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# New Model to Analyze Nonlinear Leak-Off Test Behavior

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## Abstract and Figures

A leak-off test (LOT) is a verification method to estimate fracture pressure of exposed formations. After cementing each casing string, LOT is run to verify that the casing, cement and formation below the casing seat can withstand the wellbore pressure required to drill for the next casing string safely. Estimated fracture pressure from the test is used as the maximum pressure that may be imposed on that formation. Critical drilling decisions for mild weights, casing setting depths, and well control techniques are based upon the result of a LOT. Although LOT is a simple and inexpensive test, its interpretation is not always easy, particularly in formations that give nonlinear relationships between pumped volume and injection pressure. The observed shape of the LOT is primarily controlled by the local stresses. However, there are other factors that can affect and distort LOT results. Physically the LOT, indeed, reflects the total system compressibility, i.e., the compressibility of the drilling fluid, wellbore expansion, or so-called borehole ballooning, and leak (filtration) of drilling fluids into the formation. There is, however, no mathematical model explaining the nonlinear behaviour. Disagreement on determining or interpreting actual leak-off pressure from the test data among the operators is common. In this paper, a mathematical model using a well-known compressibility equation is derived for total system compressibility to fully analyze nonlinear LOT behavior. This model accurately predicts the observed nonlinear behavior in a field example. The model also predicts the fracture pressure of the formation without running a test until formation fracture.
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Summary
Aleak-off test (LOT), commonly known as a formation-pressure-
integrity test, is a verification method to estimate the fracture pressure
of exposed formations. After cementing each casing string, a LOT is
run to verify that the casing seat can withstand the wellbore pressure
required to drill safely to the next casing setting depth. Fracture
pressure determined from this test is used as the maximum pressure
that may be imposed on that formation. Critical drilling decisions for
subsequent casing setting depths are based on LOT results.
Although a LOT is a simple and inexpensive test, its interpreta-
tion is sometimes difficult, particularly in formations that give non-
linear relationships between the pumped volume and the observed
pump pressure. Ideally, a straight line is obtained that reflects the
total system compressibility (i.e., the drilling fluid, the casing
expansion, and the wellbore expansion.) Nonlinear LOT behavior
is thought to be caused by gas in the system, by borehole failure,
or by leakage of drilling fluid into the cemented casing/borehole
annulus. There is, however, no mathematical model explaining
nonlinear LOT behavior.
In this study, a mathematical model is derived to assist in ana-
lyzing nonlinear LOT behavior. The model has been used to pre-
dict the observed nonlinear behavior of field examples. In some
cases of a nonlinear LOT, the model can be used to predict the
maximum fracture pressure of the formation.
Introduction
Safety concerns indicate that wellbore pressure at any depth must
be kept between naturally occurring formation pore pressure and
the maximum wellbore pressure that the formation can withstand
without losing integrity. Knowledge of fracture pressure, which
varies with depth, is as important as knowledge about formation
pore-pressure variation with depth. When abnormal formation
pressure is encountered, the density of the drilling fluid must be
increased to maintain the overbalance to prevent possible fluid
flow from permeable formations. However, there is a maximum
limiting drilling-fluid density that can be tolerated to avoid fracture
in the exposed shallow and weak zones below the casing shoe. This
means that there is a maximum safely drillable depth into an abnor-
mally pressured zone without running another casing string.
Fracture pressure is defined as the pressure at which an
exposed formation will rupture and accept whole drilling fluid
from the wellbore. Lost circulation, or lost returns, is the conse-
quence of fractured formations. Formation fracture resistance is
related directly to the weight of the formation overburden, also
called the geostatic load, at a given depth of burial, the intergran-
ular pressure of the formations, and the formation type. Thus,
knowledge of formation-fracture pressure as a function of depth is
an imperative requirement to plan today’s deep wells in onshore
and offshore environments.
Methods for determining formation-fracture pressure fall into
two groups: predictive methods and verification methods. Initial
well planning requires formation-fracture data based upon predic-
tive methods, generally empirical correlations such as the Eaton
correlation, the Hubbert and Willis equation, the Christman corre-
lation, etc.1Well-design results from predictive methods must be
confirmed by a verification method while drilling a well. Because
the primary objective of this study is LOT analysis, which is a
verification method, predictive methods will not be covered.
The usefulness of this model lies in its ability to indicate to the
engineer that an apparent fracturing (increased pump volume with-
out pressure increase) may simply be a flow channel through the
cement, and remedial operations could possibly repair the problem.
It also indicates that nonlinear behavior is caused by a flow path of
some sort, which may or may not warrant remedial efforts. The
model does not identify the fracture point, but rather predicts the
maximum attainable pressure for a nonlinear LOT, which is more
a question of the magnitude of the flow path and its response to
increased pressure. With the loss of drilling fluid during the test, it
is obvious that a fracture has occurred, either at the casing shoe, or
at a shallower depth by means of a path behind pipe. It is still an
engineering judgment to decide whether the formation at the shoe
actually has been fractured.
