Approximation of Gas Well Deliverability

The steady-state relationship developed from Darcy’s law for an incompressible fluid (oil) was presented as Equation. A similar relationship can be derived for a natural gas well by converting the flow rate from STB/d to MSCF/d and using the real gas law to describe the PVT behavior of the gas. Beginning with the differential form of Darcy’s law for radial flow,

where q_act denotes the actual volumetric flow rate at some temperature and pressure in the reservoir. From the real gas law, the volumetric flow rate, q_act, at any temperature and pressure is related to the volumetric flow rate at standard conditions, q, by

Bringing this expression to Equation 4-36 and separating variables yields

For steady-state flow with a pressure p_e at the drainage boundary, r_e, and a bottomhole flowing pressure, p_wf, at r_w, integrating Equation 4-38 for average values of viscosity and compressibility factor yields

Expressing the flow rate in MSCF/d, reconciling the other usual oilfield units, and adding the skin factor gives the steady-state gas inflow equation

which also results in

Equation (4-41) suggests that a gas well production rate is approximately proportional to the pressure squared difference. The properties  and  are average properties evaluated at a pressure between p_e and p_wf. (Henceforth the bars will be dropped for simplicity).

A similar approximation can be developed for pseudosteady state. It has the form

Equations (4-41) and (4-42) are not only approximations in terms of properties but also because they assume Darcy flow in the reservoir. For reasonably small gas flow rates this approximation is acceptable. A common presentation of Equation (4-42) [or Equation (4-41)] is

For larger flow rates, where non-Darcy flow is evident in the reservoir,

where 0.5 < n < 1.

A log-log plot of q versus () would yield a straight line with slope equal to n and intercept C.

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Example 4-6. Flow Rate versus Bottomhole Pressure for a Gas Well

Graph the gas flow rate versus flowing bottomhole pressure for the well described in Appendix C. Use the steady-state relationship given by Equation (4-40). Assume that s = 0 and r_e = 1490 ft (A = 160 acres).

Solution

Equation (4-40) after substitution of variables becomes

For p_wf increments of 500 psi from 1000 psi to 4000 psi, calculations are shown in Table 4-4. (Note that μ_1atm = 0.0122 cp and T_pr = 1.69 throughout, since the reservoir is considered isothermal.)

Table 4-4. Viscosity and Gas Deviation Factor for Example 4-6

As an example calculation, for p_wf = 3000, Equation (4-45) yields

If the initial μ_i and Z_i were used (i.e., not averages) the flow rate q would be 9.22 × 10² MSCF/d, a deviation of 14%. Figure 4-6 is a graph of p_wf versus q for this example.

Figure 4-6. Production rate versus flowing bottomhole pressure (IPR) for the gas well in Example 4-6.

Although this is done in the style of IPR curves as shown in this is not a common construction for gas reservoirs.

The “irregular” shape of the curve in Figure 4-6 reflects the changes in the Z factor, which for this example reaches a minimum between 2500 and 3000 psi.