The document covers reservoir engineering concepts including solutions to the diffusivity equation for radial flow of single-phase and compressible fluids. It discusses the pD and Ei-function solutions, and presents the unified steady-state flow regime equations for radial flow of single-phase and compressible fluids using the pD-function, m(p)-function, and pressure-squared approximations. It also covers the pseudo-steady state flow regime and relationships between pressure functions.

1. Reservoir Engineering 1 Course (1st Ed.)

2. 1. Diffusivity Equation
A. Solutions of Diffusivity Equation
a. Ei-Function Solution
b. pD Solution
c. Analytical Solution

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

5. Finite-Radial Reservoir
The arrival of the pressure disturbance at the well
drainage boundary marks the end of the transient
flow period and the beginning of the semi
(pseudo)-steady state.
During this flow state, the reservoir boundaries and the
shape of the drainage area influence the wellbore
pressure response as well as the behavior of the
pressure distribution throughout the reservoir.
2013 H. AlamiNia
Reservoir Engineering 1 Course: USS Regime for Radial Flow of SC & C fluids and PSS
Constant
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6. Late-Transient State
Intuitively, one should not expect the change from
the transient to the semi-steady state in this
bounded (finite) system to occur instantaneously.
There is a short period of time that separates the
transient state from the semi-steady state that is called
late-transient state.
Due to its complexity and short duration, the late
transient flow is not used in practical well test
analysis.
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Reservoir Engineering 1 Course: USS Regime for Radial Flow of SC & C fluids and PSS
Constant
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7. pD for a Finite Radial System
For a finite radial system, the pD-function is a
function of both the dimensionless time and radius,
or:
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Reservoir Engineering 1 Course: USS Regime for Radial Flow of SC & C fluids and PSS
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8. pD vs. tD for Finite Radial System
Table presents pD as a
function of tD for 1.5 < reD <
10.
It should be pointed out that
Van Everdingen and Hurst
principally applied the pD
function solution to model the
performance of water influx
into oil reservoirs.
 Thus, the authors’ wellbore
radius rw was in this case the
external radius of the
reservoir and the re was
essentially the external
boundary radius of the
aquifer.
 Therefore, the range of the
reD values in Table is practical
for this application.
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9. pD Calculation
Chatas (1953) proposed the following
mathematical expression for calculating pD:
A special case of above Equation arises when
(reD) ^2>> 1, then:
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Reservoir Engineering 1 Course: USS Regime for Radial Flow of SC & C fluids and PSS
Constant
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10. Using the pD-Function
in Pwf Determination
The computational procedure of using the pD-function
in determining the bottom-hole flowing pressure
changing the transient flow period is summarized in the
following steps:
Step 1. Calculate the dimensionless time tD
Step 2. Calculate the dimensionless radius reD
Step 3. Using the calculated values of tD and reD,
determine the corresponding pressure function pD
from the appropriate table or equation.
Step 4. Solve for the pressure at the desired radius, i.e.,
rw
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Reservoir Engineering 1 Course: USS Regime for Radial Flow of SC & C fluids and PSS
Constant
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12. pD-Function vs. Ei-Function Technique
The solution as given by the pD-function technique
is identical to that of the Ei-function approach.
The main difference between the two formulations
is that the pD-function can be used only to calculate
the pressure at radius r when the flow rate Q is
constant and known.
In that case, the pD-function application is essentially restricted
to the wellbore radius because the rate is usually known.
On the other hand, the Ei-function approach can be
used to calculate the pressure at any radius in the
reservoir by using the well flow rate Q.
2013 H. AlamiNia
Reservoir Engineering 1 Course: USS Regime for Radial Flow of SC & C fluids and PSS
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13. pD Application
In transient flow testing, we normally record the
bottom-hole flowing pressure as a function of time.
Therefore, the dimensionless pressure drop technique
can be used to determine one or more of the reservoir
properties, e.g., k or kh.
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14. Relation between pD & Ei for Td>100
It should be pointed out that,
for an infinite-acting reservoir with tD > 100,
the pD-function is related to the Ei-function by the following
relation:
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Reservoir Engineering 1 Course: USS Regime for Radial Flow of SC & C fluids and PSS
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16. Diffusivity Equation
Gas viscosity and density vary significantly with
pressure and
therefore the assumptions of diffusivity equation are not
satisfied for gas systems, i.e., compressible fluids.
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Constant
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17. Diffusivity Equation Calculation:
USS Regime for Radial flow of C Fluids
In order to develop the proper mathematical
function for describing the flow of compressible
fluids in the reservoir:
Combining the above two basic gas equations with
the other equation gives:
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18. Using M (P) for Radial Equation for
Compressible Fluids
Al-Hussainy, Ramey, and Crawford (1966) linearize
the above basic flow equation by introducing the
real gas potential m (p).
Combining above equation with below yields the
radial equation for compressible fluids:
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19. Exact Solution of the Equation for
Compressible Fluids
The radial diffusivity equation for compressible
fluids relates the real gas pseudopressure (real gas
potential) to the time t and the radius r.
Al-Hussainy, Ramey, and Crawford (1966) pointed
out that in gas well testing analysis, the constantrate solution has more practical applications than
that provided by the constant pressure solution.
The authors provided the exact solution to the Equation
that is commonly referred to as the m (p)-solution
method.
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Constant
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20. Solution Forms of the Equation for
Compressible Fluids
There are also two other solutions that
approximate the exact solution.
In general, there are three forms of the
mathematical solution to the diffusivity equation:
The m (p)-Solution Method (Exact Solution)
The Pressure-Squared Method (p2-Approximation
Method)
The Pressure Method (p-Approximation Method)
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Constant
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21. The M (P)-Solution Method
(Exact Solution)
Where
Imposing the constant-rate
condition as one of the
boundary conditions
required to solve above
Equation, Al-Hussainy, et al.
(1966) proposed the
following exact solution to
the diffusivity equation:
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 pwf = bottom-hole flowing
pressure, psi
 pe = initial reservoir pressure
 Qg = gas flow rate, Mscf/day
 t = time, hr
 k = permeability, md
 T = reservoir temperature
 rw = wellbore radius, ft
 h = thickness, ft
 μi = gas viscosity at the initial
pressure, cp
 cti = total compressibility
coefficient at pi, psi−1
 φ = porosity
Reservoir Engineering 1 Course: USS Regime for Radial Flow of SC & C fluids and PSS
Constant
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22. The M (P)-Solution Method
in Terms of the tD
The m (p)-Solution Equation can be written
equivalently in terms of the dimensionless time tD
as:
The dimensionless time is defined previously as:
The parameter γ is called Euler’s constant and given
by:
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25. Dimensionless
Real Gas Pseudopressure Drop
The solution to the diffusivity equation as given by
(p)-Solution Method expresses the bottom-hole
real gas pseudopressure as a function of the
transient flow time t.
The solution as expressed in terms of m (p) is
recommended mathematical expression for
performing gas-well pressure analysis due to its
applicability in all pressure ranges.
The radial gas diffusivity equation can be expressed
in a dimensionless form in terms of the
dimensionless real gas pseudopressure drop ψD.
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Constant
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26. Solution of Dimensionless Equation
The solution to the dimensionless equation is given
by:
The dimensionless pseudopressure drop ψD can be
determined as a function of tD by using the appropriate
expressions.
When tD > 100, the ψD can be calculated by:
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Constant
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27. The Pressure-Squared Method
(P2-Approximation Method)
The first approximation to the exact solution is to
remove the pressure- dependent term (μz) outside
the integral that defines m (pwf) and m (pi) to give:
The bars over μ and z represent the values of the
gas viscosity and deviation factor as evaluated at
the average pressure p–. This average pressure is
given by:
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Constant
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28. Converting M (P)-Solution Method
to P2-Approximation
Combining Equations:
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29. P2-Approximation Assumptions
The Pressure-Squared Method solution forms
indicate that the product (μz) is assumed constant
at the average pressure p–.
 This effectively limits the applicability of the p2-method
