REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
Author: professor Jon Kleppe
Assistant producers:
Farrokh Shoaei
Khayyam Farzullayev
NTNU
ENTER
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
Conservation of mass
Conservation of
mass
Conservation of
momentum
 Control element
u
Black Oil Model

DX
Flow Equation
Initial and
Boundary
Conditions
 Mass balance:
Multiphase Flow
Mass into the Mass out of the  Rate of change of mass 




element
at
x
element
at
x
+
D
x
inside
the
element

 
 

Non-horizontal
Flow
or
uAx  uAx  Dx 
Multidimensional
Flow
Coordinate
Systems
QUESTIONS

REFERENCES
ABOUT

ADx
t

 Au   A   
x
t
EXIT
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
Conservation of momentum
Conservation of
mass
Conservation of
momentum

Semi-empirical Darcy's equation:
Black Oil Model
u
k P
 x
Flow Equation
Initial and
Boundary
Conditions

Forchheimer equation:

Multiphase Flow
P

 u  u n
x
k
where n is proposed by Muscat to be 2.
Non-horizontal
Flow
Multidimensional
Flow

Brinkman equation:
P

 2u

u  2
x
k
x
Coordinate
Systems
QUESTIONS
REFERENCES
ABOUT
EXIT
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
Conservation of
mass

Conservation of
momentum
Constitutive equation for porous materials:
 rock compressibility:
Black Oil Model
 1   
cr    
   P T
Flow Equation
d
 c r
dP
Initial and
Boundary
Conditions
Multiphase Flow
Non-horizontal
Flow

Constitutive equations for fluids :
 fluid compressibility :
Multidimensional
Flow
 1  V 
c f   

 V  P T
Coordinate
Systems
QUESTIONS
REFERENCES
ABOUT
EXIT
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
Black Oil Model
Conservation of
mass
Conservation of
momentum
Black Oil Model

The main parameters :
Formation Volume Factor for each fluid, B :
B 
volume at reservoir conditions
volume at standard conditions
Flow Equation
Initial and
Boundary
Conditions
Multiphase Flow
Solution Gas-Oil Ratio, Rso:
Rso 
density of oil at reservoir conditions:
volume of gas evolved from oil at standard conditions
volume of oil at standard conditions
o 
Non-horizontal
Flow
 oS   gS Rso
Bo
ρoS : density of oil at standard conditions
Multidimensional
Flow
ρgS : density of gas at standard conditions
Coordinate
Systems
QUESTIONS
REFERENCES
ABOUT
EXIT
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
Conservation of
mass
Typical pressure dependencies
Conservation of
momentum
Black Oil Model
Bw
Bg
Bo
Rso
Flow Equation
Initial and
Boundary
Conditions
P
P
Pb
P
Pb
P
Multiphase Flow
Non-horizontal
Flow
w
g
o
Multidimensional
Flow
Coordinate
Systems
QUESTIONS
P
REFERENCES
ABOUT
P
P
EXIT
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
Flow Equation
Conservation of
mass
Conservation of
momentum

For a :
 Single phase flow
Black Oil Model
 One-dimensional
Flow Equation
 Horizontal system
Initial and
Boundary
Conditions

Assuming
Multiphase Flow
 Darcy's equation to be applicable.
 The cross sectional area is constant.
Non-horizontal
Flow
Multidimensional
Flow

The flow equation becomes:
Coordinate
Systems
QUESTIONS
REFERENCES
ABOUT
  k P     

  
x  B x  t  B 
EXIT
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
Initial and Boundary Conditions
Conservation of
mass
Conservation of
momentum
Black Oil Model
1.
Initial conditions (IC) :

The initial state of the primary variables of the system:
Flow Equation
P(x, t  0)  P0
Initial and
Boundary
Conditions
Multiphase Flow

For non-horizontal systems:
Non-horizontal
Flow
P(z,t  0)  Pref  (z  zref ) g
Multidimensional
Flow
Where Pref is reference pressure & ρ is fluid densities
Coordinate
Systems
QUESTIONS
REFERENCES
ABOUT
EXIT
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
Conservation of
mass
2.
Conservation of
momentum
Basic types of BC’s:

Pressure conditions (Dirichlet conditions)

Rate conditions (Neumann conditions)
Black Oil Model
Flow Equation

Dirichlet conditions:
Applied to the simple linear system described above:
Initial and
Boundary
Conditions
P( x  0, t  0)  PL
Multiphase Flow
P( x  L, t  0)  PR
Non-horizontal
Flow

Neumann conditions:
Specify the flow rates at the end faces of the system:
Multidimensional
Flow
Coordinate
Systems
QUESTIONS
REFERENCES
ABOUT
QL  
kA  P 
 
  x  x 0
QR  
kA  P 
 
  x  x  L
EXIT
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
Conservation of
mass
Before production:
Conservation of
momentum
Black Oil Model
GOC
Flow Equation
GAS
OIL
Initial and
Boundary
Conditions
OIL
WOC
Multiphase Flow
WATER
Non-horizontal
Flow
Z
Multidimensional
Flow
Coordinate
Systems
QUESTIONS
REFERENCES
ABOUT
EXIT
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
Conservation of
mass
After production:
Conservation of
momentum
BC:
Black Oil Model
BC:
kA P
0
k=0q
 x
Flow Equation
1) Pbh = constant
OIL
OIL
Initial and
Boundary
Conditions
Multiphase Flow
2) Q = constant
GAS
GOC
WOC
WATER
BC:
Non-horizontal
Flow
q=0
AQUIFER
Multidimensional
Flow
BC:
1) Pbh = constant
2) Qinj = constant
Z
Coordinate
Systems
QUESTIONS
REFERENCES
ABOUT
EXIT
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
Multiphase Flow
Conservation of
mass
Conservation of
momentum

