Q921 re1 lec9 v1

Reservoir Engineering 1 Course (2nd Ed.)

1. Solutions of Diffusivity Equation
A. pD Solution
B. Analytical Solution

2. USS(LT) Regime for Radial flow of SC Fluids:
Finite-Radial Reservoir
3. Relation between pD and Ei
4. USS Regime for Radial Flow of C Fluids
A. (Exact Method)
B. (P2 Approximation Method)
C. (P Approximation Method)

5. PSS regime Flow Constant

1. PSS Regime
A.
B.
C.
D.

Average Reservoir Pressure
PSS regime for Radial Flow of SC Fluids
Effect of Well Location within the Drainage Area
PSS Regime for Radial Flow of C Fluids

2. Skin Concept
3. Using S for Radial Flow in Flow Equations
4. Turbulent Flow

Average Reservoir Pressure in PSS
Because the pressure at every point in the reservoir
is changing at the same rate, it leads to the
conclusion that the average reservoir pressure is
changing at the same rate.
This average reservoir pressure is essentially set equal to
the volumetric average reservoir pressure p– r.
It is the pressure that is used to perform flow calculations
during the semisteady state flowing condition.

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Average Reservoir Pressure
Calculation in PSS
In the above discussion, p– r indicates that, in
principal, the above Equation can be used to
estimate by replacing the pressure decline rate
dp/dt with (pi − p– r)/t, or:
(t is approximately the elapsed time since the end
of the transient flow regime to the time of interest.)

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Volumetric Average Pressure
of the Entire Reservoir
It should be noted that when performing material
balance calculations, the volumetric average
pressure of the entire reservoir is used to calculate
the fluid properties. This pressure can be
determined from the individual well drainage
properties as follows:

Where Vi = pore volume of the ith drainage volume
p–ri = volumetric average pressure within the ith
drainage volume.
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Using the Flow Rate
in Volumetric Avg Reservoir Pressure
Figure illustrates the
concept of the
volumetric average
pressure.
In practice, the Vi’s are
difficult to determine
and, therefore, it is
common to use the
flow rate qi.

Volumetric average reservoir pressure
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Applications
of the Pseudosteady-State Flow
The practical applications of using the
pseudosteady-state flow condition to describe the
flow behavior of the following two types of fluids
are presented below:
Radial flow of slightly compressible fluids
Radial flow of compressible fluids

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Diffusivity Equation in PSS
The diffusivity equation as expressed previously for
the transient flow regime is:

For the semisteady-state flow, the term (∂p/∂t) is
constant so:

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Radial Flow of Slightly Compressible
Fluids Calculation

Where c1 is the constant of the integration and can
be evaluated by imposing the outer no-flow
boundary condition [i.e., (∂p/∂r) re = 0] on the
above relation to give:

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Flow Rate for Radial Flow of Slightly
Compressible Fluids (PSS)
Performing the above
integration and assuming
(rw 2 /re 2) is negligible
gives:

A more appropriate form
of the above is to solve
for the flow rate, to give:
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Where Q = flow rate,
STB/day
B = formation volume
factor, bbl/STB
k = permeability, md


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Q Vs. Average Reservoir Pressure for
PSS Regime
The volumetric average reservoir pressure p– r is
commonly used in calculating the liquid flow rate
under the semisteady-state flowing condition.
Introducing the p– r into previous Equation gives:
(the volumetric average pressure p–r occurs at about 47% of
the drainage radius during the semisteady-state condition.)

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Using Pd Solution (PSS)
It is interesting to notice that the dimensionless
pressure pD solution to the diffusivity equation can
be used to derive previous Equation.

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Effect of Geometry on PSS Flow
It should be pointed out that the pseudosteadystate flow occurs regardless of the geometry of the
reservoir.
Irregular geometries also reach this state when
they have been produced long enough for the
entire drainage area to be affected.

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Shape Factor
Rather than developing a separate equation for
each geometry, Ramey and Cobb (1971) introduced
a correction factor that is called the shape factor,
CA, which is designed to account for the deviation
of the drainage area from the ideal circular form.
The shape factor, accounts also for the location of
the well within the drainage area.

