UESO #200271 (EXP) [ESO/06/066] Received:? 2006 (November 26, 2006) Propagation of Radius of Investigation from Producing Well B.-Z. HSIEH G. V. CHILINGAR Z.-S. LIN QUERY SHEET Q1: Au: Please review your paper as a whole for correctness. Although you may have submitted your paper electronically, errors might have been introduced during the copy-editing and typesetting processes. It is very important that you check your paper for accuracy in all respects. Also, please check author names, affiliations, and contact information on the article opening page to be certain all is accurate and up-to-date. Thank you! Q2: Au: Please check all artwork throughout article for correctness. Images may have been obtained from electronic files supplied. Please ensure that there were no software interpretation problems. Q3: Au: Please confirm that all artwork is to appear in black and white on the web and in the print edition. If you are interested in any color artwork, please see the Instructions for Authors for a pricing breakdown, or contact me immediately for a quote: cheryl.hufnagle@taylorandfrancis.com Q4: Au: Please provide Keywords. UESO #200271 (EXP) [ESO/06/066] Received:? 2006 (November 26, 2006)
UESO #200271 (EXP) [ESO/06/066] Received:? 2006 (November 26, 2006) Energy Sources, Part A, 29:000 000, 2007 Copyright Taylor & Francis Group, LLC ISSN: 1556-7036 print/1556-7230 online DOI: 10.1080/15567030601003759 Propagation of Radius of Investigation from Producing Well Q1, Q2, Q3 B.-Z. HSIEH Department of Resources Engineering National Cheng Kung University 5 Tainan, Taiwan G. V. CHILINGAR Environmental Engineering Department University of Southern California Los Angeles, CA, USA 10 Z.-S. LIN Department of Resources Engineering National Cheng Kung University Tainan, Taiwan Abstract The purpose of this study is to estimate the pressure disturbance area, or 15 the propagation of the radius of investigation, from a producing well by both analytical and numerical methods. A linear coefficient in the relation between the square of the dimensionless radius of investigation and the dimensionless time is studied and derived. The coefficient in the equation is a constant, and varied with different criterions of radius of investigation defined, i.e., the amount of pressure change from 20 the initial formation pressure at the pressure front of the pressure disturbance area. For the dimensionless pressure defined at the pressure front changing from 0.1095 to 10 9, the coefficient varied from 4 to 71.15, respectively. The coefficient of radius of investigation is independent of the level of the flow rate for a well producing at a constant flow rate. For a well producing with variable flow rates, the coefficient is not 25 a constant for the case of larger pressure drops defined at the pressure front. The skin factor does not affect the result of the calculated radius of investigation. The wellbore storage volume will affect the propagation of the radius of investigation only at an early time, depending on the wellbore storage volume. Keywords 30 Q4 1. Introduction As fluid is produced from the reservoir by a producing well, the pressure disturbance area expands outward from the wellbore and increases as time increases. The pressure Address correspondence to Zsay-Shing Lin, Dept. of Resources Engineering, National Cheng Kung University, No. 1, University Rd., Tainan City, 701, Taiwan. E-mail: zsaylin@mail.ncku. edu.tw 1 UESO #200271 (EXP) [ESO/06/066] Received:? 2006 (November 26, 2006)
