romero, a. and hefferan, j. an application of diffusity and material-balance equations to a high...

7/29/2019 Romero, A. and Hefferan, J. an Application of Diffusity and Material-Balance Equations to a High Relief Pool

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PETROLEUM BRANCH, AmTrini ty Universal Bui lding,Dallas, 'l'uas

THIS IS A PREPRINT -- SUBJECT TO CORRECTIONPAPER 114 -GNUJl3ER

An Application Of Diffusity And Material-Balance EquationsTo A High Relief Pool

.l . Romero Juarez and J . Hefferan, Member A.IW, Petroleos lIexicanos, Mexico, D. F.- -* * * * * * * * * * * * * * * *Publication Rights Reserved

This paper i s to be presented a t the Oklahoma City Meeting of the PetroleumBrandl, American Ins t i tu te of Mining and Metallurgical Engineers, October 3-4-5, 1951,and i s considered the property of the Petroleum Brandl. Permiasion to publish i shereby restr icted to an abstract of not more than 300 words, with no i l lust ra t ions,unless the paper i s specifical ly released to th e p re ss by the Branch PublicationsChairman or the Executive Secretary on his behalf Such abstract should contain appropr ia te , conspicuous acknowledgments. Publication elsewbereaf ter publication inJournal of Petroleum Technologr i s permitted ~ request, prOVided proper credit i s,given to .lIME and to Journal ~ Petroleum Technology - - date and place of presentationand mate of pUblication in Journal of Petroleu:rn Technology.

Discussion in writ ing of this paper i s invited. Three copies of'discussion shouldbe sent to the Petroleum Branch off ice. Such discussion wi l l be presented a t the p.bovemeeting with the paper and considered fo r publication in Journal of Petroleum Technology

,ABS'l'BAC'l'A method that combines the diffusity equation and the materia l-balance equation i spresented to determine the overal l porosity and mobility of the reservoir and 'Waterintrusion.The procedure ii based on the formulas and tables developed by A. F. van Everdingen and W. Hurst, and is applied to the Poza Rica o il f ie ld which i s a high re l ie fpool. An approach to the problem of locating the datum plane i s given.A discussion of several proposed formulas for water encroachment i s included, and

comparison is made between the results given by thesee.xpressions and t hose obt ained bythe unsteady radia l flow theory.DffRODUCTION

The Poza Rica f ie ld i s located on the Gulf Coast of Mexico approximately 170 ki lo'meters (110 miles) northeast of Mexico City, almost equidistant between the harbors oflReferences given a t end of paper.


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2 114-G AN APPLICATION OF DIFFUSITY AND MATERIAL-BALANCE EQUATIONS . . . .Tampico and Veracruz, and 35 km (22 miles) from the beach. I t l ies within the geological province known as the Tampico Embayment.

The discovery well 'Was P.R. No.2 , completed in 1930 a t the to p of the structurea t a depth of 1,986 meters (6,514 f t ) .

The productive formation a t Poz.a Rica is a limestone called the Tamabra. Thestructure i s about 16 km (10 mile s) lo ng and about 5.4 km (3.4 miles) wide, and thereservoir has an average thickness of 127 m (417 f t ) . I t has a steep northeast flankand a very gentle, almost f i a t southwest flank. The northwest and southeast extremitiehave approximately the same angle of slope as t he nor thea st flank. The reservoir isl imited along the northeast f l ank , northwest and southeast extremities by sa l t w a t ~ analong the southwest flank by lack of porosity (Figure 1).PRELIMINARY CONSIDERATIONS

This paper i s concerned If'i 'ththe overall behav io r o f the Poza Rica f ie ld . The maproblem we are interested in here i s the calculation of the water influx into the resevoir , and fo r th is purpose 'the t ransient flow theory i s applied. I t i s considered thathe flowf'rom the aquifer into the o il f ie ld i s radial and that t he poros it y and permeability of the sYstem are constant. These assumptions are j u st if ied ,2 f o r the resul tobtained from their application to many o i l ,fields have been proved to be suitable forthe interpretation of o il f ie ld behavior. Even in the case of non-radial symmetry themethod affords a convenient framework to derive more exact means o f dea li ng n th theproblem of de.scribing the performance of a reservoir.

