UNSTEADY-STATE ANALYSIS IMPROVES STIMULATION SIZING

Aug. 13, 1990
David A. Rowland University of Texas of the Permian Basin Odessa, Tex. For maximizing well stimulation profitability, unsteady-state flow pressure-transient analysis methods yield more accurate predictions of productivity increases than do steady-state methods. This technique appears to be superior in representing actual field conditions.
David A. Rowland
University of Texas of the Permian Basin
Odessa, Tex.

For maximizing well stimulation profitability, unsteady-state flow pressure-transient analysis methods yield more accurate predictions of productivity increases than do steady-state methods.

This technique appears to be superior in representing actual field conditions.

In designing stimulation treatments to improve production capacity of a well, the oil and gas industry traditionally uses empirically derived graphs based on steady-state flow. These graphs usually relate increased productivity with all well damage removed. Thus, the final skin-effect value equals zero.

A more realistic approach is to relate increased productivity with any amount of well damage removed. That is, the optimum condition may be a skin effect other than zero, and may be either positive or negative.

PREDICTING PERFORMANCE

To predict reservoir performance and make economic analyses, the industry has long observed pressure performance of oil and gas wells. The theoretical basis of testing procedures and interpretations has steadily improved.

Pressure build-up and drawdown testing practices have frequently been used for evaluating both reservoir and individual well performances. Pressure testing can reveal the extent of well damage caused during the drilling process.

In most instances, some form of treatment is applied to remove this damage, either while the well is being completed or when it is producing. These treatments traditionally involve acidizing or formation fracturing, or a combination of both. Such costly treatments can greatly reduce the profitability of the well. The cost must be offset by increased producing rates and even increased recovery from the drainage area being affected by the well.

The main problem facing the completion engineer is to determine how much treatment is required for a given well or situation. Excessive treatment adds to the cost whereas too little treatment may be ineffective. More often than not the engineer lacks a method for determining the degree of the treatment and must rely on experience.

Well stimulation may be divided into two main classes: matrix acidization and hydraulic fracturing. The former is used primarily to remove well damage caused by drilling, whereas the latter is generally required to improve the productivity of low-permeability formations. Transient-pressure analysis is ignored by all of the methods currently used for predicting production increases with either treatment method.

Many authors have used the steady-state approach for evaluating productivity increases resulting from matrix-acid treatments. In general, these methods relate the increase in productivity to the removal of all well damage, where skin(s) becomes zero. Of greater interest, however, is to measure the increase in productivity with differing values of well damage reduction--not necessarily just where the skin effect is reduced to zero. Clearly, optimum profitability could very well occur at a treatment size designed for some value of s other than zero.

A general expression relating the increase in production with a decrease in skin effect may be derived by comparing the steady-state equations at varying values of s. The general form of the equation for steady-state flow with skin effect included may be used (see Equation 1 in Equation box). Vogt and Anderson 1 employed a special case of this equation by setting S2=0. Their equation (Equation 2) implies that productivity increases linearly with well damage removal for given drainage and well bore radiuses. Note that Equation 2 is analogous to the inverse of the Hawkins 2 productivity ratio equation for steady-state flow, Equation 3.

The above equations contain a serious shortcoming in that neither the drainage radius nor the well bore radius is clearly delineated. Even though the effects of using uncertain values of re and rw are masked by the logarithm, the calculation of productivities can vary considerably. Another objection to using the steady-state method is the fact that early time pressure transients are excluded.

Traditionally, well damage is evaluated using pressure-transient analysis. Including these transients allows for a more accurate estimate of productivity increases.

For the case of fracture-treatment analysis, industry commonly uses the method introduced by McQuire and Sikora 3 that involves empirically derived graphs. The ordinate of their graphs is the ratio of productivity indexes for fractured-to-unfractured cases multiplied by a scaling factor. The abscissa, designated as the relative conductivity, is defined as the ability of the fracture to transmit fluid relative to that of the formation.

