P324 08A Lec 05 (PTA) Diffusivity Eq (080113 DI) Engineering 324 —Reservoir Performance Slide — 1/17 Department of Petroleum Engineering — Texas A&M U. Diffusivity Equation Lecture

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Petroleum Engineering 324 — Reservoir PerformanceDepartment of Petroleum Engineering — Texas A&M U.

Diffusivity EquationLecture 05 — T.A. Blasingame/Dilhan Ilk (13 February 2008)

Diffusivity Equation13 February 2008

Thomas A. Blasingame, Ph.D., P.E. Dilhan IlkDepartment of Petroleum Engineering Department of Petroleum EngineeringTexas A&M University Texas A&M UniversityCollege Station, TX 77843-3116 (USA) College Station, TX 77843-3116 (USA)+1.979.845.2292 +1.979.458.1499 [email protected] [email protected]

Petroleum Engineering 324Reservoir Performance

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Petroleum Engineering 324Reservoir Performance

Diffusivity EquationDevelopment

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Diffusivity Equation: Concept (1/3)

Discussion: Diffusivity EquationConceptually, the diffusivity equation is obtained by applying mass balance over a control volume.Equation of motion (Darcy's Law) and equation of state (PVT relations) are then combined with the mass balance equation to obtain the final form of the diffusivity equation.

Governing Equation: Mass Continuity Equation (Mass Balance of the system)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

Δ=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

Δ−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

Δ ttt interval, theduring system in the

onaccumulati mass of Rate

interval, theduringsystem theofout

flow mass of Rate

interval, theduringsystem theinto

flow mass of Rate

Control Volume (System)

Mass in Mass out

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Governing Equations: (Mathematical descriptions)


t∂∂

−=•∇)() ( φρνρ

r

) ( gpk rrρ

μν +∇−=

pc

∂∂

=ρ

ρ1

Equation of motion (Darcy's Law):

Mass continuity equation:

Equation of state (isothermal compressibility for a fluid):

Discussion: Diffusivity EquationDefinitions for the velocity and density terms are inserted in the mass continuity equation.

s)coordinateCartesian (in ,,),,( (Gradient) ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

∂∂

=∇zf

yf

xfzyxf

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Diffusivity equation for the flow of a single phase fluid in a porous media with respect to time and distance:


Discussion: Diffusivity EquationDiffusivity equation is a partial differential equation.Solution of the diffusivity equation requires an INITIAL CONDITION and TWO BOUNDARY CONDITIONS.Diffusivity equation (above) applies for any geometric configuration.Diffusivity equation (above) requires the following assumptions —homogeneous, isotropic formation and 100% saturated pore space.

tp

kc

ppc t∂∂

=∇+∇φμ22)(

tp

kc

p t∂∂

=∇φμ2

■Slightly Compressible Liquid: (General Form)

■Slightly Compressible Liquid: (Small p and c form)∇

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Diffusivity Equation: Reservoir Models

Discussion: Reservoir ModelsA model is a mathematical approximation of the real system combined with the rules of physics. The success of a model depends on reproducibility of the main drives of the system.We use ANALYTICAL models for the appropriate representation of the physical drive mechanisms of the system.ANALYTICAL model requires that the physical system should be replaced by a set of relatively simple LINEAR equations which model the diffusion process within a piece of rock, assume a simple shape for the reservoir and give the initial state of the system.

—Fr

om: K

appa

Eng

inee

ring

(Dyn

amic

Flo

w

Ana

lysi

s, 2

007)

—Fractured well model

—Horizontal well model

—Horizontal well with transverse vertical fractures

—Fully penetrating slanted well

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Diffusivity equation in cylindrical coordinates (radial flow most convenient for reservoir engineering applications):

Diffusivity Equation: Radial Flow

Discussion: Diffusivity EquationIf the permeability (k) is larger, the pressure change will be smaller.If the viscosity (μ) is larger, the pressure change will be smaller.The ratio, k/μ is called the mobility ratio.If the porosity (φ) is larger, the pressure change will be smaller.If the total compressibility (ct) is larger, the pressure change will be smaller.

tp

kc

rp

rrp t

∂∂

=∂∂

+∂

∂ φμ12

2

■Slightly Compressible Liquid:

■φ, μ, ct and k are assumed to be constants (not a function of pressure).

