REVIEW OF BASIC STEPS
IN DERIVATION OF FLOW
EQUATIONS
SIG4042 Reservoir Simulation
REVIEW OF BASIC STEPS
IN DERIVATION OF FLOW
EQUATIONS
SIG4042 Reservoir Simulation
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
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Contents
Multiphase Flow
Non-horizontal Flow
Mutlidimensional Flow
Coordinate Systems
Boundary and Initial
Conditions of Multiphase
Systems
Questions
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
Introduction
Introduction
Flow equations for flow in porous materials are based on a set of mass,
momentum and energy conservation equations, and constitutive
equations for fluids and the porous material.
For simplicity, we will in the following assume isothermal conditions, so that
we not have to involve an energy conservation equation. However, in
cases of changing reservoir temperature, such as in the case of cold
water injection into a warmer reservoir, this may be of importance.
Equations are described for linear, one-dimensional systems, but can easily
be extended to two and three dimensions, and to other coordinate
systems.
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
Conservation of Mass
Conservation of mass
Again we will consider the following one dimensional slab of porous
material:
fluid
x
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
Conservation of Mass
Mass conservation may be formulated across a control element of the
slab, with one fluid of density
, flowing through it at a velocity u:
x
u
The mass balance for the control element is then written as:
element
the
inside
mass
of
change
of
Rate
Dx
+
at x
element
the
of
out
Mass
at x
element
the
into
Mass
Continue
u
A
x
u
A
x x
t
Ax
or
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
Conservation of Mass
u
A
x
u
A
x x
t
Ax
Dividing equation above by
x, and taking the limit as x goes to zero,
we get the conservation of mass, or continuity equation:
t
A
u
A
x
t
u
x
For constant cross sectional area, the continuity equation simplifies to:
Continue
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
Conservation of Momentum
Conservation of momentum
Conservation of momentum is governed by the Navier-Stokes equation, but is
normally simplified for low velocity flow in porous materials to be described by
the semi-empirical Darcy's equation, which for one dimensional, horizontal flow is:
x
P
k
u
Continue
Alternative equations are the Forchheimer equation, for high velocity flow:
n
u
k
u
x
P
where n is proposed by Muscat to be 2.
The Brinkman equation, which applies to both porous and non-porous flow:
2
2
x
u
k
u
x
P
Brinkman's equation reverts to Darcy's equation for flow in porous media, since the
last term then normally is negligible, and to Stoke's equation for channel flow
because the Darcy part of the equation then may be neglected.
In the following, we assume that
Darcy's equation is valid for flow in
porous media.
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
Constitutive Equations
Constitutive equation for porous materials
To include pressure dependency in the porosity, we use the definition of
rock compressibility:
T
r
P
c
1
Keeping the temperature constant, the expression may be written:
r
c
dP
d
Normally, we may assume that the bulk volume of the porous material
is constant, i.e. the bulk compressibility is zero. This is not always
true, as witnessed by the subsidence in the Ekofisk area.
Continue
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
Constitutive Equations
Constitutive equations for fluids
Recall the familiar fluid compressibility definition, which applies to
any fluid at constant temperature:
T
f
P
V
V
c
1
Equally familiar is the gas equation, which for an ideal gas is:
PVnRT
For a real gas includes the deviation factor, Z:
PVnZRT
The gas density may be expressed as:
g
gS
P
Z
Z
S
P
S
where the subscript S denotes surface (standard) conditions.
Continue
These equations are frequently used in reservoir engineering applications.
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
Description of Black Oil model
For reservoir simulation purposes, we normally use either so-called Black Oil
fluid description, or compositional fluid description. For now, we will
consider the Black Oil model, and get back to compositional model later on.
