Download - Well Test Mod Sol
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Model Solutions to Examination
1
Date:
1. Complete the sections above but do not seal until the examination is finished.
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3. Start each question on a new page.
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Sub ject:
INSTRUCTIONS TO CANDIDATES
8 Pages
PLEASE READ EXAMINATION REGULATIONS ON BACK COVER
No. Mk.
N A M
E :
R E G
I S T R
A T
I O N N O . :
C O U
R S E
:
Y E A R :
S I G N
A T U
R E : C o m
p l e t e t h i s s e c
t i o n b u t d o n o
t
s e a l u n
t i l t h e e x
a m
i n a t i o n
i s f i n i s h e d
Reservoir Engineering - Well Test Analysis
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Model Solutions to Examination
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ANSWER 1
The general solution is:
P tD Dxf = π
Convert dimensionless variables to actual real variables:
PPi Pff kh
qt
kt
x
Pi Pkh
q
kt
x
D Dxf t f
ff t f
= − =
− =
( )
( )
2
2
2
2
πµ ϕµ
πµ
πϕµ
C
C
Now rearrange for Pff:
( )
( )
Pi Pq
kh
kt
x
Pi Pq
kh
kt
x
ff t f
ff t f
− =
− =
µ
π
π
ϕµ
µ
π
π
ϕµ
2
2
2
2
C
C
This can be easily rearranged to the following:
P Piq
kh
k
xtff
t f
= −
−µ
ππ
ϕµ2 2
2
C
Which is the equation of a straight line:
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y = mx + c
m = -C
q
kh
k
xt f
µπ
πϕµ2 2
Thus, if a plot of Pff, measured by the downhole pressure transducer,
against t , the square root of the flowing time, is made, the gradient
m, can be evaluated.
P Pq
kh
k
xtff
t f
=
−1 22
µπ
πϕµ C
Pff - measured by pressure transducer
t - recorded over flow period
Pi - (not important) but measured by RFT or early DST
Plot Pff v t
Evaluate gradient through data, m
mq
kh
k
xt f = −
µπ
πϕµ2 2C
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Model Solutions to Examination
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q - held constant through test and recorded at surface
µ - known from PVT test data
h - known from geological modelling
φ - known from petrophysics/core tests etc
t C - known from rock mechanical testing of cores
k - known - as given in question
= = −
xf
q
m kh
k
u t
µπ
πϕ2 C
The analysis of the plot of Pff v t should be carefully carried out, i.e.
select appropriate data that is not affected by early time effects e.g.
wellbore storage, or late time effects e.g. influenced by boundaries to
flow.
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ANSWER 2
a) Gas material balance is given by:
Mg = M
gi - M
gp (1)
i.e. current mass of gas = initial mass - produced mass
in reservoir of gas in reservoir of gas
From equation of state:
(PV = ZnRT)
Mg = V
gρ
g ρg
MW P
ZRT
.=
Mass = volume x density MW = molecular weight of gas
Define G = Volume of gas at standard conditions
= = = . . ..
VG
g
G M W P
Z RT
ZRT
MW Pg
sc sc
sc sc
ρρ
at standard conditions ZSC
= 1
= = .V G P TZ
T Pg
sc
sc
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Model Solutions to Examination
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This equation can be written, taking account of gas material balance
equation (1) as:
VG G P T
T Pg
i p sc z
sc
( )
= −
Also, by the same argument:
VG P T
T PZigi
i sc
sc i
= T = constant reservoir pressure
For volumetric depletion
V V
G G P Tz
T P
G P T
T P
g gi
i p sc
sc
i sc zi
sc i
( )
=
= −
=
= − =
= = −
= = −
( )
( )
G GZ
PG
Z
P
P
Z
G G
G
P
Z
P
Z
P
Z
P
G ZG
i p i
i
i
i p
i
i
i
i
i
i
i i
p
Thus, plotting - against Gp
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where P = measured reservoir pressure
Z = compressibility of gas at P
Gp = cumulative gas produced at standard conditions
yields a straight line:
Pi
Zi
P
Z
0
The data points can be plotted, as above, and the line
extrapolated to = 0.
At this point, Gp = G
i i.e. the initial gas-in-place.
(b) In most situations the plot will not be valid since
depletion will not be entirely volumetric. The expansion of rock
and connate water will lead to errors in the process, sometimes
of 100%.
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Model Solutions to Examination
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This is most apparent in abnormally pressured reservoirs where
over-pressure results in gas compressibility and rock and water
compressibility being of similar magnitude. Under such
circumstances, these effects cannot be ignored.