These curves are used by some service companies offering fracturing services.4 They do not take into account pressure-transient analyses.

THEORETICAL DEVELOPMENT

The early work of Van Everdirgen and Hurst 5 provided an approximate solution to the diffusivity equation representing the flow of fluids through porous media. In separate papers, 6 7 they also introduced the concept of skin effect and presented a general equation of the form listed as Equation 4.

For given reservoir fluid and rock characteristics, Equation 4 represents that the bottom hole flowing pressure varies linearly with the logarithm of time. Thus, a plot of pwf vs. In t will result in a straight line having a slope equal to m.

For pressure drawdown tests, the slope m is negative. Fig. 1 is a schematic diagram representing a typical pressure drawdown plot. Note that the presence of well damage tends to lower the level of the curve by a constant amount equal to _ps. The term _ps represents the pressure loss due to well damage and is given by Equation 6.

For convenience, we will rearrange Equation 4 to obtain Equation 7. The symbol Cb represents a constant for a particular reservoir system and is defined as shown in Equation 8. It is customary to select the time t as unity. However, caution is noted that the value of pwf at this time must be read from the extrapolated portion of the straight line on the drawdown plot.

Matthews 8 introduced a useful representation of the effect well damage has on well performance. He defined "flowing efficiency" as the ratio of actual productivity index (Ja) to ideal productivity index (Ji), the latter being the case with no well damage.

Equation 10 is the expression of flowing efficiency evolving from this definition. However, this relationship suffers from the shortcoming of being time dependent, as do most other representations of well damage extent, such as damage factor, damage ratio, completion factor, completion ratio, etc. This dependence can be easily illustrated by combining Equations 7 and 10, leading to an alternate expression for flowing efficiency given by Equation 11.

It is clear that as time increases, and _pt increases, the second term in Equation 11 decreases. Thus, the value of flowing efficiency increases proportionately because the pressure loss due to well damage remains constant at uniform flowing rate.

It should be pointed out that well damage, hence _ps, may be either positive or negative, For positive well damage, the flowing efficiency increases with time, whereas for negative well damage it decreases with time. In both instances the value of flowing efficiency approaches unity asymptotically. Furthermore, in the case of zero well damage (s = 0) flowing efficiency is unity for all times.

As mentioned above, the main purpose of stimulating a well is to increase its productivity by reducing or removing the damage to the formation caused by drilling. To determine the optimum treatment, we need to know how much the production will increase with successive reductions in well damage. This increase can be arrived at by comparing the results of a pressure drawdown test performed prior to treatment with what might be expected after treatment.

Fig. 2 illustrates typical results of pressure drawdown tests performed on the same well before (Curve 1) and after (Curve 2) treatment. Equation 7 can be used to compare the two curves if we rearrange it as given in Equations 12 and 13. At any given time, tx, we can divide the two expressions as shown in Equation 14. Furthermore, by using Equation 6 we may write equations for pressure loss due to well damage for the before and after treatment cases given by Equations 15 and 16, respectively. Combining Equations 12-16 leads to Equation 17.

There are three unknowns in this equation: q2, Pwf2, and S2. Equation 17 represents q2 in terms of Pwf2 and S2. The other two unknowns can be expressed as given in Equations 18 and 19.

Because Equation 17 is cumbersome to use in its present form, it can be modified to Equation 22. It is significant to note that the constant Cm depends only on the oil viscosity and the formation capacity or conductivity, defined as the permeability-thickness product.

The above derived equations are expressed in the darcy system of units. For practical oil field units, where pressure and rates are express in psi and b/d, respectively, Equation 22 is modified to Equation 24.

It is significant to note that the _p/Q terms represent the inverse of the well known productivity index, PI. For convenience, we will designate these reciprocal productivity index terms to be "Resistivity index" and use the label RI. Equation 24 is then written as Equation 26.