2 B.-Z. Hsieh et al. disturbance area created by a producing well is the area enclosed by the pressure front where the disturbed pressure (or the pressure drop from the original pressure) being 35 defined an extremely small value (a value very close to zero). The radius of the pressure disturbance area is called the radius of investigation (r i ), or the radius of drainage. The radius of investigation created by a producing well is a function of time, such as a linear relationship between the square of the dimensionless radius of investigation (r 2 id ) and the dimensionless time (t D ) was found in the literature (Muskat, 1934; Tek et al., 40 1957; Jones, 1962; Van Poolen, 1964; Lee, 1982). The radius of investigation equation with a dimensionless form is r 2 id = αt D, where α is a coefficient. The radius of investigation with the coefficient (α) of 4 is used very often in well test analysis (Muskat, 1934; Van Poolen, 1964; Matthews and Russell, 1967; Lee, 1982; Chaudhry, 2004). Some other studies obtained the coefficient of 16 and 18.4 (Jones, 45 1962; Tek et al., 1957). These radius of investigation equations are derived for the case of constant flow rate. And no mention in the literature is made on the condition at the pressure front or the boundary of the disturbed area. It is infrequent that the coefficient (α) in the equation is obtained for the case of variable flow rates. Also, skin factor and wellbore storage are not considered in the studies of radius of investigation in the 50 literature. The purpose of this study is to estimate the pressure disturbance area (i.e., the propagation of the radius of investigation) from a producing well using both analytical and numerical methods. Also, this study is going to establish the radius of investigation equation in terms of dimensionless radius and dimensionless time for different criterion 55 of dimensionless pressure defined at the pressure front. The effects of skin and wellbore storage to the linear coefficient are also included in this study. 2. Basic Theory 2.1. Analytical Solution for Estimation of Radius of Investigation For an isotropic porous medium that is isothermal and homogeneous, with uniform thick- 60 ness, constant porosity and constant permeability, the dimensionless equation describing single phase fluid flow in a circular reservoir is (Lee, 1982): 2 p D r 2 D + 1 r D p D r D = p D t D (1) where p D = kh(p i p) 141.2qµB t D = 0.000264kt µcφr 2 w (2) (3) r D = r r w (4) For a well producing at a constant production rate with zero wellbore radius in an infinite cylindrical reservoir with uniform initial pressure before production begins, 65
Radius of Investigation from Producing Well 3 the analytical solution of the diffusivity equation for an infinite cylindrical reservoir is (Earlougher, 1977): where ( ) p D = 1 2 Ei r2 D 4t D (5) E i ( x) = E 1 (x) = e u x n u du = 0.5772 ln x + ( 1) k+1 x k k(k!) k=1 (6) In a well producing at variable flow rates, the pressure drop in the formation at the nth flow rate (n >2) can be calculated by using the superposition of Eq. (5) as follows 70 (Earlougher, 1977): ( p) total = 70.6µB kh { ( ) q 1 Ei r2 D + 4t D ( )} n rd 2 (q i q i 1 )Ei 4t D [t t i 1 ] i=2 (7) or where and { ( ) p D1 = 1 Ei r2 D + 2 4t D n ( ) ( qi q i 1 Ei i=2 q 1 p D1 = kh( p) total 141.2q 1 µb t D [t t i 1 ]= 0.000264k(t t i 1) µcφrw 2. r 2 D 4t D [t t i 1 ] )} (8) When a well is producing with a constant or variable flow rates, the pressure disturbance area gradually extends outward in the formation as producing time increases. The radius of investigation (r i ) is the distance from the center of wellbore to the pressure 75 front where the pressure difference between the initial pressure and formation pressure, or pressure drop ( p) at the pressure front, is less than the defined small value. For the case of constant flow rate, the value of rd 2 /4t D in Eq. (5) can be estimated by defining a small dimensionless pressure (p D ) value, which is proportional to the pressure drop ( p) at the pressure front for the defined small value. Thus, a linear 80 relationship between the square of the dimensionless radius of investigation (rid 2 ) and the dimensionless time (t D ) can be obtained from the analytical Ei solution (Eq. (5)). Note that the coefficient (α) of the radius of investigation equation is dependent on a criterion of p D value chosen. For the case of variable flow rates, the relationship between the dimensionless radius 85 of investigation (r id ) and the dimensionless time (t D ) can be estimated from Eq. (7) or Eq. (8) by specifying ( p) total or p D1 value.