Fracture-Pressure Verification Method
Fracture pressures are verified by closing the well at the surface
using a blowout preventer and pumping mud at constant rates into
the closed well. This procedure is continued until a predetermined
pressure value is reached or the well begins to take whole mud,
indicating a significant departure from the straight-line pressure
trend. The pump is stopped then, and the pressure is observed for
at least 10 minutes to determine the pressure-decline rate. Because
sand is weaker than shale, it is a common practice in the Gulf of
Mexico (GOM) to run the test in the first sand below the casing
shoe. Estimated fracture pressure from the test is used as the max-
imum pressure that may be imposed on the formation.
Atypical LOT plot for a well with a short openhole section is
shown in Fig. 1. Early test data fall on a relatively straight line,
resulting from constant pressure increase for incremental drilling
fluid pumped. The straight-line trend continues until Point Awhere
the formation grains begin to lose integrity and allow mud to enter
the formation. Pressure at the departure point from the straight line
at Point Ais the leak-off pressure (LOP) and is used to calculate the
formation-fracture gradient. However, in some cases, pumping is
continued until a maximum test pressure is observed. Pumping is
stopped then at Point B, and the well is shut in to observe the pres-
sure decline caused by mud or mud-filtrate loss.
Some of the main factors influencing the LOT are pre-existing
cracks and faults, cement channels, plastic behavior of formations,
casing expansion, test equipment, pressure gauges, injection rates,
and pump efficiency.2, 3 LOT behavior is examined and interpreted
based on experience, but it does not provide analytical or numeri-
cal models to support these interpretations.3It is concluded that if
observed data points in a LOT depart significantly from the mini-
mum volume line (MVL), a cement channel is suspected.3
Conversely, a computer program that predicts LOT behavior of the
formations is proposed.2However, this computer model requires
several parameters that are not easily obtained. Ref. 4 presents a
LOT procedure and considers the effects of mud gel strength on a
LOT. It suggests obtaining this value from field-circulation data
instead of a viscometer. However, this work does not consider the
nonlinear LOT behavior. Wellbore compressibility is calculated
along with drilling-fluid compressibility from the LOT.5This work
considers an elastic borehole-deformation effect, but not the leak
effect and casing expansion. Hazov5does not provide any model to
calculate borehole-expansion volume due to elastic deformation
and also does not consider nonlinear LOT behavior.
108 June 2001 SPE Drilling & Completion
This paper (SPE 72061) was revised for publication from paper SPE 56761 first presented
at the 1999 SPE Annual Technical Conference and Exhibition, Houston, 3–6 October.
Original manuscript received for review 31 January 2000. Revised manuscript received 31
August 2000. Paper peer approved 27 March 2000.
Application of a New Model
To Analyze Leak-Off Tests
G. Altun, SPE, Istanbul Technical U.; J. Langlinais, SPE, Louisiana State U.;
and A.T. Bourgoyne Jr., SPE, Bourgoyne Enterprises Inc.
June 2001 SPE Drilling & Completion 109
In some environments, particularly in shallow marine sedi-
ments in the GOM, it is difficult to define the endpoint of the
straight-line trend because of the nonlinear LOT behavior of the
formations. At times, a straight-line section is not observed
because previously mentioned factors mask it. Although a LOT is
a simple and inexpensive test, interpretation is not always easy,
particularly in formations showing nonlinear relationships between
the observed pump pressure, P, and the pumped volumes, V. Some
companies have operational rules of thumb. For a given LOT plot,
some choose the highest-pressure value reached, and others select
some smaller pressure value based on the rate of decrease in slope.
We believe that a mathematical model and reasonable assumptions
are needed to assist in interpretation. This would also enable the
engineer to understand better the possible reasons for nonlinear
behavior in LOT’s.
Mathematical Model
Amathematical model using the well-known compressibility
equation together with the material balance concept will be given
to analyze nonlinear LOT behavior. Before obtaining the general
solution, the compressible system is decomposed as: (1) compres-
sion of drilling fluid, (2) expansion of casing string, (3) openhole
expansion, and (4) fluid leakage. With a small section of open hole
at the time of a LOT, calculations have indicated that fluid loss to
filtration for a reasonable mud system is a very small volume, par-
ticularly in the time frame of a LOT. Thus, filtration losses are not
considered. Leakage is modeled by considering an arbitrary annu-
lar channel to provide a nonlinear component. Such a channel
could be the result of poor cement placement or, as recently pos-
tulated by investigators such as Zhou,6a channel caused by the
separation of the formation from the cement, due to the applied
pressure. Our work has shown that a low-volume leak through a
highly viscous narrow flow channel will yield a nonlinear LOT,
up to the point where fracture occurs.
Fig. 2 shows each component of the system, including the gen-
eral case. The model also allows investigating the effect of each
individual component of the compressible system. In other words,
the volume pumped in must be equal to the summation of the four
component volumes at any time during the test. General assump-
tions for the model are a homogenous compressible system, isother-
mal nonpenetrating drilling fluid, cylindrical borehole and casing
expansion, no end effects at the bottom of the hole, isotropic and
elastic rock, and one principal stress parallel to the borehole axis.
The derivation of the model equations is given in the Appendix.
