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2 Geothermal Resources Council TRANSACTIONS, Vol 9 - PART 11, August 1985 INTERFERENCE BETWEEN CONSTANT RATE AND CONSTANT PRESSURE WELLS Abraham Sageev and Roland N. Home Stanford University Stanford CA 9435 ABSTRACT A pressure transient analysis method is presented for interference between wells producing at constant rate and wells producing at constant pressure. The wells are modeled as two line source wells in an infinite reservoir. The first well'produces at a constant rate, and the second well remains at a constant pressure. Dimensionless semilog pressure type curves are presented together with instantaneous rates and cumulative dimensionless injection for the constant pressure well. The effects of the relative size of the two wells on the pressure response of the constant rate well and on the rate of injection at the constant pressure well are discussed. The rate-pressure model may also be applied in analyzing pressure interference between communicating geothermal reservolrs. The distance between the reservoirs may be estimated If the reservoirs are approximated as two line sources. INTRODUCTION Pressure transient analysis methods are used to estimate reservoir properties so that exploitation schemes may be evaluated. The method presented in this paper permlts the interpretation and design of a special case, which will be referred to as the rate-pressure doublet. In this twowell system, one well produces at a constant rate and the other well produces at a constant pressure. Such systems may be encountered in common kinds of reservoir flooding operations in which fluid is injected into one well at a constant pressure and produced from another well at a constant rate. Other configurations, such as as isolated five spot pattern with a constant pressure injector and four constant rate producers, may be treated with the same method. There are three time dependent parameters in the rate-pressure doublet model: the pressure response of the constant rate production well, the rate of injection and the cumulative injection at the constant pressure well. These three parameters are described in the theory section. A constant rate well is approximated as a line source, since the reservoir is much larger than the finite radius well. The constant rate production or injection line source well has been used as a building block for calculating the response of various reservoir systems. Theis (1 935) presented the lir?e source pressure solution in an infinite domain. Carslaw and Jaeger (196) and Van Everdingen and Hurst (1949) presented the pressure solution for a finite radius well in an infinite system. Mueller and Witherspoon (1 965) showed the geometrical and time conditions under which the line source and the finite radius solutions are practically identical. They concluded that for observation wells located at a distance twenty times the wellbore radius the line source approximation is applicable. Also, this approximation is applicable for any observation well after a dimensionless time of ten. Since the diffusivity equation describing the flow in the system is linear, superposition in space of constant rate line sources may be used. Stallman (1 952) presented the superposition of two constant rate line sources replicating the effects of constant pressure or impermeable linear boundaries. In the same way, superposition of arrays of rate sources (Kruseman and De Ridder 197, and Ramey et al 1873) are used to generate the effects of combinations of rectangular boundaries around a well. The response of a reservoir to line source production is known, yet, superposition of the constant rate and constant pressure line source solutions will lead to a violation of the inner boundary condition at the constant pressure source. Hence, the mathematical solution for the ratepressure system must be assembled without the use of the superposition theorem. The rate-pressure model is a particular case of a system where a constant rate line source produces near a constant pressure circular boundary, described by Sageev and Home (1 983). THEORY Prior to describing the rate-pressure doublet model, a short description of the source-sink doublet model is presented. Next, the pressure solution for the ratepressure model ls presented and expressions for the instantaneous rate of injection and the cumulative injection from the constant pressure well are derived. Pressure Solution: Source-Sink Doublet Model This discussion of the source-sink doublet model is presented in order to amplify the similarities and difficulties in developing the rate-pressure model. The source-sink doublet model uses an imaging method to generate a constant pressure linear boundary (Carslaw and Jaeger 196, Kruseman and De Ridder 197, and Ramey et al 1873). The dimensionless pressure solution for this model is: where T& xi = - 4tU and the dimensionless terms are defined in the Nomenclature. 573

3 Equation 1 is a superposition of two line sources in an Infinite system, one source produces at a constant rate and the other source injects at the same constant rate. This system approaches a steady state condition at late time, and the dimensionless pressure drop between the production well and the initial constant pressure linear boundary is: which for large values of TD becomes: pd = - I) (31 pd = In(ri) (4) The dimensionless steady state pressure at observation polnts is: Here, the center of the coordinate system is the center of the constant pressure source. Vel I Pressure Point Cons tan t Rote General Pressure Point Cons ton t Pressure n A where r2 is the distance between the observation point and the image well, and r1 is the distance between the observation point and the production well. Brigham (1979) showed that at steady state, the isopressure lines form eccentric clrcles around the source and