STEADY
STATE
FLOW
PGG 321 GROUP 4
PRESENTATIONMODULE 3
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Table of contents
01
02
Linear Flow of
Incompressible Fluid
Linear Flow of Slighlty
Compressible Fluids
03
04
Linear Flow of
Compressible Fluid
Fluid Flow through
Linear beds in series
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Table of contents
05
Fluid Flow through
Linear beds in parallel
06
Poiseuille’s Law of
Capillary Flow & Flow
through Fractures
Darcy’s
Law
Darcy's law has been reviewed and the classification of flow
systems has been discussed in the previous classes. We would be
moving into the first classification; Steady State Flow. This
section covers both linear and radial flow geometries, as they find
widespread applications across various systems. Equations are
derived for all three fluid types: incompressible, slightly
compressible, and compressible, within both linear and radial
geometries
Linear Flow of Incompressible Fluids,
Steady State
In the linear system, it is assumed the flow occurs through a constant crosssectional area A, where both ends are entirely open to flow. It is also assumed
that no flow crosses the sides, top, or bottom as shown in the figure below.
Linear Flow of Incompressible Fluids,
Steady State
If the fluid is incompressible, or essentially so for all engineering purposes, then
the velocity is the same at all points, as is the total flow rate across any cross
section, so that
Separating variables and integrating over the length of the linear system gives;
Linear Flow of Incompressible Fluids,
Steady State
where
q = flow rate (bbl/day)
k = absolute permeability
p = pressure (psia)
= viscosity (cp)
L = distance (ft)
A = Cross-sectional area (ft2)
Linear Flow of Incompressible Fluids,
Steady State
Examples:- An incompressible fluid flows in a linear porous media with the followin
properties.
L = 2000ft
h = 20ft
width = 300ft
k = 100md
O = 15%
= 2cp
p1 = 2000psi
p2 = 1990psi
a.) Flow rate in bbl/day
b.) Apparent fluid velocity in ft/day
c.) Actual fluid velocity in ft/day
d.) Flow rate in stb/day if give as 1.127bbl/day
Linear Flow of Incompressible Fluids,
Steady State
Solution
A =h x width = 20 x 300= 6000ft2
a.)
=
𝑞
0.001127 𝑥 100 𝑥 6000 (2000 − 1990)
= 1.6905bbl/day
2 𝑥 2000
1.6905 𝑥 5.605
= 0.0016 ft/day
6000
𝑞
1.6905 𝑥 5.615
c.) ∅ 𝑎 = 0.15 𝑥 6000 = 0.0105 ft/day
1
𝑏𝑏𝑙
𝑆𝑇𝐵
𝑆𝑇𝐵
d.) q x 𝐵 = 1.6905 𝑑𝑎𝑦 x 0.8873 𝑏𝑏𝑙 = 1.4999 𝑑𝑎𝑦
b.) 𝑎 =
Linear Flow of Incompressible Fluids,
Steady State
Here are some key characteristics of linear flow of incompressible fluid in a steady
state:  Constant Flow Direction
 Uniform Velocity Profile (However, in real-world scenarios, velocity may vary due
to factors such as friction with the walls of the conduit.)
 - Conservation of Mass
 - Conservation of Momentum
 - No Mixing or Cross-Flow
Linear Flow of Slightly Compressible
Fluids, Steady State
The equation governing the flow of slightly compressible fluids differs from the one
derived in the previous section, primarily due to the tendency of slightly
compressible fluids to expand as pressure decreases.
As discussed earlier the previous class, outlines the correlation between pressure
and volume for such fluids. Additionally, the product of the flow rate, measured in
SIB units, and the formation volume factor follows a comparable pressuredependent pattern and is expressed as;
Linear Flow of Slightly Compressible
Fluids, Steady State
Where qR is the flow rate at some reference pressure, pR. If Darcy's law is written
for this case, variables separated, and the resulting equation integrated over the
length of the porous body, the following is obtained
This integration assumes a constant compressibility over the entire pressure drop.
Linear Flow of Slightly Compressible
Fluids, Steady State
The above calculation shows that q1 and q2 are not largely different which is due to the fact that
the liquid is slightly incompressible and its volume is a function of pressure
Linear Flow of Compressible Fluids,
Steady State
In a steady-state, linear system, the rate of gas flow, measured in standard cubic feet
per day, remains consistent across all cross-sections. However, due to gas expansion
as pressure decreases, the velocity increases towards the downstream end compared
to the upstream end. Consequently, the pressure gradient intensifies towards the
downstream end. At any given cross-section 'x' (as illustrated in Fig. 1.0) with pressure
'p', the flow can be expressed in terms of standard cubic feet per day by incorporating
the definition of the gas formation volume factor.
Linear Flow of Compressible Fluids,
Steady State
Linear Flow of Compressible Fluids,
Steady State
• We assumed constant z and µ over the specified pressures,
i.e., 𝑃1 and 𝑃2 while integrating the eqn.
• Where 𝑄𝑠𝑐=gas flow rate at standard conditions, scf/day;
z=gas compressibility factor; 𝑇𝑠𝑐, 𝑃𝑠𝑐= standard temperature
and pressure in oR and psia respectively; T= temperature, °R;
k= permeability, md; µ= gas viscosity, cp;
A= cross-sectional area, ft2; L= total length of the linear
system, ft
• Setting 𝑃𝑠𝑐=14.7 psi and 𝑇𝑠𝑐= 520 °R, the equation becomes
𝑄𝑠𝑐 =
0.1119244𝐴𝐾(𝑃12 −22)
𝑍𝑇𝜇𝐿
Linear Flow of Compressible Fluids,
Steady State
• But z and µ are a very strongly dependent on pressure,
and we removed them from the from the integral to
simplify the final form of the gas flow equation.