In order to apply the diffusi ty method to the Poza Rica o il f ie ld , the l a t te r i sconsidered to be l imited by two circular bo.undaries If'hich, according to the above menI t ioned f ie ld data, can be calculated to have a radius o f 13.2 km and an angular ampli tude of 740

I t i s assumed fo r the PurPOse of the calculation tha t water encroachment onlytakes p lace a long the northeast flank.The aquifer i s assumed to be one of semi-infinite extension, since i t is not possible to assign to it ~ prior i any dimensions because of the fac t tha t , even in wellsdri l led a t great d is tances from the center of the o i l f ie ld , sa l t water has been found.Furthermore, the conclusion r e a c h e ~ in th e p re se nt work, in which the formula ;for

inf in i te or non....l im i ted rese rvo i rs i s used, supports !. posteriori the hypothesistha t t he aqu if er may be considered as inf in i te .Since no pressure surveys were made between 1930 and 1941, it i s assumed tha t thepressure drop i s l inear during th i s period. From 1942 to 1949 seven pressure SUl!Vey$were made, and pressures were measured a t a 2,100 m subsea datum plane. The pressurehistory i s shown in Figure 2.The choic e o f the -2,100 m datum plane was made only because the wells dri l led inthe early years of f ie ld development were completed a t tha t average depth. In recentyears many wells have been dri l led to greater depths and; }n diverse areas, thus leadingto a broadening of the l imits and rock volume of the f ie ld . Furthermore an applicationof the material-balance equation taking the datum plane a t tha t level leads to completely unacceptable resul t s .The diff icul ty encountered in locating the datum plane for a high re l ie f pool iswell known. This dif f icul ty i s even greater fo r Poza Rica because of the great thi.ckness of the producing formation. Previous to the present paper we had proposed tomeasure bottom hole pressures a t the le ve l corresponding to the mean pressure obtained


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l14-G AN APPLICATION OF D!WUSITY AND Ml\.TERIAL-B1\LA!!lCE EQUATIOliS :3by weighing volumetrical ly the variable (with depth) pressure of th.e reservoir. The

~ e v e l thus obtained is -2,.300 m. I t wil l be seen that by reducing pressures to th is levefor the purpose of applying the material-balance equation no inconsistency i s obtainedin the results and the pool performance i s better described when th is -2,.300 m datumplane i s taken.

Calculations by means of the dif fus ity equation are independent of the datum planelevel s ince pressure enters only as pressure decl ines ( in time) and these are assumed tobe the same for similar in tervals of tim e, the variation (between a reasonable range)of depth notwithstanding. This amounts to saying tha t a const an t p re ssure g radi en t i sassumed. Of course, this i s not t rue, but the way in which the datum plane level i sdetermined j us ti fi es t hi s assumption.-Pressures a t -2,.300 m are given in Table 1 ascalculated from those for -2,100 m, and assuming a constant pressure gradient of0.07.38 kg/cm2/m as prertously estimated.FUNDAMENTAL THEORY

The theoret ical basis on which th is work i s founded is to be found in reference 2,Chapter XI, mainly in Sec tions 11.3 and 11.5, and in the paper ci ted in reference 1, wheuse i s made of the Laplace transform to find solutions of the diffusityequations undervarious in i t ia l and boundary conditions.The fundamental equation that governs the flow of a f luid in a porous medium, ordiffusity equat ion , obtained by means o f D ar cy 's equation and the continuity equation i sthe following:*

ffI< .e.aT .) (1)

If the equation of the state isCP

~ _ ~ ~ eEquation (1) transforms into

(2 )

