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# Flow-after-flow tests for gas wells

This page discusses the implementation and analysis of flow-after-flow testing for gas well deliverability assessment. Both the Rawlins and Schellhardt and Houpeurt analysis techniques are presented in terms of pseudopressures.

## Flow-after-flow test procedure

Flow-after-flow tests, sometimes called gas backpressure or four-point tests, are conducted by producing the well at a series of different stabilized flow rates and measuring the stabilized BHFP at the sandface. Each different flow rate is established in succession either with or without a very short intermediate shut-in period. Conventional flow-after-flow tests often are conducted with a sequence of increasing flow rates; however, if stabilized flow rates are attained, the rate sequence does not affect the test. ^{[1]} The requirement that the flowing periods be continued until stabilization is a major limitation of the flow-after-flow test, especially in low-permeability formations that take long times to reach stabilized flowing conditions. **Fig 1** illustrates a flow-after-flow test.

## Rawlins-Schellhardt analysis technique

Recall the empirical equation that forms the basis for the Rawlins-Schellhardt analysis technique:

Taking the logarithm of both sides of **Eq. 1** yields the equation that forms the basis for the Rawlins-Schellhardt analysis technique:

The form of **Eq. 2** suggests that a plot of log (Δ*p*_{p}) vs. log (*q*) will yield a straight line of slope 1/*n* and an intercept of {–1/*n*[log(*C*)]}. The AOF potential is estimated from the extrapolation of the straight line to Δ*p*_{p} evaluated at a *p*_{wf} equal to atmospheric pressure (sometimes called base pressure). This analysis technique is illustrated with **Example 1**.

## Houpert analysis technique

Flow-after-flow tests require stabilized data or data measured during pseudosteady-state flow. Houpeurt^{[2]} gives the theoretical equation for pseudosteady-state flow, which was derived from the gas-diffusivity equation, as

The coefficients *a* and *b* have theoretical bases and can be estimated if reservoir properties are known or they can be determined from flow-after-flow test data. Dividing both sides of **Eq. 3** by the flow rate, *q*, and rearranging yields the equation that is the basis for the Houpeurt analysis technique:

The form of **Eq. 4** suggests that a plot Δ*p*_{p}/*q* vs. *q* will yield a straight line with a slope *b* and an intercept *a*. The AOF is estimated in the Houpeurt deliverability analysis by solving **Eq. 3** for *q* = *q*_{AOF} at *p*_{wf} = *p*_{b}.

## Example: Analysis of a flow-after-flow test

Estimate the initial stabilized AOF potential of a well^{[3]} with the well and reservoir properties listed. Use both the Rawlins-Schellhardt and the Houpeurt analysis techniques. In addition, estimate the AOF potential 10 years later when the static drainage area pressure has decreased to 350 psia. Evaluate the AOF potential at *p*_{b} = 14.65 psia. **Table 1** summarizes the flow-after-flow test data. *L* = 3,050 ft, *r*_{w} = 0.5 ft, *M*_{a} = 20.71 lbm/lbm-mole, *T* = 90°F = 555°R, *A* = 640 acres, *ϕ* = 0.25, *C*_{A} = 30.8828, and *h* =200 ft.

Current = 407.6 psia, *p*_{p}( = 407.6) = 1.617 × 10^{7} psia^{2}/cp. after 10 years = 350 psia, *p*_{p}( = 350) = 1.2239 × 10^{7} psia^{2}/cp. *p*_{b} = 14.65 psia, *p*_{p}(*p*_{b}) = 2,674.8 psia^{2}/cp.

The pseudopressure in this example (and all others in this section) were calculated using the methods suggested by Al-Hussainy *et al.*^{[4]} These methods, which involve numerical evaluation of the integral in **Eq. 5** and which require computational routines to estimate gas viscosity, *μ*, and deviation factor, *z*, are widely available in basic reservoir fluid flow analysis software.

### Solution

*Rawlins-Schellhardt analysis*. Plot Δ*p*_{p} vs. *q* on log-log graph paper (**Fig. 2**). **Table 2** gives the plotting functions. Construct the best-fit line through the data points. All data points lie on the best-fit line and will be used for all subsequent calculations.

Next, determine the deliverability exponent using least-squares regression analysis. Alternatively, because Points 1 and 4 both lie on the perceived "best" straight line through the test data, the reciprocal slope is estimated to be

Now, calculate the AOF of the well. Because 0.5 ≤ *n* ≤ 1.0, calculate C using either regression analysis or a point from the best-fit straight line through the test data. Estimating C with regression analysis results in log(*C*) = *α* =-3.09 . Thus,

Estimating *C* using Point 4 from the best-fit line and **Eq. 1**:

Therefore, the AOF potential of this well is

To update the AOF to a new average reservoir pressure, recall that for pseudopressure analysis, neither *C* nor *n* changes as drainage area pressure decreases. The AOF for the new drainage area pressure becomes

*Houpeurt analysis*. Plot Δ*p*_{p}/*q* vs. *q* on Cartesian graph paper (**Fig. 3**). **Table 3** gives the plotting functions. Construct the best-fit line through the last three data points. The first point, corresponding to the lowest flow rate, does not follow the trend and will be ignored in subsequent analyses.

Determine the deliverability coefficients, *a* and *b*, from a least-squares regression analysis, excluding the first point. The result is

Alternatively, use Points 2 and 4 from the line drawn through the test data to calculate *a* and *b*:

Then,

To update the AOF, note that for pseudopressure analysis neither *a* nor *b* changes as drainage area pressure changes. Therefore, the AOF for the new drainage area pressure is

A comparison (**Fig. 4**) of the results from the two parts of this example shows that the Rawlins-Schellhardt equation appears to be valid for this range of test data; however, the line representing the Houpeurt equation deviates from the Rawlins-Schellhardt equation as BHFP decreases. Although the Rawlins-Schellhardt method is valid under many testing conditions, this deviation suggests that extrapolating the empirical equation over a large interval of pressure may not predict well behavior correctly.

## Nomenclature

## References

- ↑ Rawlins, E.L. and Schellhardt, M.A. 1935. Backpressure Data on Natural Gas Wells and Their Application to Production Practices, Vol. 7. Monograph Series, USBM.
- ↑ Houpeurt, A. 1959. On the Flow of Gases in Porous Media. Revue de L’lnstitut Francais du Petrole 14 (11): 1468.
- ↑ Jennings, J. W. et al. 1989. Deliverability Testing of Natural Gas Wells. Prepared for the Texas Railroad Commission, Texas A&M U., College Station, Texas, August.
- ↑ Al-Hussainy, R., Jr., H.J.R., and Crawford, P.B. 1966. The Flow of Real Gases Through Porous Media. J Pet Technol 18 (5): 624-636. http://dx.doi.org/10.2118/1243-A-PA

## Noteworthy papers in OnePetro

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## See also

Deliverability testing of gas wells

Single-point tests for gas wells

Isochronal tests for gas wells

Modified isochronal tests for gas wells