4 B.-Z. Hsieh et al. 2.2. Numerical Solution for Estimation of Radius of Investigation In this study, the radius of investigation equation is derived not only from the analytical solution, but also from a numerical solution. The IMEX simulator (CMG, 2004) used in 90 this study is basically a three-phase black-oil simulator with a Cartesian or cylindrical grid system. The simulator is also capable of modeling two-phase (oil-water or gas-water) fluid flow. One phase of oil flow can be simulated by using two-phase oil-water flow with zero water relative permeability. The numerical simulation is started with dividing a reservoir into grids. After rock 95 and fluid properties are assigned to each grid block, the numerical simulation can be run to model a well to produce at specific flow rates, either at constant rate or variable rates. Formation pressures for each grid block at each time step from the result of the simulation run can be used to track the pressure front changes as function of time. Note that the pressure front is a point or a line where the pressure drop in the formation is equal to a 100 certain small value. The radius of investigation is the distance from the wellbore to the pressure front. 3. Results An oil reservoir used in this study has a uniform thickness of 60 ft, constant porosity of 0.2, and the constant permeability of 150 md. The initial reservoir pressure is 3,000 psi 105 (Table 1). The oil PVT data and rock fluid properties shown in Table 1 are used in the Table 1 study of the radius of investigation for both the analytical solution and the numerical solution. 3.1. Radius of Investigation Studies from Analytical Solution In this study, the pressure front is defined as a line or a curve where the pressure drop 110 from the original pressure is equal to a certain small value. The following criterions are used to define the pressure front: (i) p id = 0.1095 (criterion I), (ii) p id = 0.01095 (criterion II), and (iii) p id = 0.001095 (criterion III). The pressure front in the reservoir is changing as function of time due to a well is produced. Thus, the radius of investigation, or pressure front, is a function of time, and is derived from the solution of the diffusivity 115 equation. Applying criterion I of the radius of investigation (or p id defined at pressure front of 0.1095) to Eq. (5), we can obtain the results of rid 2 /4t D. In other words, the relationship between the dimensionless radius of investigation and the dimensionless time Table 1 Basic reservoir parameters used in this study Parameters, unit Values Parameters, unit Values p i, psi 3,000 k h, md 150 q, stb/day 100 k v, md 150 µ o, cp 13.2 φ, fraction 0.20 B o, rb/stb 1.06 h, ft 60 c t, psi 1 2.01 10 6 r w, ft 0.35
Radius of Investigation from Producing Well 5 is r 2 id = 4t D. The value of the linear coefficient (α) is 4. This equation is the same as the radius of investigation equation derived by Lee (1982) and can be expressed as 120 r i = (kt/948µcφ) 1/2. When criterion II of the radius of investigation is applied, i.e., p id defined at pressure front is 0.01095, the relationship between radius of investigation and time is r 2 id = 10.39t D. The radius of investigation equation is r 2 id = 17.82t D for criterion III (p id at pressure front is 0.001095). 125 By applying different criterions of dimensionless pressure (p id ) at the pressure front, we obtain equations of the radius of investigation with different coefficients (α) (Table 2). The coefficient (α) increases with decreasing dimensionless pressure (p id ) defined at Table 2 the pressure front. In other words, at the same producing time, the estimated pressure disturbance area is larger for the criterion of dimensionless pressure (p id ) defined as the 130 pressure front becomes smaller. The coefficient (α) is varied from 4.00 to 71.15 when the criterion value of the dimensionless pressure (p id ) defined at the pressure front is changed from 0.1095 to 10 9 (Table 2). The above results are obtained for the well producing at constant flow rate. To study the propagation of the radius of investigation for the well producing at 135 variable flow rates, the result of superposition, Eq. (8), based on constant flow rate of Eq. (5) is used. For the flow rate increasing from an initial flow rate (q 1 = 100 stb/day) to a higher flow rate (q 2 = 150 stb/day) (Figure 1), the relationship between the radius Figure 1 of investigation and time is obtained for the dimensionless pressures at the pressure front of 0.1095, 0.01095, 0.001095, and 1.0 10 9. 