PNG 301
Introduction to Petroleum and Natural Gas Engineering

5.4.1.4: The Rawlins and Schellhardt Backpressure or Deliverability Equation

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One final relationship that is often used in the oil and gas industry is the empirical Rawlins and Schellhardt Backpressure or Deliverability Equation[1]. This equation has the form:

q g =C ( p ¯ 2 p wf 2 ) n
Equation 5.25

As with all observation-based empirical relationships, to use this equation it must be tuned with appropriate data to determine the values of the tuning parameters, C and n . To do this, the gas well must be produced at several flowing pressures, p wf , and the resulting stabilized rates measured. Once these tests (Deliverability Tests) are performed, the equation parameters, C and n , are used to fit the equation to the test results. Table 5.01 a-g summarizes all of the Stabilized Inflow Performance Relationships for gas wells discussed to this point.

Table 5.01a:Stabilized Inflow Performance Relationships for Gas Wells
In Terms of Pressure ( p e ) including Damage or Stimulation:
All pressures greater than 3,000 psi
Steady-State Flow Regime Pseudo Steady-State Flow Regime
Drawdown p e p wf p e p wf
Productivity
Index
0.03976  T SC T r   p SC ( p μ g  Z ¯ ) k g  h  lo g e ( r e r w )+S 0.03976  T SC T r   p SC ( p μ g  Z ¯ ) k g  h  lo g e ( r e r w ) 1 2 +S
IPR q g = 0.03976  T SC T r   p SC ( p μ g  Z ¯ ) k g  h( p e p wf )  lo g e ( r e r w )+S q g = 0.03976  T SC T r   p SC ( p μ g  Z ¯ ) k g  h( p e p wf )  lo g e ( r e r w ) 1 2 +S
Table 5.01b:Stabilized Inflow Performance Relationships for Gas Wells
In terms of Pressure ( p ¯ ) including Damage or Stimulation:
All pressures greater than 3,000 psi
Steady-State Flow Regime Pseudo Steady-State Flow Regime
Drawdown p ¯ p wf p ¯ p wf
Productivity
Index
0.03976  T SC T r   p SC ( p μ g  Z ¯ ) k g  h  lo g e ( r e r w ) 1 2 +S 0.03976  T SC T r   p SC ( p μ g  Z ¯ ) k g  h  lo g e ( r e r w ) 3 4 +S
IPR q g = 0.03976  T SC T r   p SC ( p μ g  Z ¯ ) k g  h( p ¯ p wf )  lo g e ( r e r w ) 1 2 +S q g = 0.03976  T SC T r   p SC ( p μ g  Z ¯ ) k g  h( p ¯ p wf )  lo g e ( r e r w ) 3 4 +S
Table 5.01c:Stabilized Inflow Performance Relationships for Gas Wells
In terms of Pressure-Squared ( p e ) including Damage or Stimulation:
All pressures less than 2,000 psi
Steady-State Flow Regime Pseudo Steady-State Flow Regime
Drawdown p e 2 p wf 2 p e 2 p wf 2
Productivity
Index
0.01988  T SC T r   p SC   μ g  Z k g  h  lo g e ( r e r w )+S 0.01988  T SC T r   p SC   μ g  Z k g  h  lo g e ( r e r w ) 1 2 +S
IPR q g = 0.01988  T SC T r   p SC   μ g  Z k g  h( p e 2 p wf 2 )  lo g e ( r e r w )+S q g = 0.01988  T SC T r   p SC   μ g  Z k g  h( p e 2 p wf 2 )  lo g e ( r e r w ) 1 2 +S
Table 5.01d:Stabilized Inflow Performance Relationships for Gas Wells
In terms of Pressure-Squared ( p ¯ ) including Damage or Stimulation:
All pressures less than 2,000 psi
Steady-State Flow Regime Pseudo Steady-State Flow Regime
