Tarek Ahmad - Advanced Reservoir Engineering

Sorina Dumitrascu

Tarek Ahmad - Advanced Reservoir Engineering

Sorina Dumitrascu

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Abstract

Calculate the bottom-hole flowing pressure after 4 hours by using:

Figures (569)

Figure 1.2 Fluid density versus pressure for different fluid types.

Figure 1.8 Hemispherical flow in a partially penetrating well.

Figure 1.7 Spherical flow due to limited entry.

Figure 1.10 Pressure gradient in radial flow.

The above calculations show that g, and q2 are not largely different, which is due to the fact that the liquid is slightly incompressible and its volume is not a strong function of pressure. [1.2.6]

For an incompressible system in a uniform formation, Equation 1.2.14 can be simplified to: where:

Step 1. Rearrange Equation 1.2.15 and solve for the pressure p at radius r: Step 2. Calculate the pressure at the designated radii: Solution

Figure 1.14 Pressure profile around the wellbore. definition of the gas formation volume factor Bg to gg, as: Assuming a slightly compressible fluid, calculate the oil flow rate. Compare the result with that of an incompressible fluid. Solution For aslightly compressible fluid, the oil flow rate can be calculated by applying Equation 1.2.21:

Step 1. Calculate the term 26/j,Z for each pressure as shown below:

The well is producing at a stabilized bottom-hole flowing pressure of 3600 psi. The wellbore radius is 0.3 ft. The following additional data is available: Example 1.7 The PVT data from a gas well in the Anaconda Gas Field is given below:

Figure 1.16 Real-gas pseudopressure data for Example 1.7 (After Donohue and Erekin, 1982). Step 4. Calculate the flow rate by applying Equation 1.2.30: At pe = 4400 psi: gives We = 1089 x 10° psi?/cp

Figure 1.17 Pressure disturbance as a function of time.

Table 1.1 Values of —Ei(—x) as a function of x (After Craft et al. 1991)

Craft et al. (1991) presented the values of the Ei function in tabulated and graphical forms as shown in Table 1.1 and Figure 1.19, respectively. The mathematical function, Ei, is called the exponential integral and is defined by:

“As calculated from Equation 1.2.17. ’From Figure 1.19. Figure 1.19 Ej function (After Craft et al., 1997).

Step 2. Perform the required calculations after 1 hour in the following tabulated form: Solution Step 1. From Equation 1.2.66:

Figure 1.21 indicates that as the pressure disturbance moves radially away from the wellbore, the reservoir @ As calculated from Equation 1.2.17. ’From Figure 1.19.

Step 5. Results of step 4 are shown graphically in Figure 1.21.

Figure 1.21 Pressure profiles as a function of time on a semi-log scale.

Figure 1.20 Pressure profiles as a function of time.

Notes: For tp < 0.01: pp = 2ztp/x. For 100 < tp < 0.2572D: pp ~ 0.5 (Intp + 0.80907). Table 1.2 Pp versus tp—infinite radial system, constant rate at the inner boundary (After Lee, J., Well Testing, SPE Textbook Series, permission to publish by the SPE, copyright SPE, 1982) the dimensionless pressure drop Pp is strictly a function of the dimensionless time fp, or:

Table 1.3 Pp vs. tp—finite radial system, constant rate at the inner boundary (After Lee, J., Well Testing, SPE Textbook Series, permission to publish by the SPE, copyright SPE, 1982)

Assuming that the initial total isothermal compressibility is 3 x 10 psi@!, calculate the bottom-hole flowing pressure after 1.5 hours. Example 1.13 A gas well with a wellbore radius of 0.3 ft is producing at a constant flow rate of 2000 Mscf/day under transient flow conditions. The initial reservoir pressure (shut-in pressure) is 4400 psi at 140°F. The formation per- meability and thickness are 65 md and 15 ft, respectively. The porosity is recorded as 15%. Example 1.7 documents the properties of the gas as well as values of m(p) as a function of pressures. The table is reproduced below for convenience: Solution

Combining Equation 1.2.102 with 1.2.96, 1.2.97, or 1.2.99, gives: The above approximation solution forms indicate that the product (wZ) is assumed constant at the average pressure p. This effectively limits the applicability of the p? method to reservoir pressures of less than 2000. It should be pointed out that when the p? method is used to determine py; it is perhaps sufficient to set 7Z = p;Z.

Figure 1.22 Plot of 1/Bg vs. pressure. Fetkovich (1973) suggested that at high pressures above 3000 psi (/ > 3000), 1/uB, is nearly constant as shown schematically in Figure 1.22. Imposing Fetkovich’s condition on Equation 1.2.107 and integrating gives:

Assuming that the initial total isothermal compressibility is 3 x 10~‘psi-!, calculate, the bottom-hole flowing pressure after 1.5 hours by using the p approximation method and compare it with the exact solution. as values of m(p) as a function of pressures are tabulated below: Solution Step 1. Calculate the dimensionless time fp:

The flow rates are measured on a routing basis through- out the lifetime of the field, thus facilitating the calculation of the volumetric average reservoir pressure p,. Alterna- tively, the average reservoir pressure can be expressed in terms of the individual well average drainage pressure decline rates and fluid flow rates by: Figure 1.24 Volumetric average reservoir pressure.

Step 5. Present the results of step 4 in graphical form as shown in Figure 1.25. Step 4. Calculate py, at different assumed times, as follows:

Figure 1.25 Bottom-hole flowing pressure as a function of time. Step 2. Convert the area A from acres to ft?: in the center of 40 acre square-drilling pattern. Given the following additional information:

Figure 1.26 Near-wellbore skin effect. It is not unusual during drilling, completion, or workover operations for materials such as mud filtrate, cement slurry, or clay particles to enter the formation and reduce the perme- ability around the wellbore. This effect is commonly referred to as “wellbore damage” and the region of altered perme- ability is called the “skin zone.” This zone can extend from a few inches to several feet from the wellbore. Many other wells are stimulated by acidizing or fracturing, which in effect increases the permeability near the wellbore. Thus, the permeability near the wellbore is always different from the permeability away from the well where the formation has not been affected by drilling or stimulation. A schematic illustration of the skin zone is shown in Figure 1.26.

Figure 1.27 Representation of positive and negative skin effects.

The above expression for determining the additional pres- sure drop in the skin zone is commonly expressed in the following form: Hawkins (1956) suggested that the permeability in the skin zone, i.e., Rskin, is uniform and the pressure drop across the zone can be approximated by Darcy’s equation. Hawkins proposed the following approach:

Example 1.19 Calculate the skin factor resulting from the invasion of the drilling fluid to a radius of 2 ft. The per- meability of the skin zone is estimated at 20 md as compared with the unaffected formation permeability of 60 md. The wellbore radius is 0.25 ft.

Pseudosteady-state flow (accounting for the skin factor) For slightly compressible fluids Introducing the skin factor into Equation 1.2.123 gives:

Similar to the above modification procedure, Equations 1.2.32 and 1.2.33 can be expressed as: Example 1.20 Agas well has an estimated wellbore dam- age radius of 2 feet and an estimated reduced permeability of 30 md. The formation has permeability and porosity of 55 md and 12% respectively. The well is producing at a rate of 20 MMscf/day with a gas gravity of 0.6. The following additional data is available:

where Qo1, Qo2, and Qo3 refer to the respective producing rates of wells 1, 2, and 3. Applying the above expression to calculate the additional pressure drop due to two wells gives:

Figure 1.28 Well layout for Example 1.21. presented by Equation 1.2.134, or:

Step 1. Calculate the pressure drop at well 1 caused by its own production by using Equation 1.2.134: Step 2. Calculate the pressure drop at well 1 due to the production from well 2: Solution

Step 3. Calculate the pressure drop due to production from well 3:

It is essential to notice the change in the rate, i.e., (new rate - old rate), that is used in the above equation. It is the change in the rate that causes the pressure disturbance. Further, it should be noted that the “time” in the equation represents the total elapsed time since the change in the rate has been in effect. The first contribution results from increasing the rate from 0 to Q, and is in effect over the entire time period t,, thus:

Using the same concept, the two other contributions from Q» to Q3 and from Q3 to Q, can be computed as:

The above approach can be extended to model a well with several rate changes. Note, however, that the above approach is valid only if the well is flowing under the unsteady state flow condition for the total time elapsed since the well began to flow at its initial rate. Example 1.22 Figure 1.29 shows the rate history of a well that is producing under transient flow conditions for 15 hours. Given the following data:

Figure 1.30 Method of images in solving boundary problems.

Figure 1.31 Well layout for Example 1.23. step 2. Determine the additional pressure drop due to the first fault (i.e., image well 1):

Figure 1.33 Semilog plot of pressure drawdown data.

Figure 1.34 Earlougher’s semilog data plot for the drawdown test (Permission to publish by the SPE, copyright SPE, 1977). The following reservoir data are available: Solution Step 1. From Figure 1.34, calculate p pr: \ssuming that the wellbore storage effect is not significan alculate:

Figure 1.35 Cartesian plot of the drawdown test data (Permission to publish by the SPE, copyright SPE, 1977).

This equation is an equation of a straight line that can be simplified to give: which indicates that a plot of m (pws) vs. log(t) would produce a semilog straight line with a negative slope of:

Figure 1.36 Idealized pressure buildup test. and the amount of other information available for correlation purposes.

Figure 1.37 Horner plot (After Earlougher, R. Advances in Well Test Analysis) (Permission to publish by the SPE, copyright SPE, 1977).

For pressure-squared approach: where the gas flow rate Q, is expressed in Mscf/day.

It should be noted that for a multiphase flow, Equations 1.3.6 and 1.3.9 become:

Table 1.5 Earlougher’s pressure buildup date (Permission to publish by the SPE, copyright SPE, 1977.) Table 1.5 Earlougher’s pressure buildup data (Permission to publish by the SPE, copyright SPE, 1977.) Example 1.274 Table 1.5 shows the pressure buildup data from an oil well with an estimated drainage radius of 2640 ft. Before shut-in, the well had produced at a stabilized rate of 4900 STB/day for 310 hours. Known reservoir data is:

Figure 1.38 Earlougher’s semilog data plot for the buildup test (Permission to publish by the SPE, copyright SPE,

Figure 1.39 Typical pressure buildup curve for a well in a finite (After Earlougher, R. Advances in Well Test Analysis) (Permission to publish by the SPE, copyright SPE, 1977).