The resultant material balance equation is
In Eq. 2, the first term accounts for mud compression, the sec-
ond term accounts for casing expansion, and the last term accounts
for leaks. Nonlinearity is caused by the last term, leak volume. Eq.
2 also represents the approximate solution for the overall com-
pressible system. A more exact solution can be written in the form
In Eq. 3, the first term on the right side represents drilling-
fluid compression and fluid leaks, and the second represents
casing expansion.
With the assumption that all obvious opportunities for leaks, such
as surface equipment, pumps, etc., have been eliminated, the volume
attributable to mud, casing, and hole expansion is subtracted from the
volume pumped. This is done at each data point, and the difference is
.. . . . . . . . . . . . . . . . . . . (3)
()( )
22
22
22
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RR
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é
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ê
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csg
11 2
cP
oi
P
D
VVe P hR
qE
π
éù
æö
=−+
êú
ç÷
êú
èø
ëû
. . . . . . . . . . . . . . . . . . . . . . (2)
23
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...
oo o
DD D
cV P cV P cV P
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or
. . . . . . . . . . . . . . . . . . . . . . (1)
Volume Volume Volume Volume
Pumped to Mud to Casing to Leaks
æöæöæ öæö
=+ +
ç÷ç÷ç ÷ç÷
èøèøè øèø
,
Fig. 1—Typical leak-off behavior.
0
200
400
600
800
1,000
1,200
1,400
1,600
1,800
2,000
2,200
2,400
2,600
2,800
3,000
02 46
Volume Pumped, bbl
05 1015
Time, min
A
B
Injection Pressure, psi
Fig. 2—Possible subsystems for modeling leak-off test: (a) closed system (mud compression); (b) casing expansion; (c) borehole
expansion; (d) leak; and (e) general solution case.
(a) (b) (c) (d) (e)
attributed to a leak of some sort. By assuming a leak-path length
value, L, then a consistent value of path width, W, is found by forcing
the same value for all data points. Thus a flow path is obtained that
generates the appropriate leak flow and frictional pressure loss to
predict each data point. Assuming a different length initially will
calculate a different width, which in turn generates the appropriate
leak flow and friction losses. The actual leak path is unknown and
cannot be measured; only the effects of such a leak are needed.
The model requires calculation of each component in Eq. 1,
using either Eq. 2 or 3. The pumped volume and the observed
pump pressure are known parameters during the test. In addition,
the mud-compression and the casing-expansion terms are easy to
determine from the casing, drilling-fluid, and well geometry data.
Thus, the leak volume is the only parameter to be evaluated by dif-
ference. Strictly speaking, the leak constant, D, shown in Eqs. 2
and 3 is required to be determined. A leak is modeled by
Poiseuille’s law, which is used to model flow through channels. D
is a function of the channel’s width, area, length, and drilling-fluid
viscosity, and its equation form is given by Eq. A-29. Once Dis
evaluated using the early part of the data, the behavior of the LOT
at any Pand Vcan be determined by extrapolation.
Verification of Model Using Field Data
Three nonlinear LOT behaviors observed from three different
wells were used to verify the model. Reasonable assumptions or
assumptions are indicated in the tables. Basic properties of the ana-
lyzed wells supplied with the LOT data are listed in Table 1.
Observation of nonlinear LOT behavior was the main characteristic
of the tests. These three wells, one in the GOM, one in Montana,
and another offshore Trinidad, were selected for presentation
because the model results indicated that each was distinctly different
from the others; the differences will be explained later.
Table 2 lists the additional required input parameters to imple-
ment the model. The formation Young’s modulus of the tested for-
mations was calculated from Lama and Vutukuri’s7correlation.
Vertical stress is assumed equal to the confining stress when cal-
culating the formation Young’s modulus. Because Young’s modu-
lus is used to calculate borehole expansion, which is negligible,8its
accuracy is less important. Also, determination of the drilling-mud
compressibility requires a knowledge of mud composition (solid
and liquid fraction) and density. These fractions can be obtained
directly from charts or equations. Using the data in Tables 1 and 2,
additional model parameters are calculated, such as overburden
pressure, pore pressure, vertical stress, mud compressibility,
Young’s modulus, etc., and are tabulated in Table 3. A 30-ft-long
microannulus was used as the default-assumed condition in Table
2. A slot approximation of this geometry was used to allow a ficti-
tious channel width, W, to be determined. If the calculated Wcon-
verges to a constant value, a channel leak is indicated as the cause
of the nonlinear behavior. Once the input data are prepared, the
model is applied by using spreadsheet software.
