Mohamed SolimanHalliburton Services

Duncan, Okla.

Kevin PetakHalliburton Reservoir Services

Dallas

Pressure build-up after short producing times and pressure surges can be analyzed.

The method employs the derivative plot and does not require prior knowledge of initial reservoir pressure that may be calculated using a Cartesian plot.

Field use of the new technique has proven its superiority to previously used methods, and has shown the necessity of using highly accurate data to achieve desired results.

### BACKGROUND

In a build-up test, a production (drawdown) period is followed by a shut-in (build-up) period. By using the principle of superposition in time and long-time logarithmic approximation, the well known Horner1 equation is derived and listed as Equation 1 (see Equation box).

Equation 1 indicates that a plot of shut-in pressure vs. log (tp + _t)/_t should yield a straight line whose slope is a function of reservoir parameters. The intercept of this straight line is initial reservoir pressure.

Because Equation 1 is a long-time solution, it is expected that early time data will deviate from the straight line. Only when the logarithmic approximation is valid will data follow the expected straight line.

Consequently, the determination of the time at which data follow a straight line becomes important. Drawing a straight line through the wrong data would lead to misestimation of formation permeability and reservoir pressure.

Matching with type curves is used to determine accurately the location of the correct semilog straight line. Although these type curves have been developed for a drawdown test, they may be used to analyze a buildup test provided that producing time is much longer than shut-in time. Thus, type curves in their existing form may not be used to analyze a build-up test when producing time is short.

Four different techniques have been presented in literature to overcome this problem and analyze tests with short producing time.2-5 Three of these four techniques depend on type curve matching techniques.2-4

The fourth technique uses a specially developed method for analyzing build-up tests with short producing time.5 This technique does not require the knowledge of producing time. It only requires the accurate determination of cumulative production during the test.

Consequently, this method was generalized to surge tests where a flow period cannot be accurately identified. Reference 6 gives theoretical development and field cases for use in surge tests.

Because the type of tests discussed here involves producing a fairly small volume of fluid, it is expected that pressure does not significantly drop below initial reservoir pressure (sometimes less than 100 psi at well bore). Because only the long-time data are analyzed, and because in some cases these data may differ from the initial reservoir pressure by only a few psi, the accurate knowledge of initial reservoir pressure may be critical.

A small error in initial reservoir pressure may translate into larger error in calculated reservoir parameters. In some cases, such as field cases in Reference 6, an accurate measurement of initial reservoir pressure is available. However, in many drill stem tests (DST), an accurate estimate of initial reservoir pressure is not available.

Although the technique presented in References 5 and 6 has been successful, the requirement of an accurate initial reservoir pressure may prevent its application to some tests.

In this article a modification to that technique is offered. The new technique does not require knowledge of initial reservoir pressure to determine flow regime and formation permeability. Coupling the derivative and Cartesian plots may also yield initial reservoir pressure.

Mathematical development and field examples with interpretation using the modified technique are covered in this article. This derivative technique is applied in a somewhat different fashion from its previous applications.7-9

### MODEL DEVELOPMENT

Several assumptions are necessary for the development of flow models.

- The well is producing from an isotropic, homogeneous reservoir of permeability k, porosity f, and thickness h, and is situated in an infinite reservoir.
- The reservoir contains a slightly compressible fluid of compressibility c and viscosity m, independent of pressure.
- Gravity effects are negligible, and pressure gradients are very small throughout the reservoir.
- The reservoir has a uniform pressure before the start of the test.
- The fluid flow in the formation is laminar.
- Skin and well bore storage effects are considered.
- Well is produced at a constant rate for a period of tp, and then shut-in.

Although this last assumption is necessary in the mathematical development, it has been shown6 that it may be replaced by the following assumption: "The well produced a small amount of fluid during a short period, then was shut in." Determination of amount of flow is necessary while accurate determination of flow period is not. Using the above assumption, the equation governing fluid flow in a porous medium may be developed. References 5 and 6 give equations for flow regimes including detailed derivation of the radial and spherical flow regimes.

The long-time solution for these flow regimes is given as Equations 1-8.

### DERIVATIVE TECHNIQUE

By taking the derivative of Equation 6 with respect to In tD, Equation 9 is obtained. By multiplying both sides of this equation by -1, then taking its logarithm, Equation 10 is the result.

Equation 10 indicates that a type curve of derivative of dimensionless pressure vs. time will have a slope n, directly giving the type of flow regime.

It should be noticed that both (PD/ln tD) and n are negative quantities. This forced the multiplication of Equation 9 by -1 before taking the logarithm.

The derivative technique, as developed earlier, is graphically presented in Fig. 1 for radial, spherical, and bilinear flow regimes. Figs. 1a and 1b show the effect well bore storage and skin have on the shape of the type curve.