Figure 1.40 Miller--Dyes—Hutchinson plot for the buildup test (After Earlougher, R. Advances in Well Test Analysis) (Permission to publish by the SPE, copyright SPE, 1977).

Figure 1.41 Miller-Dyes—Hutchinson dimensionless pressure for circular and square drainage areas (After Earloughe: R. Advances in Well Test Analysis) (Permission to publish by the SPE, copyright SPE, 1977).

Figure 1.42 Matthews—Brons—Hazebroek dimensionless pressure for a well in the center of equilateral drainage areas (After Earlougher, R. Advances in Well Test Analysis) (Permission to publish by the SPE, copyright SPE, 1977).

Figure 1.43 Matthews—Brons—Hazebroek dimensionless pressure for different well locations in a square drainage area. (After Earlougher, R. Advances in Well Test Analysis) (Permission to publish by the SPE, copyright SPE, 1977).

Figure 1.44 Matthews-Brons—Hazebroek dimensionless pressure for different well locations in a 2:1 rectangular drainage area (After Earlougher, R. Advances in Well Test Analysis) (Permission to publish by the SPE, copyright SPE, 1977).

Figure 1.47 Dimensionless pressure for a single well in an infinite system, no wellbore storage, no skin. Exponential-integral solution (After Earlougher, R. Advances in Well Test Analysis) (Permission to publish by the SPE, copyright SPE, 1977).

Figure 1.48 Illustration of type curve matching for an interference test using the type curve (After Earlougher, R. Advances in Well Test Analysis) (Permission to publish by the SPE, copyright SPE, 1977).

Figure 1.49 Type curves for a well with wellbore storage and skin in a reservoir with homogeneous behavior (Copyright ©1983 World Oil, Bourdet et al., May 1983). Equations 1.4.10 and 1.4.12 indicate that a plot of the actual drawdown data of log(Ap) vs. log(#) will produce a parallel curve that has an identical shape to a plot of log (pp) vs. log (tp/Cp). When displacing the actual plot, ver- tically and horizontally, to find a dimensionless curve that coincides or closely fits the actual data, these displacements are given by the constants of Equations 1.4.9 and 1.4.11 as: groups that Gringarten et al. used when developing the type curve:

Table 1.6 Pressure buildup test with afterflow (After Sabet, M. A. “Well Test Analysis” 1991, Gulf Publishing Company)

Figure 1.50 Log-log plot. Data from Table 1.6 (After Sabet, M. A. Well Test Analysis, 1997, Gulf Publishing Company).

Figure 1.51 The Horner plot: data from Table 1.6 (Copyright ©1983 World Oil, Bourdet et al., May 1983).

Figure 1.52 Buildup data plotted on log-log graph paper and matched to type curve by Gringarten et al. (Copyright © 1983 World Oil, Bourdet et al., May 1983).

Results of the example show a good agreement between the conventional well testing analysis and that of the Gringarten type curve approach. Step 4. From the match, calculate the following properties:

shapes for high values of Cpe*s which lead to the problem of finding a unique match by a simple comparison of shapes and determining the correct values of k, s, and C. Equation 1.5.2 indicates that a plot of Dy (tp/Cp) vs. tp/Cp in log—log coordinates will produce a unit-slope straight line during the wellbore storage-dominated flow period.

Figure 1.54 Pressure derivative type curves (Copyright ©1983 World Oil, Bourdet et al., May 1983)

Figure 1.55 Differentiation algorithm using three points. Figure 1.56 Type curve matching. Data from Table 1.6 (Copyright ©1983 World Oil, Bourdet et al., May 1983)

Table 1.7 continued 4 (778.9 — 778.4) / (28.5 — 27.25) = 0.40. (0.40 + 0. 24)/2 = 0.32. €27.25 — 0.32 — (15 + 27. 25)/15 = 24.56.

Figure 1.57 Log-log plot. Data from Table 1.7. Step 5. Calculate C and Cp:

Figure 1.61 Pressure drawdown according to the model by Warren and Root (Copyright ©1969 SPE, Kazemi, SPE Dec. 1969).

Example 1.34 The pressure buildup data as presented by Najurieta (1980) and Sabet (1991) for a double-porosity system is tabulated below:

The following additional reservoir and fluid properties are available:

Figure 1.63 Semilog plot of the buildup test data (After Sabet, M. A. Well Test Analysis 1997, Gulf Publishing Company).

Figure 1.64 Dual-porosity behavior shows as two parallel semilog straight lines on a semilog plot, as a minimum on a derivative plot.

Figure 1.65 Type curve matching (Copyright ©1984 World Oil, Bourdet et al., April 1984) Bourdet et al. (1984) extended the practical applications of these curves and enhanced their use by introducing the pressure derivative type curves to the solution. They devel- oped two sets of pressure derivative type curves as shown in Figures 1.65 and 1.66. The first set, i.e., Figure 1.65, is based on the assumption that the interporosity flow obeys the pseudosteady-state flowing condition and the other set (Figure 1.66) assumes transient interporosity flow. The use of either set involves plotting the pressure difference Ap and the derivative function, as defined by Equation 1.5.4 for draw- down tests or Equation 1.5.5 for buildup tests, versus time with same size log cycles as the type curve. The controlling variables in each of the two type curve sets are given below. the fissures and the second describes the combined behav- ior of the double-porosity system. Figure 1.64 shows, at early time, the typical behavior of wellbore storage effects with the deviation from the 45° straight line to a maximum represent- ing a wellbore damage. Gringarten (1987) suggested that the shape of the minimum depends on the double-porosity behavior. For a restricted interporosity flow, the minimum takes a V-shape, whereas unrestricted interporosity yields an open U-shaped minimum.

Figure 1.66 Type curve matching (Copyright ©1984 World Oil, Bourdet et al., April 1984).

Table 1.8 Pressure Buildup Test, Naturally Fractured Reservoir. After Sabet, M. A. “Well Test Analysis” 1991, Gulf Publishing Company

Figure 1.67 The Horner plot; data from Table 1.8 (After Sabet, M. A. Well Test Analysis 1991, Gulf Publishing Company). shows that Ap is about 11 psi and Equation 1.5.10 gives an erroneous value of:

Figure 1.68(b) Log-log plot of A p vs. Ate (After Sabet, M. A. Well Test Analysis 1991, Gulf Publishing Company).

Figure 1.69 Type curve matching (Copyright ©1984 World Oil, Bourdet et al., April 1984).

Figure 1.70 Theoretical pressure buildup curve for two-layer reservoir (Copyright ©1961 SPE, Lefkovits et al., SPEJ, March 1961). Figure 1.70 Theoretical pressure buildup curve for two-layer reservoir (Copyright ©1961 SPE, Lefkovits et al., SPEJ, March 1961).

The fracture has a much greater permeability than the formation it penetrates; hence it influences the pressure response of a well test significantly. The general solution for the pressure behavior in a reservoir is expressed in terms of dimensionless variables. The following dimension- less groups are used when analyzing pressure transient data in a hydraulically fractured well:

Notice that the above equations are written in terms of the pressure drawdown tests. These equations should be modified for buildup tests by replacing the pressure and time with the appropriate values as shown below:

Figure 1.71 Flow periods for a vertically fractured well (After Cinco and Samaniego, JPT, 1981).

Figure 1.72 Graph for analysis of pressure data of bilinear flows (After Cinco and Samaniego, 1987). fracture tip begins to affect wellbore behavior. If the test is not run sufficiently long for bilinear flow to end when Fcp > 1.6, it is not possible to determine the length of the fracture. When the dimensionless fracture conductivity Fop < 1.6, it indicates that the fluid flow in the reservoir has changed from a predominantly one-dimensional linear flow to a two-dimensional flow regime. In this particular case, it is not possible to uniquely determine fracture length even if bilinear flow does end during the test.

Figure 1.73 Bilinear flow graph for data of Example 1.36 (After Sabet, M. A. Well Test Analysis 1991, Gulf Publishing Company). or in terms of real pressure and time, as:

evidence of a fracture. When properly applied, these plots are the best diagnostic tools available for the purpose of detecting a fracture. In practice, the 5 slope is rarely seen except in fractures with high conductivity. Finite conductivity fracture responses generally enter a transition period after the bilinear flow (the } i slope) and reach the infinite: -acting pseudoradial flow regime before ever achieving a 5 5 slope (linear flow). For a long duration of wellbore storage effect, the bilinear flow pressure behavior may be masked and data analysis becomes difficult with current interpretation methods.

straightening to a line of proper slope that represents the fracture linear flow. The duration of the curved portion that represents the transition flow depends on the fracture flow capacity. The lower the fracture flow capacity, the longer the duration of the curved portion. The beginning of formation linear flow, “blf,” depends on F¢p and can be approximated from the following relationship:

Figure 1.76 Use of the log-log plot to approximate the beginning of pseudoradial flow.

Figure 1.77 Log-log plot, drawdown test data of Example 1.37 (After Sabet, M. A. Well Test Analysis 1997, Gulf Publishing Company). Example 1.37 The drawdown test data for an infinite conductivity fractured well is tabulated below:

Figure 1.78 Linear plot, drawdown test data of Example 1.37 (After Sabet, M. A. Well Test Analysis 1991, Gulf Publishing Company).

in Figure 1.79) and determine the slope of the line, to give:

Figure 1.81 Vertically fractured reservoir, calculated pressure buildup curves (After Russell and Truitt, 1964)

Figure 1.82 Dimensionless pressure for vertically fractured well in the center of a closed square, no wellbore storage, infinite conductivity fracture (After Gringarten et al., 1974).

Figure 1.83 Dimensionless pressure for vertically fractured well in the center of a closed square, no wellbore storage, uniform-flux fracture (After Gringarten et al., 1974).

Example 1.38 Tabulated below is the pressure buildup data for an infinite conductivity fractured well: To illustrate the use of the Gringarten type curves in ana- lyzing well test data, the authors presented the following example: The above two approximations are only valid for x./x; >> 1 and for infinite conductivity fractures. For uniform-flux fracture, the constants 30.358 and 0.01648 become 107.312 and 0.001648.