_dP-dT (3 )The equation for the pressure can be shown to be

2 2\1 P + c ('\7 P) = fflCf(

dP-aT (4)or, if the f luid is s l ight ly compressible, one obtains for radia l flaw,

where t = aT , R = ~ r , a = Kf ~ < R :* Nomenclature given a t end of paper.

dPdt J (5 )


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4 l1.4-G AN APPLICATION OF DIFFUSI'l'Y AND MATERIAL-BALANCE EQUATIONS The solution of Equation (5) for the case of cons tan t pressure a t the border Iq,can be found by means of the Laplace transform,2,3 and from here the followingformula for the volume influx of water through Iq, into the o il f ie ld i s found:

(6 )

where4 J0 I _ e_v.2t durr2. 0 .u.z,[ J@h" 1" Y.,2(AJ)r

The foregoing formula for ~ can be extended to the case of variable (in time)pressure a t the boundary; the formula is the following:

(8)

and i s given in the paper of van Everdingen and Hurst cited in reference 1. In the sampaper a table of Qt i s given, of which we have made extensive use. (We have made anapplication of these tables and have, by no means, repeated those calculations.)We write Equation (8) in the form,

Q1" -== 2 0( feR: h [ ~ P Q + Ll'P Q + ... + ~ P QJ (9 )() t' \ I 1'\-1 rt-t Iwhere n is the number of years in time T.

Equation (9 ) can be writ ten as Qr =A Fn , Fn standing for the expression in brack(10)

and A for the coeff icient :

(ll)PROCEDURE

A complete analytic treatment of the problem can not be con tinued fur ther onaccount of the great number of unknowns. These are K , JA ,or bet ter s t i l l Kip ,often referred to as the mobili ty, r , c ,N and ~ , but one can resor t to graphicalmethods.In works writ ten on the s ub je ct up to now the point of view is expressed that ,because of the simplifying assumptions underlying the dif fusity theory, and because of tinherent uncertainties in the geometrical characteristics of t he r es er vo ir , it is not


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l14-G AN APPLICATION OF DIFFUSITY AND MATERIA.L-BALA.!fCE EQUATIONS

possible to determine i t s phys ical constan ts and also tha t it i s preferable to le t the"alue of a, the parameter given by Equation (5-) , be indeterminate to a certain extent;if some value of a i s chosen for specific application, values of geometric and physicalconstants a re ad ju ste d as convenient.

We believe that it is possible to take further advantage of the dif fus i ty methodthan just predicting influx or pressure histor ies , and tha t some of the phys ica l cons tan tof the reservoir can be determined when use i s made of the values of other bet ter knownconstants.

In the present application we t ry to determine f and K ~ ; the constants reasonably assumed to be bet ter known are those of the aquifer, which are (1) i t s thickness,which for our purpose should be taken as equal to the thickness h of the o i l reservoir ,and (2 ) the compressibil i ty c, which wi l l be taken as that o f wate r, equal to4.56 x 10's 0.';,.. We have then proceeded in the following manner. Making use of Equation (9 ) of the

foregoing section we have calculated QT for several values of the parameter K//4 and theporosity f ; th e calculations have been performed for every year during the period 1930-49Using production data and f ield constants given in Table 1, we have app li ed th e

material-balance equation to calculate Qr for each year of the period 1942-49, takingagain several values of thE:: porosity f ; Qtr depends on f through the original o il inplace N. Figure 3 shows graphica lly the variation of QT against f and KIp for 1949 asgiven by both the diffusity and material-balance equations. The l a t t e r was applied takin~ o levels of the datum plane, tha t of -2,300 m corresponding to the volumetricallyIreighted mean pressure, which i s the one we propose, and tha t of -2,200 m, depth of thevolumetric center.