Equation 26 expresses in a compact and useful way the relationship between productivity increases and changes in well damage, expressed as skin effect. There are a variety of ways for representing this equation graphically. One of the simplest and possibly most useful forms is to plot RI2 VS. the Cm_s product on Cartesian coordinate graph paper.

The result is a family of straight lines having a slope equal to minus unity as illustrated in Fig. 3. Thus, for any incremental value of well damage removed, the resulting flow rate can be predicted.

This increase in production can then be used to evaluate the expected increase in income and compared with the cost of the workover operation. An optimum workover size can then be predicted using appropriate economic factors for a given operation.

As indicated above, the values of _P1 and _P2 Must be selected at the same time, tx. The question arises as to when is the ideal time to make this calculation. In a reservoir being treated as infinite, some value of maximum time should be stipulated. In a finite reservoir the drawdown will eventually be affected by the boundaries of the flow regime, at which time the flow will transit to pseudo-steady state.

After this time the respective drawdowns, _P1 and _P2, will remain constant until the bottom hole flowing pressure reaches some minimum value. Thereafter the rate will decline as a result of continued depletion of the drainage area.

Note that the time to reach this declining rate condition will vary with both the amount of drawdown and the producing rate. Consequently, cash-flow calculations will necessarily have to include this factor.

In selecting the best time to use for tx, we propose using the end of the unsteady-state period for finite drainage areas. If the drainage pattern is unknown, we recommend assuming some limiting value, say 1,000 acres. The time it takes to reach pseudosteadystate condition for various drainage configurations is well documented in the literature.9 10

Matthews, Brons, and Hazebroek 9 have shown that for many well-centered regular drainage patterns, such as a circle or a square, the pseudosteady-state condition begins at a dimensionless time approximately equal to 0.1. Moreover, their charts indicate that the end of unsteady-state flow occurs at a dimensionless time approximately equal to 0.02.

Dimensionless time is defined in Equation 27, where A is the drainage area. The time tx, may be obtained by solving this equation for t, which in practical oil field units is approximated by Equation 28 where A is in acres.

Using the value tx computed from Equation 28, the bottom hole flowing pressure is computed using Equation 29. In the unlikely event that both m and tx are large, the calculated value of px could be negative. In such instances, a value of px equal to 100 psia could be assumed.

It is not necessary that the production increase calculations be made using the same drawdown pressure used in evaluating the pressure test data. That is, the preferred drawdown after completing the well may be more or less than that used during the testing procedure.

To correctly establish the relationship between production increases with well damage removal, we must first compute the new rate expected if no treatment were applied, but with the new drawdown. To do this we first calculate a base rate Qo by setting _s equal to zero and then solve Equation 24 for Q2 as indicated by Equation 30. The incremental production is then the difference between Q2, from Equation 24, and this base rate (Equation 31). In this manner the production increases resulting from varying stages of well-damage removed are determined.

APPLICATION

To demonstrate the utility of Equation 24, we have selected the results of both drawdown and build-up tests conducted on a West Texas well producing from the San Andres formation. The well produced at a constant rate for 72 hr, and then it was shut in for a like period. Pertinent reservoir and pressure data are listed in Tables 1, 2, and 4.

Pressure drawdown The pressure drawdown data listed in Table 2 are plotted on semilog paper as shown in Fig. 4. The slope of the straight line is calculated to be equal to - 101 psi/log cycle and the pressure at t = 1 hr (obtained from the straight line) is 2,903 psia. Calculations of pressure loss due to well damage, skin effect, and flowing efficiency are made using a computer model. The sequence of the calculations made by the model are illustrated in the box of sample calculations using actual field data listed in Table 1.

Results of calculations of flowing efficiency at various times are listed in the example calculations box. These values of E* are included in Fig. 4 for illustration.

Calculations of RI, producing rate, and incremental increase in production are made for various treatment sizes. These values are measured as a function of incremental well-damage removed.