Step 2. Calculate k and x; by using Equations 1.5.43 and 1.5.44: Step 3. Calculate the skin factor by applying Equation 1.5.45:

Figure 1.84 Type curve matching. Data from Example 1.38 (Copyright ©1974 SPE, Gringarten et al., SPEJ, August 1974).

Figure 1.86 Type curve for vertically fractured gas wells graph (After Cinco and Samaniego, 1987). Figure 1.85 Horner graph for a vertical fracture (infinite conductivity).

Figure 1.87 Effective wellbore radius vs. dimensionless fracture conductivity for a vertical fracture graph (After Cinco and Samaniego, 1981).

Figure 1.88 Type curve matching for data in bilinear and transitional flow graph (After Cinco and Samaniego, 1987). Example 1.39 The buildup test data as given in Example 1.36 is given below for convenience:

The concept of the pressure derivative can be effectively employed to identify different flow regime periods associ- ated with hydraulically fractured wells. As shown in Figure 1.89, a finite conductivity fracture shows a i straight-line slope for both the pressure difference Ap and its derivative; however, the two parallel lines are separated by a factor of 4. Similarly, for an infinite conductivity fracture, two straight parallel lines represent Ap and its derivative with a 5 slope and separation between the lines of a factor of 2 (as shown in Figure 1.90).

Figure 1.89 Finite conductivity fracture shows as a ; slope line on a log-log plot, same on a derivative plot. Separation between pressure and derivative is a factor of 4. Te, Fans Re aos SATS Ree Ee ee me ere In tight reservoirs where the productivity of wells is enhanced by massive hydraulic fracturing (MHF), the result- ing fractures are characterized as long vertical fractures with finite conductivities. These wells tend to produce at a constant and low bottom-hole flowing pressure, rather than constant flow rate. The diagnostic plots and the conventional analysis of bilinear flow data can be used when analyzing

Figure 1.90 Infinite conductivity fracture shows as a 3 slope line on a log-log plot, same on a derivative plot. Separation between pressure and derivative is a factor of 2.

The following additional data is available: Example 1.40 Apre-frac buildup test was performed ona well producing from a tight gas reservoir, to give a formation permeability of 0.0081 md. Following an MHF treatment, the well produced at a constant pressure with recorded rate-time data as given below: The following example, as adopted from Agarwal et al. (1979), illustrates the use of these type curves.

Figure 1.92 Reciprocal smooth rate vs. time for MHF, Example 1.42.

Figure 1.91 Log-log type curves for finite capacity vertical fractures; constant wellbore pressure (After Agarwal et al., 1979).

Nhen both the test well and image well are shut-in for 4 yuildup test, the principle of superposition can be applied te ‘quation 1.2.57 to predict the buildup pressure at At as:

Figure 1.94 Method of images in solving boundary problems. behavior can be described mathematically by applying the principle of superposition as given by the method of images. Figure 1.94 shows a test well that is located at a distance L from a sealing fault. Applying method images, as given Equation 1.2.157, the total pressure drop as a function of time t is:

Example 1.41 A pressure buildup test was conducted to confirm the existence of a sealing fault near a newly drilled well. Data from the test is given below:

Figure 1.95 Theoretical Horner plot for a faulted system.

Figure 1.96 Estimating distance to a no-flow boundary.

Figure 1.97 Qualitative interpretation of buildup curves (After Economides, 1988).

Figure 1.99 Illustration of rate history and pressure response for a pulse test (After Earlougher, R. Advances in Well Test Analysis) (Permission to publish by the SPE, copyright SPE, 1977).

Figure 1.98 Rate history and pressure response of a two-well interference test conducted by placing the active well on production at constant rate.

Figure 1.100 Dimensionless pressure for a single well in an infinite system, no wellbore storage, no skin. Exponential-integral solution (After Earlougher, R. Advances in Well Test Analysis) (Permission to publish by the SPE, copyright SPE, 1977).

The following additional reservoir data is available:

Figure 1.102 Interference data of Well 1. (After Sabet, M. A. Well Test Analysis 1991, Gulf Publishing Company)

Figure 1.101 Interference data of Well 3. (After Sabet, M. A. Well Test Analysis 1991, Gulf Publishing Company).

Step 3. Solve for k and ¢ between Well 2 and Well 3 by applying Equations 1.6.2 and 1.6.3 Step 4. Figure 1.104 shows the match of the test data for Well 1 with the following matching points:

Figure 1.103 Match of interference data of Well 3. (After Sabet, M. A. Well Test Analysis 1991, Gulf Publishing Company).

Figure 1.105 Nomenciature for anisotropic permeability system (After Ramey, 1975).

In a homogeneous and isotropic reservoir, i.e., perme- ability is the same throughout the reservoir, the minimum area of the reservoir investigated during an interference test between two wells located a distance 7 apart is obtained by drawing two circles of radius r centered at each well.

Example 1.43 The following data is for an interference test in a nine-spot pattern with one active well and eight observation wells. Before testing, all wells were shut in. The test was conducted by injecting at —115 STB/day and observing the fluid levels in the remaining eight shut-in wells. Figure 1.106 shows the well locations. For simplicity, only the recorded pressure data for three observation wells, as tabulated below, is used to illustrate the methodology. These selected wells are labeled Well 5-E, Well 1-D, and Well 1-E. The well coordinates (x, y) are as follows:

Figure 1.106 Well locations for Example 1.43 (After Earlougher, R. Advances in Well Test Analysis) (Permission to publish by the SPE, copyright SPE, 1977)

Equations 1.6.12 through 1.6.15 contain the following fou: unknowns:

Step 3. From the pressure match point, use Equation 1.6.11 to solve for k:

Figure 1.107 Interference data of Example 1.6 matched to Figure 1.100. Pressure match is the same of all curves. (After Earlougher, R. Advances in Well Test Analysis). (Permission to publish by the SPE, copyright SPE, 1977).

Figure 1.108 Schematic illustration of rate (pulse) history and pressure response for a pulse test (After Earlougher, R. Advances in Well Test Analysis) (Permission to publish by the SPE, copyright SPE, 1977).

Figure 1.109 Schematic pulse test rate and pressure history showing definition of time lag (t_) and pulse response amplitude (A p) curves. (After Earlougher, R. Advances in Well Test Analysis) (Permission to publish by the SPE, copyright SPE, 1977).

Figure 1.110 Pulse testing: relation between time lag and response amplitude for first odd pulse. (After Kamal and Brigham, 1976).

Figure 1.111 Pulse testing: relation between time lag and response amplitude for first even pulse. (After Kamal and Brigham, 1976).

Figure 1.112 Pulse testing: relation between time lag and response amplitude for all odd pulses after the first. (After Kamal and Brigham, 1976).

Figure 1.113 Pulse testing: relation between time lag and response amplitude for all even pulses after the first. (After Kamal and Brigham, 1976).

Figure 1.114 Pulse testing: relation between time lag and cycle length for first odd pulse. (After Kamal and Brigham, 1976). Step 3. Calculate the dimensionless time lag (t,)p by apply- ing Equation 1.6.17:

The solution methodology outlined in analyzing interfer- ence test data in homogeneous anisotropic reservoirs can be employed when estimating various permeability parameters from pulse testing. 1.6.4 Pulse testing in homogeneous anisotropic reservoirs

and length of time needed for the test can be predeter- mined. To design a pulse test, Kamal and Brigham (1975) recommend the following procedure: Step 5. Using the following parameters:

Table 1.9 Pressure behaviour of producing Well. After Slider, H. C., Worldwide Practical Petroleum Reservoir Engineering Methods, copyright ©1983, Penn Well Publishing

Figure 1.117 Pulse testing: relation between time lag and cycle length for all even pulses after the first. (After Kamal and Brigham, 1976).

Figure 1.118 Pulse pressure response for Example 1.45.

“igure 1.120 Log-log data plot for the injectivity test of Example 1.47. Water injection into a reservoir at sta -onditions (After Earlougher, R. Advances in Well Test Analysis) (Permission to publish by the SPE, copyright SPE, 1977).

Figure 1.122 Idealized rate schedule and pressure response for falloff testing. Figure 1.121 Semilog plot for the injectivity test of Example 1.47. Water injection into a reservoir at static conditions (After Earlougher, R. Advances in Well Test Analysis) (Permission to publish by the SPE, copyright SPE, 1977).

shown in Figure 1.123, a plot of pys vs. log [ (tp) + At) /At] would form a straight-line portion with an intercept of p* at (t) + At) /At = 1 and a negative slope of m. Solution

Figure 1.124 Log-log data plot for a falloff test after brine injection, Example 1.48 (After Earlougher, R. Advances ir Well Test Analysis) (Permission to publish by the SPE, copyright SPE, 1977).

Figure 1.125 Horner plot of pressure falloff after brine injection, Example 1.48.

Figure 1.127 Schematic diagram of fluid distribution around an injection well (composite reservoir).

Figure 1.126 Miller-Dyes—Hutchinson plot of pressure falloff after brine injection, Example 1.48.

Figure 1.129 Relationship between mobility ratio, slope ratio, and storage ratio. (After Merrill, et al. 1974). Step 4. Calculate the following dimensionless ratios:

The assumption of a two-bank system is applicable if the reservoir is filled with liquid or if the maximum shut-in time of the falloff test is such that the radius of investigation of the test does not exceed the outer radius of the oil bank. The ideal behavior of the falloff test in a two-bank system as expressed in terms of the Horner plot is illustrated in Figure 1.128.

Figure 1.131 Falloff test data for Example 1.49. (After Merrill et al. 1974).

Solution Yeh and Agarwal (1989) presented a different approach of analyzing the recorded data from the injectivity and falloff tests. Their methodology uses the pressure derivate Ap and Agarwal equivalent time At. (see Equation 1.4.16) in performing the analysis. The authors defined the following nomenclature: a ef: SRN Sa ener, eC ee ey Ree eee Sem oe

Figure 1.133 Falloff pressure response and derivative (base case). lasts for 30 minutes. The pressure observed at the end of each injection rate is plotted versus the rate. This plot usually shows two straight lines which intersect at the frac- ture pressure of the formation, as shown schematically in Figure 1.134. The suggested procedure is summarized below:

the floodout zone, i.e., water bank, can be estimated from:

Kamal et al. (1995) conveniently summarized; in tabulated form, various plots and flow regimes most commonly used in transient tests and the information obtained from each test as shown in Tables 1-11 and 1-12. Chaudhry (2003) presented another useful “toolbox” that summarizes the pressure derivative trends for common flow regimes that have been presented in this chapter, as shown in Table 1-10. “toolbox” of graphing functions that is considered an essential part of computer-aided well test interpretation system: As pointed out by Horn (1995), data presented in graph- ical form is much easier to understand than a single table of numbers. Horn proposed the following Table 1.10 Pressure Derivative Trends for Common Flow Regimes.

