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PETE 324

PETE 324. Lecture12 Rosalind Archer. Real Gas Diffusivity Equation. The real gas diffusivity equation can be derived in a similar manner to the (slightly compressible) liquid case. Real Gas Diffusivity Equation.

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PETE 324

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  1. PETE 324 Lecture12 Rosalind Archer

  2. Real Gas Diffusivity Equation • The real gas diffusivity equation can be derived in a similar manner to the (slightly compressible) liquid case.

  3. Real Gas Diffusivity Equation • Starting by substituting the equation of state and Darcy’s Law into the continuity equation: • Cancel the constants M, R and T from each side.

  4. Real Gas Diffusivity Equation • Assume that the permeability, k, is a constant:

  5. Real Gas Diffusivity Equation • Now use the chain rule to expand the time derivative terms. The right hand side of the equation becomes:

  6. Real Gas Diffusivity Equation • Recalling the definition of the rock compressibility: • Also recall the definition of isothermal gas compressibility:

  7. Real Gas Diffusivity Equation • Therefore:

  8. Real Gas Diffusivity Equation • Substituting the definitions of compressibility into the right hand side of the diffusivity equation gives:

  9. Real Gas Diffusivity Equation • The final form of the real gas diffusivity equation is:

  10. Real Gas Diffusivity Equation • The equation is nonlinear because the terms multiplying (p, z, m) are all functions of the unknown p. • There are also nonlinearities on the right hand side as well.

  11. Pseudopressure • The equation can be linearised by introducing a pseudo-pressure:

  12. Pseudopressure • With the introduction of the gas pseudopressure the diffusivity equation becomes:

  13. Pseudopressure • The derivative of the gas pseudopressure with respect to pressure is: • Substituting this into the diffusivity equation gives:

  14. Pseudopressure • Cancelling like terms gives: • Use of pseudopressure has linearised the right hand side of the equation but the left hand side still has some pressure depend terms (m,ct).

  15. Pseudotime • A pseudotime formulation to linearise the left hand side was proposed by Agarwal: • Using pseudopressure and pseudotime gives:

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