A semi-empirical relation between static and dynamic ...

Accepted Manuscript A semi-empirical relation between static and dynamic elastic modulus Mohammad Reza Asef, Mohsen Farrokhrouz PII:

S0920-4105(17)30544-2

DOI:

10.1016/j.petrol.2017.06.055

Reference:

PETROL 4067

To appear in:

Journal of Petroleum Science and Engineering

Received Date: 18 September 2016 Revised Date:

21 May 2017

Accepted Date: 19 June 2017

Please cite this article as: Asef, M.R., Farrokhrouz, M., A semi-empirical relation between static and dynamic elastic modulus, Journal of Petroleum Science and Engineering (2017), doi: 10.1016/ j.petrol.2017.06.055. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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A semi-empirical relation between static and dynamic elastic modulus

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Mohammad Reza Asef: Kharzami University, Faculty of Earth Sciences, Applied Geology Department, Mofateh Ave. No 49, Tehran 15719-14911, Iran

Mohsen Farrokhrouz: South Zagros Oil Company, ICOFC, No. 30, West Roodsar St., Hafez St., Tehran, Iran

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Corresponding author: Kharzami University , Faculty of Earth Sciences, Applied Geology Department Mofateh Ave. No 49. Tehran 15719-14911. Iran. E-mail: [email protected]

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ABSTRACT

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Researchers have long investigated the difference between static and dynamic

3

Young’s modulus in terms of values and application. In field experiences, dynamic

4

Young’s modulus can often be calculated using logging data, while core data for

5

static experiments in the lab are scarce. On the other hand, geomechanical analyses

6

are traditionally based on static values. Therefore, it is essential to estimate static

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Young’s modulus based on dynamic data for petroleum reservoir case studies.

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A number of empirical equations have been suggested to estimate static Young’s

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modulus based on dynamic Young’s modulus. However, the validity of these

10

relations are limited and local. In this paper, 10 suggested relations were evaluated

11

against an experimental dataset from different geographic locations. The validity of

12

each relation was evaluated, and the results confirmed the dependency of these

13

equations to the original datasets.

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Based on the concept of effective Young's modulus, we suggested a relationship to

15

predict the value of static Young's modulus from dynamic Young's modulus and

16

porosity data. The approach was performed for different rock types, and the

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predictions were found to be very accurate. This research explored different views

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on this aspect and facilitated better decision making to determine the static

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Young’s modulus of rocks in the case where only dynamic data were available.

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A semi-empirical relation between static and dynamic elastic modulus

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1. Introduction

Elastic Young's modulus is a general rock mechanical parameter that is usually

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measured in all geomechanical projects. This parameter is important because it is a

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direct representative of material stiffness: the lower the elastic modulus value, the

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lower the material stiffness.

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Elastic constants (Young’s modulus and Poisson’s ratio) are considered to be

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fundamental mechanical properties of rock material that are required to be

12

measured for analysis and design of engineering projects. Additionally, they are

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extensively used in analytical equations and modeling techniques to predict the

14

stress–strain behavior of rock subjected to various loading conditions.

15

Two methods are commonly used to measure the elastic modulus of rock material:

16

static and dynamic. Static Young’s modulus (Es) is commonly obtained as the

17

tangent of the stress-strain curve at 50% of the maximum strength of rock core

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specimens. Dynamic Young’s modulus (Ed) can be calculated knowing the rock

19

density as well as compressional and shear wave velocities. Measurement of static

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Young’s modulus requires a destructive testing approach in the lab, where

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relatively large static loads are applied to rock core specimens. Dynamic Young's

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modulus, however, is obtained from non-destructive tests, where the velocities of

23

propagation of elastic waves (seismic or acoustic) are measured. It is interesting to

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note that acoustic wave data, namely, compressional and shear wave velocities (Vp

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and Vs), can be obtained both in the lab and at the project site from logs. When an

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acoustic wave propagates through a porous medium, the deformation of the grains

27

is elastic. Since core samples are not always available, laboratory measurements

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are more expensive. Therefore, dynamic measurements are often more accessible.

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In addition, using Vp and Vs log data makes it possible to obtain dynamic values

30

for the whole volume of encountered rock, while static tests are performed on a

31

sample of limited length. However, there is a meaningful difference between static

32

and dynamic Young’s moduli.

33

Several studies have shown that static and dynamic properties are different. The

34

ratio between dynamic and static moduli was found to range between 1 to 20

35

(Wang, 2000). Low ratios occur for stiff rocks, and higher ratios are obtained for

36

softer sediments. For a solid material like steel, the ratio is 1 (Weast, 1986).

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Therefore, correlating these two values has been an important issue from very early

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times. Many descriptions have been proposed to explain this difference, ranging

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from strain amplitude effects to viscoelastic behavior. Zisman (1933) provided a

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qualitative explanation of the consistently higher magnitude of Ed compared to Es.