140 The coefficients (α) obtained are 4.101, 10.411, 17.819, and 71.312 for different p id defined at the pressure front of 0.1095, 0.01095, 0.001095, and 1.0 10 9, respectively (Figure 2). The coefficients (α) of the increasing flow rates test are slightly larger than Figure 2 those obtained from the constant flow rate test (Table 3). Table 3 For the flow rate decreasing from an initial flow rate (q 1 = 100 stb/day) to a lower 145 flow rate (q 2 = 50 stb/day) (Figure 3), the coefficients (α) obtained are 3.887, 10.374, Figure 3 17.814, and 71.312 for the criterion p id defined at the pressure front of 0.1095, 0.01095, Table 2 The linear coefficient (α) values derived from different criterion p id defined at pressure front for a constant flow rate test by using analytical solution ( ) ( ) p id = 1 2 Ei = y α = r2 id 4t D α = r2 id t D 0.1095 4.00 0.01095 10.39 0.001095 17.82 10 4 26.06 10 5 34.28 10 6 42.69 10 7 51.22 10 8 59.84 10 9 71.15
6 B.-Z. Hsieh et al. Figure 1. The designed variable flow rate test (an increasing flow rate test) and the calculated bottom-hole pressure. 0.001095, and 1.0 10 9, respectively. The coefficients (α) for these cases are slightly smaller than those obtained from the constant flow rate test (Table 3). In addition to the two-rates studied above, the radius of investigation equation for a 150 three-rate test is also analyzed. The first three-rate test was designed to increase the flow rate from the initial rate (q 1 = 100 stb/day) to a higher flow rate (q 2 = 150 stb/day), then decrease that to q 3 = 100 stb/day (i.e., a middle flow rate increasing test) (Figure 4). The coefficients (α) obtained for the middle flow rate increasing test are 4.246, 10.494, Figure 4 17.849, and 71.316 for the criterion p id defined at the pressure front of 0.1095, 0.01095, 155 0.001095, and 1.0 10 9, respectively. The coefficients (α) of the middle flow rate increasing test are larger than those obtained from the constant flow rate test (Table 3). Figure 2. The integrate results of the dimensionless radius of investigation for the increasing flow rate test by using analytical solution.
Radius of Investigation from Producing Well 7 Table 3 The radius of investigation equations (r 2 id = αt D) from different r id analysis criteria by using analytical solution Flow rates r id criteria I for p id = 0.1095 r id criteria II for p id = 0.01095 r id criteria III for p id = 0.001095 q = 100 stb/day rid 2 = 4.00t D rid 2 = 10.39t D rid 2 = 17.82t D (Constant rate) R 2 = 1 a R 2 = 1 R 2 = 1 q 1 = 100 stb/day rid 2 = 4.101t D rid 2 = 10.411t D rid 2 = 17.819t D q 2 = 150 stb/day R 2 = 0.9991 R 2 = 0.9999 R 2 = 0.9999 q 1 = 100 stb/day rid 2 = 3.887t D rid 2 = 10.374t D rid 2 = 17.814t D q 2 = 50 stb/day R 2 = 0.9983 R 2 = 0.9999 R 2 = 0.9999 q 1 = 100 stb/day rid 2 = 4.246t D rid 2 = 10.494t D rid 2 = 17.849t D q 2 = 150 stb/day R 2 = 0.9981 R 2 = 0.9999 R 2 = 0.9999 q 3 = 100 stb/day q 1 = 100 stb/day rid 2 = 3.698t D rid 2 = 10.280t D rid 2 = 17.782t D q 2 = 150 stb/day R 2 = 0.9956 R 2 = 0.9998 R 2 = 0.9999 q 3 = 100 stb/day a R 2 = the coefficient of determination. The second three-rate test was designed to decrease the flow rate from the initial rate (q 1 = 100 stb/day) to a lower flow rate (q 2 = 50 stb/day), then to increase that to q 3 = 100 stb/day (i.e., a middle flow rate decreasing test) (Figure 5). The coefficients (α) 160 Figure 5 obtained for the middle flow rate decreasing test are 3.698, 10.280, 17.782, and 71.316 for the criterion p id defined at the pressure front of 0.1095, 0.01095, 0.001095, and 1.0 10 9, respectively. The coefficients (α) of the middle flow rate decreasing test are smaller than those obtained from the constant flow rate test (Table 3). Figure 3. The designed variable flow rate test (a decreasing flow rate test) and the calculated bottom-hole pressure.