Silnilar calculations and graphs were made for 1942.For a value of K / ~ there corresponds a value of f , obtained as the abscissa of theintersection of the straight l ine and the corresponding K ~ = constant curve; one valueof f for each of the two dates i s obtained and it should be expec ted that the sameporosity would be found for both dates, though in general th is i s not actual ly the case;

th is can be made to occur i f t he d if fe rence A f of f for the two dates is plottedagainst K/I't and a value of KIp i s determined fo r which A f is zero. In this way weobtain K ~ = 0.125 taking the -2,300 m plane and K/J4. = 0.100 for the -2,200 m plane.The value of f i s the same for both planes, and i s found to be f =0.082.The method of least squares can be used when more than two dates are taken in toaccount.Using the values of f and K/I't determined from the graphs, we recalculate thevalues of water intrusion a t different dates of the period 193G-49 ; these are shown inTable 2, where the resul ts for the -2,200 m datum plane are included.Original amount of o il is calculated by the formula

N = Vf (1 - Sw)uo ; (12)\r , the in te rs t i t i a l water saturation, being equal to 0.15, as determined from coreanalysis , .and V , the rock volume of the producing formation from the i n i t i a l g a s - o ~ lcontact to the water-oil c o n ~ a c j ' i s estimated trom an isopach map to be 12,740 x 10 m3;then we find ~ =3604.1 x 10 m , or 3,391 x 10 bb1 for the -2,300 m datum plane andN =602.0 x 10 m or 3,379 x 106 bbl with datum plane located a t -2,200 m.


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6 ll4-G AN APPLICATION OF DIFFUSITY AND MATERIAL-BALANCE EQUATIONS Material-balance calculations have been carried out using the equation in the form

From these resul ts we can calcu la te the water-drive, depletion-drive and segregatiodrive indices by means of the formulas5

Water-drive index(W D I) ,(14)

nDepletion-drive index( D D I ) (15)

Segregation-drive index(S D I) (16)

respectively. We find i l =0.33,The expansion of the gas cap divided by (1 - Sw) gives the corresponding rock volume, and from this and the rock volume VB. depth graph (not shown in this paper) we

determine the drop of the gas-oil contact level to be 10 m (32.8 f t) unti l 1949, original contact (in 1930) being at -2,030 m.Figure 4 shows the comparison between the d if fusion and material-balance methods;we have plotted the values of R, the ratio of the cumulative water influx for any givendate to that at 1949, for the seven pressure surveys made. We observe from this graphthat (1 ) the 1945 and 1948 values are lower than those expected from the others and fromthe general trend o:f R as given by the d if fusion method; as has been said, the lat ter isessentially unaffected by changes in the diffusity constants; and (2) taking the datumplant at -2,200 m the values of R are s t i l l more scattered and for this reason we concluthat the -2 ,300 m plane should be used to describe more correctly the reservoir per:formance. For the -2,100 m datum plane we find that R is sometimes a decreasing function

and the values for the water encroachment and the quant ity of o il originally in placeare decidedly unacceptable.Taking into account the la rge period covered and the small number of pressure survemade i t is seen that the agreement between the two curves is satisfactory. If moreaccuracy is wanted i t is clearly apparent that a greater number of bottom hole pressuresurve:rs is necessary.For this reason no attempt of applying any stat ist ical analysis has been made tosolve the problem of locating the datum plane.


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IJ.4-G AN APPLICATION OF DIFFUSITY A...?ID MA'l'muAL-BALANCE E;,JUATIONS 1As a matter of fac t , the lack of f ie ld data i s a handicap for undertaking new"."ngineering studies and. fur ther research work in Poza Rica tha t may p re va il f or some,rears. We strongly recommend making bottom hole pressure surveys a t l eas t once per year.

OTHER FORMULAS FOR WATER INTRUSIONNow we turn out at tent ion to other f o r l l l U ~ a s fo r the estimation of wate r encroachmenthat have been thus fa r proposed. These are:1 . Simplified form of Hurst 's or Diffusion formula fo r ra dia l influx:

2. Schilthuis equation:

~ = Ap1 n t dt

3. Simple approximation:

T~ =cl AP dtoQ.r = c T AP

Due to the rather troublesome calculation implied in the dif 'fusity formula and thesimplicity of the above equations, use of' the la tte r is advised but it i s desirable tosee how the resul ts given by each one of them compares with those obt ai ned from the diffus i ty theory. We present such a comparison for the concrete pressure historyof ' the PozaRica f ie ld .