Table 1.12 Plots and flow regimes of transient tests (After Kamal et al. 1995)

Table 1.11 Reservoir properties obtainable from various transient tests (After Kamal et al. 1995).

Table 1.10 Pressure Derivative Trends for Common Flow Regimes (continued

Figure 2.3 Calculating the area under the curve. When the pressure drop (; — p) is plotted versus the time f, as shown in Figure 2.3, the area under the curve represents

Step 2. Calculate the cumulative water influx after 100 days: Step 1. Calculate the total pressure drop at each time t: Solution The aquifer is under a steady-state flowing condition with an estimated water influx constant of 130 bbl/day/psi. Given the initial reservoir pressure is 3500 psi, calculate the cumulative water influx after 100, 200, 300, and 400 days using the steady-state model.

Example 2.4 The pressure history of a water drive oil reservoir is given below:

2.3.3 The Hurst modified steady-state equation One of the problems associated with the Schilthuis steady- state model is that as the water is drained from the aquifer, the aquifer drainage radius 7, will increase as the time increases. Hurst (1943) proposed that the “apparent” aquifer radius 7, would increase with time and, therefore, the dimen- sionless radius 7,/7. may be replaced with a time-dependent function as given below:

Step 1. Construct the following table. Solution Slope = 1/C = 0.020

Example 2.5 The following data, as presented by Craft and Hawkins (1959), documents the reservoir pressure as a function of time for a water drive reservoir. Using the reservoir historical data, Craft and Hawkins calculated the water influx by applying the material balance equation (see Chapter 4). The rate of water influx was also calculated numerically at each time period:

Figure 2.5 Determination of C and n for Example 2-5. = 2388 x 103+ 420.508 x 10° =2809 Mbbl It is predicted that the boundary pressure would drop to 3379 psi after 1186.25 days of production. Calculate the cumulative water influx at that time.

The dimensionless form of the diffusivity equation, as presented in Chapter 1 by Equation 1.2.78, is basically the general mathematical equation that is designed to model the transient flow behavior in reservoirs or aquifers. In a dimensionless form, the diffusivity equation is: Figure 2.6 Water influx into a cylindrical reservoir.

Figure 2.8 Dimensionless water influx Wep for several values of re/rp, i.@., fa/te (Van Everdingen and Hurst Wep Permission to publish by the SPE). The authors expressed their mathematical relationship for calculating the water influx in the form of a dimensionless parameter called dimensionless water influx W.p. They also expressed the dimensionless water influx as a function of the dimensionless time tp and dimensionless radius 7p; thus they made the solution to the diffusivity equation general- ized and it can be applied to any aquifer where the flow of water into the reservoir is essentially radial. The solutions were derived for the cases of bounded aquifers and aquifers of infinite extent. The authors presented their solution in tab- ulated and graphical forms as reproduced here in Figures 2.8 through 2.11 and Tables 2.1 and 2.2. The two dimensionless parameters tp and 7p are given by:

Figure 2.9 Dimensionless water influx Wep for several values of re/rp, i.€@., fa/te (van Everdingen and Hurst Wep values. Permission to publish by the SPE).

Table 2.1. Dimensionless Water Influx Wep for Infinite Aquifer (van Everdingen and Hurst Wep. Permission to publish by the SPE).

Figure 2.11. Dimensionless water influx Wep for infinite aquifer (van Everdingen and Hurst Wep values. Permission t publish by the SPE).

Table 2.1 Dimensionless Water Influx Wep for Infinite Aquifer (van Everdingen and Hurst Wep. Permission to publist by the SPE).

Table 2.2 Dimensionless Water Influx Wep for Several Values of re/Trp, i.€., fa/te (Van Everdingen and Hurst Wep. Permission to publish by the SPE).

Table 2.2 Dimensionless Water Influx Wep for Several Values of re/Trp, i.€., fa/te (Van Everdingen and Hurst Wer Permission to publish by the SPE) (continued)

Example 2.6“ Calculate the water influx at the end of 1, 2, and 5 years into a circular reservoir with an aquifer of infinite extent, i.e., 7p = oo. The initial and current reservoir pressures are 2500 and 2490 psi, respectively. The reser voir- aquifer system has the following properties.

Figure 2.12 Gas cap drive reservoir (After Cole, F., Reservoir Engineering Manual, Gulf Publishing Company. 1969).

Step 4. Using Table 2.1, determine the dimensionless water influx W.p: Step 5. Calculate the cumulative water influx by applying Equation 2.3.18:

Example 2.6 shows that, for a given pressure drop, dou- bling the time interval will not double the water influx. This example also illustrates how to calculate water influx as a result ofa single pressure drop. As there will usually be many of these pressure drops occurring throughout the prediction period, it is necessary to analyze the procedure to be used where these multiple pressure drops are present.

Figure 2.13 Boundary pressure versus time. Since these pressure waves are assumed to occur at dif- ferent times, they are entirely independent of each other. Thus, water expansion will continue to take place as a result of the first pressure drop, even though additional water influx is also taking place as a result of one or more later pressure drops. This is essentially an application of the prin- ciple of superposition. In order to determine the total water influx into a reservoir at any given time, it is necessary to determine the water influx as a result of each successive pressure drop that has been imposed on the reservoir and aquifer.

In calculating the cumulative water influx into a reservoir at successive intervals, it is necessary to calculate the total water influx from the beginning. This is required because of the different times during which the various pressure drops have been effective.

24 months. The predicted boundary pressure at the end of each specified time period is given below: Data from Example 2.6 is listed below:

Water influx after 24 months: i a I a a ae The first pressure drop has now been effective for the entire 24 months, the second pressure drop has been effective for 18 months, the third pressure drop has been effective for

Step 3. Calculate the dimensionless time at 365 days, as:

Edwardson et al. (1962) developed three sets of simple polynomial expressions for calculating the dimensionless water influx W.p for infinite-acting aquifers. The proposed three expressions essentially approximate the W.p data in three dimensionless time regions. Figure 2.16 Pressure drop data for Example 2.7.

12 months, and the fourth pressure drop has been effective only 6 months. A summary of the calculations is given below: where F;, is the ratio of vertical to horizontal permeability, Fy = ky/Rp

Table 2.3. Dimensionless Water Influx, Wep, for Infinite Aquifer (Permission to publish by the SPE)

Table 2.3 Dimensionless Water Influx, Wep, for Infinite Aquifer (Permission to publish by the SPE) (continued)

Table 2.4 Dimensionless Water Influx, Wep, for ny = 4 (Permission to publish by the SPE) Table 2.5 Dimensionless Water Influx, Wep, for ry = 6 (Permission to publish by the SPE)

Table 2.5 Dimensionless Water Influx, Wep, for is = 6 (Permission to publish by the SPE) (continued) Table 2.6 Dimensionless Water Influx, Wep, for is = 8 (Permission to publish by the SPE)

Table 2.6 Dimensionless Water Influx, Wep, for rn = 8 (Permission to publish by the SPE) (continued)

Table 2.7 Dimensionless Water Influx, Wep, for ny = 10 (Permission to publish by the SPE)

Table 2.7 Dimensionless Water Influx, Wep, for ry = 10 (Permission to publish by the SPE) (continued)

The boundary pressure history is given below: Calculate the cumulative water influx as a function of time by using the bottom-water drive solution and compare with the edge-water drive approach.

It should be noted that the Carter and Tracy method is not an exact solution to the diffusivity equation and should be considered as an approximation. in which the regression coefficients are given below:

where: Linear water drive As shown by van Everdingen and Hurst, the water influx from a linear aquifer is proportional to the square root of time. The van Everdingen and Hurst dimensionless water influx is replaced by the square root of time, as given by:

The above comparison indicates that the Carter and Tracy method considerably overestimates the water influx. How- ever, this is due to the fact that a large time step of 6 months was used in the Carter and Tracy method to determine the water influx. The accuracy of this method can be increased substantially by restricting the time step to one month. Recal- culating the water influx on a monthly basis produces an excellent match with the van Everdingen and Hurst method as shown below.

Step 1. For each time step , calculate the total pressure drop Ap, = ~; — p, and the corresponding fp:

The following table compares the results of the Carter and Tracy water influx calculations with those of the van Everdingen and Hurst method.

2.3.6 The Fetkovich method Fetkovich (1971) developed a method of describing the approximate water influx behavior of a finite aquifer for radial and linear geometries. In many cases, the results of this model closely match those determined using the van Everdingen and Hurst approach. The Fetkovich theory is much simpler, and, like the Carter-Tracy technique, this method does not require the use of superposition. Hence, the application is much easier, and this method is also often utilized in numerical simulation models. Equation 2.3.35 provides a simple expression to determine the average aquifer pressure p, after removing W. bbl of water from the aquifer to the reservoir, i.e., cumulative water influx.

Step 7. Calculate the cumulative water influx as shown in the following table.

Figure 2.17 Aquifer-reservoir geometry for Example 2.10. Step 6. Calculate the productivity index J of the radial aquifer from Equation 2.3.43 Problems

If the encroachment angle is 360°, calculate the water influx as a function of time by using: The aquifer-reservoir system is characterized by the following data:

The geological data of the aquifer estimates the water influx constant at 551 bbl/psi. After 1120 days of pro- duction, the reservoir average pressure has dropped to 3800 psi and the field has produced 860000 STB of oil.

The reservoir is producing under the pseudosteady-state condition. The following additional data is available: Example 3.1 The PVT properties of a gas sample taken from a dry gas reservoir are given below: = 30453 Mscf/day Calculate the gas flow rate under the following conditions:

Figure 3.4 Graph of the pressure-squared data.

been used for more than 60 years by the petroleum industry to characterize and determine the flow potential of gas wells. There are essentially three types of deliverability tests:

Figure 3.5 Graph of the pressure method data.

