derivative v2
TRANSCRIPT
Semilog Plot
Semi-Log Plot - Drawdown
p mlogtb
Even though this is the analogous MDH for a DD, the principles are equally applicable to BU analysis
‘MDH’ (Miller-Dyes-Hutchinson) plot, for a drawdown,
• Considering the ideal case, of putting on production a well with no wellbore storage and noskin, the blue curve is obtained. The straight line representing radial flow is established almost instantaneously, and from the slope of the line the permeability-thickness product, kh, is obtained.
• With wellbore storage but no skin the red curve is obtained. • With skin but no storage, the green curve shows radial flow
immediately, parallel to but offset from the ideal blue line.• A typical test will reveal both wellbore storage and skin,
corresponding to the black curve transitioning on to the green curve. The storage causes the delay, the skin the offset, and once again the final straight line slope is unchanged, as permeability is a reservoir property and is unaffected by near-wellbore effects.
S = 1.151 m – log Φµ CtrW
k2
+ 3.23
m = 162.6qµBkh
Semi-Log Plot – Drawdown
∆p1hr
Pressure vs Delta t Plot
∆p1hr = logΦµCtrW
mk
2– 3.23 + 0.87 S
m = 162.6qµBkh
Semi-Log Plot – Drawdown - MDH
Delta Pressure vs Delta t Plot
It is strictly valid only for the first ,ever drawdown on a well but can in exceptional circumstances be used for analysis of a later drawdown or even a build-up.
Semilog – Example
The data summarized at Semilog Example.xls were recorded during apressure draw down test from an oil well. Estimate the effectivepermeability to oil and the skin factor using the semilog graphicalanalysis technique for a constant-rate drawdown flow test.
Q = 250 STB/DPi = 4,412 psiaH= 46 ftPoro = 12 %Rw= 0.365 ftB=1.136 RB/STBCt= 17 E-6 psi-1Visc= 0.8 cp
Semilog Example.xls
S = 1.151 m – log ΦµCtrW
Semilog – Example
qBµmh
k 162.6∆p1hr k
2+ 3.23
Pressure Derivative
• Pressure derivative analysis is based on the observation the pressure variation that occurs during a well test is more significant than the pressure itself.
• The use of pressure derivatives makes the well test interpretation easier in a number of ways:
• Derivative type curves increase the possibility of converging to a unique model (i.e., solution).
• Derivative analysis makes it easier to identify a semilog straight line.
• Derivative analysis makes it easier to identify the type of reservoir heterogeneity, because the signatures or patterns it provides are more definitive.
Pressure Derivative
⇒
Pressure DerivativeThe basic idea of the derivative is to calculate the slope at each point of the pressure curve on the semi-log (superposition) plot, and to display it on the log-log plot.
Points 1 and 2 fall on the wellbore storage unitslope in early time and, during the transition to IARF, the derivative peaks at point 4. The transition is complete at point 6, as the derivative flattens to a value equivalent to m.
Drawdown Test and Pressure Derivative AnalysisTransient flow period:
2332
.rc
kLogt Logmppp
wtpwfi
06162
m and kh
Bq.m sc
p
p
pp
p
pp t Lnd
dt
dt
pd.
t Logd
dt
dt
pd
t Logd
pd
30262
m
dt
pdt.
t Logd
pd
pp
p
30262
In radial flow geometry, the pressure drop during the transient period is expressed by:
where
From the above equation,
or
The above development implies that a log-log plot of [d(p)/d(Log tp)] versus tp should yield a
horizontal line with an intercept equal to m (Figure 1 ).
Figure 1
Procedure for Derivative Analysis
t
ptPDrivative
.
11
11)(
ii
iiiiDrivative tt
pptP
To calculate the pressure derivative curve we need to use the formula of derivative which is:
ti-1 Pi-1
ti Pi
ti+1 Pi+1
Illustration of Pressure Derivative Method
LOG
∆p
& LOG
∆p' ∆
t
Δt0
Wellbore Storage and Skin
Wellbore Storage
0
DERIVATIVE TYPE CURVE FOR DRAWDOWN ANALYSIS IN DIMENSIONLESS TERMS:
D
DD c
tp
1
D
D
D
ctd
dp
During early times when wellbore storage effects dominate,
where c D is the wellbore storage constant. By taking the logarithm of both sides, we obtain
Log pD = Log tD -Log cD
The above equation shows that log p D versus log tD is a unit slope line when wellbore
storage effects dominate. Now we can examine the behavior of the derivative
Therefore, when wellbore storage effects dominate, the derivative of the pressure curve with
respect to tD/cD also has a unit slope (Figure 2 ,
DRAWDOWN ANALYSIS IN DIMENSIONLESS TERMS:
During the radial flow period, the dimensionless form of the drawdown equation is:
s
DD
DD ecLn.c
tLnp 2809702
1Then, 2
1
D
D
D
ctd
dp
The above equation implies that the derivative plot during radial flow will generate a horizontal line with a value of 0.5 (Figure 2).
Properties of the Derivative
Wellbore Storage & Skin ResponseDimensionless Groups:
A “rule of thumb, ” developed from the fundamental solutions of the diffusivity equation including wellbore storage and skin effect (Agarwal et al., 1970), suggests that the “transition” period lasts 1.5 log cycles from the cessation of predominant wellbore storage effects (unit slope line). Points beyond that time fall on a semi-log straight line.
Log-Log Example.xls
Log-log – Example
The data summarized at Log-log Example.xls were recorded during apressure draw down test from an oil well.Your task is to change thevalues of k, S and C until they visually fit the test data in the Log-logplot.
k(md) 8.7
S 5
C(STB/d) 0.03
Log-log Example answer
The data summarized at Log-log Example.xls were recorded during apressure draw down test from an oil well.Your task is to change thevalues of k, S and C until they visually fit the test data in the Log-logplot.
