7/26/2019 SPE-1020-PA Mueller T. and Witherspoon P.a. Pressure Interference Effects Within Reservoirs and Aquifers

1/4

/u ---/(9

Pressure Interference Effects

THOMAS D. MUELLERPAUL A. WITHERSPOON

MEMBERS AIME

ABSTRACT

For the case of an infirsi~e radirrl system operating atcotwutt terminal rate, the reservoir engineer often usesthe point source solution oj the diffusivity equation tostudy pressure interference eflects. At early times and atshort distances from the inner boundary rhese solur~onsare invalid. The amount of error is often not preciselydefined because of mathematical difficulties. Work pre-~enteri here shows that, ij dimensionless time is definedappropriately, previous solutions of the pressure equationcan be displayed as a family of curves on one chart. Thesesolutions include the point source solution (referred to inthe field of hydrology as the Theis io[ution) and othersolutions obtained with digital computer methods. Withthese curves, an exact evaluation of the pressure dropwithin a reservoir or an aqu[jer can be made by the engi-neer. Examples of field problem solutions are presented.In most reservoirs the error nvolved when the Theis s o l ut ion is empIoyed is often negligible; whereas, in the calcu.Iation of interference efleets in an aquifer, a substantialerror can occur thror+ghsuch an approach.

INTRODUCTION

Flow equations are used in petroleum enginwring tostudy the behavior of individual wells and reservoirs. In

the case of wells, the pressure response at the wellboreface is the major point of interes~ whereas, in the caseof reservoirs, the pressure response at the interface of theaquifer boundary is sought. To aid in such studies, theflow equations have beers solved in terms of the behaviorat these two inner boundaries.

Only limited work has been published in regard tothe pressure conditions away from these points, i .e., withinthe reservoir or aquifer. Theis and Mortada are amongthe few who have reported on this problem. The Theisapproach employs the exponential integral and is validfor pressure conditions thai occur some distance awayfrom the flow disturbance, It is derived from the conceptof a point source, as opposed to a flow across some 5nitearea. The Mortada results, on the other hand, are validat ali points within the reservoir or aquifer. They arepresented in terms of dimensionless ratios of the radiuswhere the pressure is desired to the radhs where the flowrate is measured. Their main use, in the past, has been inaquifer studies, The published results are presented in th eform of graphs that are limited to a maximum radius ratio

Orhrinrdrnanuacrint rwefved {m detY of Petroleuq Errslrreeraofirx-Nov. S 1964 Revlaed rnanrmrlpt rmelved March 12 19S6.Paper m?.wrted at SPE Crdlforn[a Reslonaf Meethuz held In be An 8ele a Nov.6.6 1964. ~~

~Referenceasiven at end of Diner.

Within Reservoirs and Aquifers

1 ap r Dr

STANDARD OIL CO,OF CALIFORNIA

SAN FRANCISCO, CALIF,U. OF CA 1/FORN/ABERKELEY, CALIF.

of 64, These graphical results are cumbersome to interpo-late at non-integral radius ratios, so that one may beforced to utilize a rather involved analytical expressionpresented by Mortada.

BASIC EQUATIONS

The solutions of Mortada and Theis are both basedon the diffusivity equation as applied to the case of anintlnite radial system subject to a constant terminal rate.The equation is obtained by combining the material bai-,

ante equation with Darcys flow equation. The assumptions implicit in the use of this equation are as follows:(I) a singte fluid is present that occupies the entire porevolume; (2) the reservoir is horizontal, homogeneous, uni-form in thickness, and of inthtfte radial extent; (3) com-pressibility and viscosity of the fluid remain constant atall pressures; and (4) fluid density obeys the equation

P = p.exp-k[pop) , . , . . , . (1)

Using the dtiusivity equation in siturttions where theabove conditions do not hold wiil result in errors. These errors (not discussed here) but only the errors which arisein the solution of the equation itself.

he diffusivity equation for the homogeneous reservoirconditions cited above can be written in cylindrical coor- ~dinates as

ap +IF

To obtain a dimensionless equation, so a single sol tionLan be used for applications of different porosity, (per ea-

bllity and fluid properties, the following transfo~mationsare usually made: I

%rkh(p,-p,)pu =up

. (3)

r,, =rjrm, ,., ti. . . . . . . (4]

h= . . . . . . . +pc r. (5)

After these transformations are made, Eq. 3 can be writ-ten in dimensionless form, as

1 apo.rD 8rD

.