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Shape Factors
for Various Single-Well Drainage Areas

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Solutions Using CA
Introducing CA into following Equation and
performing the solution procedure gives the
following two solutions:

In terms of the volumetric average pressure p–r:

In terms of the initial reservoir pressure pi:

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Radial Flow of
Compressible Fluids (Gases) (PSS)
 The radial diffusivity equation was developed to study the performance
of compressible fluid under unsteady-state conditions. The equation has
the following form:

 For the semisteady-state flow, the rate of change of the real gas
pseudopressure with respect to time is constant, i.e.,
 Using the same technique identical to that described previously for
liquids gives the following exact solution to the diffusivity equation:
 Where Qg = gas flow rate, Mscf/day
 T = temperature, °R
 k = permeability, md
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Approximations for
Radial Flow of Gases (PSS)
Two approximations to the above solution are
widely used. These approximations are:
Pressure-squared approximation
Pressure-approximation

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P2 Approximation
As outlined previously, the method provides us
with compatible results to that of the exact solution
approach when p < 2000.
The solution has the following familiar form:

The gas properties z– and μ are evaluated at:

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P Approximation
This approximation method is applicable at p>3000
psi and has the following mathematical form:

With the gas properties evaluated at:

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Assumptions
in Deriving the Flow Equations
In deriving the flow equations, the following two
main assumptions were made:
Uniform permeability throughout the drainage area
Laminar (viscous) flow

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Correction Factors for Assumptions
Before using any of the previous mathematical
solutions to the flow equations, the solution must
be modified to account for the possible deviation
from the above two assumptions.
Introducing the following two correction factors
into the solution of the flow equation can eliminate
the above two assumptions:
Skin factor
Turbulent flow factor

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Wellbore Damage
It is not unusual for materials such as mud filtrate,
cement slurry, or clay particles to enter the
formation during drilling, completion, or workover
operations and reduce the permeability around the
wellbore.
This effect is commonly referred to as a wellbore
damage and
The region of altered permeability is called the skin
zone.

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Skin Zone
Skin zone can extend from a few inches to several
feet from the wellbore.
Many other wells are stimulated by acidizing or
fracturing, which in effect increase the permeability
near the wellbore.
Thus, the permeability near the wellbore is always
different from the permeability away from the well
where the formation has not been affected by
drilling or stimulation.

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Near Wellbore Skin Effect
A schematic
illustration of
the skin zone
is shown in
Figure.

Near wellbore skin effect
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Skin Effect
Those factors that cause damage to the formation
can produce additional localized pressure drop
during flow.
This additional pressure drop is commonly referred to as
Δpskin.

On the other hand, well stimulation techniques will
normally enhance the properties of the formation
and increase the permeability around the wellbore,
so that a decrease in pressure drop is observed.
The resulting effect of altering the permeability
around the well bore is called the skin effect.
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Skin Types
Figure compares the
differences in the skin
zone pressure drop for
three possible outcomes:
Δpskin > 0, indicates an
additional pressure drop
due to wellbore damage,
i.e., kskin < k.
Δpskin < 0, indicates less
pressure drop due to
wellbore improvement,
i.e., kskin > k.
Δpskin = 0, indicates no
changes in the wellbore
condition, i.e., kskin = k.
positive and negative skin effects
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Skin Zone Pressure Drop
Hawkins (1956) suggested that the permeability in
the skin zone, i.e., kskin, is uniform and the
pressure drop across the zone can be approximated
by Darcy’s equation. Hawkins proposed the
following approach:

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Skin Factor
The additional pressure drop expression is
commonly expressed in the following form:

Where s is called the skin factor and defined as:

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Positive Skin Factor

Positive Skin Factor, s > 0
When a damaged zone near the wellbore exists, kskin is
less than k and hence s is a positive number.
The magnitude of the skin factor increases as kskin
decreases and as the depth of the damage rskin
increases.

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Negative Skin Factor
Negative Skin Factor, s < 0
When the permeability around the well kskin is higher
than that of the formation k, a negative skin factor exists.
This negative factor indicates an improved wellbore
condition.
a negative skin factor will result in a negative value of
Δpskin.
This implies that a stimulated well will require less pressure
drawdown to produce at rate q than an equivalent well with
uniform permeability.

