Petroleum Engineering 324
Reservoir Performance
Diffusivity Equation
Thomas A. Blasingame, Ph.D., P.E.
Department of Petroleum Engineering
Texas A&M University
College Station, TX 77843-3116 (USA)
+1.979.845.2292
t-blasingame@tamu.edu
Dilhan Ilk
Department of Petroleum Engineering
Texas A&M University
College Station, TX 77843-3116 (USA)
+1.979.458.1499
dilhan@tamu.edu
Petroleum Engineering 324 — Reservoir Performance
Department of Petroleum Engineering — Texas A&M U.
Diffusivity Equation
Lecture 05 — T.A. Blasingame/Dilhan Ilk (13 February 2008)
Slide — 1/17
13 February 2008
Petroleum Engineering 324
Reservoir Performance
Diffusivity Equation
Petroleum Engineering 324 — Reservoir Performance
Department of Petroleum Engineering — Texas A&M U.
Diffusivity Equation
Lecture 05 — T.A. Blasingame/Dilhan Ilk (13 February 2008)
Slide — 2/17
Development
Diffusivity Equation: Concept
(1/3)
Governing Equation: Mass Continuity Equation (Mass
Balance of the system)
Control Volume (System)
Mass in
Mass out
⎡Rate of mass flow ⎤ ⎡Rate of mass accumulation ⎤
⎥
⎢out of the system
⎥ = ⎢in the system during
⎥
⎢
⎥ ⎢
⎥⎦
⎢⎣during the interval, Δt ⎥⎦ ⎢⎣ the interval, Δt
z Discussion: Diffusivity Equation
„ Conceptually, 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.
Petroleum Engineering 324 — Reservoir Performance
Department of Petroleum Engineering — Texas A&M U.
Diffusivity Equation
Lecture 05 — T.A. Blasingame/Dilhan Ilk (13 February 2008)
Slide — 3/17
⎡Rate of mass flow ⎤
⎢into the system
⎥−
⎢
⎥
⎢⎣during the interval, Δt ⎥⎦
Diffusivity Equation: Concept
(2/3)
Governing Equations: (Mathematical descriptions)
Mass continuity equation:
Equation of motion (Darcy's
Law):
Equation of state (isothermal
compressibility for a fluid):
∂ (φρ )
∇ • (ρ ν ) = −
∂t
r
r
k
ν = − (∇p + ρ g )
r
μ
1 ∂ρ
c=
ρ ∂p
⎛ ∂f ∂f ∂f ⎞
(Gradient) ∇f ( x, y, z ) = ⎜⎜ , , ⎟⎟ (in Cartesian coordinates)
⎝ ∂x ∂y ∂z ⎠
„ Definitions for the velocity and density terms are inserted in the mass
continuity equation.
Petroleum Engineering 324 — Reservoir Performance
Department of Petroleum Engineering — Texas A&M U.
Diffusivity Equation
Lecture 05 — T.A. Blasingame/Dilhan Ilk (13 February 2008)
Slide — 4/17
z Discussion: Diffusivity Equation
Diffusivity Equation: Concept
(3/3)
z Diffusivity equation for the flow of a single phase fluid
in a porous media with respect to time and distance:
■Slightly Compressible Liquid: (General Form)
φμct ∂p
2
2
c(∇p ) + ∇ p =
k
∂t
■Slightly Compressible Liquid: (Small ∇p
and c form)
φμct ∂p
2
∇ p=
k
∂t
„ Diffusivity 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.
Petroleum Engineering 324 — Reservoir Performance
Department of Petroleum Engineering — Texas A&M U.
Diffusivity Equation
Lecture 05 — T.A. Blasingame/Dilhan Ilk (13 February 2008)
Slide — 5/17
z Discussion: Diffusivity Equation
— Fractured well model
— Horizontal well model
— Horizontal well with transverse
vertical fractures
z Discussion: Reservoir Models
— Fully penetrating slanted well
„ A 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.
Petroleum Engineering 324 — Reservoir Performance
Department of Petroleum Engineering — Texas A&M U.
