DRILLING / COMPLETION TECHNOLOGY

One-day gas well testing

Part 2 - Analytical procedure for the Forchheimer equation, plus preparation and procedures for one-day testing

Haldon J. Smith, Oil / Gas Consultant, Columbus, New Mexico, in association with Geo-Consultants, Ltd., Palma de Mallorca, Spain

Part 1 introduced the subject of a new, modified isochronal test procedure that allows testing of a new gas well in less than one day, regardless of how low the permeability or transmissibility is. The background of conventional testing limitations with formations that do not stabilize during a reasonable period were given. Basic features/limitations of the two types of existing tests-flow-after-flow and isochronal-were overviewed. The new, modified isochronal test method adaptable to transient flow was introduced. And the Forchheimer equation was presented, and its components were defined.

The concluding Part 2, presented here, starts with an analysis of the Forchheimer equation introduced in Part 1, including discussion of the relationships of the equation's coefficients, plus other useful relationships. One-day test procedures are then described, starting with "preparations." A step-by-step testing program is then outlined.

The conclusion and technical sidebar notes by the author add to further understanding of the coefficients, and discuss modifications to the landmark work of Rawlins-Schellhardt's Monograph 7 published by the U.S. Bureau of Mines in 1936.

ANALYTICAL PROCEDURE

To initiate this analysis, the concluding paragraphs of Part 1 which introduced the subject equation are repeated here.

In transient-flow analysis of a modified isochronal test using 30-min readings of flowrate and flowing BHP corresponding to each other, several readings are taken of each, for every flow period. Analysis of these data is best carried out by use of the Forchheimer equation:

Where:

p_s, p_f and q are the same as for Rawlins-Schellhardt, Equation 1.

a = A coefficient which is a linear function of the log of time during transient flow and becomes constant for pseudo-steady-state flow.

b = A coefficient which is constant with time for both transient flow and pseudo-steady-state flow.

Examining the form of the Forchheimer equation shown in Equation 2, it is immediately evident that it is in the form of a straight line of intercept a and slope b. So, plotting (p_s² - p_f²⁾/q against q for three choke sizes and various intermediate values of each, during given continuous transient flow periods, the following, here-generalized data curves are obtained. In Fig. 7, a 2-hr transient flow period is shown, with 30-min. readings for each choke. Note decreasing spreads of the lines of data with increasing times. Note also extrapolations to a₁, a₂, a₃ and ₄ at zero flowrate. Finally, note that slopes of the lines are all the same and, from Equation 2, that slope is equal to b (a coefficient which is constant with time for both transient flow and pseudo-steady-state, PSS, flow).

Fig. 7. Forchheimer Plot 1.

In practice, if there is some slight variation in slope, the most reliable will be that of the longest flowtime, i.e., the uppermost line on this type of plot. Thus, the value of b in the Forchheimer equation is established, and b is constant with time for both transient and PSS flow.

Next, the value of a (a coefficient which is a linear function of the log of time during transient flow and becomes constant for PSS flow) in the Forchheimer equation can be established. The fact that a is a linear function of the log of time for transient flow-and all flow now being considered here so far is transient-is used to create another data plot, Fig. 8.

Fig. 8. Forchheimer Plot 2 .

When this line is extrapolated back to a value of zero time on the log scale, a is established. This completes determination of both coefficients in the Forchheimer equation; so, for any given values of pressure and/or rate, the performance equation governing this well's behavior can be written. The above plot also fits the following equation:

Where a is the intercept, and m is the slope, derived graphically or analytically, in the transient expression for a or a_t.

The slope, m, from Fig. 8 can be used to arrive at a value for permeability, according to Poettmann:

According to Poettmann, it is not unusual for values of k to vary dramatically, depending on the method of calculation and assumptions made. The same also applies to skin factor, s, and the non-Darcy flow constant, D. It is thus advised that a final pressure buildup always be used to more closely define permeability and formation damage values. A check can then be made between the two methods for a given reservoir.

