Geertsma_Theory of Dimension Ally Scaled

8/6/2019 Geertsma_Theory of Dimension Ally Scaled

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THEORY of DIMENSIONALLY SCALED MODELS ofPETROLEUM RESERVOIRS

A B S T R A C T

J. GEERTSMAG. A. CROESN. SCHWARZ

T he dinzensionless groups, to which the variables thatgovern the di.splacenzerzt o f oil from reservoirs byl iquids can be contbi t~ed,are der ived. Three types o fdisplacement are considered, viz. cold-water drive, hot-water drive and solverzt injection.The derivation of the dinzerzsionless groups i s car-ried o ut by m eans o f the relevant basic equations ( in-spectio nal ana lysis). T h e re.slrlting sets of gro ups areafterwa rds completed by m eatls o f dimensional analysis.T he f o r m o f t he g roups i s gi ve n itz such a way thalthey can be adapted to suit th e various boundary c on-ditions that are encounter ed in practice. Th e physicalmeaning o f the groups is discussed. Th ey have all beenbrought toge ther on a chart , f rom which their mutualrelation and their corresporzdence to related dim en-sionless groups in common use in o ther engineer ingsciences can be read off .Th e limitations of dimensionally scaled mod el ex-periments a s a useful tool for studying liquid f low inpor0u.r media, as occurring in oil reservoirs, are dis-cussed.

KONINKLIJKE/SHELL LABORATORIUMAMSTERDAM

"The use of models to study fluid mechanics hasan appeal for everyone endowed with natural curi -osity. What active boy has not played with shipand ai rplane models , or crude models of damand drainage systems? Even in the most advancedtechnical engineering, such models play a funda-mental and indispensable role.- - -

'References giv en at end of naperOriginal manuscript received in Petroleum Branch office on Jull18, 1955. Paper presented at Petroleum Branch Fall Meeting inNew Orleans. Oct. 2-5. 1955 .Discussion of this and all following technical papers is invited.Discussion in writing ( 3 copies) may be sent to the offices of theJournal of Petroleum Technology. An y discussion offered after Der.3 1, 1956, should be in the form of n new paper

"And yet in few dep artm ents of the physicalsciences is there a wider gap between theory andpractice, between scientific knowledge and thestate of art , than in the use of models to studyhydrodynamic phenomena."Garret t Birkhoff '

I N T R O D U C T I O NLaboratory displacement experiments are extensivelyused to investigate, directly or indirectly, the productionbehavior of petroleum reservoirs. Such experiments arerepresentative of the reservoirs as a whole, if they arecarricd out with models that are "properly scaled."T he perfo rman ce of oi l reservoirs is governed by thevalues of a number of variables, which can be com-bined to dimensionless groups.Two gencral methods are avai lable for the derivat ionof these groups.In the first of these, dimensional analysis, combina-tion of the variables is done essentially by trial anderror. Its only premise is knowledge of the completeset of relevant variables.In the second method, inspectional analysis", the di -mensional homogeneity of the equations describing thebehavior of the system to be studied is used.The resulting dimensionless groups can be dividedi n t o : ( a ) independent (variable) groups, such as di-mensionless length, t ime, etc.; (b) dependent groups ,giving the dim ensionless for m of the variables, suchas recovery and pressure, that can, at least in principle,be measured during an experiment ; (c) similaritygroups, which are independent constant groups , such asratio of length to height of the reservoir an d ratio of--

:-The term "in spectional analysis" wa s introduced by Ruark2.


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the viscosities of the reservoir fluids. Their value isknown a priori. If' the values of these groups are thesame for a model and for a prototype, the model isproperly scaled, which means that for equal values ofthe independent variable groups in prototype andmodel, the values of the dependent groups are equaltoo. It follows that the results of experiments with sucha dintensionally st:aled model can be directly inter-preted in terms of field performance.Three types of displacement will be considered, viz.:(1) cold-water drive, i.e. the conventional water drive;( 2 ) hot-water drive, i.e. the injection of hot water withthe objective of increasing recovery from reservoirs con-taining viscous crudes (see v. Heiningen & Schwarz";and ( 3 ) solvent injection, i.e. the injection into andcirculation through a reservoir containing a viscouscrude of a miscible liquid (see Offeringa & v. d. Poel').

