUS SIMULATOR DEVELOPMENT

Previous simulator development of polymerflooding has been reported in two different generalcategories: three-phase stream tube rnod.e!s andone- or two-phase, incompressible, finite-difference

simulators.Jewett and Sc hurz 1 developed a two-phase,

multilayer Buckley -Leverett displacement simulatorcapable of modeling either linear or five-spot

patterns. A mobile gas saturation also could be

specified, but this was treated as void space and

did not affect the flow characteristics of thesystem. Gravitational and capillarity effects wereneglected. The residual resistance of the brine

following a water slug was modeled as an increasein its viscosity; the viscous fingering of the brine

through the polymer slug was treated by alteringempirical relative permeability relationships to

specify a more adverse mobility ratio.Slater and Farouq-Ali2 modeled five-spot patterns

with a two-phase, two-dimensional, finite-difference—l. .— —-—!-—.:----- ..:------ J--- :11 --:... -I-L -..simulator, neglccung gIav ILy am Lap IL Idt LLy. IIICy

obtained an empirical expression for the resistancefactor of the porous medium as a function of atime-dependent mobility ratio.

Patton, Coats and Colegrove3 developed a

finite-difference model utilizing a stream tube

approach that could be used to simulate linear orfive-spot polymer floods in either a single sand

reservoir or a reservoir with several noncommunicat-

969

ing layers. The validity of their model was checkedby comparing the results of linear tests to aBuckley -Leverett analytical solution. Very good

agreement was obtained. The simulator. was then

used to examine the effect of traiIing-edge dispersionof the polymer slug. Their results indicated that

this dispersion had a negligible effect on oilrecovery.

CONSERVATION EQUATIONS

The conservation equations solved by the

simulator are:

V . [Awbw(vpw - pwg~)] + iw =

o ‘ads~ ( @bwSw) - – — . . . . (1)

at T at

V . [ Aobo(vpo - pog~)] + do

a=—((#)boSo) . . . . . . . ...(2)

at

V . [ XoboRs( Vpo

- pog~)l + v“ [~gbg(vpg

- pggvl))] -t- ;g = : (@boRsSo + r$bgSg)

at.:, , ,.. (3)

with the additional equation for polymer solution:

v“ [ Apbw(vpw - pwgVD) ] + ~p =

@ aQads: (@bwSp) + - —,. . . . . (4)

at z at

where Sw + SP + So + Sg = 1. The polymer solution

and the water comprise two components of the

aqueous phase existing in the reservoir; Eqs. I and4 above express the conservation of each componentseparately, with the conversion of polymer solutioninto water (due to polymer adsorption on rock) takeninto account explicitly at the end of each time step.

The oil-water and oil-gas capillary pressurerelationships are used with Eqs. 1 through 4 toobtain a single equation for the oil pressure, po.

The simulator solves the pressure equationimplicitly using either the alternating-direction-implicit (ADI) or the strongly implicit procedure

(SIP). The saturations are then obtained explicitly

from the equations above.

Transmissibilities are calculated using a two-point upstream weighting technique, which isreported elsewhere. Q

AD SORPTION

Adsorption of the polymer from the leading edgeof the polymer bank can cause a deterioration ofthe polymer slug. It” is important to know how much

adsorption will occur in a particular system in

order to optimize the slug size.The adsorption model used in the program

assumes that the reservoir rock instantaneouslystrips polymer from the polymer solution upon

contact until the rock is saturated. The maximum

amount of adsorption possible at any time is givenby the product of the rock’s adsorptive capacity

and the fraction of the block in contact with the

mobile aqueous phase. The amount of polymer

actually adsorbed is calculated, and the polymerand water saturations are adjusted. The strippingof polymer from polymer solution generates water in

front of the leading edge of the polymer, and thus

forms a stripped water bank berween the connatewater and the polymer solution banks. The detailsof the calculations are described in Appendix A.

MOBILITY OF POLYMER AND WATER

polymer solutions reduce the mobility of the

aqueous phase by two mechanisms — the increasedviscosity of the polymer solution and the reductionof permeability to the aqueous phase. Polymers

such as polyacrylamide continue to reduce the

permeability to the aqueous phase even after thepolymer solution has been displaced by brine. Thisreduction in permeability is denoted as the resiciuai

resistance factor.

Aw (before contact with polymer)R,f =

Aw (after contact with polymer) “ -.(5)

The polymer solution viscosity that is used in thesimulator is

*appP

=— ,.. . . . . .

pRrf

. . . . . .(6)

where the apparent viscosity,

Ap/L

P = kw —,. . . . . . . . . .(7)app

q/A

is measured at an average flow rate in the field.

The effect of the non-Newtonian behavior of the

polymer solution is discussed later.Polymer floods are usually considered as slug

processes where the polymer slug is followed by

water. Since the water/polymer mobility ratio isunfavorable, the water tends to finger through and,hence, gradually disperse the trailing edge of theslug. A mixing parameter model developed for

miscible displacements was used to account for

this trailing-edge dispersion.

The viscosity of a completely mixed polymer-watermixture in a grid block is expressed as a linear

sOCIETY OF PETROLEUM EXGINEERS JO URXAL

function of polymer saturation fraction:

s

IJm = vw+(v. QQ...h (8)P s

aq

s =Sw+sP“””””””””””

(9)aq

(This can be modified to conform with experimentalbehavior if the mixing is not linear.)

Effective values of water viscosity and polymerviscosity in this block are given by

l-w (JIw,., OGG =V’ww”” ”””” ””””(loa)W, CJ. L lm

l-w u).(lOb)

‘p, eff=pp pm’””’”””” “

where u is an empirical mixing parameter.Obviously, ~ = 1 corresponds to complete mixing;~= Q CQ no M.ixing (fingering with no transverse

dispersion). The value ~ = 2/3 has been found to

provide a good correlation with experiments insandpacks, while ~ = 1/3 correlates miscible data

in Hele-Shaw experiments that are believed to haveabout the same transverse dispersion as actual

.reservoirs. >

In the simulator, the permeability to the aqueousphase in a grid block is related to the amount of

polymer adsorbed on the rock. The permeabilityreduction iacror is given by

Rk= 1.0+ (Rrf _ 1- O)(Qadjij . (11)

and increases from unity to the residual resistancefactor as the grid block rock becomes satu~ated

with adsorbed polymer. The fraction ( Qad~/Ad) isthe ratio of the amount of polymer adsorbed in a

grid block to the block’s adsorptive capacity.