Drawdown p ¯ 2 p wf 2 p ¯ 2 p wf 2
Productivity
Index
0.01988  T SC T r   p SC   μ g  Z k g  h  lo g e ( r e r w ) 1 2 +S 0.01988  T SC T r   p SC   μ g  Z k g  h  lo g e ( r e r w ) 3 4 +S
IPR q g = 0.01988  T SC T r   p SC   μ g  Z k g  h( p ¯ 2 p wf 2 )  lo g e ( r e r w ) 1 2 +S q g = 0.01988  T SC T r   p SC   μ g  Z k g  h( p ¯ 2 p wf 2 )  lo g e ( r e r w ) 3 4 +S
Table 5.01e:Stabilized Inflow Performance Relationships for Gas Wells
In terms of the Real Gas Pseudo-Pressure [ m( p e ) ] including Damage or Stimulation:
Valid over the entire pressure range
Steady-State Flow Regime Pseudo Steady-State Flow Regime
Drawdown m( p e )m( p wf ) m( p e )m( p wf )
Productivity
Index
0.01988  T SC T r   p SC k g  h  lo g e ( r e r w )+S 0.01988  T SC T r   p SC k g  h  lo g e ( r e r w ) 1 2 +S
IPR q g = 0.01988  T SC T r   p SC k g  h[ m( p e )m( p wf ) ] lo g e ( r e r w )+S q g = 0.01988  T SC T r   p SC k g  h[ m( p e )m( p wf ) ]  lo g e ( r e r w ) 1 2 +S
Table 5.01f:Stabilized Inflow Performance Relationships for Gas Wells
In terms of the Real Gas Pseudo-Pressure [ m( p ¯ ) ] including Damage or Stimulation:
Valid over the entire pressure range
Steady-State Flow Regime Pseudo Steady-State Flow Regime
Drawdown m( p ¯ )m( p wf ) m( p ¯ )m( p wf )
Productivity
Index
0.01988  T SC T r   p SC k g  h  lo g e ( r e r w ) 1 2 +S 0.01988  T SC T r   p SC k g  h  lo g e ( r e r w ) 3 4 +S
IPR q g = 0.01988  T SC T r   p SC k g  h[ m( p ¯ )m( p wf ) ] lo g e ( r e r w ) 1 2 +S q g = 0.01988  T SC T r   p SC k g  h[ m( p ¯ )m( p wf ) ]  lo g e ( r e r w ) 3 4 +S
Table 5.01g:Stabilized Inflow Performance Relationships for Gas Wells
The Rawlins and Schellhardt Backpressure or Deliverability Equation[1]
Steady-State Flow Regime Pseudo Steady-State Flow Regime
IPR q g =C ( p ¯ 2 p wf 2 ) n

The following parameters are used in Table 5.01 or in the development of these equations:

  • p e reservoir pressure at the external boundary of the drainage volume, psi
  • p wf flowing well pressure, psi
  • p ¯ average reservoir pressure within the drainage volume, psi
  • m( p ) the real gas pseudo-pressure, psi2/cp
  • r e external radius of the drainage volume, ft
  • r w radius of the well, ft
  • p SC pressure at standard conditions (14.7 psi in the U.S.A), psi
  • T SC temperature at standard conditions (520°R in the U.S.A), °R
  • T r temperature of the reservoir, °R
  • k g effective permeability to gas in the presence of an immobile water phase, md
  • h thickness of the reservoir, ft
  • μ g gas viscosity, cp
  • Z gas super-compressibility factor, dimensionless
  • S skin factor, dimensionless
  • q g is the oil production rate during the stabilized production period, SCF/day
  • C factor in Rawlins and Schellhardt Backpressure or Deliverability Equation[1], SCF/(day-psi(2xn))
  • n exponent in Rawlins and Schellhardt Backpressure or Deliverability Equation[1], dimensionless
  • B g is the oil phase formation volume factor, bbl/SCF