Diffusivity Equation
Lecture 05 — T.A. Blasingame/Dilhan Ilk (13 February 2008)
Slide — 6/17
Analysis, 2007)
— From: Kappa Engineering (Dynamic Flow
Diffusivity Equation: Reservoir Models
Diffusivity Equation: Radial Flow
z Diffusivity equation in cylindrical coordinates (radial flow
most convenient for reservoir engineering applications):
■Slightly Compressible Liquid:
1 ∂p φμct ∂p
+
=
2
r ∂r
k ∂t
∂r
2
∂ p
■ φ, μ, ct and k are assumed to be constants (not a function of
pressure).
„ If 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.
Petroleum Engineering 324 — Reservoir Performance
Department of Petroleum Engineering — Texas A&M U.
Diffusivity Equation
Lecture 05 — T.A. Blasingame/Dilhan Ilk (13 February 2008)
Slide — 7/17
z Discussion: Diffusivity Equation
z Behavior of the μo and Bo variables as functions of pressure for
an example black oil case. Note behavior for p>pb — both variables 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.
Petroleum Engineering 324 — Reservoir Performance
Department of Petroleum Engineering — Texas A&M U.
Diffusivity Equation
Lecture 05 — T.A. Blasingame/Dilhan Ilk (13 February 2008)
Slide — 8/17
Diffusivity Equation: Black Oil — μo and Bo vs. p
z 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.
Petroleum Engineering 324 — Reservoir Performance
Department of Petroleum Engineering — Texas A&M U.
Diffusivity Equation
Lecture 05 — T.A. Blasingame/Dilhan Ilk (13 February 2008)
Slide — 9/17
Diffusivity Equation: Black Oil — co vs. p
Diffusivity Equation: Solution Gas Drive — p<pb
ppo = [μo Bo ] p
n
∫
p
1
dp
pbase μo Bo
„ 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.
Petroleum Engineering 324 — Reservoir Performance
Department of Petroleum Engineering — Texas A&M U.
Diffusivity Equation
Lecture 05 — T.A. Blasingame/Dilhan Ilk (13 February 2008)
Slide — 10/17
z "Solution-Gas Drive" Pseudopressure Condition: (1/(μoBo) vs. p)
Diffusivity Equation: Dry Gas Relations
z Diffusivity Equations for a "Dry Gas:"
„ General Form for Gas:
∇ •[
p
μg z
∇p] =
φct p ∂p
k z ∂t
„ Diffusivity Relations:
Pseudopressure/Time:
Pseudopressure/Pseudotime:
∇ 2 pp =
φμ g ct ∂pp
∂t
k
φ
∂pp
∇ p p = ( μ g ct ) pn
∂ta
k
2
„ Definitions:
⎡ μg z ⎤
ppg = ⎢
⎥
p
⎣
⎦ pn
∫
p
p
dp
pbase μ g z
Pseudotime:
[
]∫
1
t
t a = μ g ct
dt
n
0 μ g ( p )ct ( p )
Petroleum Engineering 324 — Reservoir Performance
Department of Petroleum Engineering — Texas A&M U.
Diffusivity Equation
Lecture 05 — T.A. Blasingame/Dilhan Ilk (13 February 2008)
Slide — 11/17
Pseudopressure:
Diffusivity Equation: (Pseudotime Condition)
[
]∫
1
t
t a = μ g ct
dt
n
0 μ g ( p )ct ( 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).
Petroleum Engineering 324 — Reservoir Performance
Department of Petroleum Engineering — Texas A&M U.
Diffusivity Equation
Lecture 05 — T.A. Blasingame/Dilhan Ilk (13 February 2008)
Slide — 12/17
z "Dry Gas" Pseudotime Condition: (μgcg vs. p)
Diffusivity Equation: p2 Condition
⎡ μg z ⎤
ppg = ⎢
⎥
p
⎣
⎦ pn
∫
p
p
dp
pbase μ g z
„ Concept: IF (μgz) = constant, THEN p2-variable valid.
„ (μgz) ≅ constant for p<2000 psia.