In this context, it is not generally realized how widely the parameter of permeability can vary. To illustrate, this author has actually measured permeabilities in the field ranging from a low of 0.0124 md (an intermediate sandstone, offshore West Australia) to a high of 88 Darcies, or 88,000 md (shallow honey-combed carbonate in the Java Sea). This gives a range of over 7,000,000 to one. Against this background, values of, e.g., 300 and 400 md, by different methods of calculation, are in good agreement.

One especially valuable relationship mentioned by Poettmann is valid for both transient and pseudo-steady-state flow:

Once b is determined in the Forchheimer equation, Fig. 7, and m is determined from Fig. 8, an estimate of the non-Darcy flow constant D can be made:

This estimate may be used in verifying formation damage values derived from pressure buildup, after multirate, PSS flows. It can then be used for transient flows as well, since b does not change with type of flow.

APPLICATION

The analytical procedure outlined above can be applied to both isochronal and modified isochronal test data. Results may be slightly more accurate with isochronal test data, because intermediate pressure buildups are complete. But that very completion itself can often require uneconomically-long time periods to achieve.

Again, the modified isochronal test will have inherent error, though small, by virtue of incomplete intermediate pressure buildups. The small physical error, however, is fully justified by the great economic advantage of limiting overall test time to economically acceptable periods of time. It is, in effect, a good trade-off, in accepting a small, qualitatively-known, physical error for great economic gain. It has already been shown that such data can vary vastly, so mathematical precision is a futile goal. In any case, the performance equation(s) arrived at will be entirely adequate for normal use.

RELATIONS OF COEFFICIENTS "CONSTANTS"

A number of attempts have been made to relate the coefficient C and the exponent n of the Rawlins-Schellhardt equation to coefficients a and b of the Forchheimer equation, with varying degrees of success. This can be done, but the relationships among these coefficients are neither simple nor straightforward. It must be remembered that two different types of flow can be involved-transient and pseudo-steady-state (PSS). For these reasons, one must be careful in using the results.

All four of these coefficients are constant with time for PSS flow, and also for a given isochronal time. However, for transient flow, C is a function of time only, and a is linear with log of time. The coefficient b is constant with time for both transient and PSS flow.

Poettmann equates the derivatives on either Cartesian coordinates or log-log coordinates to derive the following:

A more direct approach is to equate expressions from both Rawlins-Schellhardt and Forchheimer, which are each equal to the difference of the squares of the pressures. The result is:

Once a and b are established from Forchheimer analysis, it is a simple matter to solve Equation 8 by iteration, knowing that n is always between 0.5 and 1.0, for a given flowrate. Similarly, Equation 9 can be solved by iteration for the performance coefficient C. The results of these calculations, however, must be used with extreme care, always keeping in mind whether flow is transient or pseudo-steady-state.

