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Two observation wells are located 1.4 km apart in the direction of flow. Head in well No. 1 is 50m and in well No. Hydraulic conductivity is 0.7 m/day. Effective porosity is 0.2. (i) What is the total flow of water through the aquifer? 1) 0, 2), 3) 1, 4), 5) Q Q z g P z k g. 10 News has the latest local breaking news and headlines from Roanoke, Virginia. Get Roanoke, Va., local TV news, Lynchburg, Va., headlines, national news, videos and more from NBC's local.
Early estimates of gas well performance were conducted by opening the well to the atmosphere and then measuring the flow rate. Such “open flow” practices were wasteful of gas, sometimes dangerous to personnel and equipment, and possibly damaging to the reservoir. They also provided limited information to estimate productive capacity under varying flow conditions. The idea, however, did leave the industry with the concept of absolute open flow (AOF). AOF is a common indicator of well productivity and refers to the maximum rate at which a well could flow against a theoretical atmospheric backpressure at the reservoir.
The productivity of a gas well is determined with deliverability testing. Deliverability tests provide information that is used to develop reservoir rate-pressure behavior for the well and generate an inflow performance curve or gas-backpressure curve.
There are two basic relations currently in use to analyze deliverability test data. An empirical relationship was proposed by Rawlins and Schellhardt[1] in 1935 and is still frequently used today. Houpeurt[2] presented a theoretical deliverability relationship derived from the generalized radial diffusivity equation accounting for non-Darcy flow effects.
Rawlins and Schellhardt[1] developed the empirical backpressure method of testing gas wells based on the analysis of tests on more than 500 wells. They noted that when the difference between the squares of the average reservoir pressure and flowing bottomhole pressures were plotted against the corresponding flow rates on logarithmic coordinates, they obtained a straight line. This led them to propose the backpressure equation:
..........(1)
where C is the flow coefficient and n is the deliverability exponent. The deliverability exponent is the inverse of the slope of the curve. Once n is determined, C can be obtained by substituting pressure and rate data read directly from the straight-line plot into Eq. 1 and solving the resulting relation.
As discussed previously, solutions for gas well performance in terms of pressure-squared are appropriate only at low reservoir pressures. As a result, Rawlins and Schellhardt’s deliverability equation can be rewritten in terms of pseudopressure as
..........(2)
where C and n are determined in the same manner as for Eq. 1. The values of n range from 0.5 to 1.0, depending on flow characteristics. Flow characterized by Darcy’s equation will have a flow exponent of 1.0, while flow that exhibits non-Darcy flow behavior will have a flow exponent ranging from 0.5 to 1.0. While the Rawlins and Schellhardt deliverability equation is not rigorous, it is still widely used in deliverability analysis and has provided reasonable results for high-permeability gas wells over the years.
Eqs. 1 and 2 can be rewritten to facilitate the development of the inflow performance curve. In terms of pressure-squared, the relationship is
..........(3)
and ..........(4)
in terms of pseudopressure. Once the deliverability exponent is determined from a multirate test and the AOF estimated, Eqs. 3 and 4 can be applied readily to estimate the rate for a given flowing bottomhole pressure.
Houpeurt developed a theoretical deliverability relationship for stabilized flow with a Forchheimer[3] velocity term to account for non-Darcy flow effects in high-velocity gas production. The resulting relationship can be written in terms of pressure-squared or pseudopressure as
..........(5)
or ..........(6)
Eqs. 5 and 6 are quadratic in terms of the flow rate, and the solutions can be written for convenience as shown in Eqs. 7 and 8.
..........(7)
..........(8)
Jones, Blount, and Glaze[4] suggested Houpeurt’s relationship be rewritten as shown in Eqs. 9 and 10 to allow the analysis of well-test data to predict deliverability.
..........(9)
..........(10)
A plot of the difference in pressures squared divided by the flow rate or the difference in pseudopressure divided by the flow rate vs. the flow rate yields a straight line on a coordinate graph. The intercept of the plot is the laminar flow coefficient a, while turbulence coefficient b is obtained from the slope of the curve. Once these two coefficients have been determined, deliverability can be estimated from the following relationships in terms of pressure-squared or pseudopressure.
..........(11)
and ..........(12)
After the coefficients of the deliverability equations have been determined, the relationships can be used to estimate production rates for various bottomhole flowing pressures. This determination of rate vs. pressure is often referred to as the reservoir inflow performance, which is a measure of the ability of the reservoir to produce gas to the wellbore. The inflow performance curve is a plot of bottomhole pressure vs. production rate for a particular well determined from the gas well deliverability equations. Fig. 1 depicts a typical gas well inflow performance curve. This curve allows one to estimate the production rate for different flowing bottomhole pressures readily.