the sink. The centers of these circles are positioned on the line fs symmetry between the source and the sink located at the points: Figure 1 : A Schematic of the Rate-Pressure Model. The dimensionless Laplace solution describing the pressure drawdown at the constant rate line source well, presented by Sageev and Horne (1 9831, is: Here the x Coordinate is the line connecting the two wells, and the y coordinate Is the constant pressure linear boundary. The radii of these isopressure circles are: where: n =,1,2,3,... e, = 1 for n = en = 2 for n > The variables are defined in the Nomenclature. For two wells having the same radii, an = 1, yielding: r 1 (7) Points with a given T ~ / ratio T ~ exhibit the same dimensionless pressure behavior as a function of reduced dimenslonless time, td/tjm This observation holds for the sourcesink doublet model and is the basis for the simple method for analyzlng interference tests presented by Stallman (1 862). Pressure Solution: The Rate-Pressure Model The solutions presented here describe the pressure and rate behavlor of two wells completed In an infinite horizontal slab reservoir. One of the wells produces at a constant rate while the other well produces at a constant pressure. The two wells are approximates as two llne sources. It Is assumed that reservoir properties such as compressibility, porosity permeability and thickness are const ant. Sagccv and Horno (1984) presented an an appllcatior! of the superposition theorem to assemble a constant rate llne source producing near a finite radius constant pressure source. The same method Is applied here. A schematic of the rate-pressure model is presented if Fig. 1. This configuration of sources cannot be assembled by the method of superpositlon due to the mixed inner boundary conditions at the wells. Because of the boundary condltlons, the diffusivity equation, which describes the pressure behavior in the system, has to include the second derivative of the pressure with respect to the angle of rotation: The dimensionless pressure is measured between the production and the injection wells. A comparison between the pressure response of the source-sink and the ratepressure models is presented in Fig. 2. The line source solution is shown for comparison as curve 1. Curve 2 represents the source-sink model response with the pressure drop measured between the production well and the linear boundary. At early time, the effect of the constant pressure.linear boundary on the pressure response of the well Is negligible, and curve 2 is identical to curve to the line source. At later time, curve 2 deviates from the line source curve and approaches a steady state condition. In order to compare this model to the rate-pressure model, the source-sink pressures of curve 2 are doubled to give the pressure drop between the wells, yielding curve 3. Curve 4 represents the pressure response of the rate-pressure model. In this case, the effect of the constant pressure well Is not noticeable at early time and the pressure at the production well has a line source response. As time progresses, the constant pressure well starts affecting the pressure response, and curve 4 deviates from the line source. Thls deviation is more gradual than the deviation for the source-sink model as the injection at the constant pressure well increases gradually from zero. Curve 4 approaches a steady state condition at late time identical to that of curve 3 for the source-sink model. 574

4 11 Line Source 21 Source - Sink In Laplace terms, Eq. 14 becomes: (1 6) The results presented in the next section, are evaluated using an algorithm developed by Stehfest (1 7) that inverts the Laplace transforms numerically ' loe lo IDimensionless) Figure 2: A Comparison Between the Rate-Pressure and the Source-Sink Models. The steady state dimensionless pressure for a constant rate line source producing near a constant pressure circle is: 1 -In 2 Letting ad To I TD 1 1, for the rate pressure model, yields: phbs = In[ ri(rh - 1) ] = 21n(ri) for large r;. The pressure drop between the constant pressure linear boundary and the production well that forms at steady state condition is one half of Eq. 1 1 and is identical to Eq. 4. Instantaneous Rate The rate equation for a constant pressure finite radius source was presented by Sageev and Horne (1984). For the rate-pressure doublet model, where ud = 1, the dimensionless Laplace instantaneous rate is: Cumulative Injection (13) The cumulative dimensionless injection at the constant pressure well is defined as the ratio between the cumulative injection at the constant pressure well to the cumulative production at the constant rate well: td (1 4) RESULTS The analysis of interference between a constant rate well and a constant pressure well requires the investigation of three time dependent parameters. The pressure response of the production well, the rate of injection and the cumulative injection at the constant pressure well. The dimensionless pressures as a function of dimensionless time for various dimensionless distances between the wells are presented in Fig. 3. The straight line is the response of an infinite acting line source. As the dimensionless distance, rh increases, the curves branch off from the line source curve at later time. All the rate-pressure model responses approach a steady state condition at late time, that may be calculated using Eqs. 11 or 12. Figure 3 may be used to estimate the presure behavior of a designed rate-pressure test. Sageev and Horne (1 983) presented a semilog type curve matching technique by which the distance between the wells may be estimated, should this distance be unknown. - I5 1 6' so 1 b3 loo ) C D 25 5 a IO Or ' " ' " " ' ' IDirnension~essl Figure 3: Semilog Type Curve for te Rate-Pressure Model. Instantaneous dimensionless injection rates at the constant pressure well as a