Log-Log Example.xls
Build-up Tests
Build-up testA well already flowing (ideally at constant rate) is shut in andpressure is measured
Practical advantage: constant (zero) rate more easily achievedAnalysis often require slight modification of the techniques fordrawdown
Build-up
Build-up Analysis
q
tp
0
t
pi
RATE
p
tp
t = t - tp
Buildup Test
• Drawdown data quality is subject to many operational problems; slugging, turbulence, rate variation, inaccurate rate measurements, instability, unsteady flow, plugging, interruptions, equipment adjustments, etc…
• Buildup is measurement of pressure and time when well is shut-in.• In high permeability reservoirs the pressure will buildup to a stabilized
value quickly, but in tight formations the pressure may continue to buildup for month before stabilization attained.
• Buildup must be preceded by flow period.• Simplified Analysis assumes constant flow rate for a duration t hours.• Shut-in time, Δt, measured from end flow.• Buildup Analysis treated as superposition of flow and injection.• Analysis of buildup data may yield the values of K, S, and the average
reservoir pressure.
Methods of analysis:
•Horner plot (1951): Infinite acting reservoir
•Matthews-Brons-Hazebroek (MBH,1954): Extension of Horner plot to finite reservoir.
•Miller-Dyes-Hutchinson (MDH plot, 1950): Analysis of P.S.S. flow conditions.
Buildup is always preceded by a drawdown and the buildup data are directly affected by this drawdown.
Behavior of Static Sandface Pressure Upon Shut-in of a Well
Reflects “kh”
Reflects the wellbore storage (afterflow)
Reflects the effects of boundaries.
S
rc
kt
kh
qBpp
wt
o
wfi87.023.3loglog
6.1622
•Flowing sandface pressure during drawdown
•Shut-in wellbore pressure: The static sandface pressure is given by the sum of the continuing effect of the drawdown rate, qsc, and the superposed effect of the change in rate(0-qsc)
Src
kt
kh
Bq
Src
ktt
kh
qBpp
wt
o
wt
o
wsi
87.023.3loglog06.162
87.023.3loglog6.162
2
2
t
tt
kh
qBtpp o
wsilog
6.162 Horner plot relationship- Infinite acting reservoir
t
tt
kh
qBtpp o
wsilog
6.162
Horner plot relationship
kh
qBm o
6.162
t
tttimeHorner
Slope of semilog straight line same as drawdown – used to calculate permeability.
Buildup test does NOT allow for skin calculation. Skin is obtained from FLOWING pressure before shut-in.
t
tt
kh
qBS
rc
kt
kh
qBtpttp po
wt
p
o
pwfpwslog
6.16287.023.3loglog
6.1622
Src
k
tt
tt
kh
qBtpttp
wtp
po
pwfpws87.023.3loglog
6.1622
hrt 1
23.3
1log151.1
2
1
wtp
pwfhr
rct
tk
m
ppS
23.3
1log151.1
2
1
wtp
pwfhr
rct
tk
m
ppS
Detecting Faults from Buildup
41
42
Pw(∆t)=pi- mlogTp+∆t
∆t
∆t
late time
m
early time
Build-Up Analysis
Log tp + ∆t
∆ t
m =162.6qBµ
kh
Infinite acting flow period
Horner time
tp+ ∆t
∆p1hr – log
m = 162.6qµBkh
S = 1.151 mk
θµCtrW2
+ log tp+ 1 + 3.23tp
Horner Plot
Horner Plot – Example
The data summarized at Horner Example.xls were recorded during apressure build up test from an oil well. Estimate the effectivepermeability to oil and the skin factor using the Horner plot graphicalanalysis technique.
tp= 25 hrQ = 542 STB/DH= 87 ftPoro = 0.07 %Rw= 0.25 ftB= 1.56 RB/STBCt= 6E-6 psi-1Visc= 0.75 cp
Horner Example.xls
Equivalent Time
Agarwal proposed;
Valid for infinite acting radial flow
Build-up Type-curve will match Drawdown type-curve
(∆t)eq =∆t
1+∆ttp
Principle of Superposition
Superposition Concept
Superposition consists of making linear combinations ofsolutions of simple problems to form the solution of acomplex one.
Superposition in time: With simple drawdownsolutions, the solution of a test with complexproduction history is constructed
Boundary effects: The response of several wells(or virtual wells) is used to construct the solutionof test that sense boundaries
Main Superposition Principles
For a diffusion problem involving the diffusivity equation,initial conditions and boundary conditions that are alllinear:
A linear combination of solutions honoring thediffusion equation also honor this diffusionequation
At any well or boundary, the flux resulting fromthe linear combination of solutions will be thesame linear combination of the correspondingfluxes.
Main Superposition Principles
If a solution composed of a linear combination ofsolutions to the diffusivity equation is found andhonors all boundary and initial conditions, it is thesolution to the problem. Elementary solutions mayor not be physical as long as they honor thediffusivity equation.
6
Superposition Rules
Derived from the superposition principles
If one considers the pressure change due to aunit rate production, the pressure change due tothe production of the same system at rate q willbe q times the unit rate solution
Superposition Rules
To simulate the sequence of a constant rate q1
from time zero to time t1, followed by theproduction q2 from time t1 to infinity, you cansuperpose the production at rate q1 from time zeroto infinity and production of rate (q2 – q1) fromtime t1 to infinity.
Principle of Superposition
∆pBU ~ ∆pDD fortp >> ∆t
Superposition in Time - Buildup