Appendix A gives the details” of the adsorptioncalculation. The permeability to oil is not influencedby the polymer. The polymer solution and water are

modeled as two miscible components of the aqueousphase with permeabilities as foIlows.

Sp kkraq(Saq)

k =— . . . . . . (12)..p

s Raq k

Sw kkraq(Saq).- /,7\

kw=— ... . . . {12)

saq ‘k

Hence, the nobilities of polymer and water in agrid block are given by

‘P ‘P ‘kraq(Saq)

1P = .— . . (14)

s Rp‘p, eff aq k p,eff

% ‘w ‘k.aq(saqj1 . .— (15)‘“w

s Rkpw eff‘w, eff aq 2

REPRESENTATION OF

NON-NEWTONIAN EFFECTS

RHEOLOGY

The theological behavior of the flow of polymersolution through porous media can be expressed in+ml-.n. -( theLG. ,.. = “. . . . . ‘~clnlww..”t ~~~co~~~y~~ Qf ~~~ p~!y.m~.-~=- . . . . .

solution. This apparent viscosity is measured as

Ap/L

v =k — . . . . . . . . .. (16)app w

q/A

Itmayalso reexpressed interms of the “resistancefactor” as

PJapp

‘WwRf, . . . . . . . . . . . (17)

where the resistance factor is defined as

b(18)

‘fe y” ”””””----

P

where the mobility to water is evaluated be/ore therock was contacted with polymer, The apparentviscosity defined in this way includes the effect of

permeability reduction due to adsorption or pluggingby the polymer.

The theological behavior of the flow of dilute

polymer solutions through porous mediaG can be

divided into four regions as shown on Fig. 1. Atthe limit of low velocities, the apparent viscosity

will approach a maximum limiting value 7-9 if thesolution is sufficiently dilute such that it does not

have a yield stress or form a gel in the pores.8~g

For a large range of velocities, the fluid is

‘/+., + PSEUDOPIASTIC~pm,n~ DllATANT

10G U

FIG. 1 — THEOLOGICAL BEHAVIOR OF POLYMERSOLUTION IN POROUS MEDIA.

OCTOBER, 1972371

pseudoplastic and the viscosity is decreasing withincreasing velocity. 7-1s At a higher velocity there~~ ~ m;m; .n,, m .t;cfiac;*y $bla$ is -“,,=1 rn ~~ g~ea~er,,,..’ .,,’-..! . .Us”u.. .yy... .“than the solvent viscosity. lo! 11,14 At very highvelocities the viscoelastic effects become importantand the viscosity increases with increasing

velocities.1 1,14-17 In this region, the theological

behavior is “dilatant”.

The pseudoplastic behavior of the flow of the,.-l., . ..-. ~fil...:f,., ●h.,.,. mh I-.,-..,.,le .“. EJ:.IJ’J’Y11’=’ ‘U’u”u” ““VW5” PJ’--- 11~~-~-

~afi ~~

modeled over a wide range of flow rates with apower-law model. The model that we have used isthe modified Blake-Kozeny model for power-law

fluids.12~ 18 This model has been used to correlatedata in packed beds over a wide range of Reynolds

numbers.12~ 13 The apparent viscosity can beexpressed with this model as

n-1P =Hu>. .o. o....- (19)

app

where u is the superficial fluid velocity. The valuesof the parameters, H and n, can either be calculated

directly from data on flow through core material orbe estimated from viscometric measurements. If thepower-law coefficient and exponent, K and n, are

calculated from viscometric data, then H can beestimated as

‘=:(9n:3)n(’’0k4=”’20)If viscometric data are used, the constants of the

equation will not include the effect of permeabilityreduction due to adsorption or plugging. Thepermeability, k ~, and porosity to water, (#w =@w),

are evaluated at the saturation existing in thepolymer swept area. This saturation is Sw = 1- So,win a region that contained original oil, and is Sw= 1.0 in any region that was originally 100 percentwater.

The modified Blake-Kozeny model was applied todata on the flow of Kelzan-M solutions through cores

with a residual oil saturation present. The core andfluid data are summarized in Tables 1 and 2.19 Theviscometric data for the three polymer solutions areshown in Fig. 2. All three solutions showedpseudoplastic behavior to a different extent. The

Core No.

1

TABLE 1 — CORE DATA

kw (red)

17.0 0,188

sorG

2 7.7 0.20 0.20

3 22.8 0.20 0.20

Core

No.

TABLE 2 — POLYMER SOLUTION DATA

Cone. ~brine(ppm) (CP) (Cp S:&-l) _n_——

1 Kelzan-M 200 0.84 7.6 0.67

2 Kelzan-M 300 0.45 2.25 0.85

3 Kelzan-M 300 0.45 5.5 0.69

power-law coefficient, K, and exponent, n, were

determined from the viscometric data (the actualVC+IIIF. C urt=r~ Ar=rerminc=d f~Q.rn. a IOQ-IOP DIOt of shear. . . . ...-” . . . . -------------- ._. --0 =-. .

stress vs rate of strain). The coefficient, H,

determined from Eq. 20 and the power-law exponent,n, were used to compute the apparent viscosity fromEq. 19. The calculated apparent viscosity using

the modified Blake-Kozeny model is compared with

the experimental values on Fig. 3. The maximum4:C1-.---- k=+..,a-.. ●L- nnl,-iiln*,aA .amA ewmarin-.t=nralULLICLC1l LC V= LWG=ll !..~~ ~=.ti-.-.~= -.’.’ -.-y ---. --&=---

values was less than 10 percent.The Blake-Kozeny model accurately reproduced

the apparent viscosity behavior of Kelzan-M

solutions for the systems considered here. Kelzan-Mis a type of polymer that provides a mobilityreduction by an increase in viscosity alone.