Fig. 1—Typical gas well inflow performance curve.
Several different deliverability test methods have been developed to collect the data for use with the basic deliverability models. These tests can be grouped into three basic categories:
The basic deliverability test method that uses all stabilized data is the flow-after-flow test. Deliverability test methods that use both transient and stabilized test data include the isochronal and modified isochronal tests. The multiple modified isochronal test consists of all transient test data and eliminates the need for stabilized flow or pressure data.
Rawlins and Schellhardt1 presented the basic deliverability test method that uses all stabilized data. The test consists of a series of flow rates. The test is often referred to as a four-point test, because many tests are composed of four rates, as required by various regulatory bodies. This test is performed by producing the well at a series of stabilized flow rates and obtaining the corresponding stabilized flowing bottomhole pressures. In addition, a stabilized shut-in bottomhole pressure is required for the analysis. A major limitation of this test method is the length of time required to obtain stabilized data for low-permeability gas reservoirs.
Table 1 provides example flow-after-flow test data, which are analyzed with the Rawlins and Schellhardt and Houpeurt deliverability equations. The traditional Rawlins and Schellhardt analysis requires that the difference in the pressures squared be plotted vs. the flow rate on logarithmic graph paper and a best-fit straight line constructed through the data points. The data should provide a straight-line plot, which serves as the deliverability curve. From this plot, the deliverability exponent, n, is the inverse of the slope of the constructed straight line. Once the deliverability exponent is determined, the flow coefficient, C, can be determined from Eq. 13 with a point taken from the straight-line plot. The same approach is used when pseudopressures are used to analyze the data, except that the differences in the pseudopressures are plotted vs. the flow rate and Eq. 14 is used to determine C.
..........(13)
..........(14)
Table 1
Table 2 shows the data to be plotted for the Rawlins and Schellhardt analysis, while Figs. 2 and 3 show the logarithmic plots for the pressure-squared and the pseudopressure analyses, respectively.
Table 2
Fig. 2—Rawlins and Schellhardt analysis of flow-after-flow test data with the pressure-squared approach.
Fig. 3—Rawlins and Schellhardt analysis of flow-after-flow test data with pseudopressure approach.
Working with the traditional pressure-squared data, draw a straight line through the four data points to yield a slope of 1.54. The deliverability exponent, n, is the inverse of the slope, or 0.651. The flow coefficient, C, can be determined from a point on the straight line. Since the third test point lies on the line, it can be used to determine C using Eq. 15 to yield 0.2874 Mscf/D/psia2n.
..........(15)
Once n and C are determined, the deliverability equation can be written and used to determine the AOF and the production rate for any given flowing bottomhole pressure. Eq. 16 is the deliverability equation for this particular example well.
..........(16)
The AOF is determined by allowing the flowing bottomhole pressure to be equal to the atmospheric pressure for the current average reservoir pressure of 3,360 psia. In this example, when the atmospheric pressure is assumed to be 14.65 psia, the AOF is 11,200 Mscf/D.
The same approach is used to analyze the data when pseudopressures are used in the analysis. Using Fig. 3, the slope of the straight line through the data points is 1.57, yielding an n of 0.637. The flow coefficient, C, is determined to be 0.0269 Mscf/D/(psia2/cp)n from Eq. 17 using the third test point.
..........(17)
The resulting deliverability equation is
..........(18)
and the AOF is calculated to be 12,200 Mscf/D using the appropriate pseudopressure values at the current reservoir pressure of 3,360 psia and atmospheric pressure of 14.65 psia.
The difference in the calculated AOF using the pressure-squared approach and the pseudopressure method is noticeable. This variation results from the inclusion of the pressure dependence of the gas viscosity and gas deviation factor in the pseudopressure term. As noted earlier, the pressure-squared approach is only suitable at low pressures, while the pseudopressure method is good for all pressure ranges. Also, the Rawlins and Schellhardt method is not theoretically rigorous, although it is widely used.
The test data can also be analyzed with the Houpeurt approach using both the pressure-squared and pseudopressure approaches. Table 3 provides the data to be plotted in the Houpeurt analysis. Fig. 4 presents the Houpeurt plot of the pressure squared data, while Fig. 5 shows the pseuodpressure data. From Fig. 1.5, one can construct a best-fit line through the data points and determine the slope and the intercept of the line. The slope, b, is 0.0936 psia2/(Mscf/D)2, while the intercept, a, is determined to be 200 psia2/Mscf/D. These deliverability coefficients can be use to develop a deliverability equation after the form of Eq. 11 as shown in Eq. 19:
..........(19)
Table 3
Fig. 4—Houpeurt analysis of flow-after-flow test data with pressure-squared approach.