Both figures indicate that regardless of the value of well bore storage coefficient or skin factor, curves will eventually join a negative unit slope straight line. Higher well bore storage and skin factor only delays the formation of negative unit slope straight line.

The increased well bore storage coefficient had a peculiar effect on the shape of the derivative type curve. The reason for this behavior is that for CD = 0, drawdown data were already in semilog approximation state. For CD = 10, drawdown data were approaching this semilog approximation state when the well was shut in.

As well bore storage increases, the drawdown starts deviating from the semilog approximation and starts approaching the unit slope. This will in turn affect the shape of the build-up curve.

Similar discussion may be applied to Figs. 1c and 1d for spherical flow and Figs. 1e and 1f for bilinear flow. However, the slope of the straight line is -1.5 and -0.75 for spherical and bilinear flow, respectively.

The derivative technique discussed above may be applied to field tests. It will be shown that the technique will enhance the application of the short producing test by not requiring prior knowledge of initial reservoir pressure. Equations 11-19 give the developed technique in terms of field units.

Equation 13 indicates that plotting log t (Pw/t) vs. log t should result in a straight line whose slope is n and intercept of log (-n C2).

Consequently, the slope of the straight line should yield the type of flow regime while intercept yields permeability.

For example, radial flow should yield Equation 14. The slope of the straight line is -1, indicating the presence of a radial flow regime. The intercept is inversely proportional to formation permeability. Equation 14 is independent of initial reservoir pressure.

Spherical and bilinear flow on the other hand are represented by Equations 15 and 16, respectively. In Equation 16, the definition for dimensionless fracture conductivity was substituted in the longtime approximation (the derivative Equation 3).

It may be noticed that fracture length cancelled out, indicating that fracture length cannot be calculated if all data show bilinear flow, which indicates that effect of fracture tip is not felt during the bilinear flow period.

The intercept of Equation 16 is a function of both permeability and fracture conductivity. The shape of the curve (through type-curve matching) may be used to estimate fracture conductivity. High-conductivity fractures will yield curves with more curvature at early time (Fig. 1f).

Another method of plotting the type curves was introduced to the petroleum industry by Raghavan.13 In this method a plot of the derivative of dimensionless pressure divided by dimensionless pressure is plotted vs. dimensionless time.

To apply this method to short producing time test, Equation 9 is divided by Equation 6, yielding Equation 17.

Equation 17 indicates that the ratio of derivative of pressure divided by pressure difference should eventually reach a constant value n, which indicates the type of flow regime. Although this method seems interesting, it requires the prior knowledge of initial reservoir pressure.

### INITIAL RESERVOIR PRESSURE

After the determination of flow regime and formation permeability using the derivative technique, initial reservoir pressure may be determined using Equation 11.

By plotting shut-in pressure Pw vs. tn, the long-time data should follow a straight line whose slope is C2 and intercept is Pi. Because C2 has already been determined using the derivative technique, this plot may serve as a confirmation of permeability and initial reservoir pressure calculations.

### FIELD EXAMPLES

Three example analyses of actual surge tests show that the proposed technique may be easily applied to diagnose and analyze data for various flow regimes.

### EXAMPLE 1

Data for this example were measured during a 10-min surge period conducted on an oil-producing sandstone. The formation was surged into a 0.5-bbl chamber following an underbalanced perforation treatment on the entire formation interval.

A pressure vs. _t plot of the data is shown in Fig. 2a. This plot is shown to verify the mechanical success of the test.

The actual analysis plots are shown in Figs. 2b, c, and d.

The derivative plot of the data, Fig. 2b, indicates a late time negative unit slope behavior representative of radial flow.

Note that this regime begins at approximately 3 min into the surge period.

This rapid occurrence is expected because the duration of storage and skin effects is small for the given chamber volume.

Also, radial flow behavior is expected because the entire formation interval has been perforated.

If only a small portion of the formation was perforated or only few perforations were open, spherical flow should result. The calculated permeability is 40 md using Equation 14.

Fig. 2c is a Cartesian plot of pressure vs. 1/t. This plot is used primarily for extrapolation to initial reservoir pressure for the radial flow regime.

The linear portion of the response shown by Fig. 2c is extrapolated to Pi = 2,662.94 psig.

This Pi value is used to construct Fig. 2d. This type of plot was discussed in References 5 and 6.

A negative unit slope is placed through the data and an effective permeability of 39.97 md is calculated from Equation 18.

This value is in good agreement with the derivative value for permeability. All input variables and calculated results are shown in Table 1, Example 1.

Note that any of the plots may be used to calculate permeability as long as the same portion of data is analyzed on each plot.

Also, note the ease of doing this analysis.

### EXAMPLE 2

Data for this example were obtained from a 5-bbl surge test conducted on a 150-ft thick unconsolidated sandstone.

A 10-ft section of the formation was perforated underbalanced at the beginning of the test. In this case, the operator wanted to take a quick look at a portion of the formation to determine whether or not it would be a viable producer.