Figure 3.8 shows the gas pseudopressure yw as a function of pressure. Generate the current IPR by using the following methods. Example 3.2 A gas well was tested using a three-point conventional deliverability test with an initial average reser- voir pressure of 1952 psi. The recorded data during the test is given below:

Figure 3.7 Conventional back-pressure test.

Step 2. Plot @, — b?,)/Qz VS. Qg on a Cartesian scale and draw the best straight line as shown in Figure 3.10. Step 3. Determine the intercept and the slope of the straight line, to give: Step 1. Construct the following table:

Step 6. Generate the IPR data by assuming various values of Pye and calculate the corresponding Q,:

Figure 3.8 Real-gas potential versus pressure.

Step 2. Plot (W, — Ww:)/Qg on a Cartesian scale as shown in Figure 3.12 and determine the intercept a2 and slope bp as: Step 1. Construct the following table:

Step 4. Generate the IPR data by applying Equation 3.1.33:

Step 2. Plot (6, — Pwe)/@Qg vs. Qg on a cartesan scale as shown in Figure 3.11. Draw the best straight line and determine the intercept and slope as: Step 1. Construct the following table:

Step 5. Construct the IPR data by assuming various values of pwr and solving for Mg by using Equation 3.1.28:

(c) Compare the gas flow rates as calculated by the four different methods. Results of the IPR calculation are

~p 4. Generate the IPR data by assuming various values of Pwr, Le., Wwe, and calculating the corresponding Q, from Equation 3.1.38: documented below: Gas flow rate (Mscf/day)

Figure 3.13 IPR for all methods. The performance coefficient C is considered a pressure- dependent parameter and should be adjusted with each change of the reservoir pressure. Assuming that the reser- voir pressure has declined from p, to p,2, the performance coefficient at p; can be adjusted to reflect the pressure drop by applying the following simple approximation: Since the pseudopressure analysis is considered more accu- rate and rigorous than the other three methods, the accuracy of each of the methods in predicting the IPR data is com- pared with that of the w approach. Figure 3.13 compares graphically the performance of each method with that of the w approach. Results indicate that the pressure-squared equation generated the IPR data with an absolute average error of 5.4% as compared with 6% and 11% for the back- pressure equation and the pressure approximation method, respectively.

Figure 3.14 compares graphically the IPR data as predicted by the above three methods. It should be pointed out that all the various well tests and inflow performance relationships previously discussed are intended to evaluate the formation capacity to deliver gas to the wellbore for a specified average reservoir pressure p, and a bottom-hole flowing pressure py. The volume of gas which can actually be delivered to the surface will also depend on the surface tubing head pressure ~; and the pressure drop from the wellbore to the surface due to the weight of the gas column and friction loss through the tubing. Cullender and Smith (1956) described the pressure loss by the following expression:

Figure 3.15 Idealized water-drive gas reservoir.

Figure 3.16 Gas material balance equation. The calculated reservoir gas volume V can be used to determine the areal extend of the reservoir from:

The value of the gas initially in place as calcu- lated form the MBE compares reasonably with the volumetric value.

The graphical representation of Equation 3.3.10 can be used to detect the presence of water influx, as shown in Figure 3.18. When the plot of p/Z vs. G, deviates from the linear relationship, it indicates the presence of water encroachment.

It should be pointed out that the successive application o: Equation 3.3.13 can yield increasing or decreasing values o: the gas initially in place G. Two different situations therefore exist: Solution

Solving for the pore volume of the water-invaded zone, (PV) water, gives: Step 3. Calculate trapped gas volume in the water-invaded zone, or:

(1949) unsteady-state model, the selected water influx model can be integrated into Equation 3.3.23; to give:

Table 3.1 Havlena—Odeh Dry-Gas Reservoir Data for Example 8-8

Figure 3.23 Havlena—Odeh MBE plot for a gas reservoir.

Figure 3.26 p/Z versus cumulative production—North Ossum Field, Lafayette Parish, Louisiana NS2B reservoir (After Hammerlindl, 1971).

Figure 3.27 shows how c and ¢; vary as a function of pressure for this overpressured Gulf Coast sandstone reser- voir. Figure 3.27 gives the proper definition of the “pore collapse” which is the condition when the instantaneous PV compressibility begins to increase at decreasing reservoir pressure. Both types of compressibilities are pressure dependent and best determined by special core analysis. An example of this analysis is shown below for a Gulf Coast sandstone as given by Fetkovich et al.:

Figure 3.27 Cumulative and instantaneous cy.

Figure 3.28 Mobil—David Anderson “L” p/Z versus cumulative production (After Roach, 1981).

Figure 3.29 Mobil—David Anderson “L” p/Z gas material balance (After Roach, 1981).

or equivalently as the equation of a straight line: Plotting a vs. £ will produce a straight line with a correct slope of 1/G and constant intercept of ER. Fetkovich et al. plot for abnormal pressure gas reservoirs Fetkovich et al. (1998) adopted the shale water influx theory and developed a general gas MBE that accounts for the total cumulative effects of the various reservoir compressibilities as well as the total water associated with the reservoir. The “associated” water includes:

Figure 3.30 Schematic of methane flow dynamics in a coal seam system (After King et al., 1986).

The volume of gas lost during this core recovery time is referred to as “lost gas.” The volume of the lost gas can be estimated by using the USBM direct method, as illus- trated in Figure 3.31. The method simply involves plotting the desorbed gas volume versus the square root of time, Vt, on a Cartesian scale and extrapolating the early-time des- orption data back to time zero. Experience has shown that this technique works adequately in shallow, low-pressure, low-temperature coals with a lost gas volume in the range of 5-10% of the total adsorbed gas content of the coal. How- ever, in higher-pressure coal seams, the lost gas volume may exceed 50% of the total adsorbed gas content. Accurate determinations of both gas content and the sorp- tion isotherm are required to estimate recoverable reserve and production profiles. An example of a typical sorption isotherm relationship is shown in Figure 3.32 as given by Mavor et al. (1990). This sorption isotherm was measured on a sample collected from a well in the Fruitland Forma- tion Coal Seam of the San Juan Basin, New Mexico. The authors pointed out that the total gas content G, of the coal was determined to be 355 scf/ton by desorption canister tests performed on whole core samples at the well location. The gas content is less than the sorption isotherm gas storage capacity of 440 scf/ton at the initial reservoir pressure of 1620 psia. This implies that the pressure must be reduced to 648 psia which corresponds to 355 scf/ton on the sorption isotherm curve. This pressure is known as the critical or des- orption pressure p,q. This value will determine whether a coal seam is saturated or undersaturated. A saturated coal seam holds as much adsorbed gas as it possibly can for the given reservoir pressure and temperature. An analogy would be an oil reservoir having a bubble point equal to the initial reser- voir pressure. If the initial reservoir pressure is greater than the critical desorption pressure, the coalbed is considered an undersaturated one as in the case of Fruitland Forma- tion Coal. An undersaturated coal seam is undesirable since more water will have to be produced (dewatering process) before gas begins to flow.

Figure 3.32 Sorption isotherm curve (After Mavor et al. 1990).

Step 1. Calculate V/p for each of the measured data points and construct the following table: Solution

straight line to give: Step 3. Determine the coefficient of the straight line, i.e., slope and intercept, to give:

Step 4. Calculate R, from either Equation 3.4.8 or 3.4.9 at different assumed pressures:

For pressures below the critical desorption pressure, the fractional gas recovery could be roughly estimated from the following relationship:

Step 2. Assume several reservoir pressures and calculate the recovery factor in the following tabulated form: Many factors influence the measured gas content G, and sorption isotherm and, consequently, affect the deter- mination of the initial gas-in-place. Among these factors are:

Figure 3.34 Effect of moisture content on gas storage capacity.

It is commonly assumed that interbedded rocks, hav- ing densities greater than 1.75 g/cm® have negligible gas storage capacity.

Each of the above two forms of the generalized MBE is the equation of a straight line and can be written as:

- The actual well production and pertinent coal data is given below: Example 3.14 A coal well is draining a homogeneous 320 acre coal deposit.

Step 1. Calculate FE, and V as a function of pressure by applying the following expressions: Solution

Figure 3.37 Graphical determination of drainage area. Step 1. Using the given values of c,, and c; in Equation 3.4.23, calculate the Y and X terms and tabulate the results as a function of pressure as follows:

Figure 3.39 Graphical determination of reservoir pressure. Figure 3.38 Straight-line relationship of y as a function of x. on the straight line, it indicates that the assumed reservoir pressure is correct, otherwise the pro- cess is repeated at a different pressure. The process can be rapidly converged by assuming three differ- ent pressure values and connecting the coordinate plotted points with a smooth curve that intersects with the straight line at (corr, Yeorr). The reser- voir pressure at the given W, and G, is calculated from:

Figure 3.40 Relationships between GWR, Qy, and Wp.

Figure 3.41(a) Real-life example of p/Z plot from Sheet IVc in the Waterton Gas Field.

The main problem with tight gas reservoirs is : the diffi- culty of accurately estimating the average reservoir pressure required for p/Z plots as a function of G, or time. If the

Figure 3.42 Schematic representation of compartmented reservoir consisting of two reservoir compartments separated by a permeable boundary.

linear equations as expressed in a matrix form:

Figure 3.44 illustrates the general shape of the three curves at different possible values of b. It should be pointed out that the above forms of decline curve equations are strictly applicable only when the well/reservoir is under pseudosteady (semisteady)-state flow conditions, i.e., boundary-dominated flow conditions. Arps’s equation has been often misused and applied to model the performance of oil and gas wells whose flow regimes are in a transient flow. As presented in Chapter 1, when a well is first open to flow, it is under a transient (unsteady-state) con- dition. It remains under this condition until the production from the well affects the total reservoir system by reaching its drainage boundary, then the well is said to be flowing under pseudosteady-state or boundary-dominated flow con- ditions. The following is a list of inherent assumptions that must be satisfied before performing rate-time decline curve analysis:

resulting curve can be straightened out by using shifting techniques. to 1 and, accordingly, Arps’s equation can be conveniently expressed in the following three forms:

Estimate: Example 3.16 The following production data is available from a dry gas field:

Example 3.17 A gas well has the following production history:

Figure 3.46 Decline curve data for Example 3.17.

Estimate the future production performance for the next [6 years. Example 3.18% The following production datais reported by Ikoku for a gas well: Solution The following example illustrates the proposed methodol- ogy for determining b and Dj.