However, with polymers such as partially

hydrolyzed polyacrylamide, 21 a significant factor

in the mobility reduction is the reduction of the

permeability to the aqueous phase. If the parametersH and n in Eq. 19 are determined from flow

experiments through a core sample of the reservoirrock through Eqs. 16 and 19, then the effect of the

permeability reduction is included in the parameters.If the parameters are estimated from viscometric

measurements through Eq. 20, then the apparentviscosity calculated from Eq. 19 will not representthe effect of the permeability reduction. However,the development of the non-Newtonian well model

in the next section requires only the ratio of

viscosities and, thus, any permeability reduction

will be canceled in the final expression. The

y(sec-]]

Ioy 102 10’I I I r 1 I I I 1 1 I I I I

8

[

-c-SYSTEM Ib -A-SYSTEM 2

-x-SYSTEM 3 1

2

1

‘I I I I 1 I I I I I I

1 10 lo~

SHEAR RATE, yhOC-’]

FIG. 2 — VISCOMETRIC BEHAVIOR OF KELZAN-M.

I I I I I I I I II I 1 I J,.-4 ,.-3

u (cmkec)

FIG. 3 — THEOLOGICAL BEHAVIOR OF KELZAN-MIN CORES.

s7a SOCIETY OF PETROLEUM ENGINEERS JOURNAL

representation of the permeability reduction in

computing the nobilities for interlock flow wasdiscussed in the section on Mobility of Polymerand Water.

The dilatant behavior at high flow rates illustrated

on Fig. 1 may be detrimental due to loss of injectivityand ~e~~ib!e degradation of the polymer. This

viscoelastic behavior can be included in themodified Blake-Kozeny model. The resultant

apparent viscosity with this model can be expressedas

Hun-1

P= , (21)

app0< u

1.

1-

(1 - c&)d Isokwl @w

where 61 is the fluid relaxation time. (Similarmodels for the viscoelastic behavior have recentlybeen developed. z“~zl) The viscoelastic effect

becomes significant as the second term in the

denominator (which is equivalent to the Deborahnumber introduced by Marshall and Metzner 16)

becomes significant compared to unity. It will beassumed in the following that the polymer floodwi!! be designed such that the detrimental effect ofthe viscoelastic behavior does not occur in thereservoir.21

The power-law model and the resulting Blake-

Kozeny model represents only the pseudoplasticrange of flow rates. The maximum and minimum

viscosities at low flow rates and high flow rates,

respectively, could either be represented empiri-

cally or with a four-parameter model such as theMeter model. 22 However, for our purpose the

apparent viscosity was modeled to be Newtonian atlow and high flow rates and to follow the Blake-

Kozeny model in the pseudoplastic region.

( P , low velocitiesmax

[

n-1v= Hu,

apppseudoplastic region

w high velocities .(22)min ‘

This representation of the apparent viscosity is

shown by the dashed line on Fig. 1.It has been recognized23 that non-Newtonian

behavior in a pattern flood can result in a mobility

ratio that will vary throughout the pattern due toI itie~. For example, the mobility ratioy~~y~fig v~.Qc-.-_=

along the line between the injector and producer ismore unfavorable than at the offset corner areas.However, a more significant change in the polymer

mobility occus near the wells where the fluid

velocities are large.To illustrate the reiative significance of the

non-Newtonian effect near the well, consider theviscosity profile of the fluid-rock System 1 shownon Fig. 3. Assume that the fluid follows the

power-law behavior over the entire velocity rangeillustrated. The resultant viscosity profile withradial distance from the injection well is shown in

Fig. 4. The figure illustrates that most of theviscosity change occurs very near the well.

WELL MODEL

Representing the non-Newtonian behavior in ordythe intergrid-block fluid transmissibility coefficientswill not properly include the much more significant

effects near the wells. Unless the flow rate is solarge that the dilatant effect becomes significant,the low polymer viscosity near injection wells will

result in a higher injectivity than if rhe injectivity

were based on an average polymer viscosity in thereservoir. Thus, all the non-Newtonian effects arerepresented in the simulator by a well injectivitymodel. The required increase in injectivity overthat based on an average Newtonian viscosity is

obtained by introducing a negative, rate-dependent

skin factor into a radial-flow equation for eachinjection well.

The well model relates the Weii rates andpressure to the grid block pressure and saturationsby assuming radial flow in an

shown in Fig. 5. The external

radius of a circle that has thegrid b!Qck.

rAx Ay

annular region asradius, re, is the

same area as the

?- 1--- . . . . . . . . . . (23)‘e IIT

The rate of injection of polymer from the well into

a grid block is modeled as

10I 1 I 1

----- -----

z

~s -

*

I1 1 1 1

200 400 600 800 1000r(ft)

FIG, 4 — VISCOSITY PROFILES.

FIG. 5 — WELL MODEL.

OCTOBER. 1972 37’3

21?b (AaJj(pwf + -@n- pl)w xl

(24)

887(ln re/r - 1/2 + s + sp)w

The apparent viscosity model that is used to

determine this skin factor is the modified i3iaite-KQ~~fi~ .rnQd~i in the pseudopiastic region and a

maximum and minimum value of viscosity for low

and high velocities, respectively, as described

earlier. This model is represented by Eq. 22 and isshown on Fig. 1 with the dashed lines. The velocityprofile for steady-state, incompressible radiaI flow

will appear as in Fig. 6A. The viscosity profilewith the viscosity model of Eq. 22 and the velocityprofile of Fig. 6A will appear as in Fig. 6B. The

radii, rl and r2, represent rhe radius where theviscosity reaches the limiting values. If theminimum value of the viscosity is not reached inrhe reservoir, then rl is not defined. The pressureprofile with the viscosity profile of Fig. 6B will

appear as in Fig. 6c. The pressure profile, if theviscosiry were everywhere equal to pmax, is shown

with the dashed lines. The effective radius, r~, isthe radius where the pressure calculated from the

constant viscosity model equals rhe rrue sand facepressure. The apparent skin factor due ro thenon-Newtonian effect is then equal to In rW/rJ. The

apparent skin factor developed in Appendix B can becomputed as

Pmox,——-- ---- —---

+4 V’’C”’’TR”F”LELEI iI ,

I I

I I

1 I 1r, r=’ r,

10G r-

FIG, 6 — RADIAL PROFILES.