The objectives of the paper are to give for each ofthese production mechanisms: (a) an unambiguousderivation of the complete sets of dimensionless groups;and (b) the limitations of the feasibility of dimension-ally scaled model experiments.SURVEY O F LITERATURE ON DIMENSIONALLYSCALED RESERVOIR MODELS

The use of dimensionless groups for the investigationof the water drive process was initiated by Leverettet al.' Engelberts and Klinkenberg" extended this work.These authors exclusively used dimensional analysis.Rapoport and Leas' reported the results of "scaled"experiments on water flooding. A detailed study of theinfluence of the principal dimensionless group govern-ing the water drive mechanism-the oil/water viscosityratio-was published by Croes and Schwarz.' Offeringaand Van der Poel" carried out scaled model experi-ments on oil recovery by injection of solvents.

The most successful attempt to arrive at an un-equivocal derivation (by means of inspectional analysis)of the similarity groups governing cold-water drive wasrecently published by Rapoport."

The present authors gladly made use of the aboveresults, where necessary.METHODS FOR THE DERIVATION O F

DIMENSIONLESS GROUPS

The theory of dimensional analysis is described ina large number of books and articles, of which we maycspecially mention Langhaar," Focken" and Birkhoff.'The first step in the dimensional analysis of a prob-lem must be to ascertain which variables are relevant tothe problem. Special care should be exercised at thisstage that all the relevant variables are included.The variables can be arranged in a set of dimension-less groups. The set is complete if all the groups in theset are independent of each other and if every dimen-sionless group containing the same variables, and notbelonging to the set, can be formed by combininggroups belonging to the set.Buckingham has put forward the rule that the num-her of dimensionless groups in a complete set is equalto the total number of variables minus the number offundamental dimensions, e.g. mass, length and time.Langhaar generalized this rule, but Buckingham's state-ment is sufficient for our purpose.

All physical equations are relations between inde-

pendent variables, dependent variables, and constants.The total number of variables minus the number ofequations which describe the process determines thenumber of independent variables. The independentvariables can be converted into independent (dimension-less) groups by dividing them by some value of thesame dimension, characteristic of the system.

As the equations are dimensionally homogeneous, thedependent variables and the constant terms can bebrought into a dimensionless form by dividing eachequation by on e or more of its constants, suitably se-lected. The dependent variables in their dimensionlessform are called dependent groups, the (dimensionless)constant terms similarity groups.

If the equations are differential equations, the relevanthoundary and initial conditions are to be handled inthe same way.

A characteristic feature of inspectional analysis isthe existence of a mathematically expressible conceptof the phenomenon. This entails the introduction of anumber of approximations, as a consequence of whichsome groups, which can be of importance under lessideal conditions, may be "forgotten". As an examplewe may mention the groups associated with pore sizeand pore size distribution, which are deleted if a Darcy-type flow equation is adopted.

Further, it is sometimes necessary to introduce em-pirical variables such as relative permeabilities, thephysical meaning of which is not always clear.On the other hand, an inspection of the equationsmay reveal that two or more similarity groups onlyoccur coupled together. This combination can then beconsidered as a single similarity group; the result is asmaller number of groups and greater flexibility as re-gards realization of a model than would follow fromdimensional analysis.

As opposed to the above, the set of groups obtainedhy means of dimensional analysis is complete and thevariables occurring in the groups have a clear physicalmeaning. However, the physical meaning of the simi-larity groups themselves, as derived by dimensionalanalysis, is generally less apparent than that of thegroups derived by inspectional analysis. Deletion ofone or more of the groups, which is necessary for therealization of a model, is therefore most convenientlydiscussed on the basis of inspectional analysis.