For purposes of illustration, calculations made assuming the value of As to be equal to 7 are included in Table 3. The results show that if the skin effect is reduced from 9.02 to 2.02 as a result of well treatment, then the producing rate increases from 117.7 to 205.1 b/d.

Thus, the producing rate for the well will increase approximately 75% for this size of treatment. If all the well damage is removed, the resulting increase in production will be around 122%.

After making these initial calculations, the model was used to compare results using the steady-state equation (Equation 1) with the unsteady-state equation (Equation 24) for various drainage areas. For this comparison, the value Of S2 was assumed to be zero. The size of drainage areas ranged from 10 to 5,160 acres.

The results are listed in Table 5 and presented graphically in Fig. 5. They indicate that a higher producing rate is predicted using the unsteady-state flow equation. The percent difference between the two methods for each drainage area are computed and included in Table 5 and Fig. 5. The increase ranges from 5% for the wide spacing to 9% for the low drainage area.

It should be pointed out that as the drainage area increases, the time to reach the end of unsteady-state flow also increases. Hence, the drawdown pressure at time t, increases proportionately. Because of this higher drawdown pressure, the calculations of PI decrease as drainage area increases.

The next series of calculations was made using the model to determine the effects different amounts of well damage removed had on producing rates. For these calculations, the drainage area was assumed to be 160 acres, and the assigned drawdown at time tx was assumed to be 2,000 psi. The results are summarized in Table 6. Again, the calculations included the case for steady-state flow using Equation 1 for comparison.

Fig. 6 is a plot of _Q vs. _s and illustrates the rapid increase in productivity as the amount of well damage removal is increased. This reflects the great importance of not only removing all well damage caused by drilling, but also creating a negative skin if economically feasible. The results illustrate also that a much greater improvement is reflected when using the unsteady-state equation (Equation 24) as opposed to the steady-state equation (Equation 1).

A plot of RI is included in Fig. 6. As expected, this parameter plots as a straight line having a slope equal to cm.

PRESSURE BUILDUP

In many instances, the engineer has only pressure build-up test data to work with. The same equations and procedures described for drawdown tests will apply for pressure build-up data as well. In this case, the Horner" plot is used to obtain the slope m. Also used is the extrapolated pressure to infinite build-up time (Horner time ratio equal to unity), p*, in place of pi.

In applying the Horner technique it is customary to use a pseudo producing time defined as the ratio of cumulative production during drawdown to the last stabilized rate. In mathematical terms, the pseudo producing time is defined by Equation 32.

Occasionally, the pressure test is conducted with a long producing time to the extent that the flowing pressure may be influenced by boundary effects such as faults or permeability pinchouts. In the absence of pressure drawdown test data, the influence of these boundaries may be indeterminable.

For this reason, we propose adopting an alternate technique for build-up calculations. First, we calculate the time tx using Equation 28. If tx is greater than tff, then the drawdown is still in unsteady-state flow and we may calculate px using Equation 33.

Conversely, if tx is less than tff, then the flowing system is beyond unsteady-state conditions and the final flowing pressure will be affected by the drainage boundaries. Fortunately, the magnitude of this effect will be reflected on the Horner plot as well. Consequently, the value of px in establishing the drawdown at the end of unsteady-state flow is the same as pwff.

it should be pointed out that some authors recommend using the average reservoir pressure PR in place of either pi or p* in the above equations. It is believed that the results will not vary significantly using either the average pressure or the extrapolated pressure.

The build-up data used in the example are listed in table 4. They appear graphically on the Horner plot, Fig. 7, indicating a slope m equal to 103 psi/log cycle. The extrapolated pressure to infinite build-up time is shown to be 4,090 psia, and the pressure at a time ratio representing a build-up time of 1 hr is determined to be 3,894 psia.

Calculations were performed using a computer model similar to the one used for the drawdown case. The sequence of calculations for this case are recorded in the example calculation box. The equation required for calculating the pressure loss due to well damage for buildup is slightly different than the one used for drawdown. It is easily derived and takes the form given in Equation 34.