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Zero Skin Factor
Zero Skin Factor, s = 0
Zero skin factor occurs when no alternation in the
permeability around the wellbore is observed, i.e., kskin
= k.

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Modification of the Flow Equations
The proposed modification of the previous flow
equation is based on the concept that the actual
total pressure drawdown will increase or decrease
by an amount of Δpskin.
Assuming that (Δp) ideal represents the pressure
drawdown for a drainage area with a uniform
permeability k, then:
The concept can be applied to all the previous flow
regimes to account for the skin zone around the
wellbore.
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S in SS Regime
for Radial Flow of SC Fluids

 Where









Qo = oil flow rate, STB/day
k = permeability, md
h = thickness, ft
s = skin factor
Bo = oil formation volume factor, bbl/STB
μo = oil viscosity, cp
pi = initial reservoir pressure, psi
pwf = bottom hole flowing pressure, psi

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S in USS Regime
for Radial flow of SC Fluids

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S in USS Regime
of Radial Flow of C Fluids

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S in PSS regime
for Radial Flow of SC Fluids

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S in PSS Regime
for Radial Flow of C Fluids

Where:
Qg = gas flow rate, Mscf/day
k = permeability, md
T = temperature, °R
(μ–g) = gas viscosity at average pressure p–, cp
z–g = gas compressibility factor at average pressure p–

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Effective (Apparent) Wellbore Radius
Matthews and Russell (1967) proposed an
alternative treatment to the skin effect by
introducing the effective or apparent wellbore
radius rwa that accounts for the pressure drop in
the skin. They define rwa by the following equation:
All of the ideal radial flow equations can be also
modified for the skin by simply replacing wellbore
radius rw with that of the apparent wellbore radius
rwa.
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Turbulent Vs. Laminar Flow
All of the mathematical formulations presented so
far are based on the assumption that laminar flow
conditions are observed during flow.
During radial flow, the flow velocity increases as
the wellbore is approached.
This increase in the velocity might cause the
development of a turbulent flow around the wellbore.

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Turbulent (Non-Darcy) Flow
If turbulent flow does exist, it is most likely to occur
with gases and causes an additional pressure drop
similar to that caused by the skin effect.
The term non-Darcy flow has been adopted by the
industry to describe the additional pressure drop
due to the turbulent (non-Darcy) flow.

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Turbulent Flow Factor
Referring to the additional real gas pseudopressure
drop due to non- Darcy flow as Δψ non-Darcy, the
total (actual) drop is given by:

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Non-Darcy Flow Coefficient
Wattenburger and Ramey (1968) proposed the
following expression for calculating (Δψ) non-Darcy:

Where:
μgw = gas viscosity as evaluated at pwf, cp
γg = gas specific gravity
h = thickness, ft
F = non-Darcy flow coefficient, psi2/cp/ (Mscf/day) 2
β = turbulence parameter
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Turbulence Parameter
Jones (1987) proposed a mathematical expression
for estimating the turbulence parameter β as:

Where:
φ = porosity, fraction

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USS Regime in Radial flow for C Fluids
(Turbulent Flow)
The gas flow equation for an unsteady-state flow
can be modified to include the additional drop in
the real gas potential as:

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Inertial or Turbulent Flow Factor
The term DQg is interpreted as the rate dependent
skin factor.
The coefficient D is called the inertial or turbulent flow
factor and given by:

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Apparent (Total) Skin Factor
The true skin factor s, which reflects the formation
damage or stimulation, is usually combined with
the non-Darcy rate dependent skin and labeled as
the apparent or total skin factor:

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USS Regime in Radial flow for C Fluids
(Turbulent Flow) (Cont.)

Where:
t = time, hr
μi = gas viscosity as evaluated at pi, cp
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PSS Regime in Radial flow for C Fluids
(Turbulent Flow)

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SS Regime in Radial Flow for C Fluids
(Turbulent Flow)

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1. Ahmed, T. (2010). Reservoir engineering
handbook (Gulf Professional Publishing).
Chapter 6

1. Superposition
A. Multiple Well
B. Multi Rate
C. Reservoir Boundary

2. Productivity Index (PI)
3. Inflow Performance Relationship (IPR)

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