Fig. 5—Houpeurt analysis of flow-after-flow test data with the pseudopressure approach.
The AOF can be estimated for the reservoir pressure of 3,360 psia to be 9,970 Mscf/D.
A similar analysis can be undertaken for the pseudopressure data shown in Fig. 5. From this plot, the intercept of the constructed best-fit line is determined to be 10,252 psia2/cp/Mscf/D, while the slope is 5.69 psia2/cp/(Mscf/D)2. These coefficients are used to write the deliverability equation as
..........(20)
From this equation for the current reservoir pressure, the AOF is estimated to be 10,700 Mscf/D. As with the Rawlins and Schellhardt analysis, the AOFs determined by the pressure-squared method and the pseudopressure approach are different because of the pressure dependence of the μz product.
Cullendar[5] proposed the isochronal test to overcome the need to obtain a series of stabilized flow rates required for the flow-after-flow test for the slow-to-stabilize well. This test consists of producing the well at several different flow rates with flowing periods of equal duration. Each flow period is separated by a shut-in period in which the shut-in bottomhole pressure is allowed to stabilize at essentially the average reservoir pressure. The test also requires that an extended stabilized flow point be obtained. The test method is based on the principle that the radius of investigation is a function of the flow period and not the flow rate. Thus, for equal flow periods, the same drainage radius is investigated in spite of the actual flow rates.
To analyze the data from an isochronal test, the flow data from the equal flow periods is plotted according to the Rawlins and Schellhardt1 or Houpeurt[2] methods. These data points are used to determine the slope of the deliverability curve. The stabilized flow point is then used to estimate the flow coefficient, C, for the Rawlins and Schellhardt method or the intercept, a, for the Houpert method by extending the slope of the multirate data to the stabilized flow point.
Table 4 details isochronal test data for a particular well in which the flow periods are one hour in duration. The Rawlins and Schellhardt approach with pressures and the Houpeurt approach with pseudopressures are used to demonstrate the analysis of isochronal test data. Table 5 presents the plotting data for both methods. Fig. 6 shows the logarithmic plot of the pressure data for the Rawlins and Schellhardt analysis.
Table 4
Table 5
Fig. 6—Rawlins and Schellhardt analysis of isochronal test data with the pressure-squared approach.
A straight line can be constructed through the three transient points to yield a slope of 1.076. The inverse of the slope defines the deliverability exponent, n, which is 0.9294 for this example. The slope through the transient points is extended to the stabilized flow point to depict the deliverability curve. The flow coefficient, C, is calculated from the stabilized flow point,
..........(21)
to be 0.0242 Mscf/D/psia2n. The flow exponent and flow coefficient are used to define the Rawlins and Schellhardt deliverability equation for this well,
..........(22)
which is used to determine the AOF. For an atmospheric pressure of 14.65 psia, the AOF is estimated to be 27,100 Mscf/D. A similar analysis can be undertaken with pseudopressures following the same method described for the pressures squared.
Applying the Houpeurt approach, the transient flow points are used to determine the slope of the best-fit straight line constructed through the data points. This slope is used to determine the intercept from the stabilized flow point. Fig. 7 shows the plot of the pseudopressure data for the Houpeurt analysis. From the plot, the slope is determined to be 0.1184 psia2/cp/(Mscf/D)2, which is used to calculate an intercept from the stabilized flow point of 8,814 psia2/cp/Mscf/D as shown in Eq. 23.
Fig. 7—Houpeurt analysis of isochronal test data with the pseudopressure approach.
..........(23)
The deliverability equation can be written in a form similar to Eq. 1.22 to yield Eq. 24.
..........(24)
This equation can be used to estimate the AOF of 25,600 Mscf/D for the well or estimate the production rate at any other flowing bottomhole pressure. As the analysis of the flow-after-flow test data showed, the Rawlins and Schellhardt and Houpeurt methods yield different estimates of deliverability.