Fig. 3a shows the pressure vs. _t plot.

Figs. 3b, c, and d show the analysis plots for the data. Fig. 3b indicates that the initial period of 36 min is dominated by plugging and well bore storage effects.

Significant amounts of sand were recovered from this test.

The final 18-min period indicates the occurrence of spherical flow (-1.5 slope). Equation 15 is used to calculate a permeability value of 15 md.

Fig. 3c is a Cartesian plot of pressure vs. 1/dt1,5. This plot is used to extrapolate to a Pi of 5,631.86 psig through data representing spherical flow.

This Pi value has been used to construct Fig. 3d.

A negative 1.5 slope drawn through the portion of data representing spherical flow yields k = 14.77 md. The permeability equation for late time spherical flow is given in Equation 19.

Note that spherical flow permeability is independent of formation thickness. Analysis of surge data from other flow regimes assumes that the entire formation thickness contributes to the pressure response.

This assumption may be invalid for various situations, and the analysis of those situations would yield an erroneous permeability.

The input variables and calculated results for this example are shown in Table 1, Example 2.

### EXAMPLE 3

Data for this example were obtained from a 2-bbl surge test conducted on a 30-ft laminated, unconsolidated sandstone.

A 2-ft section of the zone was perforated underbalanced.

As in Example 2, the operator perforated only a portion of the formation to determine its producibility. Unlike Example 2, spherical flow behavior is not indicated on the plots Possible reasons will be examined later.

A Cartesian plot of pressure vs. delta time is shown in Fig. 4a.

Analysis plots are shown in Figs. 4b, c, and d.

The derivative plot in Fig. 4b indicates an early time portion dominated by plugging and well bore storage effects and a late-time portion showing bilinear flow (-0.75 slope).

Spherical flow was expected since the well is unfractured and only a relatively thin portion of the formation was perforated.

Fig. 4c is a Cartesian plot of pressure vs. 1/dto.75. This plot is primarily used for extrapolation of Pi through the bilinear flow region.

The Pi value is 5,646.04 psig. This pi was used to construct Fig. 4d. In Fig. 4d, a -0.75 slope is drawn through the bilinear flow region.

Calculated permeability values are shown in Table 1, Example 3.

Note that the equation for permeability (Equation 20) during the bilinear flow response contains two unknowns, k and kfwf.

It is interesting that a bilinear flow regime is observed when spherical flow was expected. We submit the following hypotheses as possible reasons for this phenomenon.

- Flow at the well bore is limited to higher permeability laminations. In this case, the higher permeability laminations are simulating a fracture through which linear flow toward the well bore results. Lower permeability laminations in the reservoir feed higher permeability laminations.
- Washed out perforation channels cause linear flow. For perforation channels to cause linear flow, they must be long enough to simulate a fracture. However, this is unlikely.

### ACKNOWLEDGMENT

The authors wish to thank the management of Halliburton Services and Halliburton Reservoir Services for permission to prepare and publish this article.

### REFERENCES

- Horner, D.R., "Pressure Build-Up in Wells," Third World Petroleum Congress, Leiden.
- Raghavan, R., "Effect of Producing Time on Type Curve Analysis," J. Petroleum Technology, June 1980.
- Raghavan, R., Meng, H., and Reynolds, A.C. Jr., "Analysis of Pressure Build-Up Data Following a Short Flow Period," paper SPE 9290, SPE Annual Technical Conference and Exhibition,, Dallas, Sept. 21-24, 1980.
- Agarwal, R.G., "A New Method to Account for Producing Time Effects When Drawdown Type Curves are Used to Analyze Pressure Build-Up and Other Test Data," paper SPE 9289, SPE Annual Technical Conference and Exhibition, Dallas, Sept. 21-24, 1980.
- Soliman, M.Y., "Analysis of Build-Up Tests with Short Producing Time," SPE Formation Evaluation, August 1986.
- Soliman, M.Y., Petak, Kevin, Christenson, Jon, and Centanni, Roy, "Analysis of Sequential Formation Testing and Surge Tests Using New Technique, CIM paper 88-39-53, 39th Annual Technical Meeting of the Petroleum Society of CIM, Calgary, June 12-16, 1988.
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- Agarwal, R.G., Al-Hussainy, R., and Ramey, H.J. Jr., "An Investigation of Wellbore Storage and Skin Effect in Unsteady Liquid Flow: I, Analytical Treatment," Soc. Pet. Eng. J., September 1970, pp. 279-90.
- Carslaw, H.S., and Jaegar, J.C., Conduction of Heat in Solids, Second Edition, Oxford at the Clarendon Press, 1959.
- Ozkan, E., Raghaven, R., and Joshi, S.D., "Horizontal Well Pressure Analysis," SPE 16378, SPE California Regional Meeting, Ventura, Calif., April 8-10, 1987.

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