Technical note on gas well test analysis

This discussion refers to Rawlins-Schellhardt's Monograph 7, published by the U.S. Bureau of Mines, Washington, DC, in 1936. The method developed and delineated by the authors was indeed monumental, greatly valuable and an immense contribution to what, at that time, was a rather rudimentary technology of gas well testing. Since then, a great deal has been learned, and published, about this most interesting field of work; and it is now possible to suggest some improvements to techniques presented 64 years ago. Fig. TN 1. Conventional Rawlins-Schellhardt plot. This "note" then, is in no way intended to demean, or diminish, the great accomplishment and major contribution made by Monograph 7, but rather to suggest a modification designed to streamline and improve work flowing from use of the technique. It is believed that suggestions made here will simplify use of the backpressure testing method for gas wells, bring the procedure more into line with mathematical theory and improve ease of use and application for operating engineers in the field. It may also tend to reduce the possibility for error. So, for purposes of illustration, please consider a hypothetical gas well test, the results of which would yield the following numbers from three choke flows, assuming stabilized, i.e., pseudo-steady-state flow: Choke 1: q1 = 5; (ps2 - pf2)1 = 5.2 Smallest Choke 2: q2 = 10; (ps2 - pf2)2 = 17.7 Intermediate Choke 3: q3 = 20; (ps2 - pf2)3 = 60.0 Largest Where: q = Flowrate, units unimportant, normally Mscfd or MMscfd. ps = Static reservoir pressure, psia. pf = Stabilized flowing formation-face pressure, psia. From these data, a conventional Rawlins-Schellhardt plot would be developed, as in Fig. TN 1. Attention is now directed to the slope of this line in Fig. TN 1. Taking a vertical difference over a horizontal difference, both positive, or the tangent of the angle from the q scale, measured directly, or analytically: 1/n = [log (ps2 - pf2)3 - log (ps2 - pf2)1] / (log q3 - log q1) Equation TN 1 The slope is the reciprocal of the exponent n in the Rawlins-Schellhardt equation: q = C(ps2 - pf2)n Equation TN 2 Where: C = Performance coefficient, constant for pseudo-steady-state flow. n = Flow exponent, varying only between 0.50 and 1.0; is constant with time. Now, applying the hypothetical test numbers above in Equation TN 1 to the largest and smallest choke results: 1/n = (log 60 - log 5.2) / (log 20 - log 5.0), or 1/n = 1.76419. Therefore: n = 1/1.76419 = 0.5668. Now, solving for C in the Rawlins-Schellhardt equation: Data from Choke 2: C = q / (ps2 - pf2)n, or C = 10 / 17.70.5668 = 1.962. This results in the performance equation for the zone: q = 1.962(ps2 - pf2)0.5668 Equation TN 3 Assuming the test has been correctly carried out, and PSS flow has been achieved, this performance equation may now be used to relate the flowrate, q, to any given pressure drawdown. Also, from Fig. TN 1, the choke necessary to create this drawdown may be accurately predicted and, from these, a performance curve of choke size vs. desired production rate is established, now independent of flowing bottomhole pressure. Fig. TN 2. Plot folded from Rawlins-Schellhardt. The above serves to outline a conventional Rawlins-Schellhardt analysis for a well which has stabilized quickly, and so has achieved PSS flow. It also sets the background for the question: Why 1/n? Why not n directly? The very form of Equation TN 2 would suggest a plot of q, production rate, on the vertical scale, instead of the horizontal, as in Rawlins-Schellhardt, and pressure values on the horizontal scale instead of the vertical. If this is not so evident at first reading, consider that, partly because of the magnitude of numbers normally involved, both sets of values are normally plotted on log scales. Thus, taking logs of Equation TN 2: log q = log C + n log(ps2 - pf2) Equation TN 4 And comparing this to the type equation for a Cartesian straight line: y = (mx + b) = b + mx Equation TN 5 Where: y = Vertical scale values. x = Horizontal scale values. m = Slope, vertical cut over horizontal cut. b = Intercept on y scale, positive or negative. Log C, obviously, is the y-intercept, b; and n is the positive slope of such a plot, comparable to "type m." Such a plot is presented in Fig. TN 2, called a folded scale from the Rawlins-Schellhardt conventional plot of Fig. TN 1. The performance coefficient, C, can be read directly as the y-intercept, slightly below 2.0 on the vertical q scale, and agrees well with the analytically determined value of 1.962; or its log can be read directly, if log values are plotted on a Cartesian scale, of course. As for the flow exponent, n, it is calculated directly: n = (log 20 - log 5.0)/(log 60 - log 5.2) = 0.5668. Or, if both log scales have (numerically) the same values, one may simply measure, in real terms, the tangent of the slope angle, and this will yield n directly. But, if scales are numerically different, n is best determined analytically. This suggested procedure places n more in the nature of a conventional slope, as it is normally thought of by engineers and mathematicians. Because it can vary-for different producing zones-only between 0.50 and 1.0, its maximum value in Fig. TN 2 will be 45° of angle, and its minimum value is about 26° 34' of angle to the horizontal pressure scale. Values are more readily amenable to visual check by this modified procedure and, thus, are less amenable to error.