1 f / ~ (1-~))

L(-1

s. in vN~’ -v+l

P (1 - n) ) I

. . . . . . . . . . . (26;

if

NV<V

or

1

s=

P (1 - n)

if

NW>U

where

v =l.l’min/LLax”””””””””” “(2*)

H

(

,Wh l-nN .— 3.17 x 103 —

)“

(29)Pv

max q

The apparent skin factor is used as an integral

part of the well model. The injection rate, when the

well is constrained by the maximum bottom-hole

injection pressure, is a function of the apparent

skin due to non-Newtonian effects, skin due to well

damage, pressure difference between the well andgrid block, the grid block and well geometry, and

the total fluid mobility-thickness product of the grid

block based on Newtonian fluids.To illustrate the effect on injectivity of the

non-Newtonian solurion behavior, consider again~h~ {Ill id.rec~ ~y~~e.rn. I on Fig. j. A. SSUM_e that the----- ..-.fluid follows the power-law model over the entirereservoir. The value of H and n can be determined

directly from the graph as H = 0.14 cp(cm/sec)l-,and n = 0.67. Suppose the formation thickness andinjection rate are h = 100 ft, and q = 1,000 B/D.The viscosity profile assuming radial flow willappear in Fig. 4.

Suppose that the distance between the injectorand producer is 1,000 ft. Since enough data werenot given in Fig. 3 to estimate the maximum apparent

viscosity at low velocity, choose as the maximum

apparent viscosity the value at the midpoint betweenthe injector and producer, i.e., re = s00 ft. From

Fig. 4, the viscosity at this point is 7.3 cp. Supposethat the well radius is rw = 0.25 fr. The apparentskin factor from Eq. 27 is then sp = -4.81. This

represents an effective well radius, r;, of 31.0 ft.The ratio of injectivity with the non-Newtonian

effects and the injectivity assuming a Newtonianviscosity of pmax is

lNN in re/rw—. = 2.73

lNin re/rw + s

P

SOCIETY OF PETROLEUM ENGINEERS JO URiVAL

TABLE 3 — DATA FOR LINEAR SIMULATION

System dimensions 2ooftxloftx loft

Reservoir data:!=i~ja! ~jl saturation 0.75

Connate water 0.25

Residual oil to WF 0,22

Pore volume 1,069 bbl

Original oil in place 778.7 bbl

Waterflood movable oil 544 bbl

Po,osity, 40.3

‘b--” ‘- -- . ..-.. h.l)*w km =Wldl- F.?rqo!=””.. . . . .- 1 darcy

& at connate water 0.9

krw .at residual oil 0.3

Fluid doto:

Oil viscosity 20 Cp

Y%ter Wisccsi?y 0.64 Cp

Polymer viscosity 2.9 Cp

Concentration 300 ppm

Ad S 10 pg/g

R,, i.dGJ 0.666

Thus, the injectivity is greater by a factor of 2.73than if the non-Newtonian effect was not included.

SIMULATION RESULTS OF LINEAR TESTS

The simulator was tested on a one-dimensionallinear model to compare the adsorption and mixing

parameter models and to determine the effect ofnumerical dispersion on the predicted recovery.

For this study, a two-phase oil-water system was

modeled and tests were made using 5-, 10- and 20-

grid blocks. Model data are given in Table 3. Figs.7A, 7B, 7C and 7D present the polymer saturationfor a 0.2-PV polymer slug followed by 0.2, 0.4, 0.6and 0.8 PV water, respectively. Note that thetrailing edge of the slug is smeared, reflecting the

influence of the mixing parameter model for theunfavorable mobility ratio displacement of polymersolution by water. The influence of adsorption is

clearly indicated, since the sharp polymer front iskiggirig behind the injection front. With pistcm

----- -. ....- -.2 PV PULYMt K

.2 Pv w&TE!?

!___.6 .8 1.0

DISTANCE, X/L

FIG. 7A — EFFECT OF ADSORPTION AND DISPERSIONON POLYMER SATURATION’, LINEAR MODEL, & 4 Pv

INJECTED.

OCTOBEN, 1972

displacement and no adsorption, polymer solutionwould break through at 0.78 PV injected. With

a&sorption, breaicti-lroiigh --- . ..-uLLkLLcd at Q.g~ pv

injected.

These figures also illustrate the effect of

numerical dispersion on the polymer saturationprofile. Even though a marked effect is evident, the

indicated recovery given by the simularor is affectedto a much lesser extent; the oil recovery curves for

.2 PV POLYMER

‘!.3 .4 PV WATER

“o .2 .4 .6 .8 1.0DISTANCE, X/L

FIG. 7B — EFFECT OF ADSORPTION AND DISPERSIONON POLYMER SATURATION, LINEAR MODEL, 0.6 PV

INJECTED.

-.. .2 PV POLYMER

$ .6 PV WATER—

FIG. 7C —EFFECT OF ADSORPTION AND DIsPERSIONON POLYMER SATURATION, LINEAR MODEL, 0.8 PV

INJECTED.

a. 3 .(n

.2 PV POLYMERz-0 .8 PV WATER

tii.2 .4 .6 .8 1.0

DISTANCE, X/L

FIG. 7D — EFFECT OF ADSORPTION AND DISPERSIONON POLYMER SATURATION, LINEAR MODEL, 1.0 PV

INJECTED.

975

these three cases are given in Fig. 8. Note that the––:, - -.!–13 –-–. .––---: –-. --.:—–&––

coarser grlas ylela conservamve esumares of theoil recovered; recovery efficiencies at 1.2 PV

throughput for the 5-, 10- and 20-block runs were 72,

77 and 81 percent, respectively, of the original

waterflood movable oil.The simulation runs were undertaken using rate

constraints on both injection and production wells,.,. -/- ,(lu D/u Daianceri injection and production). Timesteps used were: 0.2 days for the first 10 days, 0.5days for the next 10, and l-day steps for theremainder of the run. Halving the time step andrepeating the run yielded results that agreed to

within 0.2 percent.

SIMULATION OF COMPLEX RESERVOIRS

A versatile reservoir simulator is most useful as

a tool that provides improved data on which to base

operational decisions. The polymer simulator

described above makes it possible to select optimal

flood patterns, determine the optimal slug size,analyze the profitability of a fieldwide polymerflood under several different operating strategies,

and predict the effect of many different reservoirand field variabIes on oil recovery and flood

performance.In many cases, simpler models (either laboratory

models or simpler simulators) will provide

sufficiently accurate information on which to baseoperational decisions. In other instances, however,

limitations inherent in the simpler models precludetheir use. In general, as either reservoir parameters

or field conditions become more complex, simplermodels become progressively more restrictive andless useful. An example will be presented in whichir was necessary to model the recovery processwith our mathematical reservoir simulator.

