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REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
NTNU
Author: professor Jon Kleppe
Assistant producers:
Farrokh Shoaei
Khayyam Farzullayev
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
REFERENCES ABOUT EXIT
Conservation of mass
Conservation of momentum
Black Oil Model
Flow Equation
Initial and Boundary Conditions
Multiphase Flow
Non-horizontal Flow
Multidimensional Flow
Coordinate Systems
QUESTIONS
Control element
Mass balance:
X
element theinside
mass of change of Rate
x+at xelement
theofout Mass
at xelement
theinto Mass
or
uA x uA
xxt
Ax
u
t
AuAx
Conservation of mass
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
REFERENCES ABOUT EXIT
Conservation of mass
Conservation of momentum
Black Oil Model
Flow Equation
Initial and Boundary Conditions
Multiphase Flow
Non-horizontal Flow
Multidimensional Flow
Coordinate Systems
QUESTIONS
where n is proposed by Muscat to be 2.
Forchheimer equation:
Semi-empirical Darcy's equation:
Conservation of momentum
x
Pku
nuk
ux
P
Brinkman equation:
2
2
x
u
ku
x
P
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
REFERENCES ABOUT EXIT
Conservation of mass
Conservation of momentum
Black Oil Model
Flow Equation
Initial and Boundary Conditions
Multiphase Flow
Non-horizontal Flow
Multidimensional Flow
Coordinate Systems
QUESTIONS
Constitutive equation for porous materials:
rock compressibility:
Tr P
c
1
rcdP
d
Tf P
V
Vc
1
Constitutive equations for fluids :
fluid compressibility :
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
REFERENCES ABOUT EXIT
Conservation of mass
Conservation of momentum
Black Oil Model
Flow Equation
Initial and Boundary Conditions
Multiphase Flow
Non-horizontal Flow
Multidimensional Flow
Coordinate Systems
QUESTIONS
Solution Gas-Oil Ratio, Rso:
ρgS : density of gas at standard conditions
ρoS : density of oil at standard conditions
Formation Volume Factor for each fluid, B :
Black Oil Model
conditions standardat volume
conditionsreservoir at volumeB
conditions standardat oil of volume
conditions standardat oil from evolved gas of volumesoR
o
sogSoSo B
R
The main parameters :
density of oil at reservoir conditions:
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
REFERENCES ABOUT EXIT
Conservation of mass
Conservation of momentum
Black Oil Model
Flow Equation
Initial and Boundary Conditions
Multiphase Flow
Non-horizontal Flow
Multidimensional Flow
Coordinate Systems
QUESTIONS
Typical pressure dependencies
Bw
P P
Bg
g
PP
w o
P
Bo
P P
Rso
Pb Pb
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
REFERENCES ABOUT EXIT
Conservation of mass
Conservation of momentum
Black Oil Model
Flow Equation
Initial and Boundary Conditions
Multiphase Flow
Non-horizontal Flow
Multidimensional Flow
Coordinate Systems
QUESTIONS
For a :
Single phase flow
One-dimensional
Horizontal system
Assuming
Darcy's equation to be applicable. The cross sectional area is constant.
The flow equation becomes:
Flow Equation
Btx
P
B
k
x
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
REFERENCES ABOUT EXIT
Conservation of mass
Conservation of momentum
Black Oil Model
Flow Equation
Initial and Boundary Conditions
Multiphase Flow
Non-horizontal Flow
Multidimensional Flow
Coordinate Systems
QUESTIONS
For non-horizontal systems:
1. Initial conditions (IC) :
The initial state of the primary variables of the system:
P(x, t 0) P0
P(z,t 0) Pref (z zref )g
Where Pref is reference pressure & ρ is fluid densities
Initial and Boundary Conditions
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
REFERENCES ABOUT EXIT
Conservation of mass
Conservation of momentum
Black Oil Model
Flow Equation
Initial and Boundary Conditions
Multiphase Flow
Non-horizontal Flow
Multidimensional Flow
Coordinate Systems
QUESTIONS
2. Basic types of BC’s:
Pressure conditions (Dirichlet conditions)
Rate conditions (Neumann conditions)
Dirichlet conditions: Applied to the simple linear system described above:
Neumann conditions:
Specify the flow rates at the end faces of the system:
LPtxP )0,0(
RPtLxP )0,(
0
x
L x
PkAQ
LxR x
PkAQ
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
REFERENCES ABOUT EXIT
Conservation of mass
Conservation of momentum
Black Oil Model
Flow Equation
Initial and Boundary Conditions
Multiphase Flow
Non-horizontal Flow
Multidimensional Flow
Coordinate Systems
QUESTIONS
O IL OIL
GAS
W ATER
Z
GOC
W OC
Before production:
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
REFERENCES ABOUT EXIT
Conservation of mass
Conservation of momentum
Black Oil Model
Flow Equation
Initial and Boundary Conditions
Multiphase Flow
Non-horizontal Flow
Multidimensional Flow
Coordinate Systems
QUESTIONS
O ILOIL
GAS
W ATER
Z
GOC
W OC
AQUIFER
BC:
1) Pbh = constant