110 June 2001 SPE Drilling & Completion
TABLE 1—BASIC WELL DATA FURNISHED WITH LOT DATA
A-2 U-2 U-3
Date June 98 November 88 NA
Mud weight , lbm/gal 14.4 8.45 8.5
Pump rate, bbl/min 0.25 0.25 0.25
Casing OD, in. 95/820 20
TVD casing, ft 8,773 1,765 1,029
TVD well, ft 8,782 1,780 1,044
Openhole length, ft 15* 15* 15*
Water depth, ft 65* 0 196
Rotary Kelly Bushing, ft 100* 30* 86
Number of test points 28 29 26
Volume pumped, bbl 6.75 7.5 7
Maximum observed pressure, psi 1,621 1,350 380
* Assumed
TABLE 2—ADDITIONAL INPUT DATA TO IMPLEMENT THE MODEL
Parameter Value Unit
Water fraction of mud function of mud weight fraction
Oil fraction of mud function of mud weight fraction
Solid fraction of mud function of mud weight fraction
Compressibility of water 3.00 ×10–6 1/psi
Compressibility of oil 5.00 10–6 1/psi
Compressibility of solids 2.00 10–7 1/psi
Casing Young’s modulus 3.00 107psi
Casing Poisson’s ratio 0.3 dimensionless
Formation Young’s modu lus function of depth psi
Mud visc osity 30* cp
Channel length 30* ft
Channel length in lateral plane 1* fraction
Pore pressure 0.465* psi/ft
Horizontal/vertic al stress ratio 1* dimensionless
Formation type Stockton shale*
*Assumed
×
×
×
The mud-compression volume using either Eq. A-3 or A-6 and
the casing-expansion volume using Eq. A-17 are calculated and
subtracted from Vto determine the leak volume for each data
point. The MVL is determined from mud compressibility and well-
bore volume calculations. Then, D is calculated from Eq. 3.
Finally, the fictitious or equivalent Wis calculated from Eq. A-29
and simultaneously plotted against volume and pressure as shown
in Figs. 3 through 5. When convergence is observed, an addition-
al three or four data points are processed to ensure that Wdid
indeed converge. It was observed in the plots that the Wvalue was
larger in the early phase of the test because the system was more
compressible as a result of trapped air or formation gases in the
wellbore. Wconverged to a constant value with continued pump-
ing, as in Well A-2, shown in Fig. 3, and in Well U-3, shown in Fig.
5. However, Win Well U-2 initially tended to converge, but failed
to stabilize, and then increased. Because the cement is strong
enough and no large erosion is possible in such a small time peri-
od, it was postulated that naturally occurring fractures were the
cause of this behavior shown in Fig. 4. In other words, a squeeze
cementing will not solve the problem in the presence of naturally
occurring fractures just below the casing shoe. Fig. 5 shows that the
model behavior of Well U-3 followed a smooth stabilization path;
however, the channel width converged to a constant value at the end
of the test. It also indicates very high fluid losses (large cement
channel), i.e., the pumping rate was not sufficient to build pressure
fast enough in the well. Table 4 summarizes the determined Wand
Dvalues from the model for the tests.
Using the leak constant value and other relevant data in Eq. 3,
test data for the three wells were regenerated at observed pump
pressures for both used and unused data points of the test. Then
these calculated data from the model were plotted together with the
observed data shown in Figs. 6 through 8. Accurate extrapolation
compared to observed behavior was obtained for Wells A-2 and U-3
in Figs. 6 and 8, respectively. The fracture pressure predicted from
the model is in good agreement with the observed data. Lack of
channel-width stabilization for Well U-2, shown in Fig. 7, is the
cause of unsuccessful extrapolation. It would require pumping 4.75
bbl of mud to observe 1,350 psi instead of 7.5 bbl if there were no
naturally occurring fractures in Well U-2.
June 2001 SPE Drilling & Completion 111
TABLE 3—CALCULATE D PAR AMETERS AND
MODEL CONSTANT S
A-2 U-2 U-3
Sediment depth, ft 8,617 1,750 762
Overbur den, psi 8,647 1,750 853
Pore pressure, psi 4,037 763 445
Vertical stress, psi 4,610 987 408
Young’s modulus, psi 1.2 4×1068.51×1056.40×105
Mud com pressibility, 1/psi 2.27×10–6 2.78×10–6 2.89×10–6
Wellbore volume, bbl 666 632 371
Fig. 3—Well A-2 equivalent channel width variation during the test.
0
1
2
3
4
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
0
300
600
900
1,200
1,500
Equivalent Channel Width, in.
Volume Pumped, bbl
Pump Pressure, psi
Width-volume Width-pressure
Because the trapped air or formation gases in the system affect
the early portion of the recorded data, their effect should be elimi-
nated or subtracted from the recorded data. Figs. 3 through 8 show
that this effect is insignificant on Well A-2, moderate on Well U-2,
and very high on Well U-3. Note that trapped air volume (0.5 bbl
for Well U-2 and 0.75 bbl for Well U-3) was subtracted from the
observed data before the analysis. The trapped air or formation
gases affect the variation of the equivalent channel width as if the
channel width were larger. However, this effect will be negligible
with continued mud pumping until the air or gas compressibility
equals the mud compressibility in the well.
MVL and maximum volume line, which is half the slope value
of MVL, as shown in Figs. 6 through 8 are a quality indicator for
a LOT. Ideally, it is expected that the data points would be
observed to fall on the MVL as shown in Fig. 1. Departure from
this line is a direct indication of fluid leakage. This behavior is com-
mon for the analyzed tests and is severe in Well U-3. Model calcu-
lations revealed that the leak volume was accounting for approxi-
mately 35% in Well A-2, 40% in Well U-2, and 85% in Well U-3
of the pumped volume throughout the test.