Continuity equation for each fluid phase :

Black Oil Model

 l u l    l S l 
x
t
l  o, w, g
Flow Equation

Darcy equation for each phase :
Initial and
Boundary
Conditions
ul  
kkrl Pl
 l x
l  o, w, g
Multiphase Flow
Non-horizontal
Flow

Oil density equation:
o 
Multidimensional
Flow
Bo
  oL   oG
oL: the part of oil remaining liquid at the surface
oG : the part that is gas at the surface
Coordinate
Systems
QUESTIONS
 oS   gS Rso
REFERENCES
ABOUT
EXIT
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
Conservation of
mass
Conservation of
momentum

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
Black Oil Model
Flow Equation

The oil continuity equation:
Initial and
Boundary
Conditions


 oLuo    oL So 
x
t
Multiphase Flow

The gas continuity equation:
Non-horizontal
Flow

Multidimensional
Flow

 g u g  oGuo    g S g  oG So 
x
t
Coordinate
Systems
QUESTIONS
REFERENCES
ABOUT
EXIT
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
Conservation of
mass

After substituting the
Darcy's equations
Black Oil fluid properties and
Conservation of
momentum
Black Oil Model
Well rate terms
The flow equations become:
Flow Equation
  kkro Po 
  S 

  qo   o 
x   o Bo x 
t  Bo 
Initial and
Boundary
Conditions
kk Po 
S 
  kkrg Pg
  S g
 Rso ro
 qg  Rso qo  
 Rso o 
x   g Bg x
oBo x 
t  Bg
Bo 
Multiphase Flow
  kkrw Pw 
  S 

  qw   w 
x   w Bw x 
t  Bw 
Non-horizontal
Flow
Where:
Multidimensional
Flow
Pcow  Po  Pw
Pcog  Pg  Po
Coordinate
Systems
S
l
1
l  o, w, g
QUESTIONS
REFERENCES
ABOUT
EXIT
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
Non-horizontal Flow
Conservation of
mass
Conservation of
momentum

One-dimensional, inclined flow:
u
Black Oil Model
a
D
Flow Equation
Initial and
Boundary
Conditions
Multiphase Flow
x

Darcy equation:
u
Non-horizontal
Flow
k  P
dD 
 g


  x
dx 
which can be written:
Multidimensional
Flow
u
Coordinate
Systems
QUESTIONS
–
–
REFERENCES
ABOUT
k  P

  sin a 

  x

dip angle: α
hydrostatic gradient : g=rg
EXIT
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
Multidimensional Flow
Conservation of
mass
Conservation of
momentum



One-phase
Three-dimensional flow
Cartesian coordinates
Black Oil Model

Flow Equation
Initial and
Boundary
Conditions


u x    u y    u z     
x
y
z
t
Corresponding Darcy equations :
Multiphase Flow
ux  
k x  P
D 



  x
x 
uy  
k y  P
D 



  y
y 
uz  
k z  P
D 



  z
z 
Non-horizontal
Flow
Multidimensional
Flow
Coordinate
Systems
QUESTIONS
REFERENCES
ABOUT
EXIT
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
Coordinate Systems
Conservation of
mass
Conservation of
momentum


Rectangular coordinates:
Cylindrical coordinates:
y
r
Black Oil Model
x
Flow Equation
Initial and
Boundary
Conditions
z
Multiphase Flow
Non-horizontal
Flow

Spherical coordinates:
j
Multidimensional
Flow
Coordinate
Systems
QUESTIONS
z
r
q
REFERENCES
ABOUT
EXIT
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
Questions
Conservation of
mass
Conservation of
momentum
1.
Write the mass balance equation (one-dimentional, one-phase).
2.
Write the most common relationship between velocity and pressure, and write an
alternative relationship used for high fluid velocities.
Black Oil Model
Flow Equation
3.
Write the expression for the relationship between porosity and pressure.
Initial and
Boundary
Conditions
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.
Non-horizontal
Flow
7.
Write Darcy equation for one-dimentional, inclined flow.
Multidimensional
Flow
8.
Write continuity equation for one-phase, three-dimensional flow in cartesian
Multiphase Flow
coordinates.
Coordinate
Systems
QUESTIONS
REFERENCES
ABOUT
EXIT
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
References
Conservation of
mass
Conservation of
momentum
Black Oil Model

Kleppe J.: Reservoir Simulation, Lecture note 2
Flow Equation
Initial and
Boundary
Conditions
Multiphase Flow
Non-horizontal
Flow
Multidimensional
Flow
Coordinate
Systems
QUESTIONS
REFERENCES
ABOUT
EXIT
REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
About this module
Conservation of
mass
Conservation of
momentum
Black Oil Model

Title: REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS (PDF)

Author:

Flow Equation
 Address:
NTNU
S.P. Andersensvei 15A
7491 Trondheim
Initial and
Boundary
Conditions
 Website
Multiphase Flow
 Email
Non-horizontal
Flow
Multidimensional
Flow
Name: Prof. Jon Kleppe

Size: 450 Kb
Coordinate
Systems
QUESTIONS
REFERENCES
ABOUT
EXIT