„ Even with numerical solutions, p2 formulation would not be appropriate.
Petroleum Engineering 324 — Reservoir Performance
Department of Petroleum Engineering — Texas A&M U.
Diffusivity Equation
Lecture 05 — T.A. Blasingame/Dilhan Ilk (13 February 2008)
Slide — 13/17
z "Dry Gas" PVT Properties: (μgz vs. p)
Diffusivity Equation: p2 Condition
z Diffusivity Equations for a "Dry Gas:" p2 Relations
„ p2 Form — Full Formulation:
2
2
∇ (p )−
∂
∂p
[ln(μ g z )]∇( p ) =
2
2 2
φμ g ct ∂
k
∂t
( p2 )
„ p2 Form — Approximation:
2
∇ (p ) =
φμ g ct ∂
k
∂t
( p2 )
Petroleum Engineering 324 — Reservoir Performance
Department of Petroleum Engineering — Texas A&M U.
Diffusivity Equation
Lecture 05 — T.A. Blasingame/Dilhan Ilk (13 February 2008)
Slide — 14/17
2
Diffusivity Equation: p Relations
⎡ μg z ⎤
ppg = ⎢
⎥
p
⎣
⎦ pn
∫
p
p
dp
pbase μ g z
„ 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).
Petroleum Engineering 324 — Reservoir Performance
Department of Petroleum Engineering — Texas A&M U.
Diffusivity Equation
Lecture 05 — T.A. Blasingame/Dilhan Ilk (13 February 2008)
Slide — 15/17
z "Dry Gas" PVT Properties: (p/(μgz) vs. p)
Diffusivity Equation: p Relations
z Diffusivity Equations for a "Dry Gas:" p Relations
„ p Form — Full Formulation:
∂ ⎡ ⎡ μ g z ⎤⎤
2 φμ g ct ∂p
∇ p − ⎢ln ⎢
⎥ (∇ p ) =
⎥
k ∂t
∂p ⎣ ⎣ p ⎦ ⎦
2
Petroleum Engineering 324 — Reservoir Performance
Department of Petroleum Engineering — Texas A&M U.
Diffusivity Equation
Lecture 05 — T.A. Blasingame/Dilhan Ilk (13 February 2008)
Slide — 16/17
„ p Form — Approximation:
Diffusivity Equation: Multiphase Case
Gas Equation:
⎡⎡ k g
ko
kw ⎤ ⎤ ∂ ⎡ ⎡ S g
So
Sw ⎤⎤
∇ • ⎢⎢
+ Rso
+ Rsw
+ Rso
+ Rsw
⎥∇p ⎥ = ⎢φ ⎢
⎥⎥
μo Bo
μ w Bw ⎥⎦ ⎥⎦ ∂t ⎢⎣ ⎢⎣ Bg
Bo
Bw ⎥⎦ ⎥⎦
⎢⎣ ⎢⎣ μ g Bg
Oil Equation:
Water Equation:
⎡ k
⎤ ∂⎡ S ⎤
∇ • ⎢ w ∇p ⎥ = ⎢φ w ⎥
⎣ μ w Bw
⎦ ∂t ⎣ B w ⎦
Multiphase Equation:
ct ∂p
ko k g k w
+
+
λt =
∇ p =φ
μo μ g μ w
λt ∂t
2
Compressibility Terms:
1 dBo B g dRso
co = −
+
Bo dp Bo dp
1 dBw B g dRsw
cw = −
+
Bw dp Bw dp
1 dB g
cg = −
B g dp
ct = co So + cw S w + c g S g + c f
Petroleum Engineering 324 — Reservoir Performance
Department of Petroleum Engineering — Texas A&M U.
Diffusivity Equation
Lecture 05 — T.A. Blasingame/Dilhan Ilk (13 February 2008)
Slide — 17/17
⎡ ko
⎤ ∂ ⎡ So ⎤
∇•⎢
∇p ⎥ = ⎢φ ⎥
⎣ μ o Bo
⎦ ∂t ⎣ Bo ⎦