As a consequence of the above considerations themethod chosen was primarily inspectional analysis. Theset of similarity groups obtained was afterwards freedfrom empirical variables and completed by means ofdimensional analysis. It was hoped that by this pro-cedure the advantages of the two methods could becombined.

As the first stage of inspectional analysis, the basicequations of the displacement process in porousmedia will now be tabulated and classified.

SURVEY OF THE TYPES OF EQUATIONSDESCRIBING FLOW PHENOMENA INPOROUS MEDIA

The equations describing flow phenomena in porousmedia can be classified as follows: those describing theconservation of basic quantities in the system; thosedescribing the dependence of the properties of singlecomponents on pressure and temperature; and thosedescribing mutual interaction of the components.


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1. Equat ion of corzservation of n~ ut te r ma terial bal-ance o r cont inui ty equat ion ). Th is equat ion expressesthe fact that the difference between the amounts ofmatter entering and leaving a source-free part of thesystem is equal to the increase of matter in this part .2 . Equation of conservation of mornerlturn, based onNewton's law, K = ma (equa t ion o f mot ion) . The mos tgeneral form of fluid flow is described by the equationof Navier-Stokes. For slow flow through porous mediait may be specialized to some form of Darcy's law (seee.g. Muskat").3. Equat ion of conrerva~io r l / heat ( thermal balanceequat io n), expressing the fa ct that the di fference be-tween the amounts of heat entering and leaving asource-free part of the system both by conduction andby convection is equal to the increase in the amountof heat in that part . This definition entails the assump-tion that ther e is no conversion of mec hanical energyinto heat energy and vice versa.For an excellent review of these equations and oftheir interrelations the reader is referred to Klinken-berg and Mooy.''PROPERTIES F SINGLEC O M P O N E N T S1. The influence of pressure and tcmpcrature on thedensity of single reservoir fluids (equations of state) ;2. The influence of temperature on the viscosity ofthe liquids (the influence of pressure is neglected).

1. T he diffusion of two miscible l iquids into e acho ther ;2. Cap illary phen ome na a t interfaces between phases.The different ial equat ions among the above relat ionsare valid for a volume element which is small comparedwith the dimensions of prototype and model, but largecompared with the diameters of the pores .For convenience it is assumed that porosity andpermeabi l i ty are uniformly dis t r ibuted and that thepermeability is isotropic. Further, only two dimensionalflow in the vertical plane will be considered, because itrepresents the main body of experimental results ob-tained hitherto.Omission of these assumptions would result in theaddition of some obvious similarity groups, fixing thedistribution of the permeability, porosity, and geometryof the reservoir.

D IS C US S IO N O F T H E E Q U A T I O N SThe cont inui ty equat ions for the s imul taneous f lowof two immiscible l iquids (oil and water) in a porousmedium are respectively

_1 2 p < > 4 s 0 )for the oil phase: div (p ,v,) = -_- .G tA ( 2 )fo r the water phase : d i v ( ~ , , v , , )= -;--tIf the displacing liquid is miscible with the oil-asin the case of a solvent o r diluent-there is onl y onephase, but a continuity equation can be given for cachof the two com ponents , o il and di luent :

A a ( p o + c 0 )for oil: div (p,,v,) = - ---- . . . .a t ( 3 )-- a ( ~ , ~ 4 ~ , , ) ,fo r di luent : div (pdv,,)= - a t

Because of the miscibili ty of the components, concen-

tratlons (C,, C,) ar e used instead of saturation s (So,S, , . ) .The pore space is assumed to be completely fil ledwith the two liquids, which is expressed by:S , = 1 - S , . . . . 5 )