Of' course, none of the foregoing formulas is su ff ic ien t i n i t se l f to calculate the'Water influx since the constant is unlmown; it i s necessary to resor t to other means todetermine i t s value; when the cumulative water intrusion for one date i s lmown, c i scomputed and the cumulative water influx may be calculated for any other date. Assumingtha t Q49 has the value given by the dif 'fusity method, that i s , 30.0 x 106 m3 , we findfor the constant in t h e s imp li fi ed form of Hurst 's equation,2

c = 1,268 m3 / ~ day. cmIntegr-als in formulas (1) and (2) were computed mechanically by means of a planimetIn Figure 5 results given by each of the above formulae are summarized. I t may beseen tha t maximum differences given by formulas (1), (2), and 0 ) from resu l t s obtainedfrom the diffusi ty theory are 4, 6, and 8 per cent respectively, of the cumulative waterinflux for 1949.

FUTURE PRESSURESRecently we have app lie d t he simplified form of the diffusion fo rmula (1) combinedwith the material balance equat ion for predicting pressures. The usual way is to determine, by t r i a l and error , a value of P t ha t s a ti sf ie s both equations. We haYe


8/12

8 114-G \AN APPLICATION OF DIFFUSrfi AND MATERI.AJ:,...BAL,ANCE EQUATIONS . . . .followed a different procedure - Hurst 's f ormula can be writ ten in different ial formas follows: ~ = cA PIf In T , ( 1- =d9r)T dTThis and the mater ial -balance equation are taken simUltaneously and the method oft ra j ec tor ie s f o r step by step integration i s applied. Values of pressures obtained bythis and other methods are in close agreement. DetailS and results of the method wil l bepresented elsewhere.SUMMARY AND CONCLUSIONS

The Poza Rica f ie ld i s a par t ia l water-drive pool whose behavior can be described inthe radia l flow scheme; extension of the aquifer can be considered inf ini te . The porosiand mobility of the system have the values f =0.082 and K/p = 0.125 darcies/centipoiseThese value s being obtained by a method that combines the diffusity and material-balanceequations, for calculations by the l a t t e r the -2,300 m datum plane should be taken.Water influx since 1930 unt i l 1949 i s 30.0 x 106 m3, or 189 x 106 bbl; the corresponding rate of flow a t end of this period i s 6,575 Jl3/daY or 41,360 bbl/day.Amount of o il in place originally i s 604.1 x 106 m.3 or 3,800 x 106 bbl stock tank oIn view of the g re at ro ck volume of the f ie ld and the desirabil i ty of having moredata available, bottom hole p re ssure surveys are recommended to be made a t least onceevery year.Further study and research are going on and work of this kind should even beencouraged.

ACKNOWLEDGMENTSWe are indebted to Mr. J . Colomo, General Sub-Director, and Mr. A. Barnetche, Produ

t ion Manager of Petroleos :Mexicanos, for permission to publish th is paper. The helpfulsuggestions made by Professor V. C. TIl ing a re a ls o g re atly appreciated.

NOMENCLATURE~

~ cfGhil

i2i3Km

One half o f angula r amplitude of o il f ie ld .Formation volume factor a t P.Formation volume factor a t original Pressure Po.Compress ib il ity of wate r.Porosity.Cumulative gas production a t standard conditions.Thickness of producing formation.Water-drive index: rat io of volume influx to volume of voids l e f t in the reservoby production.Dissolved gas-drive index: rat io of dissolved gas expansion to volume of voidsl e f t in the reservoir.Free gas-drive index: rat io of free gas-cap expansion to volume of voids l e f t inthe reservoir.Permeability.Ratio of original gas cap volume in t he r es er vo ir to original o il volume inthe reservoir.Viscosity.Amount of stock tank o i l in place original ly .