OTHER USEFUL RELATIONSHIPS

Poettmann presents expressions for both the stabilized value of a (pseudo-steady-state) and for transient values of a, or at which, involving formation damage, are themselves beyond the scope of this paper. Then, however, by equating these two expressions for a, he obtains an equation for the time at which the two values of a become the same. This, he calls pseudo-stabilization time, t_ps. It is the time, from opening, which will be required for transient flow to end and for PSS flow to begin, provided flow is continuous throughout:

Equation 10

In a field development, r_e is taken as half the distance between wells. However, in a wildcat or exploratory well, r_e simply continues to increase with time, in an "infinite" reservoir, so this relationship is of limited value, i.e., it cannot be a fixed time.

Equations 4, 5, 6 and 7 may each be useful, depending on the situation and particular requirements.

Finally, the relationship 1/ C^1/n is linear with log of time. Both of these coefficients, C and n are from the Rawlins-Schellhardt equation (Equation 1). Having plotted the Rawlins-Schellhardt data on a log-log plot, obtain C for various isochronal times. Then plot 1/ C^1/nvs. log time to obtain a straight-line slope of m. Note that 1/ C^1/n can be read directly from the Rawlins-Schellhardt plot, since p_s² - p_f² for a flowrate of 1.0 MMscfd, and for a given isochronal time, is equal to 1/ C^1/n for that time.

Then, from resulting slope m, calculate permeability using Equation 5. This value of permeability can then be used in Equation 10 to calculate pseudo-stabilization time and, ultimately, a theoretically stabilized value of C. Then, plugging this back into the Rawlins- Schellhardt equation, a theoretically stabilized performance equation can be verified for PSS flow, which will occur at or near the lapse of pseudo-stabilization time from opening.

Lastly, the relation (a + bq)/ q^1/n-1 is also linear with log of time, and will give the slope m, which can also be used to obtain a value for permeability, and for pseudo-stabilization time. This relationship has not been previously found in the literature by this author, and is believed to be presented here for the first time.

PREPARATIONS FOR ONE-DAY TEST

In view of all the material presented up to this point, it should be apparent to the attentive reader that the choice for a test procedure to be completed in one day is a modified isochronal test followed by a Forchheimer analysis. Or, as will be seen, this should be the planned alternative to provide for the possibility of low transmissibility, long stabilization time and consequent transient flow, if they do occur.

A generalized test program will be outlined below to cover all contingencies. It must be remembered that most tests are performed on newly opened wells, e.g.,wildcats, exploratory wells or development wells in a new area of an existing field. Therefore, in virtually all cases, it is impossible to predict just how a zone will act on test. That is, indeed, the very reason for the test being performed.

First, a few related points are made, as follows:

Well tests, in general, are classified as to type by the position of the control valve. If the valve is located downhole, at or near the formation face or perforations, the test is essentially a drillstem test. This is to be preferred, as opening and closing the downhole control valve has the most immediate effect on reservoir fluid flow; a well test is really a test of the reservoir at that point. Also, this facility eliminates the effect of fluid segregation in the wellbore on downhole measured pressures. Thus, pressures measured under these conditions-isolated from the tubing at shut-in-do more accurately, and more quickly, reflect reservoir pressure behavior.

Conversely, if the control valve is located at the surface, the test is essentially a production test. This is OK once a well has been completed, but is to be avoided for initial testing of a zone. A production test is sufficient for rate testing, but has much less value for analysis of reservoir pressure behavior. In the event a DST is being run and the tool experiences a function failure, what started out as a DST finishes up being a production test, because the flow must subsequently be controlled at surface.

Chokes, for testing purposes, are always located at the surface, and a dual-flow choke head with a 2-in. straight-through bypass is required so as to be able to change chokes without shutting in the well. Downhole chokes do exist, but are to be avoided if at all possible. Once run on the test string, they cannot be changed without pulling the string.

Downhole chokes do have occasional justified uses, such as mitigating inflow pressure surges in a loosely consolidated formation; but their justification, in general, is rare. If one must be run, the downhole pressure recording or indicating gauges must be located below it, i.e., between the downhole choke and the formation face. The presence of a downhole choke severely limits the ability of the test engineer to obtain a valid test on a zone.