. . . . ., = 1 - c,, respectively ( 6 )Th e equations of continuity give no informationabout the physical behavior of the particular systemand therefore no s imilari ty groups can be derived fromthem. This fact has been referred to earlier by Birk-hoff. ' Their function in inspectional analysis is to putthe rates of flow in a convenient dimensionless form.The equations of motion for multi-phase flow in aporous m edium are extensions of Darcy's law. Th islaw has been proved to be valid for slow flow of asingle phase only, but has been adapted to multi-phaseflow by introducing the relative permeability conceptand wri t ing one equat ion f or each phase.The generalized Da rcy equat ions for the s imul taneousflow of oil and wa ter may b e written as

_1 Akk"' (grad p. - peg),, = --- I" , ,and

The mutual h indrance of the two l iquids is here sup-posed to be completely accounted for by k , , and k , , .The flow of oil and diluent can be considered as thatof a single phase because there is no definite interface2between the two liquids. Thus the velocities v, and

v,, in the Eqs. 3 and 4 consist of a common part, rep-resenting the "Darcy flow":3 kv = - (g rad p p,,,g). . . .p,,, ( 9 )and an individual part , resulting from diffusion (seebelow).Thermal balance equat ions are required for the de-scription of the hot-water drive. In the presentation ofthese equations it will be assumed that: (a ) the f lu idphases and the rock grains are in such close contactwith each oth er tha t in every volume elem ent of thereservoir the temperatures of water, oil and rock areequal ; and ( b) the thermal constants are independentof pressure and temperature, and consequently of placeand t ime. The thermal balance equat ion then reads:aT A 2- PC - ( p o ~ o v o p , c ,~ , ~ ) ra d T31

. . . . . . . . . .di v X grad T = 0 ( 1 0 )in which pc :- ~ S , c , p , -k ~ S . c , p , . f (1 - 4 ) crp.. . . . . . . . . . . . . . . 1 1 )an d

/\ = X,+S, + A,@, + X,(1 - 4 ) . . ( 1 2 )Th e equat ions of s tate are s imple, because l iquids canbe regarded as imcompressible and the dependence oftheir densities on the temperature can be given withsufficient accuracy by the cubic expansion coefficient /3.For the hot-water drive they are :

for cold-water drivep, = constantp , = constant ,and for solvent injection:


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. . . . . . . ., = constant, (15')p , ~ constant . . . . 16')

The dependence of the viscosity of the liquid phaseson pressure is neglected.The influence of temperature on viscosity is given byrelations containing a number of thermodynamio prop-erties of the liquids (see e.g. Bracket"). Such relationsare not suited for inspectional analysis and a morepractical approach will be adopted by introducing thejcaling rule: "The graphs of dimensionless viscosityagainst dimensionless temperature should be congruentfor model and prototype". This scaling rule is repre-sented by:

L O J (A LL , I , . 7'/71,) . . 17)fir, "

andpw = p w t (Afi,,..T fi w. T, , T/T,) . . (18)

where A and A , are (dimensionless) symbolsp


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D E R I V A T IO N O F THE D I M E N S I O N L E S SG R O U P S BY I N SPE C T I O N A L A N A I ,Y SI SThe independen t var iab les x . y and t can he writtenin dimensionless form as:

X = x/l , Y = y /h an d H - / r+ . . . . . . ( 4 2 )r is some character is t ic t ime, the fo rm of which wil l bedefined later.

T h e Eqb. 1 th rough 41 can then be written in fulland rendered dimensionless .The procedure wil l be i l lustrated for the equations de-scr ibing the cold-water dr ive. The continuity Eqs. 1 and2 can be writ ten as fol lows (m aking use of Eqs. 5 ,15 and 16):

Th e Levere t t Eqs . 25 an d 26 are substi tuted intothe f low Eqs. 7 and 8, which are then b rought in tosuch a dimensionless form that the groups containingv, an d v , are the same as those occur r ing in Eqs . 43an d 4 4 :

k.rp.-g sin cu :"/L \\ I

r u cos O dk+ acs,,) J , I(t) b .' rls., ' ay + k , , , .- -+I, I' p ., ' h

r u cos Odj+ d J : (St , ) k~p,, .g in a ,4p,.. I' ' rls,, ax p.,!1

th I ) 'rr cos ct,:T+ . -- / J --(s,,) + k t % .1!J ,, 1- c / S , , ' a Y I1

krp,-g cos cub~, I

Th e boundary and in it ial cond i t ions o f impor tancefor inspect ional analys i s ar e here (Eqs . 28 o r 31a n d 3 9 ) :

2"The component of g in the z-direction is g sin a , tha t In the),-direction g cos a.