Black Oil Model
conditions
standard
at
volume
conditions
reservoir
at
volume
B
conditions
standard
at
oil
of
volume
conditions
standard
at
oil
from
evolved
gas
of
volume
so
R
Continue
The density of oil at reservoir conditions is then, in terms of these
parameters and the densities of oil and gas, defined as:
o
oS
gs
R
so
B
o
Continue
The standard Black Oil model includes Formation Volume Factor, B, for each
fluid, and Solution Gas-Oil Ratio, R
so
, for the gas dissolved in oil, in
addition to viscosity and density for each fluid. A modified model may
also include oil dispersed in gas, r
s
, and gas dissolved in water, R
sw
. The
definitions of formation volume factors and solution gas-oil ratio are:
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
Black Oil Model
Typical pressure dependencies of the standard Black Oil parameters are
(click on buttons):
P
B
w
P
B
g
B
w
vs. P
B
g
vs. P
P
B
o
B
o
vs. P
P
w
w
vs. P
P
g
g
vs. P
P
o
o
vs. P
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
Black Oil Model
Typical pressure dependency of the Solution Gas-Oil Ratio in Black Oil model is
(click on button):
P
R
so
R
so
vs. P
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
Flow Equation
Flow equation
For single phase flow, in a one-dimensional, horizontal system,
assuming Darcy's equation to be applicable and that the cross
sectional area is constant, the flow equation becomes:
B
t
x
P
B
k
x
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
Boundary and Initial Conditions
Dirichlet conditions
When pressure conditions are specified, we normally would specify
the pressures at the end faces of the system in question. Applied
to the simple linear system described above, we may have the
following two pressure BC's at the ends:
R
L
P
t
L
x
P
P
t
x
P
0
,
0
,
0
For reservoir flow, a pressure condition will normally be specified as
a bottom-hole pressure of a production or injection well, at some
position of the reservoir. Strictly speaking, this is not a boundary
condition, but the treatment of this type of condition is similar to
the treatment of a boundary pressure condition.
More
Boundary conditions
We have two types of BC's: pressure conditions (Dirichlet conditions) and rate
conditions (Neumann conditions). The most common boundary conditions
in reservoirs, including sources/sinks, are discussed in the following.
Continue
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
Boundary and Initial Conditions
Neumann condition
Alternatively, we would specify the flow rates at the end faces of the
system in question. Using Darcy's equation at the ends of the
simple system above, the conditions become:
For reservoir flow, a rate condition may be specified as a production or
injection rate of a well, at some position of the reservoir, or it is
specified as a zero-rate across a sealed boundary or fault, or between
non-communicating layers.
More
0
x
L
x
P
kA
Q
L
x
L
x
P
kA
Q
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
Boundary and Initial Conditions
Initial condition (IC)
The initial condition specifies the initial state of the primary variables
of the system. For the simple case above, a constant initial
pressure may be specified as:
The initial pressure may be a function of postition. For non-horizontal
systems, hydrostatic pressure equilibrium is normally computed
based on a reference pressure and fluid densities:
Continue
P(x, t 0) P
0
P(z,t 0) P
ref
(z z
ref
)
g
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
Multiphase Flow
Multiphase flow
A continuity equation may be written for each fluid phase flowing:
The corresponding Darcy equations for each phase are:
Continue
l
l
l
l
S
t
u
x
x
P
kk
u
l
l
rl
l
g
w
o
l
,
,
g
w
o
l
,
,
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
Multiphase Flow
The continuity equation for gas has to be modified to include solution
gas as well as free gas, so that the oil equation only includes the
part of the oil remaining liquid at the surface:
Where
oL
represents the part of the oil remaining liquid at the surface
(in the stock tank), and
oG
the part that is gas at the surface.
Continue
o
oS
gS
R
so
B
o
oL
oG
o
oG
g
g
o
oG
g
g
S
S
t
u
u
x
o
oL
o
oL
S
t
u
x
Thus, the oil and gas continuity equations become:
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
Multiphase Flow
After substitution for Darcy's equations and Black Oil fluid properties,
and including well rate terms, the flow equations become:
The oil equation could be further modified to include dispersed oil in the gas,
if any, similarly to the inclusion of solution gas in the oil equation.
Continue
o
o
so
g
g
o
so
g
o
o
ro
so
g
g
g
rg
B
S
R
B
S
t
q
R
q
x
P
oB
kk
R
x
P
B
kk
x
Where:
w
o
cow
P
P
P
o
g
cog
P
P
P
S
l
lo,w,g
1
w
w
w
w
w
w
rw
B
S
t
q
x
P
B
kk
x
o
o
o
o
o
o
ro
B
S
t
q
x
P
B
kk
x
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
Non-horizontal Flow
Non-horizontal flow
For one-dimensional, inclined flow:
the Darcy equation becomes:
Continue
x
D
u
dx
dD
g
x
P
k
u
or, in terms of dip angle,
, and hydrostatic gradient:
sin
x
P
k
u
where
=g is the hydrostatic gradient of the fluid.
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
Multidimensional Flow
Multidimensional flow
The continuity equation for one-phase, three-
dimensional flow in cartesian coordinates, is:
The corresponding Darcy equations are:
More
t
u
z
u
y
u
x
z
y
x
x
D
x
P
k
u
x
x
y
D
y
P
k
u
y
y
z
D
z
P
k
u
z
z
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
Coordinate Systems
Coordinate systems
Normally, we use either a rectangular coordinate system or a cylindrical
coordinate system in reservoir simulation (click the buttons):
x
Rectangular
coordinates
y
z
r
Cylindrical
coordinates
z
r
Spherical
coordinates
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
Boundary and Initial Conditions of Multiphase
Systems
Boundary conditions of multiphase systems
The pressure and rate BC's discussed above apply to multiphase systems.