9/12

114-G AN APPl.IC,A.'l'ION OF DIFFUSITY AND ft'l'ERIAL-BALANCE EQUATIONS 9. n

'"tiPAPkQtrR%rr or p?tJ'fPovvVoz

Cumulative stock tank o i l production. Number of years in time TPressure.Pressure drop.Difference between the k and k + 1 pressure va lues .Cumulative water inf lux a t time T.Radial distance. Ratio of cumulative water in trusion to tha t a t 1949.Radius of o i l f ie ld .Dimensionless distance. Dissolved gas-oi l rat io a t P.Dissolved gas-oil ra t io a t original reservoir p r e ~ s u r e .Cumulative gas-oi l ra t io = GIn.Saturation of connate water.Time.Dimensionless t ime.= fl + ( r0 - r) v.= floRock volume. .Gas volume factor a t P.Gas volume fac tor a t original reservoir pressure.Cumulative water production.

REFERENCES1. A. F. van Everdingen and W. Hurst: "The Appli ca ti on of the Laplace Transformation to

Flow Problems in Reservoirs." Pet. Tech. Dec. 1949, p. 305.M. Muskat: Physical Principles of Oil Production. New York, 1949. McGraw-HillBook Co., p. 540. - --

McGr.mr-HillTrans., AIME, (1943), 151, p.New York, 1950.

"Analysis of Reservoir Performance.". E. Old, J r . :s. J . Pirson: Elements of Oil Reservoir Engineering.Book Co., p. 390.

3. J .C . Jaeger: An Introduction to the Laplace Transformation. London, 1949. Methuen & Co.4.5.6. R. Woods and M. Muskat: "An Analysis o f Ma te ri al Balance Calculations," Trans., AM(1945) 160, p. 124.


10/12

10 l1L.-G AN APPLICATION OF DlFFUSITl .AND M A T E R I A L - B A I . A , ~ l C : i j l EQUATIOW$ TABLE 1 - FIELD CONSTANTS AND PRODUCTION DATA

Field Constants(Nomenclature at end of paper)

P1930 = 3,589.8 Ib/in2 = 252.8 kg / cm 2 at = 2,300 m.m = 0.06750 Volumetrically determined.r o = 160.31L. m3 / m3 ; Vo = 4.379 x 10-3 m3 / m3 ; Uo = 1.470 m3 / m3muo = 22.6605.Vo

Production DataMillions of m3

Date P z n G rp - r o- 2300May 1, 1942 3097.0 0.065 25.115 4979 37.934April 1, 1943 3071.5 0.071 28.341 5654 39.185May 1, 1944 3013.2 0.077 31.736 6288 37.821July 1, 1945 2932.3 0.088 35.556 7050 37.965July 1, 1946 2902.5 0.103 39.514 7714 34.908May 1, 1948 2781.8 0.149 48.514 9080 26.848March 1, 1949 27n.8 0.175 53.189 9836 24.611TABLE 2 - CUMULATIVE WATER INFLUX

Qr (millions of m3)Date1930193119321933193419351936193719381939194019411942194319441945194619481949

Datum Plane at -2,300 m.o0.300.901.502.403.304.505.706.908.409.9011.7013.5015.3017.4019.8021.9027.6030.00

Datum plane at -2,200 m.o0.260.791.312.102.893.944.996.047.358.6610.2411.8113.3915.2317.3319.1624.1526.25


11/12

I

'"!

~FIGURE N9 I

O IL ZONE ISOPACHROCK VOLUM E

POZA R IC A F IE LDCURVAS ISOPACAS

EN METROS

I i 8 0 0 i' I, I- 0 8 0 iI- u 0N' gI I IL I _ ~ I ~ .I -N-oL, ~ - l l l . _- - - 1 r - - -i F F I IGG IUJFR A A

ZA R IC HPO ,

I' 'I I- - - r - - T I I II I .

I I I I II I 1

.... 1 1

I I1 1

~ ~ - I 1II I I I

I I II III I I

II I4-""1-.... I

I L L - - l - - - ~ - -+-Ll-L

i'

82,000

81,000 a. ! ! i ! I i J i- i i- I s\l.!2 !2 !