T h e d i m e n s ~ o n ~ e s sl.oups appearing In the above setof equations are:1 . independent var iables:

X , Y , 92 . dependent var iables:, , v , , , ~ v , , , ~ T k p ~ ,- - -'7'' h ' p , , l "

3. similarity grorrps:

V T k n p i . ,h , k,,, k,,, S,,,nd e i ther - r -I ,&,.I?In exactly the same way the fol lowing similar i tygroups can be der ived for the hot-water dr ive:

I , p.,, k~ P ~ ~ S( o os 0 ) dl--.a,+,-,- -------- , . . _ - -h /",,, I ) , . . , ' ! L i l /kx.1p ,, p,?ic p , c , ['.C'


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ADDITIONAL SIMILARITY GROUPS FROMDIMENSIONAL ANALYSISWe now can apply Buckingham's rule to the sets ofsimilarity groups obtained. In the cold-water drive wehave found 12 groups, containing 19 parameters, viz.

dJ1, h, a, 7 , +, k, p.,, p.,, PO , pw , g , U P , -, 6, krO,kc,, So ,dS,and v or A p . According to Buckingham's rule theseshould give 19-3 = 16 similarity groups (three funda-mental dimensions: mass, length and time), so that fourgroups are lacking. Two of these "forgotten" groupsare 8 and +. According to the equations of the pre-vious section they only occur in the combination-

u cos t3 d + ,which suggests that this combination can be consideredas one single parameter and that 8 and 4 can beomitted as separate similarity groups, if u is replacedby u cos O &.The remaining two groups cannot be derived in thesame unambiguous manner as the above groups, but itis plausible that one of them stands for the scalingdown of the average pore diameter. A reasonable formfor this group is l/d%.

The other group that has to be introduced is equiva-lent to Reynolds' number and accounts for the inertialforces that were neglected in the differential equations.A reasonable form for this group is mentioned byShchelkachev":

pw l d k--7 p . w

In the hot-water drive we obtained 26 groups whichcontain 33 parameters (combining u and 0 , p, and c, ,pc and c, ) , viz.1, h, a, , +, k, poi^ , u w i , P Q , , p w i , g, ( u c o s o ) i , J , ' 9 kw,k r T v 9 O ! , i, Tb, Pc,, PW9O , C W I P TC I ~ c C e , X o , X w , X r , X c ,A , A

IL",T, P w , T . ,A .,,nd v or ~ p .here are in this casefour fundamental dimensions, viz. mass, length, timeand temperature, so that 33 - 4 = 29 similarity groupscould be expected by dimensional analysis. Two of thegroups that were not found by inspectional analysis are

Iagain-1nd &?I\/* . The last group lacking ap-\I' k T/*,,. ipears to be the ratio between the heat quantity con-tained in the displacing fluid and the heat producedmechanically by the fluid forces. Thus, for the sake ofcompleteness, the group

pwic ,< .T i~p.3,. i

to be found by dimensional analysis, must be added.D+If- D, is considered as one variable, inspec-Ltional analysis of the solvent injection process giveseight similarity groups containing 13 parameters. Thushere 13 - 3 = 10 similarity groups could be expected.The lacking groups are again

