PET467E-Analysis of Well Pressure Tests 2008 Spring/İTÜ HW No. 5 Solutions

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1 . Onur PET467E-Analysis of Well Pressure Tests 2008 Spring/İTÜ HW No. 5 Solutions Due date: Subject: Analysis of an dradon test ith ellbore storage and skin effects by using typecurve matching and semilog techniques as ell as the softare SAPHIR. Pressure and pressure-derivate-vs-time data given in Table 1 are obtained from a ell completed in an oil zone by using an electronic quartz gauge. The reservoir size is unknon. The objective of the test is to determine, initial pressure, permeability, skin and reservoir size if e see the boundary effects. The rock and PVT data are given in Table 2. (a) Construct a log-log plot of delta pressure (Δp), delta pressure-derivative (Δp ) verus test time, t. Identify hether there exist ellbore storage effects in pressure data and identify the best interpretation model that could be used to analyze this test. Discuss hether initial pressure measured by the gauge is correct or not, and discuss hether the test data indicate any boundary effects. (b) If you determine that ellbore storage effects are important in test data, make a Cartesian plot of pressure (not delta pressure) vs. time for the early time data and estimate initial pressure and ellbore storage coefficient from the analysis of this Cartesian plot. You should use least squares approach to determine your pi and C values. Then compare your initial pressure ith the initial pressure measured by the gauge. If these values differ, discuss hich one could be correct and hy?. (c) The ell as produced through a 7 5/8 ID inch casing. Assuming that you have compressive type of ellbore storage, determine the volume of ell and then estimate the depth of the ell based on the total compressibility of reservoir fluid given from PVT. If the knon depth of the ell is 10 ft, discuss the agreement beteen your computed value and the knon value of 10 ft. (d) Analyze data by type-curve matching using Bourdet et al. type curves to determine C, k, and s. (e) If you think that semi-log analysis of dradon pressure data can be performed based on your results obtained in part a and d, then perform semilog analysis of pressure data to determine k and S values. (f) Analyze the data using the softare Saphir. You should also use nonlinear regression option to determine the best (optimum) estimates of C, k, p i, and S. (g) Tabulate your p i, C, k, and S values obtained from various methods considered above and discuss your findings.

2 Table 1. Pressure and derivative data at an interference ell. Test duration t, hours Pressure, p, psi Pressure- Derivative, Δp, psi

3 Table 2. Reservoir, ell and fluid parameters. Net pay thickness, h, 50 ft Well radius, r,, ft Rock compressibility, c r, 4.e-06 psi -1 Oil compressibility, c o, 1.2e-05 psi -1 Reservoir temperature, T, 180 o F (82 o C) Solution gas oil ratio, R s, ft 3 /STB Bubble point pressure, pb, 1850 psi Oil saturation in the oil zone, S o, 1.0 (or %0) Viscosity, μ, cp Formation volume factor, B, RB/STB Active ell production rate, q, 500 STB/D Porosity φ (estimated from cores) 0.2 Permeability, k, (estimated from cores) -60 md Initial pressure measured by the gauge, p i, 3458 psi? SOLUTIONS (a) Shon in Fig. 1 are log-log plots of delta pressure (Δp) (based on initial pressure of 3458 psi), delta pressure-derivative (Δp ) versus time data as given in Table Δp and Δp', psi Wellbore storage period radial flo period (infinite acting behavior) 1 HW No. 4 (PET467E, Spring 2006) Δp Elapsed time t (h) Fig. 1. Log-log diagnostic plot based on incorrect initial pressure (3458 psi). Δp'