Figure 3.47 Rate-time plot for Example 3.18. coordinates of a point on the smooth graph, i.e., (t2, gz), to give:

Results of step 7 are tabulated below and shown graphically in Figure 3.49: Gentry (1972) developed a graphical method for the coef- ficients 6 and D; as shown in Figures 3.50 and 3.51. Arps’s decline curve exponent b is expressed in Figure 3.50 in terms of the ratios g,/q and G,/(tg;) with an upper limit for q,/q at 100. To determine the exponent b, enter the graph with the abscissa with a value of G,/(tq;) that corresponds to the last

The iterative method can be conveniently performed by constructing the following table:

Figure 3.48 Rate—cumulative plot for Example 3.18. Step 3. Select the coordinates of the other end point on the smooth curve as (fp, g2), to give:

Figure 3.50 Relationship between production rate and cumulative production (After Gentry, 1972). Figure 3.49 Decline curve data for Example 3.18.

Figure 3.52 Estimation of the effect of restricting maximum production rate.

Figure 3.51 Relationship betweeen production rate and time (After Gentry, 1972).

Results of the above repeated procedure are tabulated below:

Hence, a graph of log (Ap) vs. log (£) will have an identical shape (i.e., parallel) to a graph of log (pp) vs. log (tp), although the curve will be shifted by log (0. 00708kh/QB,) vertically in pressure and log (0. 0002642/@.c:72) horizontally in time. This concept is illustrated in Chapter 1 by Figure 1.46 and reproduced in this chapter for convenience. Not only do these two curves have the same shape, but if they are moved relative to each other until they coincide or “match,” the vertical and horizontal displacements required to achieve the match are related to these constants in Equa- tions 3.5.44 and 3.5.45. Once these constants are determined from the vertical and horizontal displacements, it is possible to estimate reservoir properties such as permeability and porosity. This process of matching two curves through the vertical and horizontal displacements and determining the reservoir or well properties is called type curve matching.

following table and graphically in Figure 3.53:

Values of p; — p(7,t) are presented as a function of time and radius (i.e., at 7 = 10 feet and 100 feet) in the

Figure 3.54 Pressure profile at 10 feet and 100 feet as a function of t/r’.

where the decline curve dimensionless variables gpg and tpa are defined by: where gpq and tpg are the decline curve dimensionless vari- ables as defined by Equations 3.5.47 and 3.5.48, respectively. During the boundary-dominated flow period, i.e., steady- state or semisteady-state flow conditions, Darcy’s equation can be used to describe the initial flow rate q; as:

Figure 3.55 Fetkovich type curves (After Fetkovich, 1980, JPT June 1980, copyright SPE 1980).

Figure 3.56 West Virginia gas well A type curve fit (Copyright SPE 1987).

Step 1. Using the match point, calculate q; and D; by applying Equations 3.5.57 and 3.5.58, respectively: Step 2. Calculate the permeability k from Equation 3.5.59 Solution

The significance of the decline curve exponent b value is that for a single-layer reservoir, the value of 5 will lie between 0 and 0.5. With layered no-crossflow performance, however, the b value can be between 0.5 and 1.0. As pointed out by Fetkovich et al. (1996), the further the b value is driven towards a value of 1.0, the more the unrecovered reserves remain in the tight low-permeability layer and the greater the potential to increase production and recoverable reserves through stimulation of the low-permeability layer. This suggests that decline curve analysis can be used to rec- ognize and identify layered no-crossflow performance using only readily available historical production data. Recognition of the layers that are not being adequately drained com- pared to other layers, i.e., differential depletion, is where the opportunity lies. Stimulation of the less productive layers can allow both increased production and reserves. Figure 3.57 presents the standard Arps depletion decline curves, as pre- sented by Fetkovich et al. (1996). Eleven curves are shown So eee | ETS aS SN ERAS | SEA! ete Nene Sas Se The following i is a tabulation of the values of 6 that should be expected for single-layer homogeneous or layered crossflow systems. However, the harmonic decline exponent, b = 1, cannot be obtained from the back-pressure exponent. The b value of 0.4 should be considered as a good limiting value for gas wells when not clearly defined by actual production data.

The exponent ” from a gas well back-pressure perfor- mance curve can therefore be used to calculate or estimate b and D;. Equation 3.5.70 provides the physical limits of b, which is between 0 and 0.5, over the accepted theoretical range of n which is between 0.5 and 1.0 for a single-layer homogeneous system, as follows:

Figure 3.57 Depletion decline curves (After Fetkovich, 1997, copyright SPE 1997). with each being described by ab value which ranges between 0 and 1 in increments of 0.1. All of the values have meaning and should be understood in order to apply decline curve analysis properly. When decline curve analysis yields a b value greater than 0.5 (layered no-crossflow production) it is inac- curate to simply to make a prediction from the match point values. This is because the match point represents a best fit of the surface production data that includes production from all layers. Multiple combinations of layer production values can give the same composite curve and, therefore, unrealistic forecasts in late time may be generated.

Calculate the initial gas-in-place and the drainage area.

Figure 3.58 Radial-linear gas reservoir type curves (After Carter, SPEJ 1985, copyright SPE 1985). Solution

Step 4. Estimate the initial gas-in-place by applying Equation 3.5.83. problems are well established from the traditional well test analysis approach. The new solution for the gas problem is based on a material-balance-like time function and an algorithm that allows:

Step 1. Calculate the initial gas-in-place G as outlin previously. Step 2. Construct the following table:

Figure 3.60 Palacio-Blasingame type curve. Figure 3.61 Palacio-Blasingame West Virginia gas well example.

Figure 3.62 Typical distribution of the viscosity-compressibility function (After Anash et al., 2000).

$Figure 3.63 “First-order” polynomial solution for real-gas flow under boundary-dominated flow conditions. Solution assumes a \t C; profile that is linear with pp (Permission to copy by the SPE, 2000).$

Figure 3.63 “First-order” polynomial solution for real-gas flow under boundary-dominated flow conditions. Solution assumes a \t C; profile that is linear with pp (Permission to copy by the SPE, 2000).

Figure 3.64 “Exponential” solutions for real-gas flow under boundary-dominated flow conditions (Permission to copy by the SPE, 2000).

Figure 3.65 “General polynomial” solution for real-gas flow under boundary-dominated flow conditions (Permission to copy by the SPE, 2000).

Figure 3.66 Type curve analysis of West Virginia gas well “A” (SPE 14238). “General polynomial” type curve analysis approach (Permission to copy by the SPE, 2000)

Figure 3.67 Fetkovich-McCray decline type curve—rate versus material balance time format for a well with a finite conductivity vertical fracture (F.p = 0.1) (Permission to copy by the SPE, 2003).

Figure 3.68 Fetkovich-McCray decline type curve—rate versus material balance time format for a well with a finite conductivity vertical fracture (Fop = 1) (Permission to copy by the SPE, 2003).

Figure 3.69 Fetkovich-McCray decline type curve—rate versus material balance time format for a well with a finite conductivity vertical fracture (Fp = 10) (Permission to copy by the SPE, 2003).

Figure 3.71 Fetkovich-McCray decline type curve—rate versus material balance time format for a well with a finite conductivity vertical fracture (Fep = 1000) (Permission to copy by the SPE, 2003).

Figure 3.70 Fetkovich-McCray decline type curve—rate versus material balance time format for a well with a finite conductivity vertical fracture (Fep = 100) (Permission to copy by the SPE, 2003).

Figure 3.72 Match of production data for Example 1 on the Fetkovich-McCray decline type curve (pseudopressure drop normalized rate versus material balance time format) for a well with a finite conductivity vertical fracture (Fep = 5) (Permission to copy by the SPE, 2003).

Figure 3.74 Pressure-temperature curves for predicting hydrate (Courtesy Gas Processors Suppliers Association).

Figure 3.73 Phase diagram for a typical mixture of water and light hydrocarbon.

Equation 3.6.5 was developed using data on black oil, volatile oil, gas condensate, and natural gas sys- tems in the range of 32°F to 68°F, which covers the practical range of hydrate formation for reservoir fluids transportation.

Figure 3.77 Permissible expansion of a 0.8 gravity natural gas without hydrate formation (Courtesy Gas Processors Suppliers Association).

Calculate the hydrate dissociation pressure at 45°F, ie., 505°R. Example 3.29 A gas condensate system has the follow- ing composition:

Figure 3.78 Permissible expansion of a 0.9 gravity natural gas without hydrate formation (Courtesy Gas Processors Suppliers Association).

step 5. Calculate the constants b, and bz for CO2 by applying Equations 3.6.8 and 3.6.9 to give:

Figure 3.79 Permissible expansion of a 1.0 gravity natural gas without hydrate formation (Courtesy Gas Processors Suppliers Association). Solution

Solution Step 2. Determine values of the coefficients b and k from Figure 3.80, to give:

Step 2. Interpolating linearly at }* y;/Kj@_s) = 1 gives: at these pressures, to give:

Example 3.31 Using the equilibrium ratio approach, cal- culate the hydrate formation pressure py, at 50°F for the following gas mixture:

Figure 3.81 Vapor-solid equilibrium constant for methane (Carson & Katz, 1942, courtesy SPE-AIME).

The temperature at which hydrate will form is approxi- mately 54°F, Solution The iterative procedure for estimating the hydrate-forming temperature is given in the following tab- ulated form:

Figure 3.82 Vapor-solid equilibrium constant for ethane (Carson & Katz, 1942, courtesy SPE-AIME)

Figure 3.83 Vapor-solid equilibrium constant for propane (Carson & Katz, 1942, courtesy SPE-AIME).

Figure 3.84 Vapor-solid equilibrium constant for i-butane (Carson & Katz, 1942, courtesy SPE-AIME).

“igure 3.85 Vapor-solid equilibrium constant for n-butane (Carson & Katz, 1942, courtesy SPE-AIME).

Table 3.2 Values of coefficients Ap through A17 in Slolan’s equation

Figure 3.87 Vapor-solid equilibrium constant for H2S (Carson & Katz, 1942, courtesy SPE-AIME).

Figure 3.89 Hydrate zone thickness for temperature gradient at Prudhoe Bay (Permission to copy SPE, copyright SPE 1977).

Figure 3.88 Method for locating the thickness of hydrate layer (Permission to copy SPE, copyright SPE 1977).