1 ,


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it will thus have no influence it this l aw applies formulti-phase f low. I t is plausible to assume that I / \ / %can be dele ted, if the average diameter of the pores ismuch smaller than the smallest dimension of the res-ervoir, i.e. if l / d k ( o r h / \ / k ) is very large . The valueof this group down to which the assumption is per-missible has to be found exper im entally.T h e g r o u p x can only be given its proper valuei f e i the r : ( 1 ) the sand in bo th pro to type and mode lis unconsolidated, because x is believed to have aboutthe same va lue for al l loose sands ; o r (2 ) in themodel the same porous medium is used as that ofwhich the prototype consists , in which case a core ofconsolidated material can he used. Both possibilitieswill be discussed below.I t wil l fur ther be assumed that the init ia l distr ibu-t ion of oil and water can be sim ulated. W e will nowconsider the individual production mechanisms.COLD-WATER RIVE

T h e case of a given injection ra te will be discussed.An analogous reasoning can be applied to the case ofa given pressure dif ference; the results obta ined are es-sentia l ly the same.The similar i ty groups to be considered are :

T h e g r o u p causes no dif f icult ies and therefore needPwnot be discussed. Fur ther i t is convenient to writek n p g kp,.ginstead of - bzcause then the interactionPwV Ll r i,

f ~ , , k p W g .that prob ably exists between - nd is removed.p > /.L IS i n c e l a r g e f l u c t u a t i o n s i n n p e n t a i l o n l y s m a l l

Pchanges in" this la t ter group is considerecl as in-Pvariable and will be dele ted. Finally the values ot +a n d p, will be taken t o be equal in prototype andmodel.W e define a scaling facto r y to be the ra t io of thevalue of i ts argument in the prototype to that inthe mode l . The subsc r ip ts , an d ,, will fur ther beomit ted .With the above considerations the scaling condit ionsbecome:

k n p g = ucos e d i v d k - 1y ( 7 (PJ-)(7--)an d

y ( + ) = Y ( P ) = 1.These can be wri t ten in the m ore convenien t fo rm:

UNCONSOLIDATED SANDSTaking the same l iqu ids in pro to type and mode l , i tcan be seen that the condit ions become:

y ( v 2 ) -- ( k - ' ) = y ( g d k ) = y ( I - ' )or , fo r ins tance ,y ( l ) = y ( k ) = r ( v - ' ) = y ( g - 2 ' 3 )In the following we will assume that because of theexperimental difficulties involved, y ( g ) = 1. T h e n o n eof the three group s must be dele ted.If this is chosen to be the Reynolds group !'v\/k, theI L

two rem aining condit ions a re sa tisf ied by: y ( k ) - y ( v )= y ( I - ' ) . Both th e permeab ility a nd the velocity art.larger for this model than for the reservoir by a fac-tor equal to the square of the length reduction. Thismakes i t possible to use easily obta inable loose sands.and gives a convenient t ime scaling, namely y ( t ) =y ( 1 - 7 . F u r th e r it a p p e a r s t h at y ( p ) = y ( I ) . T h u s t h e r e-ductions in pressure and length become directly pro-portional.

k n p sIf gravity forces m ay be neglected, the group " Pis dele ted and the condit ions become y ( 1 ) = r ( k ) -( v - ' ) , which in general is impossible to realize becauseof the reducticn in permeabil i ty of the model and thehigh pressures to be used, s ince y ( p ) = y ( v ) = y (1. ' ) .Deletion of cap il lary forces finally leads to:

y ( v ) = y ( k ) = I ,independent of ~ ( 1 ) .n this case experiments will lastra ther long.If it is permissible to use model liquids different fromthose in the prototype, the condit ions:. .

can be sa tisf ied without omitt ing any groups, as ap:>earsfrom the following example:y ( k ) = 1, y ( l ) = 100, y ( v ) = y ( p ) = ? L .y ( ~ p ) !A, y ( u 0 0s 0 ) = 25.