A valid test means simply and exactly that the choke is controlling rate of flow in the overall hydraulic profile of flow. With gas wells, this is particularly easy to establish, simply by maintaining a downstream flowing pressure (separator controlled) of less than half upstream flowing pressure across the choke. This ensures so-called "critical flow", in which fluid velocity through the choke is greater than velocity of sound in the fluid. If this condition is maintained, then nothing occurring downstream of the choke, such as a separator pressure change, will have any effect on flowrate, so the choke is truly controlling flow.

Newly-perforated zones should usually be cleaned up before being placed on test. Extraneous materials have probably lodged in the face of the open zone due to drilling, casing, cementing, perforating and pressure surges in the wellbore, etc. A strong initial flow for a short time will remove much of this material.

PROGRAM FOR ONE-DAY TEST

First, knowing beforehand that the test will last 24 hr, divide the 24-hr into three 8-hr periods, and each 8-hr period into two 4-hr periods. Following modified isochronal procedure, plan to flow the well each first 4-hr period, and then shut it in for each second 4-hr period-thus, the three choke flows and buildups. Procedurally, in general terms:

•  Run DST tool to depth, do not set packer
  
•  Remain at depth for 20 min. to record initial static reservoir pressure
  
•  Set packer
  
•  Open tool for 5-min. flow
  
•  Close tool for one hour, as check on initial static reservoir pressure by extrapolation
  
•  Open tool for 30 to 45 min. for clean-up flow, putting stream through 2-in. choke-head bypass, longer for deeper wells
  
•  Shut in tool for at least one hour, possibly two
  
•  During this shut-in, place first two positive chokes in choke head, carefully identifying each, and marking
  
•  Open tool for 4-hr flow on smallest choke. Take 30-min. readings, exactly to the second, of flowrate, pressures, temperatures, etc.

Intermediate note: Experience teaches that, generally speaking, if a well is going to stabilize, it will do so within 3 hr or less from opening. Some take less than 10 min., others 2 hr or more. At this time, the test engineer, after carefully observing all data for 3 hr, must make a decision. If stabilized flow is achieved, he can go to a flow-after-flow test in which flow periods need not be a full 4 hr, but can each be limited to the time necessary to achieve stabilized flow, and then a final shut-in for pressure buildup equal to total flow time. Analysis then is by Rawlins-Schellhardt.

Conversely, if the well is still not stabilized after 3 hr of flow, it probably will not stabilize at all. In this case, the test engineer must opt for a modified isochronal test, to be followed by a Forchheimer analysis. Assuming this is the case:

•  After a 4-hr flow, shut in well for 4-hr pressure buildup
•  Open well for second 4-hr choke flow on intermediate choke, continuing all readings at 30-min. intervals
•  Shut in well for second 4-hr pressure buildup, prepare third choke in choke head
•  Open well for third 4-hr flow on largest choke, continuing all readings at 30-min. intervals
•  Shut in well for third, and final, 4-hr pressure buildup. This shut-in may be extended, if considered necessary, and if time is available
•  Proceed with Forchheimer analysis of this modified isochronal test, to obtain performance equation for zone on test
•  During shut-in, balance out string to hold well
•  At termination of pressure buildup, kill zone
•  Release packer and pull tool
•  Stimulate zone if/as required from test and analysis
•  After stimulation, retest with same program, and re-evaluate, using only one additional rig-day to do so.

CONCLUSION

From the above discussion, it emerges that the coefficient C, in the Rawlins-Schellhardt equation, is some function of time for transient flow, and the expression 1/ C^1/n is linear with log of time for transient flow. However, this expression involves both coefficients, not just one, and thus cannot be quantitatively related to log of time without difficult and approximate iteration during transient flow.

Conversely, the coefficient a, in the Forchheimer equation, is a simple, single unknown which is, thus, easily and directly related quantitatively to log of time, thus greatly simplifying the quantitative analysis of transient flow.