function of dimensionless time for various distances between the wells are presented in Fig. 4. The dimensionless injection rate curves approach a value of 1, indicating that the system approaches a steady state condition at late time. As in Fig. 3, it can be seen that the closer the wells are, the earlier the deviation from the line source behavior and the earlier the approach to steady state condition. The cumulative dimensionless injection at the constant pressure well as a function of dimensionless time for various distances between the wells Is presented In Fig. 5. The cumulative dimensionless InJection approaches a value of 1 at late time, again indicating a limiting steady state condition. By use of Eqs. 13 and 14, the actual injection rate and the cumulative injection may be estimated. The relative sizes of the wells have an effect on the responses of the two wells. So far, the analysis considered 575

5 IO 2SO 5 loo 5ooo loz lo4 1' Figure 4: Semilog Type Curve of the Instantaneous Rate for the Rate-pressure Model. Figure 7: The Effect of the Diameters of the Wellbores on the Instantaneous Rate of Injection/Production of the Rate-pressure Model. I, t 1 1 * ut 25 s 5 1 r; '2 IO Figure 5: Semilog Type Curve of the Cumulative Injection~ProductIon for the Rate-Pressure Model. I (R s C.- ut.^ c '2 1 Line Source -a *. ',. a * * '.5 two wells having the same diameter. Two cases are now considered, where the diameter of the constant pressure well is either twice or a half of the diameter of the constant rate production well. Figure 3 presents the pressure response for the rate-pressure doublet model for a fixed distance between the weffs and for three diameters of the constant pressure well. As expected, the larger the diameter of the constant pressure well, the more the pressure response of the constant rate well is affected. The towermost curve in Fig. 6 represents a constant pressure well with a diameter twice that of the constant rate well The relative diameter of the wells affects the injection rate at the constant pressure well. Figure 7 presents two groups of instantaneous dimensionless rate curves. The uppermost group of curves Is for two wells with T$ = 1, and the lowermost group of curves is for 71, = 5. Each group includes three curves for various relative diameters of the two wells. In every group of curves, the uppermost curve is for a constant pressure well h&v~ng B diameter twice that of the constant rate well, and the lowermost curve is for a constant pressure well having a diameter half that of the constant rate well. The effect of the relative radii of the wells on the cumulative injection of the constant pressure well Is similar to the effect on the instantaneous injection rate. CONCLUSIONS 1. A semilog type curve method is presented that allows the ~nterpre~at~on or the design of a rate-pressure test Three time dependent parameters are considered: the pressure response of the constant rate well, the instantaneous rate and the cumulative Injection at the constant pressure well. 3. Large differences in the diameters of the wells significantly affect the pressure and rate responses roe 18 1' 4. A configuratlon of a constant pressure well and a conid IDimenoioniesol stant rate well may not be assembled using the superposition theorem, and is solved as a special case of a constant rate line source producing near a constant pressure finite radius source. Figure 6: The Effect of the Diameters of the Wellbores on the Pressure Response of the Rate-pressure Model. 576

6 NOMENCLATURE B = formation volume factor 1, = modified Bessel function, first kind, n'th order & = modified Bessel function, third kind, n'th order 8 = cumulative Injection Ei = Exponential Integral X = argument of the Exponential Integral radius of the constant pressure source compressibility formation thickness permeability pressure Laplace transform of p~ volumetric rate Laplace transform of matrix flow rate radius, centered at the pressure source distance between the pressure and rate sources radius of the constant pressure circle Laplace variable time 8 = angle of rotation p = viscosity = porosity q = diffusivity constant E = Subscripts D = dimensionless i = initial zu = well ss = steadystate 1 = at the constant rate source 2 = at the constant pressure source REFERENCES Brigham, W.E.: Stanford University Advanced Reservoir Engineering Class, Carslaw, H.S. and Jaeger, J.C.: Conduction of Heat in Solids, 2nd ed. Oxford University Press, 196. Kruseman, G.P., and De Ridder, N.A.: "Analysis and Evaluation of Pumping Test Data," International Institute of Land Reclamation and Improvement, Wageningen, The Netherlands (197). Mueller, T.D., and Witherspoon, P.A.: "Interference Effects Withln Reservoirs and Aquifers," J. Pet. Tech. (Oct. 1965) Ramey, H.J., Jr., Kumar, A. and Gulati, M.S., "Gas Well Test Analysis Under Wuter Drive Conditions", American Gas Association, Arlington, VA, (1973). Sageev, A., and Horne, R.N.: "Pressure Transient Analysis of Reservoirs with Linear or Internal Linear Boundaries," SPE 1276, Presented at the 68th Annual Technical Conference and Exhibition, San Francisco, California (Oct. 1983). Sageev, A., and Horne, R.N.: "Interference Between Constant Rate and Constant Pressure Reservoirs Sharing a Common Aquifer," In press, SPEJ (1984). Stallman, R.W., "Nonequilibrium Type Curves Modified for Two-Well Systems", U.S. Geol. Surv., Groundwater Note 3, (1952). Stehf est, H.: "Algorithm 368, Numerical Inversion of Laplace Transforms," Communications of the ACM, D-5 (Jan. 197) 13, NO. 1, Theis, C.V., "The Relationship between the Lowering of Piezometric Surface and Rate and Duration of Discharge of Wells using Groundwater Storage," Trans., AGU, 2, 519, (1935). Van Everdingen, A.F. and Hurst, W.: "The application of the Laplace Transformation Flow Problems In Reservoirs,'' Iprans., AlME (Dec. 1949) 186,