texas a&m university lecture 04: diffusivity equations 050131).pdf · pete 613 (2005a) slide —1...

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  • PETE 613(2005A)

    Slide 1Diffusivity Equationsfor Flow in Porous Media

    T.A. Blasingame, Texas A&M U.Department of Petroleum Engineering

    Texas A&M UniversityCollege Station, TX 77843-3116

    +1.979.845.2292 t-blasingame@tamu.edu

    Petroleum Engineering 613Natural Gas Engineering

    Texas A&M University

    Lecture 04:Diffusivity Equations

    for Flow in Porous Media

  • PETE 613(2005A)

    Slide 2Diffusivity Equationsfor Flow in Porous Media

    Diffusivity Equations: "Black Oil" (p>pb) "Solution-Gas Drive" (valid for all p, referenced for ppd)Multiphase Flow

    Lecture: Diffusivity Equations

  • PETE 613(2005A)

    Slide 3Diffusivity Equationsfor Flow in Porous Media

    Diffusivity Equations for a Black Oil:Slightly Compressible Liquid: (General Form)

    Slightly Compressible Liquid: (Small p and c form)

    tp

    kc

    p t

    2

    tp

    kc

    ppc t

    22)(

    Diffusivity Equation: Black Oil (p>pb)

  • PETE 613(2005A)

    Slide 4Diffusivity Equationsfor Flow in Porous Media

    Behavior of the o and Bo variables as functions of pressure foran example black oil case. Note behavior for p>pb both vari-ables should be considered to be "approximately constant" forthe sake of developing flow relations. Such an assumption (i.e.,o and Bo constant) is not an absolute requirement, but thisassumption is fundamental for the development of "liquid" flowsolutions in reservoir engineering.

    Diffusivity Equation: Black Oil o and Bo vs. p

  • PETE 613(2005A)

    Slide 5Diffusivity Equationsfor Flow in Porous Media

    Behavior of the co variable as a function of pressure exampleblack oil case. Note the "jump" at p=pb, this behavior is due tothe gas expansion at the bubblepoint.

    Diffusivity Equation: Black Oil co vs. p

  • PETE 613(2005A)

    Slide 6Diffusivity Equationsfor Flow in Porous Media

    Diffusivity Equations for Solution-Gas Drive: (p

  • PETE 613(2005A)

    Slide 7Diffusivity Equationsfor Flow in Porous Media

    1/(oBo) vs. p (pb=5000 psia, T=175 Deg F)

    "Solution-Gas Drive" Pseudopressure Condition: (1/(oBo) vs. p)Concept: IF 1/(oBo) constant, THEN oil pseudopressure NOT required.1/(oBo) is NEVER "constant" but does not vary significantly with p.Oil pseudopressure calculation straightforward, but probably not necessary.

    dpB

    pp

    Bpoobase

    poopo n 1

    Diffusivity Equation: Soln Gas Drive 1/(oBo) vs. p

  • PETE 613(2005A)

    Slide 8Diffusivity Equationsfor Flow in Porous Media

    (oco) vs. p (pb=5000 psia, T=175 Deg F)

    "Solution-Gas Drive" Pseudopressure Condition: ((oco) vs. p)Concept: IF (oco) constant, THEN oil pseudotime NOT required. (oco) is NEVER "constant" BUT, oil pseudotime would be very difficult.Other evidence suggests that ignoring (oco) variance is acceptable.

    dtpcp

    tctto

    ntoo,a )()(1

    0

    Diffusivity Equation: Soln Gas Drive (oco) vs. p

  • PETE 613(2005A)

    Slide 9Diffusivity Equationsfor Flow in Porous Media

    Solution-Gas Drive Mobility/Compressibility (Camacho)

    "Solution-Gas Drive" Behavior: ((ct/t) vs. time)Observation: (ct/t) constant for p>pb and later, for p

  • PETE 613(2005A)

    Slide 10Diffusivity Equationsfor Flow in Porous Media

    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 forthe two-phase system, and the pressure distribution maybe determined. However, it will be sufficient for thecalculation of the productivity factor to consider only thelimiting form ... (i.e., the constant property liquid relation).