However, for a production well in a reservoir, we normally specify
either an oil production rate at the surface, or a total liquid rate at the
surface. Thus, the rate(s) not specified must be computed from
Darcy's equation. The production is subjected to maximum allowed
GOR or WC, or both. We will discuss these conditions later.
See a picture
O I L
O I L
G A S
W A T E R
Z
G O C
W O C
A Q U I F E R
BC:
1) P
bh
= constant
2) Q = constant
BC:
1) P
bh
= constant
2) Q
inj
= constant
BC:
k = 0
0
x
P
kA
q
BC:
q = 0
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
Boundary and Initial Conditions of Multiphase
Systems
Initial conditions of multiphase systems
In addition to specification of initial pressures, we also need to specify
initial saturations in a multiphase system. This requires knowledge of
water-oil contact (WOC) and gas-oil contact (GOC). Assuming that the
reservoir is in equilibrium, we may compute initial phase pressures
based on contact levels and densities. Then, equilibrium saturations
may be interpolated from the capillary pressure curves. Alternatively,
the initial saturations are based on measured logging data.
O I L
O IL
G A S
W A T E R
Z
G O C
W O C
See a picture
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
Questions
1) Write the mass balance equation (one-dimentional, one-phase).
Next
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.
8) Write continuity equation for one-phase, three-dimensional
flow in cartesian coordinates.
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
A
- area, m
2
B
- formation volume factor
c
- compressibility, 1/Pa
k
- permeability, m
2
k
r
- relative permeability, m
2
L
- lenght, m
N
- number of grid blocks
n
- number of moles
O(...) - discretization error
P
- pressure, Pa
P
c
- capillary pressure, Pa
Q, q - flow rate, Sm
3
/d
R
- gas constant
R
so
- solution gas-oil ratio
r
- radius, m
S
- saturation
T
- temperature, K
t
- time, s
u
- Darcy velocity, m/s
V
- volume, m
3
x
- distance, m
x, y, z
- spatial coordinate
Z
- deviation factor
- angle
x
- lenght of grid block, m
t
- time step, s
- porosity
- viscosity, Pa·s
- density, kg/m
3
Subscripts:
0
- initiall value
e
- end of reservoir
f
- fluid
g
- gas
i
- block number
L
- left side
l
- liquid
o
- oil
R
- right side
S
- surface (standard) conditions
r
- rock
w
- water
w
- well
Nomenclature
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
General information
Title:
Review of Basic Steps in Derivation of Flow Equations
Teacher(s):
Jon Kleppe
Assistant(s):
Szczepan Polak
Abstract:
Review of basic steps in derivation of flow equations for
flow in porous materials based on set of mass, momentum
and energy conservation equations, and constitutive
equations for fluid and porous material in isothermical
conditions.
Keywords:
conservation equations, constitutive equations, black oil
model, multiphase flow, non-horizontal flow,
multidimensional flow, coordinate systems
Topic discipline:
Reservoir Engineering -> Reservoir Simulation
Level:
4
Prerequisites:
Good knowledge of reservoir engineering
Learning goals:
Learn basic principles of Reservoir Simulation
Size in megabytes:
0.9
Software requirements:
-
Estimated time to complete:
60 minutes
Copyright information:
The author has copyright to the module
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
References
Aziz, K. and Settari, A.: Petroleum Reservoir Simulation, Applied
Science Publishers LTD, London (1979)
Mattax, C.C. and Kyte, R.L.: Reservoir Simulation, Monograph Series,
SPE,
Richardson, TX (1990)
Skjæveland, S.M. and Kleppe J.: Recent Advances in Improved Oil
Recovery Methods for North Sea Sandstone Reservoirs, SPOR
Monograph, Norvegian Petroleum Directoriate, Stavanger 1992
Initial
Conditions
Flow
Initial
Conditions of
Multiphase
Systems
About the Author
Name
Jon Kleppe
Position
Professor at Department of
Petroleum Engineering and
Applied Geophysics at NTNU
Address:
NTNU
S.P. Andersensvei 15A
7491 Trondheim
E-mail:
kleppe@ipt.ntnu.no
Phone:
+47 73 59 49 33
Web:
http://iptibm3.ipt.ntnu.no/~kleppe
/
