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T.A. Blasingame, Texas A&M U. Department of Petroleum Engineering Texas A&M University College Station, TX 77843

Petroleum Engineering 613 Natural Gas Engineering Texas A&M University. Lecture 04: Diffusivity Equations for Flow in Porous Media. T.A. Blasingame, Texas A&M U. Department of Petroleum Engineering Texas A&M University College Station, TX 77843-3116

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T.A. Blasingame, Texas A&M U. Department of Petroleum Engineering Texas A&M University College Station, TX 77843

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  1. Petroleum Engineering 613 Natural Gas Engineering Texas A&M University Lecture 04: Diffusivity Equations for Flow in Porous Media T.A. Blasingame, Texas A&M U. Department of Petroleum Engineering Texas A&M University College Station, TX 77843-3116 +1.979.845.2292 — t-blasingame@tamu.edu PETE 613 (2005A)

  2. Lecture: Diffusivity Equations • Diffusivity Equations: • "Black Oil" (p>pb) • "Solution-Gas Drive" (valid for all p, referenced for p<pb) • "Dry Gas" (p>pd) • Multiphase Flow PETE 613 (2005A)

  3. Diffusivity Equation: Black Oil (p>pb) • Diffusivity Equations for a Black Oil: • Slightly Compressible Liquid: (General Form) • Slightly Compressible Liquid: (Small p and c form) PETE 613 (2005A)

  4. Diffusivity Equation: Black Oil — mo and Bo vs. p • Behavior of the mo and Bo variables as functions of pressure for an example black oil case. Note behavior for p>pb — both vari-ables should be considered to be "approximately constant" for the sake of developing flow relations. Such an assumption (i.e., mo and Bo constant) is not an absolute requirement, but this assumption is fundamental for the development of "liquid" flow solutions in reservoir engineering. PETE 613 (2005A)

  5. Diffusivity Equation: Black Oil — co vs. p • Behavior of the co variable as a function of pressure — example black oil case. Note the "jump" at p=pb, this behavior is due to the gas expansion at the bubblepoint. PETE 613 (2005A)

  6. Diffusivity Equation: Solution Gas Drive (p<pb) • Diffusivity Equations for Solution-Gas Drive: (p<pb) • Oil Pseudopressure Form: (Accounts for mo and Bo) • Oil Pseudopressure Definition: (pn is any reference pressure) PETE 613 (2005A)

  7. Diffusivity Equation: Soln Gas Drive 1/(moBo) vs. p 1/(moBo) vs. p (pb=5000 psia, T=175 Deg F) • "Solution-Gas Drive" Pseudopressure Condition: (1/(moBo) vs. p) • Concept: IF 1/(moBo)  constant, THEN oil pseudopressure NOT required. • 1/(moBo) is NEVER "constant" — but does not vary significantly with p. • Oil pseudopressure calculation straightforward, but probably not necessary. PETE 613 (2005A)

  8. Diffusivity Equation: Soln Gas Drive (moco) vs. p (moco) vs. p (pb=5000 psia, T=175 Deg F) • "Solution-Gas Drive" Pseudopressure Condition: ((moco) vs. p) • Concept: IF (moco)  constant, THEN oil pseudotime NOT required. • (moco) is NEVER "constant" — BUT, oil pseudotime would be very difficult. • Other evidence suggests that ignoring (moco) variance is acceptable. PETE 613 (2005A)

  9. Diffusivity Equation: Soln Gas Drive (ct/lt) vs. tD Solution-Gas Drive — Mobility/Compressibility (Camacho) • Camacho-V., R.G. and Raghavan, R.: "Boundary-Dominated Flow in Solution-Gas-Drive Reservoirs," SPERE (November 1989) 503-512. • "Solution-Gas Drive" Behavior: ((ct/lt) vs. time) • Observation: (ct/lt)  constant for p>pb and later, for p<pb. • pwf = constant — but probably valid for any production/pressure scenario. PETE 613 (2005A)

  10. Diffusivity Equation: Soln Gas Drive (2-phase pp) Historical Note — Evinger-Muskat Concept (1942) • Why not use liquid pseudopressure? • Evinger and Muskat (1942) note that: The indefinite integral may be evaluated, as was done for the two-phase system, and the pressure distribution may be determined. However, it will be sufficient for the calculation of the productivity factor to consider only the limiting form ... (i.e., the constant property liquid relation). PETE 613 (2005A)

  11. Diffusivity Equation: Dry Gas Relations • Diffusivity Equations for a "Dry Gas:" • General Form for Gas: • Diffusivity Relations: Pseudopressure/Time: Pseudopressure/Pseudotime: • Definitions: Pseudopressure: Pseudotime: PETE 613 (2005A)

  12. Diffusivity Equation: Pseudotime (mgcg vs. p) Dry Gas Pseudotime Condition (mgcg vs. p, T=200 Deg F) • "Dry Gas" Pseudotime Condition: (mgcg vs. p) • Concept: IF mgcg constant, THEN pseudotime NOT required. • mgcgis NEVER constant — pseudotime is always required (for liquid eq.). • However, can generate numerical solution for gas cases (no pseudotime). PETE 613 (2005A)

  13. Diffusivity Equation: p2 Relations Dry Gas — p2 Relations • Diffusivity Equations for a "Dry Gas:" p2 Relations • p2 Form — Full Formulation: • p2 Form — Approximation: PETE 613 (2005A)

  14. Diffusivity Equation: p2 Relations (mgz vs. p) Dry Gas p2 Condition (mgz vs. p, T=200 Deg F) • "Dry Gas" PVT Properties: (mgz vs. p) • Concept: IF (mgz) = constant, THEN p2-variable valid. • (mgz)  constant for p<2000 psia. • Even with numerical solutions, p2 formulation would not be appropriate. PETE 613 (2005A)

  15. Diffusivity Equation: p Relations Dry Gas — p Relations • Diffusivity Equations for a "Dry Gas:" p Relations • p Form — Full Formulation: • p Form — Approximation: PETE 613 (2005A)

  16. Diffusivity Equation: p Relations (p/(mgz) vs. p) Dry Gas p Condition (p/(mgz) vs. p, T=200 Deg F) • "Dry Gas" PVT Properties: (p/(mgz) vs. p) • Concept: IF p/(mgz) = constant, THEN p-variable is valid. • p/(mgz) is NEVER constant — pseudopressure required (for liquid eq.). • p formulation is never appropriate (even if generated numerically). PETE 613 (2005A)

  17. Diffusivity Equation: Multiphase Relations Multiphase Case — p-Form Relations (Perrine-Martin) Gas Equation: Oil Equation: Water Equation: Multiphase Equation: Compressibility Terms: PETE 613 (2005A)

  18. Petroleum Engineering 613 Natural Gas Engineering Texas A&M University Lecture 04: Diffusivity Equations for Flow in Porous Media (End of Lecture) T.A. Blasingame, Texas A&M U. Department of Petroleum Engineering Texas A&M University College Station, TX 77843-3116 +1.979.845.2292 — t-blasingame@tamu.edu PETE 613 (2005A)

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