8 B.-Z. Hsieh et al. Figure 4. The designed triple flow rates test (a middle flow rate increasing test) and the calculated bottom-hole pressure. 3.2. Radius of Investigation Studies from Numerical Solution 165 To verify the results of the radius of investigation from the analytical solution, a numerical simulation study is also used. A cylindrical oil reservoir is simulated with 5,000 grids in r direction (radial direction), 1 grid (i.e., 360 degree) in θ direction (tangent direction), and 1 single layer in k direction (vertical direction). The grid size in radial direction is small at the vicinity of the wellbore and increases gradually as the distance outward 170 from the wellbore increases. The formation parameters used in the numerical model is the same as that used in analytical model (Table 1). Numerical simulation studies conducted to investigate the propagation of the radius of investigation include production well producing at constant flow rate and at variable flow rates. The results from numerical simulation studies for both constant and variable 175 flow rates cases are the same as these from analytical solution (Tables 3 and 4). Table 4 Figure 5. The designed triple flow rates test (a middle flow rate decreasing test) and the calculated bottom-hole pressure.
Radius of Investigation from Producing Well 9 Table 4 The radius of investigation equations (r 2 id = αt D) from different r id analysis criteria by using numerical simulation Flow rates r id criteria I for p id = 0.1095 r id criteria II for p id = 0.01095 r id criteria III for p id = 0.001095 q = 100 stb/day rid 2 = 3.986t D rid 2 = 10.363t D rid 2 = 17.799t D (Constant rate) R 2 = 0.9999 R 2 = 0.9999 R 2 = 0.9999 q 1 = 100 stb/day rid 2 = 4.082t D rid 2 = 10.381t D rid 2 = 17.804t D q 2 = 150 stb/day R 2 = 0.9991 R 2 = 0.9999 R 2 = 0.9999 q 1 = 100 stb/day rid 2 = 3.868t D rid 2 = 10.344t D rid 2 = 17.799t D q 2 = 50 stb/day R 2 = 0.9983 R 2 = 0.9999 R 2 = 0.9999 q 1 = 100 stb/day rid 2 = 4.227t D rid 2 = 10.464t D rid 2 = 17.833t D q 2 = 150 stb/day R 2 = 0.9981 R 2 = 0.9999 R 2 = 0.9999 q 3 = 100 stb/day q 1 = 100 stb/day rid 2 = 3.679t D rid 2 = 10.249t D rid 2 = 17.767t D q 2 = 150 stb/day R 2 = 0.9956 R 2 = 0.9998 R 2 = 0.9999 q 3 = 100 stb/day 3.3. Radius of Investigation Affected by Skin Factor We investigated the effect of skin factor (s) to the propagation of the radius of investigation in simulation studies by varying skin factors. The results show that the radius of investigation (r id ) is independent of skin factor (s). For the different skin factors (s = 2, 180 5, 8, and 10), the entire linear coefficients (α) are 3.986, 10.363, and 17.799 for the criterion p id defined at the pressure front of 0.1095, 0.01095, and 0.001095, respectively (Figure 6). The radius of investigation equations for different skin factors are the same Figure 6 as result from the equations of no-skin factor (s = 0). 3.4. Radius of Investigation Affected by Wellbore Storage 185 The effect of wellbore storage volume, in terms of dimensionless wellbore storage volume (C D ), on the propagation is also investigated in this study. By using numerical simulation studies, the radius of investigation equation for different wellbore storage volume (C D = 10 2, C D = 10 3, C D = 10 4, and C D = 10 5 ) are obtained and plotted the results of rid 2 versus t D on linear coordinates (Figure 7) for different criterion p id defined at the 190 Figure 7 pressure front. The results are very close to these with no wellbore volume. The results show that the coefficients (α), except at an early time (or small t D ), are 3.98, 10.36, and 17.80 for the criterion p id defined at the pressure front of 0.1095, 0.01095, and 0.001095, respectively, for all wellbore storage volumes studied (Figure 7). From the result of dimensionless radius of investigation (r id ) versus dimensionless 195 time (t D ) plotted on log-log plot axis for different wellbore storage volumes (C D = 0, C D = 10 2, C D = 10 3, C D = 10 4, and C D = 10 5 ) with criterion of p id = 0.1095, the propagation of the radius of investigation influenced by the wellbore storage effect