POLYMER FLOODING IN A VISCOUS OIL

RESERVOIR WITH A DESATURATED ZONE

Waterflooding a viscous oil reservoir with a

continuous gas zone (due to gravity drainage), Fig.

9, is characterized by the tendency of the gas zoneto act as a thief zone until most of the gas hasbeen displaced by water. Such a waterflood results

?5

.2

I/

oo~PORE VOLUMES INJECTED

FIG. 8 — OIL RECOVERY EFFICIENCY, LINEARMODEL.

TABLE 4 — DATA FOR CROSS-SECTIONAL SIMULATION

System dimensions 180 ftx20ftx10fi

Reservoir data:

Gas zone height 9 ft

Oil zone height llft

Porosity, + 0.3

Absolute permeability k 1 darcy

krO at c.annate water 0.9

& at residual oil 0.3R=5~~u=! =~! .!”*,,.”* ;-”-“.”,”, ,“. , Q, ~~

Connate water saturation 0,25

Initial saturation distribution:

In gas zone:

Sw 0.25

so 0.22

%0.53

In oil zone:

Sw 0.25

so 0.75

F!g~~ d“+”!-----

Oil viscosity, cp 20

Water viscosity, cp 0.64

Polymer viscosity, cp 2.9

Polymer concentration 300 ppm

Adsorption 10 ~g/gm

R,f 1.4U 0.666

Constant iniection rate 5.7 B/D

(for both polymer and water)

in very early water breakthrough, followed by verylong production at high water cut. In such a reservoir

polymer flooding might prevent the thief behavior ofthe gas zone by decreasing the transmissibility ofthe gas zone to the aqueous phase. The polymerwould generate an oil bank and provide productionat higher oil cuts.

SIMULATION OF A CROSS-SECTIONAL MODEL

An experimental program was undertaken to

evaluate the effectiveness of polymer flooding insuch a reservoir. A cross-sectional bead pack modelwas constructed that was designed such thatnondimensional parameters of the system were the

same in both model and prototype. A series ofexperiments were performed for both waterflooding

PROOUCING

WELL A WELL B WELL

‘U ‘‘ v-

FIG, 9 — RESERVOIR WITH DESATURATED ZONECAUSED BY GRAVITY DRAINAGE.

FIG. 10 — EXPERIMENTAL WATERFLOOD, CROSS-SECTIONAL MODEL.

S76 SOCIETY OF PET ROLE[!M ES GISEERS JOURNAL

and polymer flooding. The polymer flood of interesthere was run for 0.25 PV of polymer solution~fije~~e~, ~~~ &~~ f~!!~w~~ @ w~~~~<

A two-dimensional cross-sectional simulation wasmade of the waterflood and of the 0.25 PV polymer

iiooci. Data used are given in Tabie 4. ~~le ~~,m.tiiator

confirmed the qualitative behavior of theexperimental model.

Fig. 10 illustrates an experimental waterflood in

the cross-sectional model. Figs. 11 and 12 are.water and oil saturations obtained by the simularedwaterflood at the same time. The overrunning of theoil by the water in the desaturated zone is evident..No oil bank is formed by the flood water, and very

lirtle oil has been moved.Fig. 13 illustrates an experimental 0.25 PV

polymer flood where the slug has been followed bywater. Figs. 14 and 15 are polymer and oilsaturations obtained by the simulation of this

poiymer fiOOd. Note the formation of an oii banitahead Qf ~he PQi~rne~, NQte aisQ Lhat 50rne Oii iS

moved upward into the desaturated zone; if thedesaturated zone has a residual oil saturation lessthan waterflood residual, there will be resaturationlosses even in the swept portions of the reservoir.

Observed and calculated production curves are

presented in Fig. 16 for floods terminated at 98percent water cut. The extent of the vertical andhorizontal lines on the experimental data points

indicates the standard deviation of the experimentalresults. The agreement between experiment andsimulation is reasonable. Note that, while bothstudies indicate that the waterflood will recovermore of rhe movable oil in place (68 percent vs 61------- *L- --1.? --- ficc~ nh,n; ”c ~~~ Qii ~~ ~.~l-’--u), ‘“= PJJY lll=L ““. a.L.s

FIG. 11 — WATER SATURATION DISTRIBUTION,CROSSSECTIONAL SIMULATION.

I I 1 f 1~ FLOW

FIG. 12 — OIL SATURATION DISTRIBUTION, CROSS-SECTIONAL SIMULATION.

CONNAIE WAIE* f,ooo WAIE*

GAS PO, ”MER

=i

SLUG

FIG. 13—EXPERIMENTAL 0.25 PV POLYMER FLOOD,CROSS-SECTIONAL MODEL.

PV injected, while the waterflood needs 2.8 PV of

injected fluid to obtain the same recovery.A cornParisQn of the simularor results for this

model was made to examine the effect of a five-grid-layer vs a one-grid layer simulation. Fig. 17:- J:___ -:”’.l.?..l-. >-. -AA-Illlulca Les that a sJil&c-.ayc, ...”=-.. c;m,,larinu ~~~-...s -.- . . ..~

desaturated gas zone as a dispersed phase, andassuming negligible gravity segregation of the

aqueous and oil phases, yields results that arecomparable to the five-Iayer model in which the gas

zone is distinct. For this problem a single-layer

model using the same relative permeability curvesas the five-layer model is sufficient.

SIMULATION OF A FLOOD IN ARESERVOIR WITH A DESATURATED ZONE

A simulation study was carried out for a reservoir

with six noncommunicating layers, each with a

desaturated zone that varied areaiiy. Tine study~--m :.-,-1 rha ~ff,=rr ~f che &~~rgK~t~~ z~e on oil&Aca . . . .. L&u . ..s ------

recovery and on the oprimal flood pattern, andincluded the variable reservoir continuity and

heterogeneity. We include here the results of afive-spot pattern flood that was modeled.