    Diffusivity Equation: Soln Gas Drive (2-phase pp)

  • PETE 613(2005A)

    Slide 11Diffusivity Equationsfor Flow in Porous Media

    t

    p

    k

    cp ptgp

    2

    tp

    zp

    kc

    pz

    p tg

    ][

    a

    pptgp t

    pc

    kp

    n

    )(2

    dpz

    pppp

    zp

    gbasep

    gpg

    n

    dt

    pcptct

    tgntga )()(

    1

    0

    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)

    Slide 12Diffusivity Equationsfor Flow in Porous Media

    Dry Gas Pseudotime Condition (gcg vs. p, T=200 Deg F)

    "Dry Gas" Pseudotime Condition: (gcg vs. p)Concept: IF gcg constant, THEN pseudotime NOT required.gcg is NEVER constant pseudotime is always required (for liquid eq.).However, can generate numerical solution for gas cases (no pseudotime).

    dtpcp

    tcttgn

    tga )()(1

    0

    Diffusivity Equation: Pseudotime (gcg vs. p)

  • PETE 613(2005A)

    Slide 13Diffusivity Equationsfor Flow in Porous Media

    Dry Gas p2 Relations

    Diffusivity Equations for a "Dry Gas:" p2 Relationsp2 Form Full Formulation:

    p2 Form Approximation:

    )()()][ln()( 222222 p

    tk

    cpz

    pp tgg

    )()( 222 ptk

    cp tg

    Diffusivity Equation: p2 Relations

  • PETE 613(2005A)

    Slide 14Diffusivity Equationsfor Flow in Porous Media

    Dry Gas p2 Condition (gz vs. p, T=200 Deg F)

    "Dry Gas" PVT Properties: (gz vs. p)Concept: IF (gz) = constant, THEN p2-variable valid. (gz) constant for p

  • PETE 613(2005A)

    Slide 15Diffusivity Equationsfor Flow in Porous Media

    Dry Gas p RelationsDiffusivity Equations for a "Dry Gas:" p Relationsp Form Full Formulation:

    p Form Approximation:

    tp

    k

    cp

    p

    z

    pp tgg

    22 )(ln

    tp

    k

    cp tg

    2

    Diffusivity Equation: p Relations

  • PETE 613(2005A)

    Slide 16Diffusivity Equationsfor Flow in Porous Media

    "Dry Gas" PVT Properties: (p/(gz) vs. p)Concept: IF p/(gz) = constant, THEN p-variable is valid.p/(gz) is NEVER constant pseudopressure required (for liquid eq.).p formulation is never appropriate (even if generated numerically).

    Dry Gas p Condition (p/(gz) vs. p, T=200 Deg F)

    dpz

    pppp

    zp

    gbasep

    gpg

    n

    Diffusivity Equation: p Relations (p/(gz) vs. p)

  • PETE 613(2005A)

    Slide 17Diffusivity Equationsfor Flow in Porous Media

    Multiphase Case p-Form Relations (Perrine-Martin)Gas Equation:

    Oil Equation:

    Water Equation:

    Multiphase Equation:

    w

    wsw

    o

    oso

    g

    g

    ww

    wsw

    oo

    oso

    gg

    g

    BS

    RBS

    RB

    S

    tp

    Bk

    RB

    kR

    B

    k

    o

    o

    oo

    oBS

    tp

    Bk

    w

    w

    ww

    wBS

    tp

    Bk

    tpc

    pt

    t

    2

    Compressibility Terms:

    dpdR

    B

    B

    dpdB

    Bc so

    o

    go

    oo

    1

    dpdR

    B

    B

    dpdB

    Bc sw

    w

    gw

    ww

    1

    dp

    dB

    Bc g

    gg

    1

    fggwwoot cScScScc ww

    g

    g

    o

    ot

    kkk

    Diffusivity Equation: Multiphase Relations

  • PETE 613(2005A)

    Slide 18Diffusivity Equationsfor Flow in Porous Media

    T.A. Blasingame, Texas A&M U.Department of Petroleum Engineering

    Texas A&M UniversityCollege Station, TX 77843-3116

    +1.979.845.2292 t-blasingame@tamu.edu

    Petroleum Engineering 613Natural Gas Engineering

    Texas A&M University

    Lecture 04:Diffusivity Equations

    for Flow in Porous Media(End of Lecture)

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