10 B.-Z. Hsieh et al. Figure 6. The integrate results of the dimensionless radius of investigation for different skin factor (S = 0, 2, 5, 8, 10) by using numerical simulation. only in the early dimensionless time is observed (Figure 8). A linear relationship exists Figure 8 between the dimensionless radius of investigation (r id ) and dimensionless time (t D )in log-log plot when the wellbore storage effect is ended (C D = 0) (Figure 8). The curves 200 of dimensionless radius of investigation (r id ) versus dimensionless time (t D ) for different wellbore storage volumes (C D = 10 2, C D = 10 3, C D = 10 4, and C D = 10 5 ) diverge from the straight line of C D = 0. The degree of deviation is increased as the wellbore storage volume (C D ) increases (Figure 8). In other words, the propagation time required to reach the specific boundary or distance is longer for the large wellbore storage volume 205 than for the small wellbore storage volume. Figure 7. The integrate results of the dimensionless radius of investigation for different wellbore storage volumes (C D = 0, 10 2,10 3,10 4,10 5 ) by using numerical simulation.
Radius of Investigation from Producing Well 11 Figure 8. Dimensionless radius of investigation vs. dimensionless time log-log plot for different wellbore storage volumes (C D = 0, 10 2,10 3,10 4,10 5 ) at the criterion p D = 0.1095. 4. Discussion A linear relationship between the square of the dimensionless radius of investigation and dimensionless time (r 2 id = αt D) was studied in this study for both constant flow rate and variable flow rate tests. The coefficient (α) of4,orr 2 id = 4t D, was obtained from the 210 results of all tests with constant flow rate for the criterion p id defined at the pressure front of 0.1095. In other words, the widely-used radius of investigation equation r 2 id = 4t D is derived by the assumption of the dimensionless pressure defined at the pressure front of 0.1095. Using different criterion p id defined at the pressure front, we obtained different 215 coefficients (α), which vary from 4 to 71.15 for the pressure front varied from 0.1095 to 10 9, respectively. The results from our study shows that radius of investigation is independent of any constant flow rate. This result is the same as from Lee (1982), mentioning that in principle, any flow would suffice time required to achieve a particular radius of 220 investigation is independent of flow rate. However, the radius of investigation affected by a rate change in a well for the criterion p id defined at the pressure front of 0.1095 is observed (Figure 9). When flow rate increases from previous constant flow rate for Figure 9 two-rate cases, the coefficient (α) increases and vice versa in the latter time (Figure 9). For the pressure front defined as p id = 0.01095, the slope in the plot of the square of 225 dimensionless radius of investigation verses dimensionless time is not affected much by the rate changes (Figure 10). The results are very close to the case of constant flow rate, Figure 10 in which the coefficient is 10.39. When the criterion p id defined at the pressure front decreases, such as p id = 0.001095, the rate change affecting the slope in the plot of the square of dimensionless radius of investigation versus dimensionless time also decreases 230 (Figure 11). Figure 11
12 B.-Z. Hsieh et al. Figure 9. The integrate results of the dimensionless radius of investigation for entire constant flow rate and variable flow rate tests at the criterion p D = 0.1095. In the radius of investigation studies by both analytical solution and numerical solution for injecting well, the results of the dimensionless radius of investigation varied with dimensionless time are the same as those from producing well (Figure 12). Figure 12 5. Conclusions The propagation of the radius of investigation from a production well has been studied 235 by both analytical and numerical methods. The radius of investigation equations in dimensionless terms are derived with different criterions of radius of investigation defined. The conclusions of this study are as follows: Figure 10. The integrate results of the dimensionless radius of investigation for entire constant flow rate and variable flow rate tests at the criterion p D = 0.01095.