Conclusions
Amathematical model for nonlinear LOT behavior has been
developed and applied using field data. Mud compression, casing
expansion, and leak volumes are the major factors affecting LOT
behavior. Leak volume was found to be a plausible source of non-
linear LOT behavior. The degree of nonlinearity increases with
increasing leak volume. However, borehole expansion volume
was found to be negligible. The model also makes it possible to
observe the individual effects on the nonlinear LOT behavior.
The model postulates the existence of naturally occurring
fractures from the analysis of test behavior. This behavior is
determined by the leak model, which indicates progressive equivalent
channel size development throughout the test phase.
The model requires a precise record of mud-volume pumped
and observed pump-pressure data because the model relies on only
these observed records. More data points make the analysis easier
and more reliable using this model.
Nomenclature
Axs =cross-sectional area of channel, L2, in.2
c=compressibility, Lt2/m, 1/psi
d=differential operator
D=leak constant, L4t/m
e=exponential
E=Young’s modulus, m/Lt2, psi
h=openhole length, L, ft
hcsg =casing length, L, ft
L=channel length, L, ft
n=number
o=initial
P=pressure, m/Lt2, psi
P
i=inner-casing pressure, m/Lt2, psi
P
o=outer-casing pressure, m/Lt2, psi
q=injection rate, L3/t, bbl/min
ql=leak rate, L3/t, bbl/min
r =displacement, L, in.
t=time, t, min
T=temperature, T, °F
V=volume pumped, L3, bbl
V
e=borehole-expansion volume, L3, bbl
V
ec =casing-expansion volume, L3, bbl
V
f=leak volume, L3, bbl
V
o=system volume, L3, bbl
W=channel width, L, in.
x=variable
z=direction
D=difference
¶=partial differential
e=strain, L/L, in./in.
¥=infinity
m=fluid viscosity, m/Lt, cp
n=Poisson’s ratio, dimensionless
p=constant, 3.141592654
112 June 2001 SPE Drilling & Completion
TABLE 4
STABILIZED DUMMY VARIABLES OBTAINED
FROM THE M ODEL
A-2 U-2 U-3
Channel width, W, in. 0.0253 0.0121 0.0333
Channel area, Axs, in.20.7 659 0.7529 2.0913
Leak const ant, D8.00 ×10–5 3.50×10–5 7.50×10–4
Fig. 6—Extrapolated prediction from the model for Well A-2.
0
200
400
600
800
1,000
1,200
1,400
1,600
1,800
2,000
012345 678
Pump Pressure, psi
Volume Pumped, bbl
Observed data Min. volume line Max. volume line Model prediction Used data
Fig. 4—Well U-2 equivalent channel width variation during the test.
0
1
2
3
4
0.000 0.010 0.020 0.030
0
300
600
900
1,200
Equivalent Channel Width, in.
Volume Pumped, bbl
Pump Pressure, psi
Width-volume pumped Width-pressure
Fig. 5—Well U-3 equivalent channel width variation during the test.
0
1
2
3
4
5
6
7
0.02 0.03 0. 04 0.05 0.06 0.07
0
50
100
150
200
250
300
350
400
Width-volume Width-pressure
Equivalent Channel Width, in.
Volume Pumped, bbl
Pump Pressure, psi
s=stress, m/Lt2, psi
sq=tangential or diametral stress, m/Lt2, psi
sz=vertical stress, m/Lt2, psi
Acknowledgments
This study was supported by the Minerals Management Services
(MMS). Views expressed in this paper are those of the authors and
not the MMS. The authors express their appreciation, not only to
MMS for making this work possible, but also to Unocal and
Amoco for providing the LOT data.
References
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presented at the 1997 SPE/IADC Drilling Conference, Amsterdam,
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4. Chenevert, M.E. and McClure, L.J.: “How to run casing and open-hole
pressure tests,” Oil & Gas J. (1978) 6 March, 66.
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bore compressibility,” Oil & Gas J. (1993) 29 November, 71.
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SI Metric Conversion Factors
bbl ´1.589 873 E - 01 =m3
cp ´1.0* E - 03 =Pa×s
ft ´3.048* E - 01 =m
ft2´9.290 304* E - 02 =m2
ft3´2.831 685 E - 02 =m3
in. ´2.54* E + 00 =cm
psi ´6.894 757 E + 00 =kPa
*Conversion factor is exact.
AppendixDerivation of Model Equations
Presented here is the derivation of the total system compressibility
equation using the material balance concept, which is the mathe-
matical model for an observed physical phenomenon from a LOT
for the following subcases and general solution.
