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Pressure Transient Analysis
Basic concepts
Hassan Bahrami
2013
Well Models
Vertical well Deviated well
Horizontal well
Well Models
Hydraulic fractured well
Longitudinal fracture
Transverse fracture
Reservoir Flow Regimes
• Radial flow:
• Spherical flow:
• Linear flow:
&
Bilinear flow
• Radial flow:
(1/r) d/dr[ r dp/dr ] = (phi*mu*c/k) [dp/dt]
• Spherical flow:
(1/r^2) d/dr[ r dp/dr ] = (phi*mu*c/k) [dp/dt]
• Linear flow:
[d^2p/dx^2] = (phi*mu*c/k) [dp/dt]
Diffusivity equations for the reservoir
flow regimes
Reservoir models
Radial flow
Reservoir models
Spherical flow
Reservoir models
Linear flow in a channel
Elliptical Flow
Reservoir models
Wellbore Storage Effect
Early Time Data are affected by WBS effect
Question:
- What flow regimes may be observed with
passage of time, in a horizontal well ?
Answer:
Question:
- What flow regimes may be observed in a
horizontal well with a single hydraulic fracture
along the wellbore?
Welltest Analysis Basics
Diffusivity Equation for Radial Flow:
(1/r) d/dr[ r dp/dr ] = (phi*mu*c/k) [dp/dt]
Solution for constant rate
drawdown test See chapter 7 of L.P.Dake,
reservoir engineering book
Radial flow equation
Pressure Draw-Down Test Analysis
Time
Flow rate
Pressure Pwf
Pi
P = m (log[t]) + b
Slope: m
Radial flow analysis: Semi-Log Plot
Pi-Pwf
t-t0
Pressure Build-Up Analysis Example
Semi-Log
Plot
Logarithmic
curve fit ! (Not linear)
To make the plot, do
not take log of the
numbers, just show
the time values as
they are, on the log
scale !
Early
Time
Late
Time
Pressure Draw-Down Exercise
Calculate the slope (m) and ‘Pi-P1hr’ for the data
Pi=5000 psia
Answer
A
B
M= -181
psi/cycle
Answer
P1hr = 4080 psia
Pi – P1hr = 5000 – 4080
= 920 psia
P1hr
Question: Considering the following points A and B,
the slope of radial flow straight line (m) is defined as
follows:
m=PB-PA (Psi/Cycle) P
ressure
, psia
∆t,
hrs
Semi-Log Plot
1 10 100
A B
P: Pressure, t: Time
1. True
2. False
Question:
• For radial flow equation in the case of constant
flow rate (pressure declines with time), how can
permeability be determined from the transient
data?
DP = (log[Dt] + n ) kh
BQ o6.162
Question:
• For radial flow equation in the case of constant
bottom-hole pressure (rate decline with time),
how can permeability be determined from the
transient data?
DP = (log[Dt] + n ) kh
BQ o6.162
Question:
• For radial flow equation in the case that bottom-
hole pressure and flow rate are not constant and
change with passage of time, how can
permeability be determined from the transient
data?
DP = (log[Dt] + n ) kh
BQ o6.162
P = m (log[t]) + b
Slope: m
Question: Which ‘m’ should be
considered for K and S calculation?
?
m1
m2
m3
Pressure Derivative Method / Drawdown Test
P = m (log[t]) + b
m*3.2t)(ln d
P)( d
]*3.2log[][ log0]t)(ln d
P)( d[ mtLog
P’
P
))tln(-)tP1)/(ln(-(P2:P' 12
RF
t
P
&
P’r
“0 slope on derivative curve”
Pressure Derivative Method / Drawdown Test
m IARF
hm
BQK
RF **3.2
***6.162
P
&
P’r
t
Question: Using “m” for each test, reservoir around
which well has higher permeability?
m=7E+07
m=7E+05
hm
BQK
RF **3.2
***6.162
Well A
Well B
P
&
P’r
Identifying the Reservoir Flow Regimes using
pressure derivative method:
Log-Log plot of P’, d[p]/d[ln(t)] , versus time function
• Radial flow: Slope: 0 • Spherical flow: Slope: -1/2
• Linear flow:
Slope: +1/2
• Bi-linear flow:
Slope: +1/4
• Elliptical flow:
Slope: +1/3
• Wellbore storage effect:
Slope: +1
Slopes on Log-Log plot
+1 +1/2 +1/3 +1/4
10
100
1000
0.1 1 0.01
-1/2
Estimate slope of the lines
10
100
1000
0.1 1 0.01
Question: Identify the flow regimes
What flow regimes can be detected
on this plot?
A- Linear flow regime
B-Radial flow regime
C-Spherical flow regime
D-Elliptical flow regime
What is the well model?
A- Vertical well
B- Partial perforated vertical well
C- Hydraulically fractured vertical well
D- Horizontal well
E- Cannot be identified
Nomenclature
• P Pressure
• K Permeability
• Q Flow rate
• C Compressibility
• t Time
• h Layer thickness
• r Radius
• Porosity
• Density
• Viscosity
• B Formation volume factor
• NTG Net to Gross ratio
• HC Hydrocarbon
• STOIIP: Stock tank oil initially in place
• GIP: Gas in place
M: 1000
MM: 10^6
MMM: 10^9
1 Acres: 43,560 ft^2
FVF units are rbbl/stb (oil)
and r.cu.ft/scf (gas)
(r for reservoir and s for
standard)