For some low-permeability wells, the time required to obtain stabilized shut-in pressures may be impractical. To overcome this limitation, Katz et al.[6] proposed a modification to the isochronal test by requiring equal shut-in periods. The modified isochronal test is essentially the same as the isochronal test, except the shut-in periods separating the flow periods are equal to or longer than the flow periods. The method also requires the extended stabilized flow point and a stabilized shut-in bottomhole pressure. The modified isochronal test method is less accurate than the isochronal method because the shut-in pressure is not allowed to return to the average reservoir pressure. In the analysis of the collected data, the measured bottomhole pressure obtained just before the beginning of the flow period is used in Eqs. 1 and 2 or Eqs. 9 and 10 instead of the average reservoir pressure.
The analysis of the data is exactly the same as that used to analyze the isochronal test data. With the Rawlins and Schellhardt data, the transient flow points are used to construct a best-fit straight line through the data points. The inverse of the slope of this line yields the deliverability exponent, n. The deliverability exponent is then used with the data of the stabilized flow point to estimate the flow coefficient , C, with Eqs. 1 or 2 depending on whether pressure or pseudopressure data is used. In the Houpeurt analysis, a best-fit straight line is constructed through the transient flow points to yield the slope, b. Once the slope is determined, it is used with the stabilized flow point in the appropriate equation for pressure or pseudopressure (Eqs. 9 and 10) to determine the intercept, a. Once the flow coefficients are determined by either analysis method, the deliverability equation can be written and used to estimate the AOF and production rates for the well.
The multiple modified isochronal test consists of all transient test data and eliminates the need for stabilized flow or pressure data. The analysis method requires estimates of drainage area and shape along with additional reservoir and fluid property data that are not required with the previous deliverability test methods. As a result, the analysis techniques are more complex than for flow-after-flow, isochronal, or modified isochronal test data. However, the method provides a means to estimate deliverability of slow-in-stabilizing wells and consists of running a minimum of three modified isochronal tests with each test composed of a minimum of three flow rates. To analyze the test data, modifications to the Rawlins and Schellhardt analysis have been proposed by Hinchman, Kazemi, and Poettmann[7] while modifications to the Houpeurt pressure-squared technique have proposed by Brar and Aziz,[8] Poettmann,[9] and Brar and Mattar.[10] These modifications have been extended to the pseudopressure analysis technique by Poe.[11] Several authors[7][8][9][10][11] give details on estimating deliverability from transient test data.
The petroleum engineer is required to forecast or predict gas well performance as the reservoir pressure depletes. There are several methods to assist in making these future performance estimates, including the direct application of the appropriate analytical solution to provide estimates of rate vs. pressure for different average reservoir pressures. However, the use of Eqs. 25 through 28 requires that one estimate rock and fluid properties for the well of interest.
..........(25)
and ..........(26)
..........(27)
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and ..........(28)
Another technique also requires knowledge of rock and fluid properties by estimating the flow coefficients, a and b, in Houpeurt’s relationships (Eqs. 7 and 8). When Houpeurt’s method is used in terms of pressure-squared, a and b are
..........(29)
and ..........(30)
where the non-Darcy flow coefficient
..........(31)
The value of β, the turbulence factor,[12] can be estimated from
..........(32)
When Houpeurt’s relationship is used in terms of pseudopressure, a and b are estimated from
..........(33)
and ..........(34)
The variables D and β are estimated with Eqs. 31 and 32. Once the flow coefficients, a and b, are determined at new average reservoir pressures, Eqs. 11 and 12 can be used to estimate rates for different pressures to generate the inflow performance curve.
Russell et al.[13] studied the depletion performance of gas wells and proposed a technique to estimate gas well performance that was dependent on gas compressibility and viscosity. From this study, Greene[14] presented a relationship to describe the well performance.
..........(35)
In this equation, C1 is a constant that is a function of permeability, reservoir thickness, and drainage area, which can be estimated from a single-point flow test with knowledge of gas compressibility and viscosity. This value is not the same as the flow coefficient C in Eqs. 1 and 2. C1 will remain constant during the life of the well, assuming no changes in permeability. Once C1 is determined, one can estimate future performance from Eq. 35 with the gas compressibility and viscosity estimated at the average bottomhole pressure defined as
..........(36)
A technique that does not require the use of rock and fluid properties assumes that the deliverability exponent, n, remains essentially constant during the life of the well.[15] While this assumption may not be accurate, many gas wells have exhibited behavior such that the deliverability exponent has varied slowly over the life of the well. Under this assumption, future performance can be predicted with the following relationships in terms of pressure-squared and pseudopressure, respectively.