Behavior of System Allowing Fluid Compression
The basic equation used to calculate annular-pressure response
due to an applied hydrostatic pressure change is the isothermal-
compressibility equation. The system in this case is assumed as a
totally closed or isolated borehole, indicating that throughout the
LOT, the system boundary is essentially rigid and fixed. The pressure
change is obtained by pumping the drilling fluid into the system
steadily. This situation consists of only drilling-fluid compression in
the well. The fluid compressibility is calculated from the well-known
compressibility equation1in differential form,
between the pump pressure and the pumped volume. It tells that if
the volume of the drilling fluid in the system is decreased due to
injection, the pressure of the drilling fluid increases. Because the
decrease in fluid volume due to compression is essentially equal to
out. The subscript Twill be dropped while deriving the following
model equations with the understanding that temperature is held
constant in the system during the LOT. Separating variables and
integrating Eq. A-1 gives
An exact solution of Eq. A-2 is obtained in terms of pumped volume as
V=V
o
(
ecP-1
)
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-3)
Eq. A-3 is not the familiar form of the compressibility equation.
The more useful form is known as the approximate solution
and is obtained using the relationship of series expansion of
logarithmic function,
Using Eq. A-4 in Eq. A-2 gives
Because V/V
oare small, their squared terms will be even smaller.
Thus, the approximate solution is written by keeping the first term,
. . . . . . . . . . . (A-5)
234
....
oo o o
VV V V
cP VV V V
æöæöæö
=− + − +
ç÷ç÷ç÷
èøèøèø .
.. . . . . . . . . . (A-4)
()
234
ln(1 ) .... 1
234
xxx
xx x+=− + −+− <
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-2)
ln(1 )
o
V
cP V
=+
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-1)
1
oT
V
cVP
æö
=− ç÷
èø
SPEDC
Fig. 7—Extrapolated prediction from the model for Well U-2.
0
200
400
600
800
1,000
1,200
1,400
1,600
1,800
0123 45678
Pump Pressure, psi
Volume Pumped, bbl
Observed data
Model prediction
Min. volume line
Used data
Max. volume line
Corrected for trapped air
Fig. 8—Extrapolated prediction from the model for Well U-3.
0
50
100
150
200
250
300
350
400
450
500
0123456789
Observed data Corrected for trapped air
Min. volume line Max. vo lume line Model prediction
Pump Pressure, psi
Volume Pumped, bbl
June 2001 SPE Drilling & Completion 113
114 June 2001 SPE Drilling & Completion
V=cV
oP.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-6)
Eq. A-6 is the well-known compressibility equation written for V.
Behavior of System Allowing Only
Casing Expansion
Stresses acting on a uniform casing shown in Fig. A-1 will cause
strain or displacement and will result in volume change. Because of
symmetry, casing keeps its cylindrical shape when applied pressure
displaces all points of the casing wall by the same amount. Thus, no
shearing stresses can take place on transverse planes. The principal
stresses are radial, sr, tangential or hoop, sq, and vertical or longi-
tudinal, sz. Vertical stress is calculated from the condition of plain-
strain case, indicating no strain in the vertical direction. The sign
convention is that compression and contraction are positive while
tension and elongation are negative. Because the change in pressure
and stress is of interest rather than the absolute value of these
parameters, the radial and the tangential stresses are written9as
respectively.
In addition, if no pressure change occurs outside of the casing, the
outer pressure, DP
o, is dropped. Also, the pressure change in the casing,
DPi, can be represented as DP. Then, Eqs. A-7 and A-8 are rewritten as
respectively.
The equations predicting the change in the vertical or longitudinal
stress can be derived using Hooke’s law, which relates the principal
stresses and strains to each other using the linear-elasticity concept.
This relationship for szwith plain-strain case, ez=0, is
Dsz=n
(
Dsr+Dsq
)
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-11)
Diametral strain on a casing is equal to the tangential or the radial
strain. The diametral strain, eq, caused by a change in inside pres-
sure is obtained from Hooke’s law,
casing wall thickness. The radial and the tangential stresses on
inner-casing wall can be calculated from Eqs. A-9 and A-10 by
replacing rwith Ri. Then, the radial stress, tangential stress, and
vertical stress become
respectively. Diametral strain, eq, is calculated from Eq. A-12. Once
the diametral strain on the inner-casing wall is determined, the casing-
expansion volume is calculated from the following equation.
Substituting Eqs. A-13, A-14, A-15, and A-12 in Eq. A-16, the casing-
expansion volume related to observed pump pressure is obtained as
Note that because eqis small, its square will be even smaller.
Therefore, the square term of eqin Eq. A-16a was neglected while
deriving Eq. A-17. Negative volume is obtained from Eq. A-17
because the casing expansion is caused by tension stresses whose
sign convention was assumed as negative. Therefore, the casing-
expansion volume is taken as positive as in Eqs. 2 and 3. The vol-
ume needed to compress the volume created by casing expansion
is obtained by substituting Vec from Eq. A-17 instead of Voin Eq.
