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New Model to Analyze Nonlinear Leak-Off Test Behavior

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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
Copyright © 2001 Society of Petroleum Engineers
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
1
oi
oi
RR
RR ννν
é
ù
+−++
ê
ú
ë
û
()
2
csg
11 2
cP
oi
P
D
VVe P hR
qE
π
éù
æö
=−+
êú
ç÷
êú
èø
ëû
. . . . . . . . . . . . . . . . . . . . . . (2)
23
234
...
oo o
DD D
cV P cV P cV P
qq q
éù
æö æö æö
êú
++++
ç÷ ç÷ ç÷
êú
èø èø èø
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()( )
22
222
csg 22
21
oi
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PR R
VcVP hR
ER R
πννν
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+
=+ −++
êú
ëû
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
approximations had to be made when data were unknown. These
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
Location GOM Montana Trinidad
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
Overbur den gradient 1* psi/ft
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.
ro=initial wellbore radius, L, in.
Ri=inner-casing radius, L, in.
Ro=outer-casing radius, 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
sr=radial 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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4. Chenevert, M.E. and McClure, L.J.: “How to run casing and open-hole
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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,
In Eq. A-1, the minus sign indicates an inverse relationship
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
the volume pumped, the minus sign in the equation is cancelled
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,
The radial and tangential stresses vary with radial location in the
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
Borehole expansion caused by loading is examined in this special
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
... Once the cement slurry is in place, and it has hardened, the casing shoe and the cement are normally tested through a pressure integrity test or a leak-off test [17,18]. As SCP records and vertical interference tests suggest that realistic microannuli and effective wellbore permeabilities are of the order of tens of micrometer or about 1 darcy or less [19][20][21], conventional pressure integrity tests cannot be relied upon for detecting such features behind casing. ...
Article
Full-text available
The cement behind casings is an important barrier element in wells that should provide zonal isolation along the well. The hardened cement does not always isolate permeable formations, either due to placement issues or loads that over time compromise the integrity of the barrier. The modern method used to characterize the annular material is ultrasonic logging which provides essential information concerning the type of material behind casing, but no measurement of the annular permeability. This study provides permeability characterization of a casing-cement sandwich joint retrieved from a 33 years old production well that has been logged at surface using a state-of-the-art ultrasonic tool. The joint contains an interval of low-permeable cement that previously has prevented permeability measurement by gas injection. A pressure-pulse decay test method has now been performed that is based on monitoring the evolution of a pressure pulse through the joint. Long-term pressure measurements show communication through the entire joint and are in qualitative agreement with the log. A pressure diffusion model is used to estimate local permeability along the joint, enabling comparison of log response and permeability. The low-permeable region is relatively short, situated directly on top of a casing collar, and has permeability that is orders of magnitude lower than the cement above and below. In the longer term, results from this and related studies can be used as input for future sustained casing pressure evaluations or for quantifying seepage risk behind casings for abandonment designs.
... The extent and consequences of tensile failure are increased when drilling in deep offshore basins, depleted formations, or when planning highly deviated wellbores. Information from leak off tests (LOTs) is often used to find the upper bound for wellbore fracturing [13][14][15][16]. The predicted pressure from this test is used as the upper bound mud weight that can be used to drill the formation. ...
Article
Full-text available
This paper presents finite-element simulation for hydraulic fracture's initiation, propagation, and sealing in the near wellbore region. A full fluid solid coupling module is developed by using pore pressure cohesive elements. The main objective of this study is to investigate the hypothesis of wellbore hoop stress increase by fracture sealing. Anisotropic stress state has been used with assignment of individual criteria for fracture initiation and propagation. Our results demonstrate that fracture sealing in "wellbore strengthening" cannot increase the wellbore hoop stress beyond its upper limit when no fractures exist. However, this will help to restore part or all of the wellbore hoop stress lost during fracture propagation.
Article
Typical analysis of the Leak-off testing (LOT) in wells assumes elastic wellbore and involves identification of diversion points from linear trends of the recorded plots. However LOTs from wells in the shallow marine sediments (SMS) are inherently nonlinear and their analysis becomes a problem. The paper presents mathematical models of the pressure-volume behavior for two different possible failures around the casing shoe, annular crack (cement-rock parting) and formation fracture. The study submits that these two failure modes would control abnormal LOT patterns. A general pressure-volume model of LOT has been developed including volumetric effects of wellbore expansion, mud loss into the rock, and propagation of an annular crack or plastic fracture. A diagnostic method is proposed to identify LOT-control mechanisms (i.e., formation fracture, annular crack, or mud loss) by analyzing the shut-in section of the LOT's plot.
Article
This article is intended to present new procedures being used today to simplify and help quantify the leak-off test. Testing the well bore for maximum pressure limits can be easy and accurate if certain specific procedures are followed. These procedures include correcting for mud gel strength, displaying pressure and volume limits on the test graph, having the proper pump rate, and running the test long enough. Post-test analysis is also a critical factor.
Article
Review of three steps for applying the most widely used fracture gradient prediction method shows how to use it correctly and incorporate new data for calculations in deep water.