..........(37)
..........(38)
Once the new AOF at the future reservoir pressure has been determined, the inflow performance curve can be constructed with a modified version of the deliverability equation as shown in Eqs. 3 and 4.
| a | = | laminar flow coefficient, m2/L5t3, psia2/Mscf/D or m/L4t2, psia2/cp/Mscf/D or mL4/t, psia/STB/D |
| b | = | turbulence coefficient, m2/L8t2, psia2/(Mscf/D)2 or m/L7t, psia2/cp/(Mscf/D)2 or mL7, psia/(STB/D)2 |
| B | = | formation volume factor, dimensionless, RB/STB |
| ct | = | total compressibility, Lt2/m, psia–1 |
| C | = | flow coefficient, L3 + 2nt4n–1/m2n, Mscf/D/psia2n or L3 + nt3n–1/mn, Mscf/D/(psia2/cp)n or L3 + 2nt4n–1/m2n, STB/D/psia2n |
| C1 | = | flow coefficient in Eq. 35, L3 + 2nt4n–2/m2n–1, cp-Mscf/D/psia2n |
| Cd | = | discharge coefficient, dimensionless |
| Cp | = | specific heat capacity at constant pressure, L2/t2T |
| Cv | = | specific heat capacity at constant volume, L2/t2T |
| d | = | pipe diameter, L, in. |
| D | = | non-Darcy flow coefficient, t/L3, D/Mscf |
| El | = | energy loss per unit mass, L2/t2, ft-lbf/lbm |
| f | = | friction factor, dimensionless |
| g | = | gravitational acceleration, L/t2, ft/sec2 |
| gc | = | conversion factor, dimensionless, 32.2 ft-lbm/lbf-sec2 |
| h | = | formation thickness, L, ft |
| J | = | productivity index, L4t/m, STB/D/psia |
| k | = | permeability, L2, md |
| L | = | length, L, ft |
| M | = | molecular weight, m, lbm/lbm-mole |
| n | = | deliverability exponent, dimensionless |
| p | = | pressure, m/Lt2, psia |
| = | average bottomhole pressure, m/Lt2, psia | |
| pb | = | bubblepoint pressure, m/Lt2, psia |
| pe | = | external boundary pressure, m/Lt2, psia |
| pn | = | node pressure, m/Lt2, psia |
| pp | = | gas pseudopressure, m/Lt3, psia2/cp |
| pp | = | average reservoir pseudopressure, m/Lt3, psia2/cp |
| pp(pwf) | = | flowing bottomhole pseudopressure, m/Lt3, psia2/cp |
| = | average reservoir pressure, m/Lt2, psia | |
| ps | = | separator pressure, m/Lt2, psia |
| psc | = | standard pressure, m/Lt2, psia |
| pwf | = | bottomhole pressure, m/Lt2, psia |
| pwfs | = | sandface bottomhole pressure, m/Lt2, psia |
| pwh | = | wellhead pressure, m/Lt2, psia |
| q | = | flow rate, L3/t, STB/D or Mscf/D |
| qb | = | oil flow rate at the bubblepoint pressure, L3/t, STB/D |
| qg | = | gas flow rate, L3/t, Mscf/D |
| qg,max | = | AOF, maximum gas flow rate, L3/t, Mscf/D |
| qL | = | liquid flow rate, L3/t, STB/D |
| qo | = | oil flow rate, L3/t, STB/D |
| qo,max | = | maximum oil flow rate, L3/t, STB/D |
| qw | = | water flow rate, L3/t, STB/D |
| qw,max | = | maximum water flow rate, L3/t, STB/D |
| r | = | radius, L, ft |
| re | = | external drainage radius, L, ft |
| rw | = | wellbore radius, L, ft |
| R | = | producing gas/liquid ratio, dimensionless, scf/STB |
| s | = | skin factor, dimensionless |
| t | = | time, t |
| T | = | temperature, T, °R |
| Tsc | = | standard temperature, T, °R |
| Twh | = | wellhead temperature, T, °R |
| v | = | velocity, L/t, ft/sec |
| W | = | work per unit mass, L2/t2, ft-lbf/lbm |
| y | = | ratio of downstream pressure to upstream pressure, p1/p2, dimensionless |
| z | = | gas compressibility factor, dimensionless |
| Z | = | elevation, L, ft |
| α | = | kinetic energy correction factor, dimensionless |
| β | = | turbulence factor, L–1, ft–1 |
| γg | = | gas specific gravity, dimensionless |
| ε | = | absolute pipe roughness, L, in. |
| μ | = | viscosity, m/Lt, cp |
| ρ | = | fluid density, m/L3, lbm/ft3 |
| ϕ | = | porosity, fraction |
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