MORTADA SOLUTION

One solution of Eq. 6 has been given by Mortada?

1: :

Al It IL, 1965 Revrintud from AwII, 1965,Issueof JOURNAL OF PETROLEUM TECHNOLOGY 471

,. .. . ..- ..--- .- ,.. .... . . -., . ,: -;. - ..-.:.,...-- ., -. _: _L._:

.. .

7/26/2019 SPE-1020-PA Mueller T. and Witherspoon P.a. Pressure Interference Effects Within Reservoirs and Aquifers

2/4

---

-- . .. --

where he presented dimensionless pressure drop as a func-tion of dlmensionlms time. Hk graphical results are re-produced in Fig. 1. The dimensionless time is given interms of the radius at the inner boundary, i.e., the well-bore or the aquifer boundary, Results of the dimension-less p~sure drop vs dimensionless time are given at thefollowing radius ratios: 1, 2, 4, 8, 16, 32 and 64, a -though results at much higher ratios are available fromMortada, As will be shown there is no need to carry outsuch determinations for radius ratios above 20,

van Everdingen and Hurst have also presented resultsof the dimensionless pressure drop at the wellbore inter-face (rn = 1) as a function of dimensionless time definedin the same manner as above. Their results correspondto Mortada results at the sanle r~ = 1. More recentlY>Driscoll has also used the concept of dimensionless pres-sure vs dimensionless time at various radius ratios.

THEIS SOLUTION

The mathematical formulation of the point source soh-tion and its resultant. exponential integral are due to LordKelvin? Theis, huwever, is the first, to our knowledge,to demonstrate how the point source solution could beemployed in the analysis of non-steady-state flow prob-lems. In recognition of his early work, the exponentialintegral solution is normally referred to in the fieId ofhydrology a.s the Theis solution, and that term is adoptedhere.

In this solution, the variable X is defined as a dimen-sionless quantity inversely reJated to time, X is the inde-pendent variable in the Theis solution, and the integralvalue or dependent variable is related to the dimension-less pressure drop, The definitions of the dependent andindependent variables are compared with those of Mortadaand van Everdingen and Hurst in Table 1. The Theissolution of the exponential integral is shown in 13g. 2.

If we alter the definition of dimensionless time givenin Eq. 5 to be based on any radius in the infinite sys-tem, we then have

kt= ,. . ,,. . . .

0 + llc(7)

The dimensionless time of Mortada is related to that ofEq. 7, by

IMENSIONLESS TIM b

FIG. 1MOEITADASPOINT SOURCESOLUTION.

rASLE I OMPAR15GN OP iEPEt4DENTANDINDEPEt40iNTVARkES-

DlmendonleN Dlmmslonless Dlms&wdlndependont Vorlabla Oopm?ont Variable Pm surt Drop

.Mortado, and ktvan EVMIIWM tD= Ap,,

Ip/.ww~% APD

an d Hu m rkh

t.( Mortada)t(Eq. 7)= r; . . . . . . . . (8)

From Table 1, it can also k, seen that, with reference tothe Theis solution,

@q.7=+x, . . . . . . . . . . (9)

and

Ei( -X)AP. = ~. . . . . . .,. , (lo)

Fig. 3 represents the Theis results of Fig. 1 with thedefinitions of dimensionless time and dimensionless pres-sure as given in Eqs. 9 and 10, respectively. By adjustingthe dimensionless time of the Mortada solutions in ac-cordance with Eq. 8, it is apparent that the array ofcurves on Fig. 1 becomes a family of curves on Fig. 3that converge on the Tbeis solution. Other radius ratiosnot given in Mortadas work were obtained from the digi-tal calculations performed to obtain the results given byMueller?