Figure 3.90 Production history for a typical Medicine Hat property (Permission to copy SPE, copyright SPE 1995).

Use of the EMB method in the Medicine Hat shallow gas makes the fundamental assumptions (1) that the gas pool depletes volumetrically (.e., no water influx) and (2) that all wells behave like an average well with the same deliverability constant, turbulence constant, and BHFP, which is a reason- able assumption given the number of wells in the area, the homogeneity of the rocks, and the observed well production trends.

Estimate: 12. The following production data is available from a dry gas field:

a: (MMscf/day) G, (MMscf) g; (MMscf/day) G, (MMscf) 10. A 3000 foot horizontal gas well is draining an area of approximately 180 acres, given:

13. A gas well has the following production history:

Figure 4.1 Solution gas drive reservoir. (After Clark, N.J., Elements of Petroleum Reservoirs, SPE, 1969).

Figure 4.2 Production data for a solution gas drive reservoir. (After Clark, N.J., Elements of Petroleum Reservoirs, SPE, 1969).

Figure 4.3 Gas cap drive reservoir (After Clark, N.J., Elements of Petroleum Reservoirs, SPE, 1969).

Figure 4.5 Effect of gas cap size on ultimate oil recovery (After Cole, F., Reservoir Engineering Manual, Gulf Publishing Company, 1969). Figure 4.4 Gas cap drive reservoir (After Cole, F., Reservoir Engineering Manual, Gulf Publishing Company, 1969).

“igure 4.7 Reservoir having artesian water drive (After Clark, N.J., Elements of Petroleum Reservoirs, SPE, 1969). Figure 4.6 Production data for a gas cap drive reservoir (After Clark, N.J., Elements of Petroleum Reservoirs, SPE, 1969. Courtesy of AP)).

Cole (1969) presented the following discussion on the characteristics that can be used for identification of the water-driving mechanism. Figure 4.8 Aquifer geometries.

Ultimate oil recovery is also affected by the degree of activ- ity of the water drive. In a very active water drive where the degree of pressure maintenance is good, the role of solution gas in the recovery process is reduced to almost zero, with maximum advantage being taken of the water as a displac- ing force. This should result in maximum oil recovery from the reservoir. The ultimate oil recovery normally ranges from 35% to 75% of the original oil-in-place. The character- istic trends of a water drive reservoir are shown graphically in Figure 4.10 and summarized below:

Figure 4.11 Initial fluids distribution in an oil reservoir.

Figure 4.10 Production data for a water drive reservoir (After Clark, N.J., Elements of Petroleum Reservoirs, SPE, 1969. Courtesy of AP)).

Figure 4.12 Gravity drainage reservoir (After Cole, F., Reservoir Engineering Manual, Gulf Publishing Company, 1969).

Figure 4.13 Combination-drive reservoir (After Clark, N.J., Elements of Petroleum Reservoirs, SPE, 1969).

Figure 4.14 Tank-model concept. The above nine terms composing the MBE can be deter- mined separately from the hydrocarbon PVT and rock properties, as follows.

The following additional information is available: Example 4.1 The Anadarko Field is a combination drive reservoir. The current reservoir pressure is estimated at 2500 psi. The reservoir production data and PVT information are given below: Volume of bulk gas zone = 20000 acres-ft

fraction of the production above the bubble point. Ranges of compressibilities are given below: PV occupied by the injection gas and water Assuming that Ginj volumes of gas and Wing volumes of water have been injected for pressure maintenance, the total PV occupied by the two injected fluids is given by:

Example 4.2 A combination drive reservoir contains 10 MMSTB of oil initially in place. The ratio of the original gas cap volume to the original oil volume, i.e., m, is estimated as 0.25. The initial reservoir pressure is 3000 psia at 150°F The reservoir produced 1 MMSTB of oil, 1100 MMscf of gas of 0.8 specific gravity, and 50000 STB of water by the time the reservoir pressure dropped to 2800 psi. The following PVT data is available:

An important ‘factor i in determining the effectiveness ofa gas cap drive is the degree of conservation of the gas cap gas. As a practical mater, it will often be impossible, because of When the reservoir has a very weak water drive, but has a fairly large gas cap, the most efficient reservoir produc- ing mechanism may be the gas cap, in which case a large gas cap drive index is desirable. Theoretically, recovery by gas cap drive is independent of producing rate, as the gas is readily expansible. Low vertical permeability could limit the rate of expansion of the gas cap, in which case the gas cap drive index would be rate sensitive. Also, gas coning into producing wells will reduce the effectiveness of the gas cap expansion due to the production of free gas. Gas con- ing is usually a rate-sensitive phenomenon: the higher the producing rates, the greater the amount of coning.

Calculate the initial oil-in-place by using the MBE and compare with the volumetric estimate of N. The field production and PVT data is summarized below:

Example 4.3 The Virginia Hills Beaverhill Lake Field is a volumetric undersaturated reservoir. Volumetric calcula- tions indicate the reservoir contains 270.6 MMSTB of oil

41P. Dake, The Practice of Reservoir Engineering, 1994, Elsevier.

Figure 4.18 F vs. E, + E¢w for Example 4.3.

Step 2. Calculate N using the undersaturated reservoir data: Solution Calculate the initial oil-in-place by using the MBE and compare with the volumetric estimate of N.

As the solution gas evolves from the oil with declining reservoir pressure, the gas saturation (assuming constant water saturation S,;) is simply given by: Another important function of the MBE is history matching the production—-pressure data of individual wells. Once the reservoir pressure declines below the bubble point pressure, it is essential to perform the following tasks:

Example 4.5 In addition to the data given in Example 4.4 for the chalk reservoir, the oil-gas viscosity ratios a func- tion of pressure are included with the PVT data as shown below:

Figure 4.20 Gas recovery factor versus oil recovery factor.

Estimate the gas-oil volume ratio m and compare with the calculated value The production history in terms of parameter F and the PVT data are given below:

This relationship shows that a plot of the term (F/N — E,) vs. EF, would produce a straight line with a slope of m. One advantage of this particular arrangement is that the straight line must pass through the origin

N and m are unknown If there is uncertainty in both the values of N and m, Equation 4.4.16 can be re-expressed as: straight line through the origin with a slope of N, as shown in Figure 4.21. In making the plot, the underground withdrawal F can be calculated at various times as a function of the production terms N, and Rp.

The initial gas solubility Rj is 975 scf/STB. Estimate the initial oil-and gas-in-place. Example 4.8 The production history and the PVT data of a gas cap drive reservoir are given below: Step 5. Calculate m:

The above tabulated results appear to confirm the volu- metric m value of 2.9; however, the results also show the sensitivity of the m value to the reported average reservoir pressure.

Step 4. Plot F/E, vs. Eg/E, as shown in Figure 4.24, to give: Step 3. Calculate F/E, and E,/E,:

Step 2. Calculate F, E,, and E, from: Step 1. Calculate the cumulative produced gas-oil ratio R, Solution

Figure 4.24 Calculation of m and N for Example 4.8.

Plotting F/(E, + mE) vs. tF (b; — p)dt/(E, + mE,) results in a straight line with an intercept that represents the ini- tial oil-in-place N and a slope that describes the water influx constant C.

Assume that the water influx could be properly described by using the simple pot aquifer model as described by Equation 2.3.3, 88: Since the ability to use Equation 4.4.22 relies on knowledge of the aquifer properties, i.e., Cw, cr, W, 7, and 6, these prop- erties could be combined and treated as one unknown K in Equation 4.4.22, or:

Example 4.9 The material balance parameters, the underground withdrawal F, and the oil expansion E, of a saturated oil reservoir (i.e., m = 0) are given below:

Figure 4.30 Solution of the material balance for the pressure. Where ©, ®g, and ®, are considered PVT-related proper- ties that are functions of pressure and defined by:

A plot of the left-hand side of Equation 4.4.29 versus the sec- ond term on the right for a selected aquifer model should, if the choice is correct, provide a straight line with unit slope whose intercept on the ordinate gives the initial oil- in-place N. After having correctly determined N and W., Equation 4.4.9 can be solved directly for m, to give:

The calculated values of the PVT functions are given below:

Example 4.10 The production history of a saturated oil reservoir is as follows:

Solution The calculations can be conveniently performed in following tabulated form using: The above results show that the original oil in place in this reservoir is approximately 50 MMSTB of oil. The calculation at 1600 psia is a good example of the sensitivity of such a calculation near the bubble point pressure. Since the last two values of the original oil-in-place agree so well, the first calculation is probably wrong.

1. You have the following data on an oil reservoir: Problems

The production history of the field is given below:

Figure 5.1 Characteristics of solution gas drive reservoirs.

The initial reservoir pressure is 2925 psia with a bub- ble point pressure of 2100 psia. Calculate cumulative gas produced G, and cumulative GOR at each pressure. Example 5.1 The following production data is available on a depletion drive reservoir:

Figure 5.3 Relationship between GOR and Gp. Equation 5.1.5 simply indicates that the cumulative gas production at any time is essentially the area under the curve of the GOR vs. N, relationship, as shown in Figure 5.3.

Step 1. Construct the following table by applying Equations 5.1.4 and 5.1.7: It should be pointed out that the crude oil PVT prop- erties used in the MBE are appropriate for moderate-low volatility “black oil” systems which, when produced at the surface, is separated into oil and solution gas. These properties; as defined mathematically below are designed to relate surface volumes to reservoir volumes and vice versa. Solution

Oli saturation aaqjustment for combination arive For a combination drive reservoir, i.e., water influx and gas cap, the oil saturation equation as given by Equation 5.1.14 can be adjusted to account for both driving mechanisms, as: a a oe The control of the gas cap size is very often a reliable guide to the efficiency of reservoir operations. A shrinking gas cap will cause the loss of a substantial amount of oil, which might otherwise be recovered. Normally, there is little or no oil saturation in the gas cap, and if the oil migrates into the original gas zone there will necessarily be some residual oil saturation remaining in this portion of the gas cap at aban- donment. As pointed out by Cole (1961), the magnitude of this loss may be quite large and depends on:

Figure 5.6 Pressure voidage relationship.

The following additional data is available:

Example 5.4 The following PVT data characterizes a solution gas drive reservoir. The relative permeability data is shown is Figure 5.7.