The results for the build-up test data assuming a drainage area of 160 acres are presented in Table 7. It may be observed that these results are very similar to those obtained for the drawdown case, confirming that the method applies for build-up test data as well as for drawdown data.

GAS RESERVOIRS

The technique described for oil wells will apply for gas wells as well. There are three methods for analyzing pressure transient data that apply for the particular conditions for which they were designed. These are p-method, p2_method, and the m(p)method. The appropriate equations may be developed using the same procedure as outlined above for oil production.

ECONOMIC EVALUATION

It was pointed out earlier that Equation 24 may be presented in several different forms. For economic evaluations, two forms of this equation appear to be useful. In the first instance we may rearrange Equation 24 to read as listed in Equation 35, Taking the derivative with respect to _s yields Equation 36. This equation plots as a straight line through the origin as illustrated in Fig. 8.

The slope of the line is a dimensionless constant given by the right-hand side of Equation 36. Using the drawdown data for the example problem, the value of the slope is calculated to be 0.061.

It is important to note that in applying Equation 36 there appears to be no limiting value for the well treatment size. Obviously, this would not be a valid conclusion when treatment costs are included in the analysis.

A more useful expression in an economic analysis may be developed by first rearranging Equation 24 leading to Equation 37. Again, taking the derivative with respect to _s and performing the necessary algebraic steps leads to Equation 38. Integrating Equation 38 yields Equation 40 which plots as a straight line on log-log scales having a slope of -1. If the y-intercept is assumed to be where a -_s = 1, then the constant b is represented by the value of Q2 at this point.

Fig. 9 illustrates this plot for the drawdown case for the example problem given above. Values of a and b for the example problem are computed to be 16.4 (dimensionless) and 1,934 b/d, respectively. These calculations are illustrated in the example calculations box.

Still another useful variation of Equation 40 is given by Equation 42. In this expression it may be observed that as _s increases, the producing rate Q2 approaches infinity. Again, it is clear that there must be some limiting value of _s which controls the size of the treatment. That is (O _s

The value of a for the example problem is 16.4. Thus, the design of the treatment size for this well should be for _s to be something less than 16.4. As mentioned above, the optimum well treatment design is governed by economics.

ACKNOWLEDGMENT

Special thanks are due Paul E. Hodges, associate professor of economics, University of Texas of the Permian Basin.

REFERENCES

  1. Vogt, T. C., and Anderson, M.L., "Optimizing the Profitability of Matrix Acidizing Treatments," JPT. September 1984.

  2. Hawkins, Murray F., Jr. "A Note on the Skin Effect," Trans. AIME 1956. p. 207.

  3. McGuire, W. J., and Sikora, V. J., "The Effect of Vertical Fractures on Well Productivity," Trans. AIME, 1960, p, 219.

  4. "Production Stimulation for the Permian Basin," Halliburton Services Manual.

  5. Van Everdingen, A. F., and Hurst, W., "The Application of the Laplace Transformation to Flow Problems in Reservoirs," Trans. AIME, 1949, p. 186.

  6. Van Everdingen, A. F., "The Skin Effect and Its Influence on the Productive Capacity of a Well," Trans. AIME, 1953, p. 198.

  7. Hurst, William, "Establishment of the Skin Effect and Its Impediment to Fluid Flow Into a Well Bore," Petroleum Engineering, October 1953.

  8. Matthews, C. S., "Analysis of Pressure Buildup and Flow Test Data," JPT September 1961.

  9. Matthews, C. S., Brons, F., and Hazebroek, P., "A Method for Determination of Average Pressure in a Bounded Reservoir," Trans. AIME, 1954, p. 201.

  10. Matthews, C. S., and Russell, D. G., "Pressure Buildup and Flow Tests in Wells," Monograph Series Vol. 1, SPE, 1967.

  11. Horner, D. R., "Pressure Buildup in Wells," Proc. Third World Pet. Cong., E. J. Brill, Leiden, 1951.

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