A-7. Then, it becomes
V=cVecP.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-18)
Behavior of System Allowing Only
Borehole Expansion
case. Although the system is closed, the system boundary is not
constant but expands; i.e., the overall system volume, Vo, changes
with time during the loading and increases a new value, Vo+Ve. The
volume increment, Ve, is the volume increment or variable volume
of the system is due to the borehole expansion caused by the pump
pressure. The strain relationship for an elastic material is given10 as
Using Hooke’s law for an elastic perfectly plastic rock-constitutive
model shown in Fig. A-2, relationships between the strain and the
stress, including well-radius change (enlargement) due to the pump
pressure, are written as
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-19a)
dr
dr
ε=
or
,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-19)
rr
r
ε+∆
=
.. . . . . (A-17)
()
2
ννù
−+ ú
û
()
22
22
22
21
oi
ec i
oi
PRR
VhR
ERR
πν
é+
=−−
ê
ë
.. . . . . . . . . . . . . . . . . . . . (A-16a)
22
2
ec i
VhR
θθ
πεε
é
ù
=+
ë
û
or
,. . . . . . . . . . . . . . . . . . . (A-16)
()
22
ec i i
VhRrRπ
é
ù
=+
ë
û
,. . . . . . . . . . . . . . . . . . . (A-15)
()
22
22
1oi
z
oi
RR
PRR
σν
é
ù
+
ê
ú
∆=∆ −
ê
ú
ë
û
and
,. . . . . . . . . . . . . . . . . . . . . . . . (A-14)
()
22
22
oi
oi
RR P
RR
θ
σ+
∆=− ∆
,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-13)
rPσ∆=
.. . . . . . . . . . . . . . . (A-12)
()
1
drZ
E
θθ
εε σνσ σ
éù
== ∆∆+
ëû
,. . . . . . . . . . . . . (A-10)
222
22 22 2
()1
iio
oi oi
RP RR P
RRRr
θ
σ∆∆
æö
∆=− − ç÷
−−
èø
and
,. . . . . . . . . . . . . . . (A-9)
222
22 22 2
()1
iio
r
oi oi
RP RR P
R
RRRr
σ∆∆
æö
∆=− + ç÷
−−
èø
,. . . . (A-8)
22
22 2
()1
io i o
oi
RR P P
R
Rr
∆−æö
ç÷
èø
22
22
ii oo
oi
R
PRP
RR
θ
σ∆− ∆
∆=
and
,. . . . . (A-7)
22
22 2
()1
io i o
oi
RR P P
R
Rr
∆−æö
+ç÷
èø
22
22
ii oo
r
oi
R
PRP
RR
σ∆− ∆
∆=
Fig. A-1—Stresses acting on a casing string.
P
o
Pi
Ri
Ro
r
r
σ
σθ
June 2001 SPE Drilling & Completion 115
dP=Ede,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-20)
Integration of Eq. A-20 relates the wellbore displacement with the
pump pressure as
The expansion volume increment, Ve, owing to expansion can be
calculated easily using borehole geometry and Eq. A-21. Assuming
cylindrical borehole enlargement,
Using Eq. A-22 together with Eq. A-21, the exact solution of the
borehole enlargement volume is calculated as
The approximate solution for the borehole enlargement volume is
obtained by a series form of exponential function. The exponential
function is given by
Applying Eq. A-24 in Eq. A-23 and ignoring the cubic powered
P/Eterms and later terms, the borehole volume expansion becomes
The volume needed to compress the volume created by the bore-
hole expansion is obtained by substituting the volume term Vefrom
Eq. A-25 in place of Voin Eq. A-6 to give
V=cVeP.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-26)
Behavior of System Allowing Only
Leak (Filtration)
The system is closed in this case; i.e., the system volume is constant,
or the system boundary is fixed. However, the system wall allows fluid
losses. The leak volume, Vf, can be modeled using different flow
models, but the leak volume vs. the observed pump pressure rela-
tionship is the same in all models. The only difference is the constant
term, which has different parameters because of the different
geometry assumptions. The general relationship for the leak volume is
dV
f=DDPdt ,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-27)
or V
f=DDPt.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-27a)
Time, t, in Eq. A-27 is the ratio of pumped volume to flow rate and
is written as
The constant Din Eqs. A-27 and A-27a takes various forms
depending on the leak model. In this study, leak is modeled by
Poiseuille’s law, which is used to model flow through channels. If
the channel shape is considered as rectangular form, the constant
Dbecomes11
In Eq. A-29 W=the width of the fracture; Axs=the cross-sectional
area of the fracture, which equals the product of Wand the lateral
extent of the fracture. Note also that Din Eq. A-29 is in field units.
If the system allows only leak and drilling-fluid compression, the
pumped volume is calculated as
V=cV
oP+V
f+cV
fP,. . . . . . . . . . . . . . . . . . . . . . . . . . . (A-30)
The last term on the right side is very small and will be neglected.
Substituting leak volume in Eq. A-30 and solving for Vgives
Solving for V, it yields
The term DP/qis less than one. Thus, we can use the relationship,
Using Eq. A-33 in Eq. A-32 gives the approximate solution as
The exact solution is obtained by substituting Eq. A-4 for the first
term in Eq. A-30.
