ORIGINAL PAPER - PRODUCTION ENGINEERING
Development of a new correlation to determine the static Young’s
modulus
Salaheldin Elkata tny
1
•
Mohamed Mahmoud
1
•
Ibrahim Mohamed
2
•
Abdulazeez Abdulraheem
1
Received: 23 September 2016 / Accepted: 5 January 2017 / Published online: 30 January 2017
The Author(s) 2017. This article is published with open access at Springerlink.com
Abstract The estimation of the in situ stresses is very
crucial in oil and gas industry applications. Prior knowl-
edge of the in situ stresses is essential in the design of
hydraulic fracturing operations in conventional and
unconventional reservoirs. The fracture propagation and
fracture mapping are strong functions of the values and
directions of the in situ stresses. Other applications such as
drilling require the knowledge of the in situ stresses to
avoid the wellbore instability problems. The estimation of
the in situ stresses requires the knowledge of the Static
Young’s modulus of the rock. Young’s modulus can be
determined using expensive techniques by measuring the
Young’s modulus on actual cores in the laboratory. The
laboratory values are then used to correlate the dynamic
values derived from the logs. Several correlations were
introduced in the literature, but those correlations were
very specific and when applied to different cases they gave
very high errors and were limited to relating the dynamic
Young’ modulus with the log data. The objective of this
paper is to develop an accurate and robust correlation for
static Young’s modulus to be estimated directly from log
data without the need for core measurements. Mul tiple
regression analysis was performed on actual core and log
data using 600 data points to develop the new correlations.
The static Young’s modulus was found to be a strong
function on three log parameters, namely compressional
transit time, shear transit time, and bulk density. The new
correlation was tested for different cases with different
lithology such as calcite, dolomite, and sandstone. It gave
good match to the measured data in the laboratory which
indicates the accuracy and robustness of this correlation. In
addition, it outperformed all correlations from the literature
in predicting the static Young’s modulus. It will also help
in saving time as well as cost because only the available
log data are used in the prediction.
Keywords Static Young’s modulus Dynamic Young’s
modulus Log data Hydraulic fracturing In situ stresses
Correlation
List of symbols
E Young’s modulus
Dt Transit time
e Strain
m Poisson’s ratio
q
b
Bulk density of the rock
r Stress
Subscripts
s Shear
p Compressional
dynamic Dynamic value
static Static value
Introduction
The terms Young’s modulus, tensile modulus, elastic
modulus, modulus of elasticity, and stiffness are refereeing
to the mechanical property that measures the stiffness of a
certain material. Young’s modulus is the ratio of the stress
applied on the material to strain associated with the applied
& Salaheldin Elkatatny
elkatatny@kfupm.edu.sa
1
King Fahd University of Petroleum and Minerals, Dhahran,
Saudi Arabia
2
Advantek Waste Management Services, Houston, TX, USA
123
J Petrol Explor Prod Technol (2018) 8:17–30
https://doi.org/10.1007/s13202-017-0316-4
stress (Chen 2011). Young’s Modulus can be calculated
using Hook’s Law (Nguyen et al.
2009) as follows:
E ¼
r
e
ð1Þ
where E = Young’s modulus (GPa), r = stress (GPa), and
e = strain.
Young’s modulus is different for different rock type; the
value depends on the rock properties including porosity,
lithology, temperature, pore pressure, fluid saturation, and
the rock consolidation (William
1969). Soft formations like
shale have a low Young’s modulus value
(100,000–1,000,000 psi) comparing to medium formations
like sandstone (2,000,000–10,000,000) and hard formation
limestone (8,000,000–12,000,000 psi) as presented by Nur
and Wang (
1989).
The ranges presented above show that there is no typical
value of Young’s modulus for certain rock, and measuring
the Young’s modulus is a must in order to conduct the
geomechanical analysis for the formation in interest.
Building a geomechanical model is essen tial for several
applications related to mechanical rock failure during well
drilling, completion, and stimulation, which include esti-
mation of the formation breakdown pressure, fracture
simulation, wellbore stability, and formation compaction
(Ciccotti and Mulargia
2004).