2) Q = constant
BC:
1) Pbh = constant
2) Qinj = constant
BC:
k = 0 0
x
PkAq
BC:
q = 0
After production:
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
REFERENCES ABOUT EXIT
Conservation of mass
Conservation of momentum
Black Oil Model
Flow Equation
Initial and Boundary Conditions
Multiphase Flow
Non-horizontal Flow
Multidimensional Flow
Coordinate Systems
QUESTIONS
Multiphase Flow
llll St
ux
gwol ,,
x
Pkku l
l
rll
Continuity equation for each fluid phase :
Darcy equation for each phase :
gwol ,,
Oil density equation:
o oS gSRso
Bo
oL oG
oL: the part of oil remaining liquid at the surface oG : the part that is gas at the surface
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
REFERENCES ABOUT EXIT
Conservation of mass
Conservation of momentum
Black Oil Model
Flow Equation
Initial and Boundary Conditions
Multiphase Flow
Non-horizontal Flow
Multidimensional Flow
Coordinate Systems
QUESTIONS
The oil continuity equation:
ooLooL St
ux
The gas continuity equation:
ooGggooGgg SSt
uux
the oil equation only includes the part of the oil remaining liquid at the surface which is: ρOL
The continuity equation for gas has to be modified to include solution gas as well as free gas which is: ρG & ρOG
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
REFERENCES ABOUT EXIT
Conservation of mass
Conservation of momentum
Black Oil Model
Flow Equation
Initial and Boundary Conditions
Multiphase Flow
Non-horizontal Flow
Multidimensional Flow
Coordinate Systems
QUESTIONS
After substituting the
Darcy's equations
Black Oil fluid properties and
Well rate terms
The flow equations become:
o
oo
o
oo
ro
B
S
tq
x
P
B
kk
x
o
oso
g
gosog
o
o
roso
g
gg
rg
B
SR
B
S
tqRq
x
P
oB
kkR
x
P
B
kk
x
w
ww
w
ww
rw
B
S
tq
x
P
B
kk
x
wocow PPP
ogcog PPP
Sl
lo, w, g 1
Where:
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
REFERENCES ABOUT EXIT
Conservation of mass
Conservation of momentum
Black Oil Model
Flow Equation
Initial and Boundary Conditions
Multiphase Flow
Non-horizontal Flow
Multidimensional Flow
Coordinate Systems
QUESTIONS
– dip angle: α– hydrostatic gradient : g=rg
Non-horizontal Flow
One-dimensional, inclined flow:
dx
dDg
x
Pku
x
u D
Darcy equation:
which can be written:
sinx
Pku
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
REFERENCES ABOUT EXIT
Conservation of mass
Conservation of momentum
Black Oil Model
Flow Equation
Initial and Boundary Conditions
Multiphase Flow
Non-horizontal Flow
Multidimensional Flow
Coordinate Systems
QUESTIONS
Multidimensional Flow
One-phase Three-dimensional flow Cartesian coordinates
t
uz
uy
ux zyx
x
D
x
Pku x
x
y
D
y
Pku y
y
z
D
z
Pku z
z
Corresponding Darcy equations :
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
REFERENCES ABOUT EXIT
Conservation of mass
Conservation of momentum
Black Oil Model
Flow Equation
Initial and Boundary Conditions
Multiphase Flow
Non-horizontal Flow
Multidimensional Flow
Coordinate Systems
QUESTIONS
Coordinate Systems
Rectangular coordinates:
r
z
y
x
z
r
Cylindrical coordinates:
Spherical coordinates:
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
REFERENCES ABOUT EXIT
Conservation of mass
Conservation of momentum
Black Oil Model
Flow Equation
Initial and Boundary Conditions
Multiphase Flow
Non-horizontal Flow
Multidimensional Flow
Coordinate Systems
QUESTIONS
Questions
1. Write the mass balance equation (one-dimentional, one-phase).
8. Write continuity equation for one-phase, three-dimensional flow in cartesian
coordinates.
2. Write the most common relationship between velocity and pressure, and write
an alternative relationship used for high fluid velocities.
3. Write the expression for the relationship between porosity and pressure.
4. List 3 commonly used expressions for relating fluid density to pressure.
5. Describe briefly Black Oil model.
6. Sketch typical dependencies of the standard Black Oil parameters.
7. Write Darcy equation for one-dimentional, inclined flow.
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
REFERENCES ABOUT EXIT
Conservation of mass
Conservation of momentum
Black Oil Model
Flow Equation
Initial and Boundary Conditions
Multiphase Flow
Non-horizontal Flow
Multidimensional Flow
Coordinate Systems
QUESTIONS
References
Kleppe J.: Reservoir Simulation, Lecture note 2
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
REFERENCES ABOUT EXIT
Conservation of mass
Conservation of momentum
Black Oil Model
Flow Equation
Initial and Boundary Conditions
Multiphase Flow
Non-horizontal Flow
Multidimensional Flow
Coordinate Systems
QUESTIONS
Title: REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS (PDF)
Author: Name: Prof. Jon Kleppe
Address:NTNU
S.P. Andersensvei 15A
7491 Trondheim
Website
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