Radius of Investigation from Producing Well 13 Figure 11. The integrate results of the dimensionless radius of investigation for entire constant flow rate and variable flow rate tests at the criterion p D = 0.001095. 1. The relationship between the square of the dimensionless radius of investigation and the dimensionless time (r 2 id = αt D) is linear for the case of constant flow 240 rate and may not be linear for the case of variable flow rates. 2. The constant coefficient (α) in constant flow rate cases varies from 4 to 71.15 when the defined dimensionless pressure at the pressure front of the radius of investigation is changed from 0.1095 to 10 9. 3. The radius of investigation is affected by a rate change in a well for the case of 245 dimensionless pressure defined at the pressure front less than or equal to 0.1095. The widely-used radius of investigation equation, r 2 id = 400t D, independent of flow rate is valid only for constant flow rate cases. Figure 12. The integrate results of the dimensionless radius of investigation for the injecting well and the producing well at a constant flow rate test.
14 B.-Z. Hsieh et al. 4. The radius of investigation equation for an injecting well is the same as that from a producing well. The radius of investigation is independent of skin factor. The 250 propagation of the radius of investigation is affected by wellbore storage effect only at an early time, depending on the size of wellbore storage volume. References Chaudhry, A. U. 2004. Oil Well Testing Handbook. Amsterdam: Elsevier Inc. CMG. 2004. User s Guide IMEX Advanced Oil/Gas Reservoir Simulator. Calgary, Alberta: Com- 255 puter Modelling Group Ltd. Earlougher, R. C., Jr. 1977. Advances in Well Test Analysis. Dallas, TX: Society of Petroleum Engineers of the AIME. Jones, P. 1962. Reservoir limit test on gas wells. J. Petrol. Technol. June:613 618. Lee, J. 1982. Well Testing. Dallas, TX: Society of Petroleum Engineers of the AIME. 260 Matthews, C. S., and Russell, D. G. 1967. Pressure Buildup and Flow Tests in Wells. Dallas, TX: Society of Petroleum Engineers of the AIME. Muskat, M. 1934. The flow of compressible fluids through porous media and some problems in heat conduction. Physicals 71. Tek, M. R., Grove, M. L., and Poettmann, F. H. 1957. Method for predicting the back-pressure 265 behavior of low-permeability natural gas wells. Trans. AIME 210 302. Van Poolen, H. K. 1964. Radius-of-drainage and stabilization-time equations. Oil Gas J. September 14:138 146. Nomenclature c fluid compressibility, psi 1 C D wellbore storage effect, dimensionless B formation volume factor, rb/stb h formation thickness, ft k permeability, md q flow (production) rate, stb/day p pressure, psi p D dimensionless pressure, dimensionless p D1 dimensionless pressure of variable flow rates, dimensionless p i initial formation pressure, psi p id dimensionless pressure defined at the pressure front, dimensionless r radius, ft r D dimensionless radius, dimensionless r i radius of investigation, ft r id dimensionless radius of investigation, dimensionless r w wellbore radius, ft s skin factor, dimensionless t time, hours t D dimensionless time, dimensionless α the linear coefficient in the radius of investigation equation, dimensionless p pressure drop, psi ( p) total pressure drop in variable flow rates, psi φ porosity, fraction µ viscosity, cp 270