Young’s Modulus can be either calculated from the
sonic and density logs (dynamic Young’s modulus), or
measured directly in the laboratory (static Young’s mod-
ulus). The dynamic modulus is usually significantly higher
than the static moduli as originally noted by Zisman (
1933)
and Idle (
1936). The difference between the dynamic and
static modulus is more pronounced for soft rocks (sand-
stone) than hard rock (granite) (King
1966).
Idel (
1936), Brace (1965), and Walsh (1966) stated that
the difference between the static and dynamic modulus is
strongly affected by the rock microstructure (natural frac-
tures and pores) and the confining stress. High stresses
might close the microcracks and result in increase in the
velocity waves as the rock is compacted. The faster waves
will be interpreted as a higher elastic modulus (dyna mic
modulus). Zisman (
1933) explained the difference between
the static and dynamic modulus by the loss of energy that
the wave pulse might suffer when passing through the rock
pores (intergranular pores or natural fractures) due to
reflection and refraction at the fluid/rock interfaces
(Howarth
1984). Static Young’s modulus is often used in
the wellbore stability and in situ stress applications (Led-
better
1993; Hammam and Eliwa 2013). However, col-
lecting cores from the well to measure the moduli in the
laboratory is expensive and usually not feasible.
Several equatio ns have been developed to establish a
relationship between the static and dynamic Young’s
modulus. However, the applica bility of each of those cor-
relations is limited to specific rock type under certain
conditions. Belikov et al. (
1970) has developed a correla-
tion to estimate the static Young’s modulus from the
dynamic one. However, his equation is only applicable for
microcline and granite rock. King (
1983) established a
correlation for igneous and metamorphic rocks. McCann
and Entwisle (
1992) equation is valid for Jurassic granites.
Morales and Marcinew (
1993) correlation can be used for
rocks with high permeability values. Wang (
2000) has
developed two different correlations for hard and soft
rocks. Gorjainov (
1979) introduced two relationships for
clays and for wet soils. Al so, there are some other equa-
tions that can be used for a wide range of rocks (Eissa and
Kazi
1988; Canady 2010). As the Young’s modulus value
are strongly dependent on the rock microstructure, miner-
alogy, and confining stresses, neither of the correlation
mentioned above give an acceptable match to the labora-
tory measurement for carbonate formation in Saudi Arabia
as well be shown later in ‘‘
Triaxial testing’’ section, and
development of a new correlation to calculate static mod-
ulus is essential.
The in situ stresses can be determined from the leak-off
test, and the fracture pressure of the formation behind the
casing can be determined as well as the minimum hori-
zontal principal stress. The nonlinear behavior of the leak-
off test could be due to drilling fluid loss to the formation,
fractures and cracks in the cement behind the casing (test
run before cement setting), and plastic fracturing around
the wellbore. The linear behavior during the leak-off test
resulted from the drilling fluid compression and wellbore
expansion around the well. The leak-off test should be
repeated several times to distinguish between the different
mechanisms to identify the rock fracture pressure and
minimum horizontal stress (Zhou and Wojtanowicz
2002).
The determination of in situ stresses are important
during the drilling operations to maintain the hole integrity
and wellbore stability to avoid drilling problems. Integrated
rock mechanical properties analysis will enhance the dril-
ling process because rock mechanical properties are one of
the groups that should be integrate d with petrophysical
parameters and other parameter to enhance the drilling and
avoid wellbore stability problem s. Nes et al. (
2012) have
done a complete analysis by integrating several rock and
fluid properties among them is the rock mechanical
parameters to reduce the wellbore stability issues during
drilling using different types of drilling fluids. They
developed a model that can be used to identify the drilling
problems and to design the drilling operations. They used
their model in high-pressure high-temperature drilling and
compared the data to the field observations.
The static Young’s modulus is used to identify the down
hole stresses profiles which are important for fracture
18 J Petrol Explor Prod Technol (2018) 8:17–30
123
mapping and fracture design in several rocks such as car-
bonates and unconventional shales. Li et al. (
2014) intro-
duced analytical solution to the stresses induced during the
fracturing of unconventional shale and how they can be
used in shale gas exploration. Their model can be used to
predict the induc ed stresses due to the fracturing operations
and also to predict the minimum spacing between the
fractures to prevent the communication between these
fractures which will cause loss of gas production due to the
interference between these fractures. Zhou et al. (
2015)
studied the interaction between the hydraulic fractures in
shale formations numerically. They concluded that a stress
shadow area around the fracture will be generated due to
the induced stresses, three areas of compressive stress will
be formed , and this will affect the fracture orientation from
the normal trends or direct ions.