It can be seen on Fig, 3 that, for all radius ratios great-er thrm 20, the Theis solution adequately gives the pres-sure drop after any practical time,

This can be further demonstrated by the results pre-sented in Fig. 4, which shows the relationship betweenthe per cent error rsne would get in using the Theis solu-tion for various radius ratios in place of the exact SOhr-tion. It will be seen that, after a dimensionless time of50, the Theis solution can be used with an error of only1 per cent for all radius ratios. This, of course, is alsoevident from the convergence of ail curves onto essential-ly a single line on Fig. 3. Any combination of radiusratio and dimensionless time that falls to the right ofthe 1 per cent line on Fig. 4 will have an error of lessthan 1 per cent, Correspondingly, any combination fallingto the right of the 0.1 per cent line will have an errorof less than 0.1 per cent.

0.00001 0.0001 0.001 0.10.0~L 1 I I

0.045, This is equivalent to about2 minutes. For all practical examples where the radiusratio is large, the Theis solution can be used with confi-dence.

CONCLUSIONS

1. Solutions for the diffusivity equation for the infiniteradial case at constant terminal rate are presented in theform of graphs of dimensionless pressure drop w dimen-sionless time for radius ratios from 1 to infinity.

2. For radius ratios of 20 or above, the Theis, or pointsource, solution can be used with little or no error formost practical situations. A chart is presented where onecan determine the order of the errors that will resultthrough use of the Theis solution.

3. In aquifer studies, it might be necessary to use thenew solutions of the cliffusivity equation presented herefor low radius ratios. The absolute magnitude of the pres-sure effects with low ratios and small times might, how-

ever, be insignificant;4. In well interference tests, the Theis solution can be

used for all practical lengths of time, and at all normalwell spacings, without introducing errors greater than 0.1per cent.

~,

NOMENCLATURE

c = compressibility of fluid, PSF .h = reservoir thickness, ftk = permeability; rnd, (1 perm = 158 mdjp = pressure, psia

p. = dimensionless pressurep. = fefereace pressure, psiap, = mltial pressure at s~nle givfn point, psiop, = pressure at some gwen point after an el apse of time,

psia

q = constant flow rate at well, B/Dr = radial distance, ft

)0 = dimensionless radiusvu = wellbore radius, ft

t = t ime, dayst.= dimensionless timeX = independent variable in Theis solutionf.t= fluid viscosity, cpp = fluid density at pressure p, lb/cu ft

P. = fluid density at pressure p,, , lb/cu ft+ = porosity

REFERENCES

L Theis,Charles V.: The Relationship Between the Lwerin

the Flezometric Surface and the Pcote and Duration of%:

charge Using Groundwater Storage, Trans., AGU (1935) 519.

2. Mort;da, M:: A Practical Method for Treating Interferencein Water Drive Reservoirs, Trans., AIME (1955) 204, 217.

3. van Everdingen, A. F. and Hnrst, W.: The Application of th~Laplace Transformation to Flow Problerne in Reser{oir ,Trans., AIME (1949) 186,305.

4.fMseoll, V, J.: U.SCof Well Interference and Build-Up,Datafo r Ea rly Quan titative Dete rmination o f Reserve s, Permeab ilityand W6ter Influx, Jour.Pet. Tech. (Oct., 1%3) 1127.

5. Mueller, T. D.: Transient Response of Non.Homogen ;Aquifers , Sot. Pet. Erg. our (March, 1962 ) 33.

JOURNAL OF PETROLEUM TECHNOLOGY

--- .=..- .. . . . ... . . . . . ... .-. .. . . .. --. -. .. ... .... . ..