Figs. 18A and 18B illustrate the five-spot pattern.Each producing sand was treated as a single layer

~~~o Iq _ POLYM.ER SP.TIURA.TION DISTR.IEKJTION,CROSS-SECTIONAL SIMULATION.

FIG. 1S — OIL SATURATION DISTRIBUTION, CROSS-SECTIONAL S2MULATION.

I● MATH MODEL

}O PHYSICAL MODEL ‘ATERFLOOD

e 100 A MATH MODELE }

~ PHYSICAL MODEL 1/4 PV POLYMER SNJG.

Ii

I

-+z

1●

a ‘e> 20 4~ a

we oo-1.0 2.0 3.0 4.0

PV THROUGHPUT

FIG, 16 — OIL RECOVERY EFF1C2ENCY, CROSS-SECTIONAL MODEL.

OCTOBER, 19i2 a77

with distributed gas. The results of this simulation

are presented in Figs. 19A and 19B. A 0.11 PV

slug of 4 cp polymer with an adsorption of 10 pg/g

was used. For this case, the simulator predicted a

recovery of 45 percent of the movable oil in place

for the poiymer fiOOd, in comparison with 32 percentof the movable oil in place for the waterflood. Thecumulative production curves are given in Fig. 20.

The simulation study of this reservoir is alsoinvestigating the selection of the optimal polymerflood pattern (e.g., should the injection well be

located off center to compensate for dip?), selectionof the proper reservoir intervals for injection (e.g.,

should intervals with desaturation above a maximumvalue be excluded?), and determination of theoptimum polymer slug size.

A-5 GRID B1OCKS IN Z DIRECTION

A -SINGLE GRID B1OCK IN Z DIRECTION

AA

AA&

A

O..i.0 2.0

PV THROUGHPUT

—.—FIG. 17 — COMPARISOFi OF R3ZSU”L~Sj OF SINGLE-

----—-

LAYER VS FIVE-LAYER SIMULATION.

x

YI ‘

–––––-- + -–- -– ---- -o- -– -–- -–– -@- --- --- --- -y-Q$ #!! !

“ Q’+) ,~-++ A A A&Y&!!

I

––-–-–- -o- –– -–- ---- -o- -– -–- --–. -o- –- -– -––- -0

A INJECTION WELL

● PRODUCING WELL

DIP

-13” — 400’

FIG. 18A — FIVESPOT PATTERN MODELED, AREALVIEW.

Another reservoir to which the polymer simulatorhas been applied is shown in Fig. 21. This reservoir

is limited by faults and has an oil rim between agas cap and an aquifer. Oil viscosity is 9 cp at thereservoir temperature of 131 ‘F. Gravity segregation

is significant in this iow dip reservoir. The irregularwell spacing complicates the polymer flood design.For brevity, the results will not be included.

Typical questions being asked of the simulatorinclude the following. Should polymer be substituted

for water in the current injections wells? Will

additional injectors be required? Should bothpolymer and water be injected at different points inthe reservoir? What is the effect of the gas cap,

etc. ?

CONCLUSIONS

The investigation of several polymer floodprojects for complex reservoirs required theutilization of the flexibility of a compressible,three-phase, three-dimensional mathematical reser-voir simulator to represent the reservoir features

and well performance. In adapting such a simulatorto represent polymer flooding, the conservation

equation for polymer solution has been included asa component of the aqueous phase. The simulatorwas extended further to represent the adsorption ofpolymer, the reduction of rock permeability to the

aqueous phase, the dispersion of the polymer slug,

and the effect of non-Newtonian flow behavior onthe injectivity of polymer.

The polymer flood simulator has been used to aid

in the design of a project in a multizone, heteroge-neous? dilminp reservoir with de~atur~red ~Qn~s du~==... O ---------to gravity drainage. The simulator was shown torepresent the displacement of oil in a desaturatedreservoir as observed in a physical experiment.

These applications illustrate problems that couldnot be represented adequately with stream tube

models or two-phase simulators. The two field

applications and the laboratory experiment all hadthree phases present. The influence of the gasphase on the reservoir performance was too-:-_: t:---. A. L_ —_-l--. -L L.. .L _ ..– - -f- -—. – _L. —-SIgIIIIICaIIC m De l]eglecteu Dy me use or a cwo-pnasesimulator. The crossflow in the laboratory

experiment, the areal distribution of gas saturationand reservoir heterogeneities in the pattern flood,and the irregular well spacing with a gas cap andaquifer in the fault block reservoir could not havebeen represented with a stream tube model.

378

I 40 +

zu.

20:’

10 $

3035

10 +

20

x

FIG. 18B — FIVE-SPOT PATTERN MODELED, CROSS-SECTIONAL VIEW.

SOCIETY OF PET ROLEIJM ENGINEERS JOURNAL

NOMENCLATURE

Ad = polymer adsorption, pgm/gm

A> = polymer adsorption, Pgm/bbl

b = reciprocal of formation volumeSTB/res bbl or Mcf/res bbl

C = polymer concentration, ppm

~ = polymer concentration, pgm/bbl

factor,

wATERFLOOD -RECOVERY FACTOR .31 W. EM.O.

..............wAW lNJECTloN

! OLto,w,, . ~

~& ~L~QLo-N-:: --:--”::_::;:””””””.. ~~~ ‘...-z=-------m @! -- ----

u- --,.. , ~2!/,

‘O 500 1000 1500 2000 250f3 2000 3500 4000

FIG. 19A — OIL RECOVERY RATE, WATERFLOOD.

POLYMER FLOOD - RECOVERY FACTOR .45 W. F. M.O.

WAIIR !NJfC!19?$. ........ .. ... ........ .. . . . .....=.s..~ ~.’”;

-%..lN,.,y”;E&@@’Q~-------------..,,,,.a.’ .N

J

o 500 Wo 1500 20D0 2500 3000 3500 4000

TIME, DAYS

FIG. 19B- OIL RECOVERY RATE, POLYMER FLOOD.

.2 -

.1-

00 .2 .4 .6 .8 1.0

PORE VOLUMES INJECTED

FIG, 20 — OIL RECOVERY EFFICIENCY, FIVE-SPOTPATTERN.

+/

●●

● ‘TGOC‘\“ —~ .y

“(- \A

●● PRODUCER

\ \

I9 “NO \

y WATER IN,KTOR ~>~—-

FIG. 21 — FAULT BLOCK RESERVOIR WITH GAS CAPAND AQUIFER.

OCTOBER, 1972

cads = adsorptive capacity of rock, pg/bbl ofpore volume

g = (1/144) lbf sq ft/lbn sq in.

H=

b=

K=

k=

kri =

kw =~=

Np =

p=

Fe =

Pruf =

q=~=

Qad, =

R, =

Rk .

R,, =

R. =

‘e =

Tw =/

‘w =

s=

s.

Sp =

u=

y.

Ax, Ay, Az =

(j, =

coefficie’nr in Blake-Kozeny model,

cp(cm/sec)l-n or cp(ft/day)l-n

formation thickness, ft

power-law coefficient, cp(sec)n- 1

absolute permeability, sq cm or md

relative permeability of ith phase

permeability to water, sq cm or md

power-law exponent

dimensionlesss viscosity group

pressure, psi.. ---- ----- ...-=average pLC2aU.. of f $~ !~wr @

block, psi

bottom-hole injection pressure at

datum, psi

injection rate, B/D

source term, (STB or Mcf)/D/(unitvolume)

mass of adsorbed polymer, pg/bbl ofpore volume