The modification is as follows:
Vg = V
gi - ∆V
g
∆Vg = V
pi (S
wcC
w+C
f) ∆P
here Vpi is the initial pore volume∆P is the pressure depletion
Swc
is the connate water solution
Cw is the water compressibility
Cf is the formation compressibility
Vgi
= Vpi
(1-Swc
)
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= = −−
+
= = − ∆
−( ) +
= −
=
=> −
( )
( )
( )
,( )
( )
V VV
SS C C P
V VP
S
S C C
From before VG G P T
T P
VG P T
T P
G G
g gi
gi
wc
wc w f
g gi
wc
wc w f
g
i p sc z
sc
gi sc zi
sc i
i p
1
1
1
∆
ZZP
G ZP
PS
S C C
P
Z
P
SS C C
G G
G
P
Z
P
Z
P
SS C C
P
Z
P
G ZG
ii
i wc
wc w f
wc
wc w f
i p
i
i
i
wc
wc w f i
i
i
i i
p
= − ∆−
+
=> − ∆−
+
=
−
=> − ∆−
+
= −
11
11
11
( )
( )( )
( )
( )
This will give a straight line only when Cf is constant. This is rarely
the case, since Cf is a pressure dependent quantity. Selection of C
f is
therefore a difficult and important decision if this modified plot is to
be of benefit.
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Model Solutions to Examination
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(c) The problems with water influx are significant. Not only does
influx affect the material balance
Vg = V
gi - W
e,
but there are additional effects of gas trapping in the invaded region
of the reservoir/aquifer system.
The management of such fields demands that both reservoir and
aquifer behaviour are understood. The nature of gas sales is to
arrange long term contracts with specified Daily Contracted
Quantities of gas, which are, additionally, subject to swing factors i.e.the delivery must be able to meet demand above the DCQ, or below,
depending on seasoned variation. In designing such contracts, it is,
therefore, crucially important to fully understand reservoir and
aquifer characteristics, as failure to meet contracts can prove a costly
business.
The DCQ is based on the peak production rate of the reservoir, again
accounting for saving factor. The Annual Contract Quantity = 365
DCQ i.e. saving is balanced. The reservoir model must be able to
predict future performance. In this situation of Water influx, the
material balance equation can be conveniently coupled to an analysis
method which has well testing theory as its routes. The reservoir
becomes “the well”, while the aquifer becomes “the reservoir”.
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Variable rate solution of this familiar problem is required to model the
aquifer performance.
Fetkovich showed that results derived by a quasi-semi-steady state
solution to the problem generated acceptable results, saving on
computation time.
With the material balance and Fetkovich model, the production of the
system can be predicted. The aquifer inner boundary pressure is the
gas reservoir average pressure, so required depletion rates can be
modelled.
The timing of decline will be when:
P Pg− −
= swing i.e. the gas reservoir average pressure is equal to that
required to meet the maximum gas output. Both this and the peak
production rate must be well designed to honour the gas contract.
Production problems include the trapping of gas in the region of water
influx. As water invades the reservoir it displaces gas to a residual
concentration Sgr which becomes lost in terms of production.
The water influx will then be impeded due to relative permeability
effects and pressure maintenance may be affected.
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Model Solutions to Examination
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A major benefit in the management of such fields is to have good
aquifer description which may be achieved by better data gathering
from the aquifer i.e. core sampling. Many characteristic will be
different between reservoir and aquifer not least of which is
permeability, which will be greater in the aquifer due to increased
periods of diagenesis.
With good aquifer data, simulation becomes more valuable, helping the
management process.
ANSWER 3
a) Several factors influence well test design.
Assuming wellbore storage, an estimate of CD, which can be deter
mined from very early time data (for constant compressibility (in
single phase) and skin (an estimate can be determined from anearlier well test or from corey time data using type curve matching)
can give an indication of the time needed to reach the semi-log
straight line. So, to be able to set good estimates of k and S, the
straight line on the semi-log plot must be reached (out of wellbore
storage affected region) and at least 1 log cycle of data should be
measured in this region (MTR).
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In a bounded reservoir, if the geometry of the drainage region (and
- average drainage area) is the objective of the test, then time to
reach the boundaries has to be allocated to the test (reservoir limit
test). In this case, m* is calculated from slope in SSS depletion. In
Cartesian plot m q
cth* = −
φ A and can be determined. C
A Dietz
shape factor necessary for the MBH correction. If CA is already
known, the well test only needs to reach the SSS zone. The radius at
investigation r 4 t 4kt
ci
t
= =α φµ
gives an indication of the distance that
the pressure disturbance traveled, so if the distance to boundaries
(faults, etc) is known (from geophysics, etc), an estimation of time
taken to reach boundaries can also be performed.
For this, relevant data is compressibility of oil, total compressibility
(great uncertainty), estimates of k and S, oil viscosity, porosity,
thickness of formation, and flow rates.
b) A software package can import data in real time and perform an
early time analysis to determine the existence of wellbore storage,
and from type curve matching or derivative (log-log) analysis set CD,
and type curve matching an estimate of K and S can be made and CD
stated before, this can give management an indication of time
necessary to reach MTR region. Analysis of log - log (p’ vs t) plot can
give a real time indication of when the MTR region is reached(derivative plateau) and depending on objective of test, the real time
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Model Solutions to Examination
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analysis can continue for the design of the testing in the LTR as
stated before, with some knowledge of the distance to the boundaries.