CONSOLIDATED SANDSI f consolidated sands are used, y ( k ) = 1 and thegeneral condit ions are :

wh ich, obviously , cann ot be satisfied if the m ode l liquidsare the same as the prototype l iquids, unless capil laryforces may be neglected, in which case y ( v ) = 1.If the l iquids may be dif ferent , the design of a scaled

model is feasible (compare example given above) .HOT-WATERD R I V E

Again we take. . . .Fur the r i t i s a ssumed tha t the o the r ma te r ia l con-stants are invar iable :y ( p c ) = y ( h ) = y ( A ) = . / ( P ) = 1.

The sca l ing condi t ions fo l lowing f rom the r ema in inggroups a re :

o r :0 cos 0r ( v 2 ) = y (g) = v t a p = (T) = y ( l . 2 ) ,

Th e last cond it ion, s ta t ing that . / (a cos 6)= y( 1 . I )cannot be fulf i l led, so that one of the groups has to beomit ted .I f Reynolds ' g roup is dele ted :

0 cos E) \LCY (9)Y ( pvl -) Y (A) = 1 ,o r:

P E ~ R O 1 . E I ' M I' H \ZS.\(: 'I ' lOX%. \ I W k


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and possibly other groups is permissible has to be an-.;wered experimentally.

IJhing the same liquids:

which is impossible.If different liquids are used, various possibilities ex-

ist for loose sands, e.g.y (Ap) = y(;c) = 1 , y(1) = y(v-') = y(k-') =

y ( a cos 0 )'For consolidated sands y(k) = 1 and hence

u COS tfy(v u cos 0 ) = y (f) = y(AP) = -y (T),which cannot be done.

If gravity is negligible, it can be shown thatY( ucos 8 ) = y(l-l),

if capillary forces are neglected :ytnpp.l") = 1;both requirements cannot be met.

SOLVENTN.JECTIOI\With a reasoning analogous to that given for the pre-

ceding mechanisms, the scaling rules to be discussedcan be reduced to:

. .As y ( D ) =: 1, the scaling rules are the same asthose for the hot-water drive without capillary forces,which means that model experiments are impossibleunless one of the groups is deleted.

Deleting Reynolds' g roup leads to:

The same fluids can be taken:y(Ap) = -/(p.) = I , y (v) = y(k) = ~ ( 1 . ' )Loose sand must be taken, however.Deleting gravity gives:

or:

which is only possible if p is increased appreciably.C O N C L U S I O N S

1. A complete arid detailed survey of the require-ments that dimensionally scaled models of oil reservoirsshould fulfil, viz. equality of the values of the similaritygroups in model and prototype, is presented for thethree production mechanisms investigated.

2. The procedure of deriving the similarity groupsfor displacement processes in porous media by meansof inspectional analysis, completed by dimensionalanalysis, has advantages over the procedures used hith-erto, namely those employing either inspectional ordimensional analysis.

3. Dimensionally scaled models can be realized onlyif the average pore size need not be sealed, that is ifthe group l / d x can be deleted in the range of interest.

4. A requirement for most of the practical modelsis that the influence of inertial forces is not significantin the range of interest, so that the Reynolds' groupcan be left out of consideration.

5. The question whether deletion of the Reynolds'

6. The sets of dimensionless groups derived here areformally equivalent to sets of groups that are in com-mon use in other engineering sciences. There are tworeasons why the groups derived here are preferredfor two-phase flow through porous media: (a) mostof the conventional groups are ratios of forces toinertial forces, which latter are expected to be of minorimportance in porous media; and (b) in the conven-tional groups k does not occur. Since l\/ k cannot bcscaled, introduction of k by means of this group posesa difficult problem.

A C K N O W L E D G M E N TThe authors' thanks are due to H. J. de Haan and

J. Offeringa for their valuable discussions, to B. P.Boots and J. van Heiningen for their stimulating inter-est and constructive criticism and to the manage-ment of the Koninklijke/Shell-Laboratorium,Amster-dam, for their permission to publish this paper.