resistance factor

permeability reduction factor

residual resistance factor

solution gas ratio, Mcf/STB

external radius, ft

well radius, ft

effective well radius, ft

saturation

skin factor due to well impairment orstimulation

pseudo skin to represent non-Newtonianeffects

superficial fluid velocity, cm/sec orft/day

shear rate, see-l

dimensions of grid block, ft

fluid relaxation time, sec

TASi E 5 — RELAT:VE PERMEABILITY DATA

Oil-water Relative permeability

Sw km k2G Ii5iii 0.0000.30 0.630 0.0020.35 0.438 0.0100,40 0.325 0.0200.45 0.240 0.0380.50 0,180 0.0600.55 0.130 0.0850.60 0.085 0.1200.65 0.052 001550.70 0.030 0,2050.75 0.010 0,2640.78 0.000 00300

Gas-Oil Relative permeability

‘liquid k ro ~

0.25 z 0.550.99 0.85 0,001.OO 0,90 0.OO

379

A=

x=p=

Papp =

l-%.x =

Fmin =v=

p=

p,b =pAD =

+=+W .

~.

v=

‘v. =

SUBSCRIPTS

0=

‘w.

g.

p=

f=

aq =

eff =

t=

mobility = kip, md/cp

average mobility of the fluids’ in gridblock, md/cp

viscosity, cp

apparent polymer solution viscosity, Cp

maximum viscosity, cp

minimum viscosity, cp

ratio of viscosities (Eq. 28)

density, lb/cu ft

rock bulk density, gin/cc

gravity head from datum, psi

porosity

pore space occupied by water, qbw =

+Sulempirical mixing parameter

gradient operator

divergence operator

oil

water

gas

polymer

grid block Iayer number

aqueous phase

effective

total

ACKNOWLEDGMENT

The authors would like to express theira~~reciation to Mr. G. E. Tinker of Shell Oil Co.f;;

1.

2.

3.

4.

5.

6.

7.

the use of his reservoir study.

REFERENCES

Jewett, R. L. and Schurz, G. F.: “Polymer Flooding— A Current Appraisal, “ J. Pet. Tech. (June, 1970)675-684.

Slater, G. E. and Farouq-Ali, S. M.: “Two-DimensionalPolymer Flood Simulation, ” paper SPE 3003presented at the 45th SPE Annual Fall Meeting,Houston, Oct. 4-7, 1970.

Patton, J. T., Coats, K, H, and CO!e=O}W, G. T’. :“Prediction of Polymer Flood Performance,’9 SOC,

r lnf---u.Pe:. Efig# J . (1,wun, 1971; 72-84.

Todd, M. R, O’Dell, P. M. and Hiraaaki, G. J.: “TheApplication of Two-Point Approximations for FluidInterlock Transmissibilities for Increased Accuracyin Numerical Reservoir Simulator,” paper SPE3516, presented at the 46th SPE Annual Fall Meeting,New Orleans, Oct. 3-6, 1971.

Todd, M. R, and Longstaff, W. J.: “The Development,Testing, and Application of a Numerical Simulatorfor Predicting Miscible Performance, ” J. Pet. Tech.(July, 1972) 874-882.

Savins, J. G.: ‘ ‘Non- Newtcrnian Flow through PorousMedia, ” Ind. and Eng. Chetn. (1969) Vol. 61, No. 10,18.

McKinley, R. M., Jahns, H. O., Harris, W. W. andGreenkom, R. A.: ‘ ‘Non-Newtonian Flow in PorousMedia, ” ], Am. lzrst. Chern, Engt. (1966), Vol. 12,No. 1, 17.

8.

9.

10.

11.

12.

13.

14.

15.

16,

17.

18.

19.

20.

21.

22.

23.

Gegarty, W. B.: “Theological Properties ofPseudoplaatic Fluids in Porous Media, ” Sot. pet.Etzg, j. (June, 1967) 149-160.

Sadowski, T. J.: “NOn-Newtonian Flow throughMedia, 11. Experimental, ” Trans. Sot. Rbeol. (1965),Vol. 9, No, 2, 251.

Gogarty, W. B.: “Mobility Control with PolymerSolutions, j> Sot. Pet, Eng. .1. (June, 1967) 161-173.

Smith, ‘F. W.: “The Behavior of Partially HydrolyzedPolyacrylamide Solution in Porous Media, ” J. Pet.Tech. (Feb., 1970) 148-156.

Christopher, R. H. and Middleman, S.: ‘ ‘Power-LawFlow through a Packed Tube, ” Ind. and En& Cbem.Fund. (1965) Vol. 4, No. 4, 422.

Gaitonde, N. Y. and Middleman, S.: “Flow ofViscoelastic Fluids through Porous Media, ” Ind.and Eng. Chetn. Fund. (1967) Vol. 6, No. 1, 145.

PYe, D. J.: ~qrnproved Secondary Recovery by ContrO1

of Water Mobility, ” J. Pet. Tech. (Aug, 1964) 911-916.

Dauben, D. L. and Menzie, D. E.: “Flow of PolymerSolutions through Porous Media, ” J. Pet. Tech.

(Aug. , 1967) 1065-1073.

Marshall, R. J. and Metzner, A. B.: “F1ow ofViscoelastic Fluids through Porous Media, ” Ind.and Eng. Cbem. Fund. (1967) Vol. 6, No. 3, 393.

Burcik, E. J. and Ferrer, J.: “The Mechanism ofPseudo Dilatant Flow, IJ p,od. Month/y (March? 1968)

7.

Bird, R. B., Stewart, W. E. and Lightfoot, E. N.:Transport Phenomena, John Wiley and Sons, Inc., NewYork (1960).

Herce, J. A. and Rivera, R. V.: personal communica-tion, 1971.

Wissler, E. H.: “Viacoelastic Effects in the Flow ofNon-Newtonisn Fluids through a Porous Medium, ”!rd. afid E%g. Crbi?iE. Fund. (1971) VOl. 10, No. 3, 41 i.

Jennings, R. R., Rogers, J. H. and West, T. J.:(t Factors ~fluencing Mobility cOntrO1 by pOIYmer

Solutions, ” ]. Pet. Tech. (March, 1971) 391-401.

Meter, D. M. and Bird, R. B.: “Tube Flow of Non-Newtonian Polymer Solutions: Part I, Laminar Flowand Theological Models, ” J. Am. Inst. Cbem. Eng.(1964) Vol. 10, No. 6, 878.

Lee, K. S. and Claridee, E. L.: “Areal SweepEfficiency of Pseudopla;tic Fluid. in a Five-Spo_tHele-Shaw Model, ” Sot. Pet. Eng, J, (March, 1968)52-62.

APDENllTY A. . . . e.. ”... . .

POLYMER ADSORPTION

The polymer adsorption is modeled such that therock will strip the polymer out of the polymer

solution resulting in a decrease in polymer solutionsaturation and an increase in water saturation.

Input quantities are: Ad = polymer adsorption,

pgm polymer/gm rock; C = polymer concentration,

pgm polymer/gm solution; ~,b = rock density, gmrock/cc bulk; ~ = porosity.

The adsorption and concentration are converted

into units of micrograms per barrel of pore volumeand barrel of solution volume, respectively.

SOCIETY OF PETROLEUM EXGINEERS JOURNAL

APPENDIX BWgm (polymer) Ad Prb

A> = 1.59 xlo5—

bbl (pore) @

. . . . . . . . . . . . . . . . . .(A-l)

kgm (polymer)

F = 2.544 X 103PW ~cC

STB (soln.) 9

. . . . . . . . . . . . . . . . . .(A-2)

At the end of a time step, the amount of polymer

avaiia”bie for adsorption, in micrograms per barrel

of pore volume, is

6(polymer) = Es . . . . . . . . . .(A-3)P

while the adsorptive capacity of the rock in rhe gridblock is given by the maximum adsorptive capacity

multiplied by the fraction of mobile aqueous phasepresent in the gridblock:

/

Sw+s .Swcc = AZ 1 .(A-4)

ads\ 1 -s-s

orw Wc /

1’7:.L --J2.1 L1lCL