4 As seen from Fig. 1, the behavior of delta pressure and its derivative resembles to that of a ell producing ith ellbore storage and skin effects in an infinite acting reservoir. It is also clear that derivative data exhibits a ell-defined radial flo period (zero slope or constant derivative value) in the time interval from 0.5 to 3 hours. The derivative in this time interval takes the constant value of approximately 22 psi. In fact, using this value and the folloing equation, e can directly estimate kh/μ: 70.6q sc Bμ Δ p = (S.1) kh Or rearranging the previous equation for kh/μ gives kh 70.6qscB = = = 2081 md ft / cp μ Δp 22 (S.2) Using the values of viscosity and thickness given in Table 2, e can compute an estimate of permeability as kh μ k = = 2081 = 19.6md μ h 50 (S.3) The early-time data (see the time interval from to 0.1 h) exhibit ellbore storage effects. Hoever, delta pressure (Δp) and its derivative are not equal at very early times here e expect to see unit slope line. Then e suspect that the value of initial pressure (3458 psi) used to construct the Δp shon in Fig. 1 is incorrect. In fact, delta p is smaller in magnitude than derivative data in the time interval from to 0.01 h, e suspect that the correct or more appropriate value for the initial pressure should be higher than the value of 3458 psi used. Note that there is only single data point that fits the unit slope line. One reason that e do not see a ell defined unit-slope line data is that in the test, they did not measure pressure values at times less than h. If they did, e could have seen a better unit slope line indicating ell storage period clearly. Based on the above observations from Fig. 1, e can conclude that (i) the best interpretation model for the test data is the ellbore storage and skin effects in an infinite acting homogeneous reservoir; and (ii) the initial pressure given is incorrect, actually it is smaller than the correct, unknon, initial pressure. (b) To find the correct or appropriate value of initial pressure, e can make a Cartesian plot of floing bottomhole pressure, pf vs. time data at very early times as shon in Fig. 2. Based on the log-log plot of derivative data in Fig. 1, e should best fit a straight line only through the first to points as e do not have data during unit-slope line period.

5 The straight line fitted in Fig. 2 gives the initial pressure as p i = psi and slope of psi/h. Using the value of slope in the folloing equation, e estimate the ellbore storage coefficient as: 3520 Fit Results Floing bottom hole pressure, p f, psi Fit 1: Linear Equation Y = * X Number of data points used = 2 Average X = Average Y = Residual sum of squares = E-026 Regression sum of squares = Coef of determination, R-squared = 1 Residual mean square, sigma-hat-sq'd = Elapsed time, t, h Fig. 2. Cartesian plot of floing ellbore pressure vs. time. qscb C = = = bbl / psi 24 m (S.5) Fig. 3 shos log-log plot of delta pressure (Δp) (based on initial pressure of 3489 psi), delta pressure-derivative (Δp ) versus time. As seen correct initial pressure, brings delta p and derivative data together on the unit-slope line.

6 00 0 Δp and Δp', psi Wellbore storage period radial flo period (infinite acting behavior) 1 HW No. 4 (PET467E, Spring 2006) Δp Elapsed time t (h) Fig. 3. Log-log diagnostic plot based on correct initial pressure ( psi). (c) The equation for the ellbore storage coefficient C for compressive ellbore storage is given by C = c V (S.6) Δp' Using the value of C estimated in part b, and the value of compressibility of the ellbore fluid (hich is oil and c = 1.2x -5 psi -1 ), e can compute the ellbore volume, V, from the rearrangement of Eq. S.6 as V 3 C 1.67 = = = 139 bbl 5 c 1.2 (S.7) Note the ellbore volume is given by V D 1 π = = bbl (S.8) Solving fort he ell depth D gives D = 2461 ft (S.9) If the ell-depth given is correct, then the difference beteen computed ell depth and given depth can be attributed to the compressibility of ellbore (hich also affects the value of ellbore storage coefficient). If e assume that the ell depth is 10 ft, then the corresponding ellbore volume V should be: 62 bbl. Then, using this value and the value of ellbore storage coefficient given, e can estimate the ellbore fluid compressibility as:

7 C 1.67 c x psi = = = 2.7 (S.) V 62 hich needs further verification by PVT studies. We could be observing the effects of to phase flo in the ellbore. The bubble point pressure of the system is reported to be 1850 psi, hence at any point along the ellbore, if the pressure decreases belo that value, e ill have to phase flo. (d) After obtaining a match ith the Bourdet et al. Type curve the folloing points are obtained: ( Δ p) =, ( pd ) 0.24 t D () t = 1, 504 = CD The kh product can be obtained ith the folloing equation: ( p ) ( Δp) D 141.2qscBoμ kh = = kh = 37 md ft The ellbore storage coefficient can be determined from the folloing equation: ( t) 37 1 kh C = C bbl / psi μ t = = D CD The dimensionless ellbore storage coefficient can be determined ith the folloing equation: C D 5.615C = = CD = πφchr 2π t s The type curve match has been obtained ith Ce = 00 Hence: 2s ( Ce D ) s= ln = ln s CD 2 = D (e) From log-log diagnostic plot shon in Figs. 1 and 3, it is clear that e can perform a semi-log analysis of the data in the time interval from 0.5 to 3 hours because derivative data exhibits a ell-defined zero slope indicating that the radial flo as established near the end of the test. The semi-log analysis of the floing bottom hole pressure data is used to determine permeability and skin. Note that e ill fit a best straight line (using least-squares) through data points in the time interval from 0.5 to 3 hours. Fig. 4 shos a semi-log plot of floing bottom hole pressure vs. time data.

8 3500 Floing bottom hole pressure, p f, psi Fit Results Fit 2: Log Equation Y = * ln(x) Number of data points used = 9 Average ln(x) = Average Y = Residual sum of squares = Regression sum of squares = Coef of determination, R-squared = Residual mean square, sigma-hat-sq'd = Elapsed time, t, h Fig. 4. Semi-log plot of floing bottom hole pressure vs. time. The equation for the best straight line is determined as: pf = log( t) (S.11) From this, e note that the slope m = psi/cycle and p 1hr = psi. From the folloing equations, e can determine permeability and skin as: m qscbμ qscbμ = k = (S.12) kh m h 500 *1.297* k = = 19.4 md (S.13) 51.24*50 After determining the value of permeability, e can determine skin factor as: S = p p k + i 1hr log m φctμr (S.14)

9 S = log *1.6 *0.472*0.354 = 1.1 (S.15) Note that in above skin equation, e should use total compressibility value, c t = c r +c o = 1.6x -5 psi-1 and p i = psi. (f) No, e perform the analysis by Saphir. Fig. 5 shos the match obtained by considering ellbore storage skin + infinite acting model and by using improve button. Fig. 6 and 7 sho the semi-log and history plots. As seen from these three figures, matches of the data obtained by Saphir is very good. The parameter values obtained are tabulated in Table S.1. Log-Log plot: dp and dp' [psi] vs dt [hr] Fig. 5. History match of data by Saphir; log-log plot Semi-Log plot: p [psia] vs Superposition time Fig. 6. History match of data by Saphir; semi-log plot

10 History plot (Pressure [psia], Liquid Rate [STB/D] vs Time [hr]) Fig. 7. History match of data by Saphir; history plot Table S.1 Selected odel odel Option Standard odel Well Storage + Skin Reservoir Homogeneous Boundary Infinite Results Tatch 585 [hr]**-1 Patch [psia]**-1 C bbl/psi Skin Delta P Skin psi Pi psia k.h 989 md.ft k 19.8 md Rinv 182 ft Test. Vol E+5 Barrels As a final remark, note that the match obtained ith Saphir gives the ellbore storage coefficient C as C = bbl/psi. If e ere to repeat the computations given in part-c for determining the ell height e have:

11 C C = c V V = = V = 88.33bbl = 496 ft o 5 c 1.2 Since e have a ID casing then the inner radius of the ellbore ould be r = ft. Then e could compute the height of the ell as follos: V 496 h = = h = 1560 ft A It is clear that the height determined in this case is in better agreement ith the reported value of 10ft. Hoever, still there is a 400 ft difference hich may be due to to-phase flo affect that needs to be accounted on the compressibility of ellbore fluid. 3 (g) The folloing table summarizes the results: Cartesian plot partb Type curve Semilog Saphir Kh, md-ft S C, bbl/psi Pi, psi