1. Diffusivity Equations for Flow in Porous Media PETE 613 (2005A) Slide — 1 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 Petroleum Engineering 613 Natural Gas Engineering Texas A&M University Lecture 04: Diffusivity Equations for Flow in Porous Media

2. Diffusivity Equations for Flow in Porous Media PETE 613 (2005A) Slide — 2 Diffusivity Equations: "Black Oil" (p>p b ) "Solution-Gas Drive" (valid for all p, referenced for p<p b ) "Dry Gas" (p>p d ) Multiphase Flow Lecture: Diffusivity Equations

3. Diffusivity Equations for Flow in Porous Media PETE 613 (2005A) Slide — 3 Diffusivity Equations for a Black Oil: Slightly Compressible Liquid: (General Form) Slightly Compressible Liquid: (Small p and c form) Diffusivity Equation: Black Oil (p>p b )

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

5. Diffusivity Equations for Flow in Porous Media PETE 613 (2005A) Slide — 5 Behavior of the c o variable as a function of pressure — example black oil case. Note the "jump" at p=p b, this behavior is due to the gas expansion at the bubblepoint. Diffusivity Equation: Black Oil — c o vs. p

6. Diffusivity Equations for Flow in Porous Media PETE 613 (2005A) Slide — 6 Diffusivity Equations for Solution-Gas Drive: (p<p b ) Oil Pseudopressure Form: (Accounts for  o and B o ) Oil Pseudopressure Definition: (p n is any reference pressure) Diffusivity Equation: Solution Gas Drive (p<p b )

7. Diffusivity Equations for Flow in Porous Media PETE 613 (2005A) Slide — 7 1/(  o B o ) vs. p (p b =5000 psia, T=175 Deg F) "Solution-Gas Drive" Pseudopressure Condition: (1/(  o B o ) vs. p) Concept: IF 1/(  o B o )  constant, THEN oil pseudopressure NOT required. 1/(  o B o ) is NEVER "constant" — but does not vary significantly with p. Oil pseudopressure calculation straightforward, but probably not necessary. Diffusivity Equation: Soln Gas Drive 1/(  o B o ) vs. p

8. Diffusivity Equations for Flow in Porous Media PETE 613 (2005A) Slide — 8 (  o c o ) vs. p (p b =5000 psia, T=175 Deg F) "Solution-Gas Drive" Pseudopressure Condition: ((  o c o ) vs. p) Concept: IF (  o c o )  constant, THEN oil pseudotime NOT required. (  o c o ) is NEVER "constant" — BUT, oil pseudotime would be very difficult. Other evidence suggests that ignoring (  o c o ) variance is acceptable. Diffusivity Equation: Soln Gas Drive (  o c o ) vs. p

9. Diffusivity Equations for Flow in Porous Media PETE 613 (2005A) Slide — 9 Solution-Gas Drive — Mobility/Compressibility (Camacho) "Solution-Gas Drive" Behavior: ((c t / t ) vs. time) Observation: (c t / t )  constant for p>p b and later, for p<p b. p wf = constant — but probably valid for any production/pressure scenario. Camacho-V., R.G. and Raghavan, R.: "Boundary-Dominated Flow in Solution-Gas-Drive Reservoirs,"SPERE (November 1989) 503-512. Diffusivity Equation: Soln Gas Drive (c t / t ) vs. t D

10. Diffusivity Equations for Flow in Porous Media PETE 613 (2005A) Slide — 10 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). Diffusivity Equation: Soln Gas Drive (2-phase p p )

11. Diffusivity Equations for Flow in Porous Media PETE 613 (2005A) Slide — 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:

12. Diffusivity Equations for Flow in Porous Media PETE 613 (2005A) Slide — 12 Dry Gas Pseudotime Condition (  g c g vs. p, T=200 Deg F) "Dry Gas" Pseudotime Condition: (  g c g vs. p) Concept: IF  g c g  constant, THEN pseudotime NOT required.  g c g is NEVER constant — pseudotime is always required (for liquid eq.). However, can generate numerical solution for gas cases (no pseudotime). Diffusivity Equation: Pseudotime (  g c g vs. p)

13. Diffusivity Equations for Flow in Porous Media PETE 613 (2005A) Slide — 13 Dry Gas — p 2 Relations Diffusivity Equations for a "Dry Gas:" p 2 Relations p 2 Form — Full Formulation: p 2 Form — Approximation: Diffusivity Equation: p 2 Relations

14. Diffusivity Equations for Flow in Porous Media PETE 613 (2005A) Slide — 14 Dry Gas p 2 Condition (  g z vs. p, T=200 Deg F) "Dry Gas" PVT Properties: (  g z vs. p) Concept: IF (  g z) = constant, THEN p 2 -variable valid. (  g z)  constant for p<2000 psia. Even with numerical solutions, p 2 formulation would not be appropriate. Diffusivity Equation: p 2 Relations (  g z vs. p)

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

16. Diffusivity Equations for Flow in Porous Media PETE 613 (2005A) Slide — 16 "Dry Gas" PVT Propertie s: (p/(  g z) vs. p ) Concept: IF p/(  g z) = constant, THEN p-variable is valid. p/(  g z) is NEVER constant — pseudopressure required (for liquid eq.). p formulation is never appropriate (even if generated numerically). Dry Gas p Condition (p/(  g z) vs. p, T=200 Deg F) Diffusivity Equation: p Relations (p/(  g z) vs. p)

17. Diffusivity Equations for Flow in Porous Media PETE 613 (2005A) Slide — 17 Multiphase Case — p-Form Relations (Perrine-Martin) Gas Equation: Oil Equation: Water Equation: Multiphase Equation: Compressibility Terms: Diffusivity Equation: Multiphase Relations

18. Diffusivity Equations for Flow in Porous Media PETE 613 (2005A) Slide — 18 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 Petroleum Engineering 613 Natural Gas Engineering Texas A&M University Lecture 04: Diffusivity Equations for Flow in Porous Media (End of Lecture)