Article
SPE/IADC Member Abstract A Pressure Integrity Test (PIT), sometimes called a leak-off test, is a field practice used to evaluate cement jobs and estimate the formation fracture gradient. Interpretation of the PIT becomes the basis for critical well decisions such as cement job evaluation, casing setting depths, mud weights, and well control alternatives. Incorrect interpretation of the PIT can lead to unnecessary squeeze jobs, premature setting of casing, lost circulation, or other costly problems that could jeopardize well progress. The shape of the PIT plot and the leak-off pressure are primarily governed by the local stresses at the wellbore wall. However, a number of other factors can distort PIT results and lead to interpretation difficulties. This paper explains how these factors can influence the PIT and shows how to interpret PIT plots and differentiate between cement problems and formation effects. Field examples are given to demonstrate the practical application of simple guidelines for PIT interpretation. The paper also presents guidelines for procedures to help ensure a valid PIT. Introduction A pressure integrity test is essentially a measurement of the strength of the formation. Formation strength is governed primarily by the natural compressive stresses exerted on most underground rock. Figure 1 illustrates these stresses on an element of underground rock. Vertical stress in the rock is a function of the overburden, or the weight of the earth above. Horizontal stresses are affected by rock properties and vertical stress. Under the influence of vertical stress, rock tends to shrink vertically and expand horizontally. Horizontal expansion is resisted by the surrounding rock, which creates horizontal stress. It is important to know the pressure integrity of downhole formations when planning and drilling a well. The well-planning process uses pressure integrities from offset wells to calculate formation fracture gradients. Critical drilling decisions such as casing setting depths, mud weights, and well control alternatives are based on these values. The PIT is a relatively simple field procedure to determine fracture gradient. It is also used at times to evaluate the integrity of the cement at the casing shoe. Incorrect interpretation of the PIT can lead to a variety of problems and/or unnecessary expense on a well. For example, if a low leak-off value is interpreted as a cement channel, the operator may conduct a squeeze job in an attempt to increase the leak-off pressure. If the low leak-off is simply caused by a lower-than-expected fracture gradient, the operator will have wasted time and money on the squeeze job. Conversely, if a PIT is interpreted as an indication of a low fracture gradient when it is really caused by a cement channel, the operator may use an unrealistically low value as an upper limit to mud weight. This could lead to prematurely setting the next string of casing or improper mud weight choices in a well control situation. If a PIT is misinterpreted to show a higher leak-off rather than a correct lower value, an operator may use an unrealistically high value as a mud weight guideline. This could lead to lost circulation problems. An "unusual" PIT may or may not be cause for concern. PIT behavior is affected by a variety of factors, the most influential of which are discussed below. Almeida presents a good summary review of all the factors that influence leak-off pressure. Basic PIT Theory The same basic procedure is used in all PITs: a BOP is closed and fluid is slowly pumped into the well. At a certain pressure, the pump is stopped. Shut-in pressure is monitored for a short-time to check for leaks, then is released. The data are plotted and interpreted to determine the formation pressure integrity. P. 169
Book
This book is a reference to the application of significant technological advances in hydraulic fracturing. It features illustrative problems to demonstrate specific applications of advanced technologies. Chapters examine pretreatment formation evaluation, rock mechanics and fracture geometry, 2D and 3D fracture-propagation models, propping agents and fracture conductivity, fracturing fluids and additives, fluid leakoff, flow behavior, proppant transport, treatment design, well completions, field implementation, fracturing-pressure analysis, postfracture formation evaluation, fracture azimuth and geometry determination, and economics of fracturing.
Thesis
In this study, the development of a computer simulation model for leak-off tests has been accomplished. This model is more realistic than the one currently used, but is sufficiently simple that it can be applied with data normally available during leak-off test operations in the field. The model includes the many factors that affect pressure behavior during the test, and can predict with reasonable accuracy what the pressure curve will look like. In addition, test interpretation using the computer model is easily achieved using a curve matching technique. The first step toward the development of the computer model was to subdivide the leak-off test into four phases: (1) pressure increase due to overall compressibility of the system, (2) fracture initiation, (3) fracture expansion, and (4) pressure decline and fracture closure after the pump is shut-in. The second step was the development of mathematical models for each phase separately. The mathematical model that predicts pressure increase before fracture initiation includes the most important variables affecting overall compressibility of the system. The modeling of fracture initiation is based on the classical elasticity theory. The modeling of fracture expansion and closure is based on the solution of the continuity equation for flow into a vertical-elliptical fracture with constant height. A computer program that predicts the pressure behavior during the leak-off test was written. This computer model was then verified using field data furnished by Tenneco Oil Company.
Book
This book reports on single-phase gas reservoirs; gas-condensate reservoirs; undersaturated oil reservoirs; saturated oil reservoirs; and single-phase oil reservoirs.
Book
This book discusses petroleum engineering. Engineering science fundamentals and engineering applications involving these fundamentals are presented. Subjects covered include rotary drilling, drilling fluids, cements, drilling hydraulics, rotary drilling bits, formation pore pressure and fracture resistance, casing design, directional drilling and deviation control, plus two appendices and numerous examples.
Article
Well bore compressibility and hydraulically formed fractures can contribute to elastic well bore deformation in unstable shale formations. During leak-off tests in a basin near the Terek River in eastern North Caucasus in the former Soviet Union, the mud and well bores had anomalous, high compressibilities. Subsequent analyses indicated the system compressibility was related to elastic hydrofracture behavior, with the fracture being open without additional pressure applied at the surface. The paper discusses fluid compressibility, well bore deformation, the leak-off tests, and similar problems which occurred in the Maikop shales in the eastern North Caucasus.
Article
This printing includes corrections made since the original publication in 1986. The text presents petroleum engineering science fundamentals as well as example engineering applications involving those fundamentals. Subjects covered include rotary drilling, drilling fluids, cements, drilling hydraulics, rotary-drilling bits, formation pore pressure and fracture resistance, casing design, directional drilling, and deviation control.