12/12

!TE'-m----fI- - - ~ f --8't t t i - - ~ t t - ~ ~ - - - ~ ' . ~IT J ~ . . . - .,.111: -.I---------L ! _ ~- - - . - - J ~ - - -t---+--+,-- -H-t- . - - / 1j-+-ttt---.tcri'::r . ,- ------ -- - - - - l - - - - ~ -- -- - - ~ + - : - -

04

" - -,. ~ ~ + - I - - - + ~ I - - - - +

t

~ . _ .

1-

r-=:

&

1-

o

RESERVOIR PREKG/CM1 AT 210

----t-- --.-- -- .----1---.------

1- ..

8

1--

g

t

v n 3M .:l0 l:t39wnN

- - - t - - - - - - - - - - - - ! - - ~ - , ~ . . . . : - - ~ - -

i\ :;; I:' - - f __ l+ - - ; i \\ , ~ -I... I J : ~ _ -0 i \\ l' r1- 0 1 -? I i. I \::: + ;'J -+ ''t. 1- - ] . Ii ~ . ~ ~ =,i=JI ="j( ~ \ . . - ~ -

~ ' - 1 - - - ~ \ - 1 , , 1 - - ~ ' ~ I i - -;f'-- S ~ l ~ : ~ j ; {'\ ,l[: .. - ~ J ~ l ~::,i ~ ,\1 ~ '1"' . l'i'ji:\J+. +Q '" ," I I .. I'" - - --- I \ '\_ .... " t ~ . ~ I \" . hg!::-=-=- =-- ! ---: -t r- L IT; I : ~ - - I - - - ' - - .-\ 1

, i ~ : ~ I ~ ~ ~ ,'II, : ;;: ;-;;'~ l : : r - - - - - - + - - ==:= +=. ~ t r - - [ I J - I l : J = ~ t ~

I I ~ ~ ' ~ =f= ~ ~ + r T J -f . -++-- .--IE :::f__- r-t-- ---I--+-+-r--' ----

,ATE,--+---

riGf./RCN!*-.!j

-----f---P-

l ~ - -

- - - ! . - ~ - - ....----+----

I-- - - 1 - - - + - - - - 1 - - - - 1 - - - - - - l - - 1 - ~ - - - ~ - ~ -

///1':/1./[/17

1----1----1-----+ -+--1---+---

COMPARISON DIFFU!>LON AND ....ATERiAL BAL.A.NCE METHODS.- M A . TER I A L &ALANe-: EQUATION

= ~ ; ~ ~ : ~ : ~ ~ : ; l ~ ~ : ~ ; ~ ~ ~ t : r:: ~ : : ~ 4 " : ~ : ~ ~ ~ : ~II DISCARDED VALUE$, -2.300t. t?LANE (1945ANO 1948)C!lI DATUM PLANE AT-2,2OO '".. DATUM PLANE AT -2JOO M

i TT I I I \l\ I J I = t . = F t ~ T I j l l ~ l ; ' -

:>A1 f 19-49CUM-)LATtVE WATER INFLU)' Q. IN MILLION!!> Qf CU. METERS ..0.& f"LJNCTIQNCf" PO"lOS;T'o' f A N 0 M O V IL l T v -- DArUM P L A ~ l AT METERS, BELOW 5EA LEVEL.-- DATUM P...ANE AT 2.200 Mi::TERS BELOW SEA L.(V(L

"""

R=3L.a..

,.." ~ r + + - _ _/- .' = ~ 1 + l - H - - - - -, . / - - - - . --- ---1- -

- - - - ! - -T - - ------ tr

T

I---+ I, I 1---+-+..,..--1-----1---- . ~ - I - - _ _ _ _ _ _ _ t - - - -; " ~ f f O O f t o 19)-0

romero, a. and hefferan, j. an application of diffusity and material-balance equations to a high...

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