If this distance is not known, then testing should be extended to
“feel” them : normally performed after well is drilled.
c) Flow rate is dependent on type of fluid and associated conditions,
such as:
- In oil wells, qo is normally limited by test separator capacity
- In gas wells (exploration), gas rate can be limited by the capacity of
the flaring system
- In High Pressure, high temperature wells, rate can be limited bythe maximum allowable temperature at the wellhead (rating of
elastomers seals).
Production software performing a nodal analysis taking in consideration
all this (separator, chokes, tubing lift performance and reservoir in
flow - needs k and S estimates) can give a good indication of the well-test flow rate or ranges at allowable rates.
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ANSWER 4
a) This is a very useful plot. Allows the identification of 3 flow
regimes (this diagnostic role is a very important application of the
derivative response) and also allows a precise identification of the
boundaries (in time) of those regimes (another important role of the
log-log derivative diagnostic plot).
MTR) In this case, the existence of the derivative plateau
indicates the existence of a middle time region (MTR) in radial flow.
ETR) After a transition period that may be shorter or longer
(depends on geometry of system and well location within it). A periodof half slope is reached, indicating the existence of linear flow.
Following this period, another transition is reached, when the far
boundaries of the system start to affect the pressure response.
When the period of unit slope is reached, the pressure response is
dominated by the bounded reservoir model, being semi-steady-state
depletion beyond that. This transition period is also affected by the
distance to the reservoir boundaries, and if they are reached or not at
the same time. A possible geological interpretation could be a bounded
fluvial channel
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Model Solutions to Examination
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Unit slope period other
boundaries are reached
(SSS depletion)
Linear flow peridRadial flow perid
For the correct unique interpretation, input needs to be received from
other disciplines such as geology, geophysics, Petrophysics, etc →
multidisciplinary task!
b) Not counting a possible wellbore storage dominated period that
would be analysed in the log-log plot, the following specialist plots arenecessary.
Semilog-plot → to perform the analysis of the MTR region (derivative
plateau), giving the best estimates of k and S. Flow in the MTR region
(radial) is a straight line in a semilog plot.
I assume the flow periods were already defined in the log-log
derivative plot.
Linear flow plot ( ) → after the MTR period analysis, the relevant
parameters are stored (k,S) and the linear flow plot can give estimates
of the geometry of the system, L1, L
2 and W
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L1
L2
W
Cartesian flow plot _ in SSS regime, pwf
vs t is a straight line at slope
− qc hAt φ
and assuming that ct is known, an estimate of the connected
pore volume can be achieved. This allows the estimation of reserves.
Also the extrapolation of the pressure to t=0 yields a relation with the
Dietz shape factor of the drainage area.
ANSWER 5
i) The lack of uniqueness can be overcome by the input of geologists,
geophysicists, petrophysicists, etc, giving information about the
depositional environment, likely geological structure, faulting, layering
(cores, WFTs) etc.
ii) In some situations, these can be overcome using the Hawkins
equation. AC>υ the resistivity or equivalent logs can help.
iii) Analysis of cores (permeameter) or WFT measurements can allow
the estimation of vertical profiles for the permeability and identify
layering, etc.
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Model Solutions to Examination
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iv) This can be overcome using an approximation such as:
Q
qsq
tp t
tp =Q
qs
Where Q is cumulative production, qs is surface rate and T p is a
production time accounting for variation in the rate during the test.
The limitation is that before shut in a stabilised rate qs should exist.
This can overcome some variability in the rate (uncertainty). The
continuous development of more precise flowmeters (multiphase) can
also help in reducing uncertainty.
v) A material balance using the pressure decline and estimated
reserves from geology can help in identifying if there is some
pressure support from an aquifer.
b)
i) Lack of uniqueness is an important issue but assuming that there is
some previous geological or geophysical study, the effects of it can be
reduced. In new exploration areas this problem can be more serious,
where data input from other disciplines is still reduced (lack of
information).
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ii) This is also important because when designing a stimulation
treatment to reduce or overcome the damaged region (acidising,
fracturing), the knowledge of the depth of damage would help in
optimising this.
iii) As long as WFT data (MDT) and/or cores are available, this shouldbe a minor problem.
iv) This is very important as the transient well test analysis normally
assumes constant and known rate. If this is not the case, then the
analysis is incorrect. Maybe that’s why such a large effort is being put
on the development of accurate and reliable multiphase flowmeters.
v) This can be very important because in the exploration & appraisal
stage of a field, especially offshore, the knowledge of whether there
is aquifer support or not would be extremely valuable in optimising the
development cost. (Accounting for water injection support or not and
all the related facilities).
c) The fact that when there are no measurements of initial Pi by W
Ft
or MDT, some error can be introduced if the pressure transducer is
not at perforation depth. Pi = Ptranducer
+ ρh and this ρh may notbe well
known if there is some cushion fluid going down through valve.
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→ The calculation of the wellbore storage coefficient assumes
constant single phase compressibility. This can lead to serious errors
in estimating CD because prior to shut-down most oil wells would be in
Z phase flow.
→ Very important limitation → deferred production while test isbeing performed (no production at all during buildups) →maybe the
largest limitation of well testing from the managers point of view!
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