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Diffusivity Equation: Black Oil — μo and Bo vs. p

Behavior of the μo 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., μo and Bo constant) is not an absolute requirement, but this assumption is fundamental for the development of "liquid" flow solutions in reservoir engineering.

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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.

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Diffusivity Equation: Solution Gas Drive — p<pb

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

[ ] dpB

pp

Bpoobase

poopo n μμ 1

∫=

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Diffusivity Equation: Dry Gas Relations

Diffusivity Equations for a "Dry Gas:"General Form for Gas:

Diffusivity Relations:Pseudopressure/Time:Pseudopressure/Pseudotime:

Definitions:Pseudopressure: Pseudotime:

tp

zp

kcp

zp t

g ∂∂

=∇•∇φ

μ][

a

pptgp t

pc

kp n ∂

∂=∇ )(2 μφ

dpz

pppp

zp

gbasep

gpg

nμ

μ∫⎥

⎦

⎤⎢⎣

⎡= [ ] dt

pcptct

tgntga )()(1

0 μμ ∫=

tp

kc

p ptgp ∂

∂=∇φμ2

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

[ ] dtpcp

tcttgntga )()(

10 μ

μ ∫=

Diffusivity Equation: (Pseudotime Condition)

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Diffusivity Equation: p2 Condition

"Dry Gas" PVT Properties: (μgz vs. p)Concept: IF (μgz) = constant, THEN p2-variable valid.(μgz) ≅ constant for p<2000 psia.Even with numerical solutions, p2 formulation would not be appropriate.

dpz

pppp

zp

gbasep

gpg

nμ

μ∫⎥

⎦

⎤⎢⎣

⎡=

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

p2 Form — Approximation:

)()()][ln()( 2222

22 ptk

cpz

pp tg

g ∂∂

=∇∂

∂−∇

φμμ

)()( 222 ptk

cp tg

∂∂

=∇φμ

Diffusivity Equation: p2 Condition

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7



Diffusivity Equation: p Relations

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

dpz

pppp

zp

gbasep

gpg

nμ

μ∫⎥

⎦

⎤⎢⎣

⎡=

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Diffusivity Equation: p Relations

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

p Form — Approximation:

tp

kc

pp

zp

p tgg∂∂

=∇⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡∂∂

−∇φμμ 22 )(ln

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Diffusivity Equation: Multiphase CaseGas Equation:

Oil Equation:

Water Equation:

Multiphase Equation:

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡++

∂∂

=⎥⎥⎦

⎤

⎢⎢⎣

⎡∇⎥⎥⎦

⎤

⎢⎢⎣

⎡++•∇

w

wsw

o

oso

g

g

ww

wsw

oo

oso

gg

gBSR

BSR

BS

tp

BkR

BkR

Bk

φμμμ

⎥⎦

⎤⎢⎣

⎡∂∂

=⎥⎦

⎤⎢⎣

⎡∇•∇

o

o

oo

oBS

tp

Bk φ

μ

⎥⎦

⎤⎢⎣

⎡∂∂

=⎥⎦

⎤⎢⎣

⎡∇•∇

w

w

ww

wBS

tp

Bk φ

μ

tpcp

t

t∂∂

=∇λ

φ2

Compressibility Terms:

dpdR

BB

dpdB

Bc so

o

go

oo +−=

1

dpdR

BB

dpdB

Bc sw

w

gw

ww +−=

1

dpdB

Bc g

gg

1−=

fggwwoot cScScScc +++=w

w

g

g

o

ot

kkkμμμ

λ ++=

Documents

P324 08A Lec 05 (PTA) Diffusivity Eq (080113 DI) Engineering 324 —Reservoir Performance Slide — 1/17 Department of Petroleum Engineering — Texas A&M U. Diffusivity Equation Lecture