• The equation is valid for applications when the pressure <
2000 psi.
• The gas properties must be evaluated at the average
pressure ത𝑃 as defined in the equation below. This is reffered to as
pressure-squared method
𝑃12 + 𝑃22
2
• P=√
Linear Flow of Compressible Fluids,
Steady State
Examples:- A linear porous media is flowing a 0.72 specific gravity gas at
600°R. The upstream and downstream pressures are 2100 psi
and 1894.73 psi, respectively. The cross-sectional area is
constant at 4500 𝑓𝑡2. The total length is 2500 feet with an
absolute permeability of 60 md. Calculate the gas flow rate in
scf/day (psc= 14.7 psia, Tsc= 520°R, z = 0.78, µ= 0.0173 cp).
Solution
𝑄𝑠𝑐 =
0.1119244𝐴𝐾(𝑃12 −𝑃22)
𝑍𝑇𝜇𝐿
=
0.1119244 𝑥 4500 𝑥 60(21002 − 1894.732)
0.78 𝑥 600 𝑥 0.0173𝑥 2500
= 122428.238𝑠𝑐𝑓/𝑑𝑎𝑦
Fluid Flow through Linear beds in series
Consider linear flow for flow units in series as shown in the diagram below:
Fluid Flow through Linear beds in series
Flow going through flow units in series can be treated as resistors in series, where flow rate (q)
represents current (I), pressure (P) represents voltage (V), and flow resistance (
𝑳
) represents
𝒌𝑨
resistance (Re). For resistors in series, the following rules apply:
The current is the same across each resistor.
The sum of the potential differences across individual resistors is equal to the total voltage drop across
all resistors.
This analogy can be extended to fluid flow, i.e.:
flow rate is the same across each flow unit for flow units in series.
the sum of the pressure drops across individual flow units is equal to the total pressure drop across all
flow units
Knowing these rules, we will proceed to the derivation of average permeability for this system.
Using Rule 2 the following expression can be derived:
1. ∆𝑃𝑡 = 3𝑖=1 ∆𝑃𝑖 = ∆𝑃1+ ∆𝑃2+ ∆𝑃3
Fluid Flow through Linear beds in series
Using Darcy’s law, the total pressure drop (∆𝑃𝑡=Pi-Pf )across the entire system is given by the
following expression:
2.
Using Darcy’s law, the total pressure drop across a flow unit () is given by the following expression:
3.
Substituting Equations (2) and (3) into Equation (1) we can develop the following expression:
4.
Fluid Flow through Linear beds in series
Using Rule 1 and realizing the cross sectional area of each flow unit is identical,
Equation (4) can be reduced to the following in terms of the average permeability:
5.
A more general equation that describes the average permeability of flow units
arranged in series is the following:
Fluid Flow through Linear beds in
parallel
Consider linear flow for flow units in parallel as shown in the diagram below
Fluid Flow through Linear beds in
parallel
Again we apply the same analogy. For resistors in parallel, the following rules
apply:
1. The voltage drop is the same across each resistor.
2. The sum of the current across individual resistors is equal to the total current
across all resistors.
This analogy can be extended to fluid flow, i.e.:
1.
2.
Pressure drop is the same across each flow unit for flow units in parallel.
the sum of the flow rates across individual flow units is equal to the total flow
rate across all flow units when they are in parallel.
Fluid Flow through Linear beds in
parallel
Knowing these rules, we will proceed to the derivation of average permeability for
this system. Using Rule 1 the following expression can be derived:
6.
Using Darcy’s law, the total flow rate (q
the following expression:
7.
t) across the entire system is given by
Fluid Flow through Linear beds in
parallel
Using Darcy’s law, the total flow rate across a flow unit (qI) is given by the
following expression:
(8)
Substituting Equations (7) and (8) into Equation (6) we can develop the following
expression:
(9)
Fluid Flow through Linear beds in
parallel
Using Rule 1 and assuming the width of each flow unit is identical, Equation (9)
can be reduced to the following in terms of the average permeability:
(10)
A more general equation that describes the average permeability of flow units
arranged in parallel is the following:
Poiseuille’s Law of Capillary Flow & Flow
through Fractures
Poiseuille's Law describes the flow of a fluid through a
cylindrical pipe or capillary. It states that the rate of flow
(Q) is directly proportional to the fourth power of the radius
(r) of the pipe and the pressure difference (∆P), and
inversely proportional to the length (L) of the pipe and the
viscosity (η) of the fluid.
Poiseuille’s Law of Capillary Flow & Flow
through Fractures
Poiseuille's Law, typically applied to fluid flow through pipes
or capillaries, can be conceptually extended to describe fluid
flow within porous reservoir rocks. Here, the porous medium
acts as a network of interconnected capillaries, influencing
the flow behavior. The law underscores the importance of
factors like permeability, fluid viscosity, pressure
differentials, and rock properties in determining flow rates.
Poiseuille’s Law of Capillary Flow & Flow
through Fractures
Fluid flow through fractures plays a critical role in
various natural and engineered systems, including
geological formations, hydrocarbon reservoirs, and
geothermal reservoirs. Understanding the behavior of
fluid flow in fractures is essential for optimizing
resource extraction, groundwater management, and
geotechnical engineering projects.
Poiseuille’s Law of Capillary Flow & Flow
through Fractures
Fractures, which are natural or induced discontinuities in
rock formations, serve as preferential pathways for fluid
migration. Unlike porous media, fractures exhibit
complex flow dynamics characterized by non-linear
behavior, heterogeneous flow paths, and variable
aperture sizes.Key factors influencing fluid flow through
fractures include fracture geometry, aperture distribution,
roughness,