Examples of Transient Calculations cont'd

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Examples of Transient Calculations (cont'd)
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Infinite acting, constant rate, line source
Wellbore pressure relations
Exponential Integral Solution ("Ideal", Field
Units)
-Slope pwf vs ln(t)
"Log Approximation" Solution ("Ideal")
"Log Approximation" Solution ("Real")
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Calculation of pressure in the wellbore

• q 20 STB/D
• Assume s 5
• Calculate pressure in the wellbore
• At time 1 hr, 10 hr

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End of Infinite Acting Period
In hour
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Example End of Infinite Acting Period

• Calculate end of "Infinite Acting" period (or
  end of validity of the transient solution) if re
  2100 ft

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Diffusivity Equation for Gas
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PreviouslyDiffusivity equation for a slightly
compressible liquid

• Laminar flow (non-laminar flow effects are not
  considered)
• Homogeneous and isotropic porous media (uniform
  permeability)
• Isothermal flow
• Negligible gravity forces (horizontal flow only)
• AND
• no z
• (mct) is constant by definition

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Now Diffusivity equation for Gas

• Laminar flow (non-laminar flow effects are not
  considered)
• Homogeneous and isotropic porous media (uniform
  permeability)
• Isothermal flow
• Negligible gravity forces (horizontal flow only)
• AND VARIOUS ASSUMPTIONS ON
• (mz) and
• (mct)

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Linearizing the Gas Diffusivity Equation

• Because the compressibility is not constant, we
  need p/z
• This is non-linear. To linearize the LHS, we
  introduce the "gas pseudo-pressure"

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Three forms of the diffusivity equation for gas
(assumptions on mz)

• The pressure (p) form
• p/mz constant
• The pressure-squared (p2) form
• mz constant
• The pseudo-pressure (pp) form
• p/mz no assumption

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mz (Cartesian)
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mz (log-log)
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p/mz (Cartesian)
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p/mz (log-log)
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mz (Cartesian)
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The Pseudo-pressure function can be "prepared"
for a certain gas (for a field)
We have to be able to calculate the viscosity
and the z-factor of the gas and carry out a
numerical integration (trapezoidal rule).
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Linearizing the right-hand side

• More tricks are needed Agarwal pseudotime
• Finally