Chan and Board (
2009) used finite elements calculations to
determine the induced stresses to heating and thermal effects.
They found out that the in situ induced stresses due to thermal
heating and the rock displacement are primarily affected by
the temperature relation with the rock thermal expansion
coefficient. They considered this coefficient to be the main
factor that controls the prediction of the rock stresses.
Based on the literature survey, it can be said that till now
there is no such general equation to calculate the static
Young’s modulus from the well log data. All the correla-
tions reported in the literature are based on the laboratory
measurements to develop the relation between dynamic
and static Young’ modulus which is time-consuming and
expensive. The relationship should be picked carefully to
satisfy the validation conditions of each equation. Other-
wise, generalization will always give the wrong answer.
The objective of this study is to develop a new correlation
to estimate the static Young’s modulus from the well log
data for carbonate rocks (limestone and dolomite). This
correlation can be used directly to estimate the static
Young’s modulus from log data (density, compressional,
and shear transit times).
Triaxial testing
In the laboratory, Young’s modulus can be calculated from
the stress–strain curve, and the experiment can be con-
ducted under uniaxial or triaxial stress conditions. It is
always preferred to run the test under triaxi al conditions
because uniaxia l test might overestimate the static Young’s
modulus value due to closing the fractures parallel to the
stress direction (Thill and Peng
1974). The triaxial test is
used to measure the mechanical properties of a cylindrical
rock sample. The fluid pore pressure, drainage conditions,
axial load, and confining stresses can be controlled to
simulate the actual formation conditions.
A triaxial test is conducted by loading the sample axially
while applying a constant all-around (cell or confining)
pressure equivalent to the effective reservoir pressure. The
prepared samples wer e 3.81 cm in diameter and 7.62 cm
long. The end faces of all samples were cut and ground
parallel. The samples were then cleaned using toluene and
were vacuum-dried in the oven at 60 C. All samples were
tested under dry condition. The testing was conducted
under room temperature, and confining pressure was kept
constant during the loading phase. The confining pressure
was calculated using Eq. (
2).
r
0
3
¼
m
1 m
r
v
aP
p
þ r
tectonic
ð2Þ
where r
0
3
is the effective confining pressure; m is the
Poisson’s ratio (assumed 0.3); r
v
is the vertical stress; a is
Biot’s constant (assumed 1); and P
p
is the pore pressure.
Tectonic component is estimated using leak-off or micro-
frac tests conducted in the field.
The confining pressure was increased gradually from 0
to the required level with an increment of 0.02068 MPa/s.
For the determination of E and m of the rock sample, the
sample was jacketed using heat shrink tubing. The jacketed
sample was then placed between the hardened steel plates,
and the sample was tightly secured with the platens using
steel wires. The sample was then instrumented with
LVDTs (Linear Variable Differential Transformer). Two
LVDTs were used for recording axial displacement. These
two LVDTs were mounted on the steel platens opposite to
each other using LVDT holders. The radial displacement
was recorded using an LVDT mounted directly on the
sample. An LVDT—instrumented rock sample is shown in
Fig. 1 A rock sample instrumented with LVDTs for measuring
deformation in the sample
J Petrol Explor Prod Technol (2018) 8:17–30 19
123
Fig. 1. The stress–strain response was plotted for all tested
samples, and the elastic constants (Young’s modulus and
Poisson’s ratio) were computed at 50% of the peak stress
(for example, it will be at 80 MPa for Fig.
2). A tangent
will be done at stress = 80 MPa, and the static Young’s
modulus will be calculated from the slope of the stress –
strain line tangent at 80 MPa. The left curve (radial one)
the slope of the tangent at 80 MPa will yield the static
Young’s modulus divided by the static Poisson’s ratio.
