Royal Helium Ltd. V.RHC

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[Margin321]

Frostback: You write:  Please elaborate on the math used to determine what flow rate to expect at the well head.  Here is a small piece to get you started.

Gas well deliverability tests are used to predict the flow rate of a gas well during reservoir depletion (Canadian Energy Resources Conservation Board, 1975; Beggs, 1984; Ahmed, 2000; Lee, 2007). Measurements of reservoir pressures and their corresponding flow rates are obtained during the test and are subsequently analyzed. Selection of a deliverability test depends on the length of time needed to stabilize the rate of pressure decline. Pressure is stabilized when pressure fluctuations are minimized. An estimate of stabilization time is given by

(9.2.1)$t_s=\mathrm{1,000}\frac{\mathrm{\mu }{\text{r}}_{\text{e}}^2}{\text{k}{\text{P}}_{\text{r}}}$

where

t_s = stabilization time (hr)

P_r = stabilized reservoir pressure (psia)

μ = gas viscosity at P_r (cp)

= porosity (fraction)

k = effective permeability (md)

r_e = outer radius of drainage area (ft)

Stabilized reservoir pressure P_r is obtained by shutting in the well until reservoir pressure stabilizes.

Single well gas deliverability tests include the conventional backpressure test, the isochronal test, and the modified isochronal test. Each flow rate for the conventional backpressure test should be maintained until the rate of pressure decline stabilizes. As reservoir permeability decreases, t_s increases. The stabilization time for the backpressure test may be excessive for some reservoirs, such as low-permeability reservoirs. In this case, the isochronal test may be used. The isochronal test consists of several flow periods of equal duration. Each flow period begins at static reservoir conditions and uses a different flow rate. The final flow rate is maintained (or extended) until a stabilized pressure is reached.

Although the isochronal test is faster than the conventional backpressure test, production time is lost while the well is shut in until the reservoir returns to static conditions between each flow period. An alternative approach, which is faster but less accurate, is the modified isochronal test. In this case, the shut-in period is the same duration as the flow period, and unstabilized shut-in pressures must be used to evaluate test results.

The applicability of a Simplified Backpressure Analysis (SBA) or a Laminar-Inertial-Turbulent (LIT) analysis depends on the gas flow regime. Lee (2007) pointed out that the SBA method is valid at low pressures (<2,000 psia), while the LIT (pseudopressure) method is valid for all pressures.

9.2.1 The Simplified Backpressure Analysis Method

The backpressure equation is

(9.2.2)$q_{\mathit{sc}}=C{(P_r^2-P_{\mathit{wf}}^2)}^n\equiv C{(\mathrm{\Delta }{P}^2)}^n$

where

C and n = empirical parameters

q_sc = gas flow rate at standard conditions (MMSCFD)

P_r = stabilized reservoir pressure (psia)

P_wf = flowing wellbore pressure (psia)

Taking the logarithm of the backpressure equation for measurement i yields

(9.2.3)$\log q_{\mathit{sci}}=n\log {(\mathrm{\Delta }{P}^2)}_i+\log C$

If we plot log q_sci versus log(ΔP²)_i, we obtain the equation for a straight line where n is the slope and log C is the intercept. The absolute open flow (AOF) of the well is the rate corresponding to P_wf = 0. Rates can be calculated from the backpressure equation using the values of C and n determined by a least squares fit of test data.

9.2.2 The Laminar-Inertial-Turbulent Method

The LIT method is applicable to any gas deliverability test. Test data are fit to the LIT equation

(9.2.4)$m\left(P_r\right)-m\left(P_{\mathit{wf}}\right)\equiv \mathrm{\Delta }m=a{q}_{\mathit{sc}}+b{q}_{\mathit{sc}}^2$

where

q_sc = gas flow rate at standard conditions (MMSCFD)

m(P_r) = pseudopressure corresponding to P_r (psia²/cp)

m(P_wf) = pseudopressure corresponding to P_wf (psia²/cp)

aq_sc = laminar flow

bq_sc = inertial and turbulent flow

The LIT equation is second order in rate. Dividing Eq. (9.2.4) for measurement i by q_sci lets us write the LIT equation as

(9.2.5)$\frac{\mathrm{\Delta }{m}_i}{q_{\mathit{sci}}}=b{q}_{\mathit{sci}}+a$

Plotting Δm_i/q_sci versus q_sci gives a straight line with slope b and intercept a. Values of a and b can be determined by a least squares fit of test data, and flow rate can be calculated by solving Eq. (9.2.4) for q_sc using the quadratic formula

(9.2.6)$q_{\mathit{sc}}=\frac{-a+\sqrt{a^2+4b\mathrm{\Delta }m}}{2b}$

Absolute open flow occurs when m(P_wf) = 0.