to reservoir pressures < 2000.
It should be pointed out that when the p2-method is
used to determine pwf it is perhaps sufficient to set μ–
z– = μi z.
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Constant
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31. The Pressure Method
(P-Approximation Method)
The second method of approximation to the exact
solution of the radial flow of gases is to treat the
gas as a pseudoliquid.
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32. The Pressure Method Equation
Fetkovich (1973)
suggested that at high
pressures (p > 3000),
1/μBg is nearly
constant as shown
schematically in Figure.
Imposing Fetkovich’s
condition gives:
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Constant
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33. Converting M (P)-Solution Method
to P-Approximation
Combining Equations:
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34. Average Pressure
in P-Approximation Method
It should be noted that the gas properties, i.e., μ,
Bg, and ct, are evaluated at pressure p– as defined
below:
Again, this method is only limited to applications
above 3000 psi.
When solving for pwf, it might be sufficient to evaluate
the gas properties at pi.
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Constant
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36. Flash Back: USS
In the unsteady-state flow cases discussed
previously, it was assumed that a well is located in a
very large reservoir and producing at a constant
flow rate.
This rate creates a pressure disturbance in the reservoir
that travels throughout this infinite-size reservoir.
During this transient flow period, reservoir boundaries
have no effect on the pressure behavior of the well.
Obviously, the time period where this assumption can be
imposed is often very short in length.
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Constant
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37. Pseudosteady (Semisteady)-State
Flow Regime
As soon as the pressure disturbance reaches all
drainage boundaries, it ends the transient
(unsteady-state) flow regime.
A different flow regime begins that is called
pseudosteady (semisteady)-state flow.
It is necessary at this point to impose different
boundary conditions on the diffusivity equation and
derive an appropriate solution to this flow regime.
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Constant
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38. Semisteady-State Flow Regime
Figure shows a well in radial
system that is producing at
a constant rate for a long
enough period that
eventually affects the entire
drainage area.
During this semisteadystate flow, the change in
pressure with time
becomes the same
throughout the drainage
area.
Section B shows that the
pressure distributions
become paralleled at
successive time periods.
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Constant
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39. PSS Constant
The important PSS condition can be expressed as:
The constant referred to in the above equation can
be obtained from a simple material balance using
the definition of the compressibility, thus:
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Constant
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40. Pressure Decline Rate
Expressing the pressure decline rate dp/dt in the
above relation in psi/hr gives:
Where q = flow rate, bbl/day
Qo = flow rate, STB/day
dp/dt = pressure decline rate, psi/hr
V = pore volume, bbl
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41. Pressure Decline Rate
for Radial Drainage System
For a radial drainage system, the pore volume is
given by:
Where A = drainage area, ft2
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Constant
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42. Characteristics of
the Pressure Behavior during PSS
Examination reveals the following important
characteristics of the behavior of the pressure
decline rate dp/dt during the semisteady-state flow:
The reservoir pressure declines at a higher rate with an
increase in the fluids production rate
The reservoir pressure declines at a slower rate for
reservoirs with higher total compressibility coefficients
The reservoir pressure declines at a lower rate for
reservoirs with larger pore volumes
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43. Semisteady-State Condition
Matthews, Brons, and Hazebroek (1954) pointed
out that once the reservoir is producing under the
semisteady-state condition,
Each well will drain from within its own no-flow
boundary independently of the other wells.
For this condition to prevail:
 The pressure decline rate dp/dt must be approximately
constant throughout the entire reservoir,
Otherwise, flow would occur across the boundaries
causing a readjustment in their positions.
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Constant
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44. 1. Ahmed, T. (2006). Reservoir engineering
handbook (Gulf Professional Publishing). Ch6

45. 1. PSS
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