Correlation development
Formation characterization
The selected formation comprising of carbonates and
anhydrite. The overall petrographic characteristics of seven
samples showed that the studied section is divided into
three units: dolomitized grainstone, lime mudstone, and
peloidal bioclas tic–intraclastic dolograinstone.
Dolomitized grainstone facies were observed in two
samples that consist of medium-to-coarse-grained, moder-
ately sorted grainstone. All the matrix and debris were
dolomitized showing dolomitized rhombs, Fig.
3. The
grainstone was partly leached and shows both intercrys-
talline and intracrystalline porosity, which ranges from 9 to
10%.
The lime mudstone facies were observed in three sam-
ples consi sting of gray colored compact limestone with
scattered dolomite rhombs and anhydrite, Fig.
4. The rock
is very tight, and the porosity ranges from 0.5 to 1.0%.
Peloidal bioclastic–intraclastic dolograinstone facies
were observed in two samples consisting of medium-to-
fine-grained peloidal grainstone with some skeletal debris
and intraclast, Fig.
5. At some places, anhydrite is found
replacing some of the leached peloidal and skeletal grains.
Leaching of the grains has resulted in good porosity
development, which ranges from 9 to 10%.
Log data analysis
Figure
6 shows the available log data for the selected
formation. The data contain the neutron porosity, bulk
density, and sonic time (compressional and shear times).
The available data were used to estimate the dynamic
parameters; Poisson’s ratio, and Young’s modulus. Fig-
ure
6 shows the results of the dynamic geomechanical
Fig. 2 Determination of static Young’s modulus from stress–strain
curves. E
static
= static Young’s modulus and m
static
= static Poisson’s
ratio
Fig. 3 Thin section photomicrograph, X25 under cross-polarized
light, of carbonate sample (Depth: XX43.8 m)
Fig. 4 Thin section photomicrograph, X25 under cross-polarized
light, of carbonate sample (Depth: XX33.2 m)
Fig. 5 Thin section photomicrograph, X25 under cross-polarized
light, of carbonate sample (Depth: XX44.4 m)
20 J Petrol Explor Prod Technol (2018) 8:17–30
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parameters Young’s Modulus (E
dynamic
) and Poisson’s
Ratio (m
dynamic
).
m
dynamic
¼
2
Dt
s
Dt
p
2
2 2
Dt
s
Dt
p
hi
2
ð3Þ
E
dynamic
¼ 21 m
dynamic
1000q
b
ðÞ1000
0:3048
Dt
s
2
"#
ð4Þ
where Dt
s
= Shear transit time, lsec/ft, Dt
p
= compres-
sional transit time, lsec/ft, q
b
= bulk density, g/cc,
m
dynamic
= dynamic Poisson’s ratio, dimensionless, E
dy-
namic
= dynamic Young’s Modulus, GPa.
Environmental corrections
The environmental corrections for bulk density, neutron
porosity, and sonic time were performed. Corrections such
mud cake correction and lithology correction were applied
for all logs used. The lithology correction was done using
the cross-plot.
Depth shifting
The log parameters for each static Young’s modulus
measured in the laboratory were obtained at the corre-
sponding core depth after adjusting the depth between the
log and core data (depth shifting). This correction of depth
shift is necessary, as the depth measurement for log values
using cables is not the same as that measured from the
number of drill strings and core lengths for core data. Core
porosity values were also used to estimate the depth shift
by comparing them with the corresponding log porosity
and density values. From the core and log porosity profiles,
it is cla rified that the coring depth shift was 1.46 m for the
base case, and the core depth was increased by this value to
match the log depth. Hence, no depth shift was applied to
convert laboratory-based sample depth to corresponding
log depth. This was done for all cases tested using the new
correlation.
Static Young’s modulus
Table
1 lists the static Young’s modulus for five core
samples that were measured in the laboratory. To obtain
the static Youn g’s modulus for all the depth range, a
Fig. 6 Log data for the base case. The top depth of the formation is represented by the zero value here, and the bottom depth is the 91.44 m
Table 1 Core depth and static Young’s modulus from laboratory
measurements
True vertical depth (m) E
static
(GPa)
XX35 36
XX41 37
XX45 31
XX46 26
XX51 35
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