Behavior of Whole System (General Solution)
The general solution of the total system is the summation of all
subsystems’ solutions. Total system solution can be written in
material balance equation form as
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-35)
(1)
1
cP
o
Ve
VDP
q
=
.. . . . . . (A-34)
2
23
...
oo o
DD
VcVP cV P cV P
qq
é
ù
æö æö
ê
ú
=+ + +
ç÷ ç÷
ê
ú
èø èø
ë
û
.. . . . . . . . . . . . (A-33)
23
0
11...1
1
n
n
xxxxx
x
=
==++++<
å
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-32)
1
o
cV P
VDP
q
=
.. . . . . . . . . . . . . . . . . . . . . . . . . . . (A-31)
o
V
VcVPDP
q
=+
.. . . . . . . . . . . . . . . . . . . . . . . . . (A-29)
()
2
9
8.7 10 xs
WA
D
L
µ
=
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-28a)
V
tq
=
or
,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-28)
dV
dt q
=
.. . . . . . . . . . . . . . . . . . . . . (A-25)
2
2
2
eo
PP
Vhr
EE
π
éù
æö
=+
êú
ç÷
èø
êú
ëû
. . . . . . . . . . . . . . . . . . (A-24)
234
1....
2! 3! 4!
xxxx
ex=+ + + ++
.
.. . . . . . . . . . . . . . . . . . . . . . . . . . . (A-23)
2
21
P
E
eo
Vhreπéù
=−
êú
ëû
.. . . . . . . . . . . . . . . . . . . . . . (A-22)
()
22
eoo
Vhrrrπéù
=∆+
ëû
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-21)
1
P
E
o
rre
æö
∆= −
ç÷
ç÷
èø
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-20a)
0
o
o
rr
p
r
dr
dP E r
+∆
=
òò
or
Fig. A-2—Borehole expansion and constitutive rock model.
Casin g
ro
ro+r
h
Py
P
E
ε
Substituting Eqs. A-6, A-17, A-18, A-25, A-26, and A-34 on the
right side of Eq. A-36, the desired solution for the total system
behavior in approximate form is obtained as
After sensitivity analysis, it was found that the terms of the vol-
ume to compress the casing-expansion-volume fluid, the volume to
expand the borehole, the volume to compress the borehole expan-
sion volume fluid, and the volume to compress the leakage volume
given in Eqs. A-36 or A-37 are negligible and ignored. These terms
account for less than 1% of the volume pumped into the system,
even in extreme cases.8Then, Eq. A-37 is reduced to Eq. 2. The
exact solution form of the Eq. A-37 is obtained by substituting Eqs.
A-3, A-17 and A-35, and is given in Eq. 3.
SI Metric Conversion Factors
bbl ´1.589 873 E -01 =m3
ft ´3.048* E -01 =m
ft2´9.290 304* E -02 =m2
°F ´(°F -32)/1.8 C
gal ´3.785 412 E -03 =m3
in. ´2.54* E +00 =cm
in.2´6.451 6* E +00 =cm2
lbm ´4.535 924 E -01 =kg
psi ´6.894 757 E +00 =kPa
*Conversion factor is exact.
Gursat Altun is an assistant professor at Istanbul Technical U.,
Turkey. e-mail: galtun@itu.edu.tr. Altun holds an MS degree
from Istanbul Technical U. and a PhD degree from Louisiana
State U., both in petroleum engineering. Julius Langlinais is a
professor of petroleum engineering and Associate Dean for
the College of Engineering, Louisiana State U., Baton Rouge,
Louisiana. e-mail: eglang@eng.lsu.edu. He was employed by
Conoco and Superior Oil companies. Langlinais holds a PhD
degree in physics from Louisiana State U. Adam T. Bourgoyne
Jr. is Professor Emeritus of petroleum engineering and former
Dean of the College of Engineering, Louisiana State U., Baton
Rouge, Louisiana. e-mail: ted@BourgoyneEnterprises.com.
Currently he is a petroleum industry consultant. Bourgoyne
holds a PhD degree in petroleum engineering from the U. of
Texas, Austin, Texas.
SPEDC
. . . . . . . . . . . . . . . . . . . . (A-37)
.
23
234
...
oo o
DD D
cV P cV P cV P
qq q
éæö æö æö
++++
ç÷ ç÷ ç÷
èø èø èø
ù
êú
êú
ëû
22
21 2 1
oo
PP P P
hr cP hr
EE E E
ππ
éù
éù æöæ ö
+++ +
êú
ç÷ç ÷
êú
ëû èøè ø
ëû
()( )
2
22
222
22
1
oi
csg i
oi
PR R
cP h R ER R
πννν
æö
éù
+
+−++
ç÷
êú
ç÷
ëû
èø
()( )
22
222
22
21
oi
ocsgi
oi
PRR
VcVP hR ERR
πννν
éù
+
=+ − −+
êú
ëû
.. . . . . . . . . . . . . . . . . . . . . . (A-36)
Volume to Compress
Leaks Volume
æö
+ç÷
èø
()
Volume to Compress Volume to Leaks
Borehole Expansion Volume
èø
æö
++
ç÷
èø
()
Volume to Compress Volume to Expand Borehole
Casing Expansion Volume
æö
++
ç÷
èø
Volume Volume Volume
Pumped to Mud to Casing
æöæöæ ö
=+
ç÷ç÷ç ÷
èøèøè ø
116 June 2001 SPE Drilling & Completion