~~~ ~i jii St etloii~!l G$ t~~ poly,mer to

saturate the rock will be adsorbed. The cumulativeadsorption of eachpolymer saturation

!f

cads - Qads

block is stored, and the loss ofis converted to water saturation.

> 6(polymer) ,

NON-NEWTONIAN INJECTIVITY MODEL

The expressions for the rate-dependent skin torepresent the non-Newtonian effects, Eqs. 26 and

27, will be derived. It is assumed that the viscositycan be represented by a power-law model with the

uPPer and lower Newtonian limits, Eq. 22.

The radial velocity profile can be expressed as

~~u =— —.. . . . . . . . . .(B-l)

2* r

The viscosity profile with the velocity profile of

Eq. B-1 is

()n-1

~ rl-n~(r) = H — J

, 211hI

IJMin ~ w(r) < IAmax “ “

.(B-2)

The radii, r, and r~. where the viscosity reachesthe limiting ~alues c-an be determined from Eq. B-2.

1

1~ \x ~

H _ . . . . .(B-3)‘1 =

w 211hmi n

then1

ti(sp)= sP

and

6 (Q ads) = 6(polymer)

/ H \n-l q /-,.‘2 =

H

— . . . . .( B-4)

P 2 rrhma x

The pressure drop from 12 to ~1 is

and if

c -Q ~d5 < 6(polymer)ads

thenc -Q

6(SP) = ads ads

E

and

6(Q ~d5)= cad5 - Qads “

In both cases,

6(SW) = -6(SP) .

()qn H

P2 -pi. .—

211h k(l - n)

●

✻

(l-n) 1-r(l-n) . .(B-j)‘2 1

The pressure drop from rl to rw is

q IJmin ()rlpl-pw =-— —in—.

2rrh k rw

{n <N. . . . . . . . . . . . . . . . (u-u)

The total pressure drop between r2 and rw can be

expressed in terms of an effective well radius, r;,

OCTOBER, 1972 Ssl

The apparent skin factor can be expressed as

1. [ in (u”N~l-v)) - u + l])

(1 - n)

. . . . . . . . . . . . . . . . . . (B-8)

where

wminv=—

IJJmax

1-nH

()

2 rrrwh

N .— . . . . (B-9)Pp

maxc1

The condition that rl L Tw is sarisfied when

NW < V . . . . . . . . . . . . .( B-1O)

CASE 2 — MINIMUM VISCOSITYNQ’r ATTAENED IN =n=p A~TfiN. . .. ... . . . . . .

The minimum limiting viscosity, ~min, is not

attained in the formation if rl < rw or

NW> V . . . . . . . . . . . .. (B-11)

The pressure drop from r2 to rw can be expressedas

/\ qn H

P2-Pw=-

\–1211h k(l - n)

[

(l-n) 1-#n)..(B-12)●

‘2 w

The skin factor can be expressed as

r

s =ln SP r

rw

11.

[

Ln NJ

-Nw+l .

(1 - n) PJ

. . . . . . . . . . . . . . . . . (B-13)

***

SS!2 sOCIETY OF PETROLEUM EXGIXEERS JO URKAL