N O M E N C L A T U R ELATINA = symbolic representation of a set of thermo-

dynamic properties.A . , stands for the scaling rule: "the graphs ofdimensionless a against dimensionless b shouldbe congruent for the prototype and for themodel" (dimensionless) ;b = distribution factor of the interfacial force (di-mensionless) ;c = heat capacity per unit mass (J/kg OK);C = volume concentration (dimensionless) :

D = diffusion constant (m2/sec);F = functions describing the relation between the

properties of fluids and dependent variables:they always contain A,,, as a variable andhave corresponding subscripts f dimension-less) ;

g = acceleration of gravity (m/sec3) ;k = thickness of the sand (m);J = Leverett's function (dimensionless) ;k = absolute permeability (m');

k, = relative permeability (dimensionless) :1 = length of the sand (m)

L = lithologic factor (dimensionless) :p = pressure (N/m2)

Ap = pressure difference between inflow and outflowfaces of the sand (N/m2);

q = production rate (m3//msec):_1r = geometrical vector fm);S = saturation (dimensionless) :2 = time (sec) ;T = temperature (OK);v = flow or injection rate (mVm2.sec);x = coordinate in direction of I(m):

XX = - dimensionless) ;Iy = coordinate in direction of h(m) :

YY = -- (dimensionless) :kz = coordinate in direction perpendicular to x and

y(m) .


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n - angle of inclination of the sand with respect tothe horizontal (dimensionless) ;/3 = thermal cubic expansion coefficient (OK ') ;y = scaling down factor (dimensionless);p = viscosity (N.sec/mZ)O = wetting angle of fluid interface (dimensionless):

t8 = - dimensionless)79X = thermal conductivity (J/m.sec0K) ;a =. interfacial tension (N/m);p = density (kg,/m";T = characteristic time (sec) ;+ = porosity (dimensionless) ;x = similarity group for pore size distribution (di-

mensionless).

,, = injection water;,. = cap rock;= concentration;, = diluent,= diffusion;= outflow end of the sand;

, = initial;:,,= mixture:= oil;

,. = rock (except for D, and k,);,.= temperature;, = water;= coordinate in direction of 1;= coordinate in direction of h .

fi = viscosity;a = interfacial tension.

K E F E R E N C E SBirkhoff, G.: Hydrodynamics , a Study in Logic ,Fact and Similitude, Princeton University Press(1950).Ruark, A. E.: J. Eliha Mitchell Soc. (1935) 51,127.Van Heiningen, J., and Schwarz, N.: "RecoveryIncrease by 'Thermal Drive' ", Preprint II/E-1,4th World Petr. Congress, Rome (1955).Offeringa, J., and van der Poel, C.: Trans. AIME(1954) 201, 310.Leverett, M. ., Lewis, W. B., and True, M. E.:Trans . AIME (1942) 146, 175.Engelberts, W. F., and Klinkenberg, L. J.: Proc .Third W orld Petr . Congress, The Hague ( 1951)Section 11, 544.Rapoport, L. A., and Leas. W. J.: Trans . AIME(1953) 198, 139.Croes, G. A., and Schwarz, N.: Trans . AIME(1955) 204, 35.Rapoport, L.A.: Trans . AIME (1954) 201, 143.Langhaar, H. .: Ditnensional Analysis and Theoryo f M o d e ls , John Wiley & Sons, New York (1951).Focken, C. M.: Dimensional Methods and TheirApplications, Arnold & Co., London (1953) .Muskat, M.: Physical Principles of O il Produc-tion, New York ( 1949).Klinkenberg, A., and Mooy. H. H.: Che m . Eng .Progress ( 1948) 44, 17.Bracker, A. V.: T he I nd . Che m . (March, 1954)112.Klinkenberg, L. J.: Bull . GSA (1951) 62, 559.Leverett, M. C.: Trans . AIME ( 1941) 142, 152.Shchelkachev, V. N.: (in Russian), Nef tyanoeK h o z ~ ~ a i s t v o1948) 26, 24. M

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Geertsma_Theory of Dimension Ally Scaled