It follows that the Rawlins-Schellhardt equation is best applied to pseudo-steady-state or true-steady-state flow (stabilized flow), but is not readily applicable to transient flow. It also follows that the Forchheimer equation is best applied to analysis of transient flow, but it also applies to stabilized flow conditions.

Because the Forchheimer equation also applies to pseudo-steady-state or true-steady-state flow, this equation is the best to use for all testing of gas wells; it can be applied quantitatively regardless of transient or stabilized flow.

In the technical note numerical example worked out in the sidebar presentation, the stabilized, pseudo-steady-state or true-steady-state form of the Forchheimer equation would be:

(p_s² - p_f²⁾/q = 0.3867 + 0.1307q, or

p_s² - p_f² = 0.3867q + 0.1307q²

The coefficient b, or 0.1307 above, will be constant for both types of flow. The coefficient a above, 0.3867, will become constant when transient flow finishes and becomes pseudo-steady-state, or true-steady-state, flow. At this point, elapsed time may be equated to stabilization time, Equation 10, but still does not yield a reliable value to use in calculating permeability unless/until true-steady-state flow is achieved, and drainage radius, r_e, ceases to increase with time.

Because true-steady-state flow rarely, if ever, occurs during the brief time of testing, permeability is best calculated from a pressure buildup, via Horner, even with the inherent errors due to intermittent flow. And, of course, both calculations may be made and compared.

So, a final pressure buildup should always be taken, preferably with time equal to total test flow time - greater for very or extremely "tight" zones. If nothing else, this will provide a permeability value independent of any formation damage which may exist in the test zone, and so allow evaluation of true formation damage apart from the non-Darcy flow constant.

BIBLIOGRAPHY

Poettmann, F. H. and H. J. Smith, Personal correspondence on Forchheimer analysis, Islamabad, Pakistan, April 28, 1989.

Poettmann, F. H., "Gas well deliverability testing: Myths and realities," unpublished notes, 1988.

Poettmann, F. H., H. Kazemi and S. B. Hinchman, "Further discussion of the analysis of modified isochronal tests to predict the stabilized deliverability of gas wells without using stabilized flow data," JPT, January 1987.

Poettmann, F. H., "Discussion of analysis of modified isochronal tests to predict the stabilized deliverability potential of gas wells without using stabilized flow data,"JPT, October 1986, pp. 1122- 1124.

Brar, G. S. and K. Aziz, "Analysis of modified isochronal tests to predict the stabilized deliverability potential of gas wells without using stabilized flow data," JPT, February 1978, pp. 297- 304, Trans. AIME 265.

Theory and practice of the testing of gas wells, Third edition, Alberta Energy Resources Conservation Board, Calgary, 1975.

Odeh, A. S., E. E. Moreland and S. Schueler, "Characterization of a gas well from one flow test sequence," JPT, December 1975, pp. 1500-1504, Trans. AIME 259.

Poettmann, F. H. and R. E. Schilson, "Calculations of the stabilized performance coefficient of low permeability natural gas wells," JPT, September 1959, pp. 240- 246, Trans. AIME 216.

Cullender, M. H., "The isochronal performance method of determining the flow characteristics of gas wells," Trans. AIME, 1955, 204, p. 137.

Rawlins, E. L. and M. A. Schellhardt, "Backpressure data on natural gas wells and their application to production practices,"Monograph 7, U.S. Bureau of Mines, Washington, D C, 1936.

The author

Haldon J. Smith holds a professional PE degree from Colorado School of Mines (1953), a master of PE degree from the University of Tulsa (1956), and an MBA diploma from Alexander Hamilton Institute, of New York University. He worked 6-1/2 years in Venezuela for Creole Petroleum, followed by several years in France, Libya, Indonesia, Saudi Arabia, etc. - in all, more than 50 countries - mainly on reservoir studies / well testing. He has tested about 600 oil / gas wells, analyzed some 2,000 such tests and performed about 125 reservoir studies. He has been a full-time consultant since 1980, and has served 66 clients. His most recent significant work was 2-1/2 years in Bulgaria; he continues to be active.