Muskat method Step 2. Estimate (assume) a value for the GOR at 4150 psi: Muskat (1945) expressed the MBE for a depletion drive reservoir in the following differential form: Complete results of the method are shown below: Step 3. Calculate the average GOR:

Fluid property p* =2500 psi p = 2300 psi The following additional information is available: property data is listed by Craft and his co-authors and given here for only two pressures:

It is apparent from the three predictive oil recovery meth- ods; i.e., Tracy’s, Muskat’s, and Tarner’s, that the relative permeability ratio k,g/k,o is the most important single factor governing the oil recovery. In cases where no detailed data are available concerning the physical characteristics of the reservoir rock in terms of k,g/kro relationship, Wahl et al. (1958) presented an empirical expression for predicting the relative permeability ratio in sandstones: The final results as summarized below show the cumulative gas and oil production as the pressure declines from the bubble point pressure:

Figure 5.9 Productivity index during flow regimes. SAGES ls Ie RS UNE RE = AN a ile a ee It is is s important t to note that the productivity index is a valid measure of the well productivity potential only if the well is flowing at pseudosteady-state conditions. Therefore, in order to accurately measure the productivity index of a well, it is essential that the well is allowed to flow at a constant flow rate for a sufficient amount of time to reach the pseudosteady state as illustrated in Figure 5.9. The figure indicates that during the transient flow period, the calculated values of the productivity index will vary depending upon the time at which the measurements of py; are made.

Figure 5.10 Q, vs. Ap relationship. Equation 5.2.6 indicates that the relationship between Q, and Ap is a straight line passing through the origin with a slope of J as shown in Figure 5.10.

Alternatively, Equation 5.2.1 can be written as:

Figure 5.15 Effect of reservoir pressure on IPR. Figure 5.14 k,o/\1)Bo as a function of pressure.

Figure 5.13 Effect of pressure on Bo, > and Kyo.

Example 5.8 A well is producing from a saturated reser- voir with an average reservoir pressure of 2500 psig. Stabi- lized production test data indicates that the stabilized rate and wellbore pressure are 350 STB/day and 2000 psig, respectively. Calculate:

Figure 5.16 Stabilized flow test data. The vertical well IPR in undersaturated oil reservoirs Beggs (1991) pointed out that in applying the Vogel method for undersaturated reservoirs, there are two possible out- comes of the recorded stabilized flow test data that must be considered, as shown schematically in Figure 5.16:

Case 2 Pwr <Pp When the recorded py¢ from the stabilized flow test is below the bubble point pressure, as shown in Figure 5.16, the following procedure for generating the IPR data is proposed: Step 2. Calculate the oil flow rate at the bubble point pres- sure by applying Equation 5.2.10:

Step 3. Generate the IPR data by applying Equation 5.2.6 when pyr > pp and Equation 5.2.11 when pyt < pp: Step 1. Solve for J by applying Equation 5.2.12: olution Notice that the stabilized py is less than py.

Generate the future IPR for the well at 3000 psig by using the Standing method. Example 5.13 A well is producing from a saturated oil reservoir that exists at its saturation pressure of 4000 psig. The well is flowing at a stabilized rate of 600 STB/day and a pw of 3200 psig. Material balance calculations provide the following current and future predictions for oil saturation and PVT properties. Solution Step 1. Calculate the current (Q,) max from Equation 5.2.18:

Standing method = 989[1 — 0.52(Py¢/2200) — 0.48 (Pyt/2200)7] Standing (1970) essentially extended the application of the Vogel method to predict the future IPR of a well as a func- tion of reservoir pressure. He noted that Vogel’s equation (Equation 5.2.9) can be rearranged as:

Fetkovich (1973) suggested that the pressure function f(p) can basically fall into one of the following two regions:

Step 5. Generate the IPR by using Equation 5.2.22: It should be noted that one of the main disadvantages of Standing’s methodology is that it requires reliable perme- ability information; in addition, it also requires material bal- ance calculations to predict oil saturations at future average reservoir pressures.

Results show a reasonable match between the two approaches. However, it should be noted that several uncer- tainties exist in the values of the parameters used in Equation 5.2.31 to determine the productivity index. For example, changes in the skin factor k or drainage area would change the calculated value of J. Calculate the productivity index by using both the reser- voir properties (i.e., Equation 5.2.31) and flow test data (i.e., Equation 5.2.1):

Step 1. Construct the following table: Step 2. Plot G. — p?,) vs. Qo on log-log paper as shown in Figure 5.19 and determine the exponent 1, or:

Figure 5.19 Flow-after-flow data for Example 5.15 (After Beggs, D., Production Optimization Using Nodal Analys permission to publish by the OGCI, copyright OGCI, 1997).

The authors correlated C/C, and n/m, to the dimensionless pressure by the following two expressions:

Figure 5.23 (kio/\t Bo) vs. pressure for case 3. Substituting Equations 5.2.27 and 5.2.28 into the above expression gives:

The computational steps of the Klins and Clark method are summarized below: Klins and Clark (1993) proposed an inflow expression sim- ilar in form to that of Vogel’s and can be used to estimate future IPR data. To improve the predictive capability of Vogel’s equation, the authors introduced a new exponent d to Vogel’s expression. The authors proposed the following relationships:

All reservoir performance techniques show the relationship of cumulative oil production and the instantaneous GOR as a function of average reservoir pressure. However, these techniques do not relate the cumulative oil production N, and cumulative gas production G, with time. Figure 5.26 shows a schematic illustration of the predicted cumulative oil production with declining average reservoir pressure.

Example 5.21 A horizontal well 1000 foot long is drilled in a solution gas drive reservoir. The well is producing at a stabilized flow rate of 760 STB/day and wellbore pressure of 1242 psi. The current average reservoir pressure is 2145 psi. Generate the IPR data of this horizontal well by using the Cheng method. Petnanto and Economides (1998) developed a generalized IPR equation for a horizontal and multilateral well in a solu- tion gas drive reservoir. The proposed expression has the following form:

The following methodology can be employed to correlate the predicted cumulative field production with time f:

The initial reservoir temperature was 133°F, the initial pressure was 220 psia, and the bubble-point pressure was 1850 psia. There was no active water drive. From 1850 psia to 1300 psia a total of 720 MMSTB of oil were produced and 590.6 MMMscf of gas.

Figure 6.7 Cash flow diagram depicting the present worth of a uniform series of equal payments. This formula is called the uniform series present worth for- mula. It is an equivalent value formula used to determine the present value (P) of a uniform series of equal payments (A) made at the end of each period of a series of interest com- pounding periods (m) at a given periodic interest rate (i) as shown in Figure 6.7. This formula could be used to deter- mine the present value of an annuity, the right to receive

In Example 6.2, the present value of receiving $500 at the end of the fifth year assuming 5% interest compounded annually was equal to $391.76. F, = P(1+ ic)!

Table 6.1 time value of money, the present values of the two options are within 1% of each other, so other issues may need to be considered before choosing. If 15% is the rate that defines the time value of money, then option A may be your preference since its present value is almost 15% higher than option B.

Table 6A.1_ Periodic Interest Rate (i) = 0.5%

Table 6A.2 Periodic Interest Rate (i) = 1%

Table 6A.1 Periodic Interest Rate (i) = 0.5% (continued)

Table 6A.3 Periodic Interest Rate (i) = 2%

Table 6A.4 Periodic Interest Rate (i) = 3%

Table 6A.4 Periodic Interest Rate (i) = 3% (continued)

Table 6A.5 Periodic Interest Rate (i) = 4%

Table 6A.6 Periodic Interest Rate (i) = 5%

Table 6A.5 Periodic Interest Rate (i) = 4% (continued)

Table 6A.7 Periodic Interest Rate (i) = 6%

Table 6A.6 Periodic Interest Rate (i) = 5% (continued)

Table 6A.8 Periodic Interest Rate (i) = 7% Table 6A.9 Periodic Interest Rate (i) = 8%

Table 6A.9 Periodic Interest Rate (i) = 8% (continued)

Table 6A.10 Periodic Interest Rate (i) = 9%

Table 6A.11 Periodic Interest Rate (i) = 10%

Table 6A.10 Periodic Interest Rate (i) = 9% (continued)

Table 6A.12 Periodic Interest Rate (i) = 11%

Table 6A.11 Periodic Interest Rate (ij) = 10% (continued)

Table 6A.13_ Periodic Interest Rate (i) = 12% Table 6A.14 Periodic Interest Rate (i) = 13%

Table 6A.15 Periodic Interest Rate (i) = 14%

Table 6A.14 Periodic Interest Rate (i) = 13% (continued)

Table 6A.16 Periodic Interest Rate (i) = 15% Table 6A.17 Periodic Interest Rate (i) = 20%

Table 6A.15 Periodic Interest Rate (i) = 14% (continued)

Table 6A.17 Periodic Interest Rate (i) = 20% (continued) Table 6A.18 Periodic Interest Rate (i) = 25%

Table 6A.20 Periodic Interest Rate (i) = 40%

Table 6A.18 Periodic Interest Rate (i) = 25% (continued) Table 6A.19 Periodic Interest Rate (i) = 30%

Table 6A.21 Periodic Interest Rate (i) = 50% Table 6A.22 Periodic Interest Rate (i) = 70%

Table 6A.20 Periodic Interest Rate (ij) = 40% (continued)

Table 6A.22 Periodic Interest Rate (ij) = 70% (continued)

Table 6A.23 Periodic Interest Rate (i) = 90%

Table 6A.24 Periodic Interest Rate (i) = 110%

Table 6A.25 Periodic Interest Rate (i) = 130%

Table 6A.27 Periodic Interest Rate (i) = 200%

Table 6A.26 Periodic Interest Rate (i) = 150%

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4350 1323 49.182 1576 43.6 7.25 4060 1143 58.383 1788 56.8 16.6 3840 1037 64.812 1992 57.3 21.0 3600 958 69.562 2158 56.7 23.8 3480 882 74.572 2383 54.4 25.6 3260 791 78.400 2596 53.5 28.3 3100 734 81.275 2785 51.6 29.2 2940 682 83.879 2953 50.0 29.9 2800 637 86.401 3103 48.3 30.3 Step 2. Calculate the PV at the bubble point pressure from: (PV) b = (N i -N pb )B ob + N i B oi 1 -S References TLFeBOOK REFERENCES
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