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He suggested that the wave pulse or packet of energy passing through rock upon

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entering a cavity (crack, pore) suffers a loss of energy (due to reflection and

43

refraction) at the fluid/rock and rock/fluid interfaces. A small amount of wave

44

energy is transmitted across the boundary and a large amount is scattered around

45

the cavities. Therefore, this difference was explained as static measurements being

46

more influenced by the presence of fractures, cracks, cavities, planes of weakness,

47

or foliation (Ide, 1936; Sutherland, 1962; Coon, 1968). Investigation of such

48

differences is still an active area of research to understand the various contributing

49

parameters and to better interpret the mechanical properties from wave velocity

50

measurements (Asef and Najibi, 2013; Najibi and Asef, 2014). In this paper, a

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comprehensive study of suggested prediction equations is presented and the range

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of predicted values based on different equations is compared. Afterwards, a new

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approach is implemented according to the poroelastic nature of rocks, especially

54

sedimentary rocks, to present theoretical and practical views on the conversion of a

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dynamic modulus to a static modulus.

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2. Previously Suggested Relations 2.1 General Equation Form

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The general form of previously suggested equations for predicting static Young’s

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modulus is as follows:

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E s = aE d + b

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where ‘a’ and ‘b’ are the constants that are highly dependent on the rock type. A

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list of suggested equations for predicting static Young’s modulus is shown in Table

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1. As stated before, dynamic Young’s modulus is larger than the corresponding

64

static modulus. The difference is large for weak rocks and is reduced with

65

increases of confinement (King, 1970). The static modulus is affected by existing

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micro-cracks, while the dynamic modulus appears to be less affected by micro-

67

cracks (Tutuncu and Sharma, 1992; Morales and Marcinew, 1993; Yale and

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Jamieson, 1994). However, since dynamic Young’s modulus is calculated by wave

69

velocities, some parameters, such as porosity, can influence the dynamic modulus

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to a large extent (Asef et al, 2010).

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Table 1

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Suggested equations for prediction of static Young’s modulus

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(1)

Equation

Unit

R2 *

Rock Type

Reference

(2)

Es =1.26Ed − 29.5

GPa

0.82

Igneous and metamorphic rocks

King (1983)

(3)

Es = 0.74Ed − 0.82

GPa

0.84

Different Rock Types

Eissa and Kazi (1988)

No.

Es =1.05Ed − 3.16

GPa

0.99

Sedimentary rocks

Christaras et al (1994)

(5)

Es = 0.83Ed

GPa

---

Concrete

Neville (1995)

(6)

Es =1.25Ed −19

GPa

---

Structural Design

CP110 (1972)

(7)

Es = 0.2807Ed

GPa

0.60

Sedimentary rocks

Brautigam (1998)

(8)

Es = 0.8069Ed − 29.5

GPa

0.92

Composite Resin

Helvatjoglu et al (2006)

(9)

Es = 0.7707Ed − 5854

MPa

0.96

Mokovciakova (2003)

(10)

E s = e (0.0477 Ed )

GPa

0.72

Different Rock Types

Fahimifar and Soroush (2003)

(11)

Ed = 0.5087Es − 6 ×106

psi

0.60

Sandstone

Al-Tahini (2003)

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(4)

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Different Rock Types

*R2 reported in the original references

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2.2 Prediction Verification

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A large amount of experimental data were gathered and classified into four

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categories: limestone, sandstone, shale, and tuff. Some of these data have been

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previously published by others, and some data were obtained experimentally by the

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authors. Among these data, 19 limestone values were obtained from Al-Shayea

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(2004) and Farquhar et al (1994) and 38 values were obtained from the current

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study. Additionally, 15 shale values were reported by Lashkaripour (1996), and 16

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shale values were from the current study. Al-Tahini (2003) published some

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sandstone data from deep core samples in an oil field, and 15 other specimens were

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tested in the current study. The scatter of the gathered data versus different

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equations is illustrated in Figure 1.

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As is seen in Figure 1, the data points are highly scattered versus the predicted

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equations. This analysis also revealed that large errors imply predictions that are

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far from reality. Some of the equations give negative values for Young’s modulus,

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which is obviously not true. Some of them may be suitable for a specific rock type,

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while others give highly erroneous values. Altogether, application of the prediction

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equations listed in Table 1 requires calibration of the constant values, and any

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calibration means a new correlation. Accordingly, in this research, the

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methodology suggested for predicting Young's modulus of any porous material

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(ceramics, biological tissues, foams and other sintered materials) was selected for

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natural porous material (rock and/or soil). The mechanical theory behind this

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methodology is introduced in poromechanics and, in this paper, it is modified into

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poroelasticity.

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Figure 1: Scatter of data points versus different suggested equations listed in Table

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3. New Approach for estimation of the static modulus

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Within the last few years, Phani and Niyogi (1987) have proposed a power-law

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empirical relationship for predicting porous material properties over an entire

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range of porosity (Equation 12):

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 ϕ E = E 0 1 − ϕ c 

   

f

(12)

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where E is effective Young’s modulus of porous material with porosity φ, E0 is

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Young’s modulus of solid material, φc is the porosity at which the effective

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Young’s modulus becomes zero, and ‘f’ is the parameter dependent on the grain

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morphology and pore geometry of porous material (Phani and Niyogi, 1987).

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Later, it was noted that fitting experimental data to Equation (12) often gives φc = 1

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(Wagh et al. 1991). Some other studies suggest that either φc = 1 (Boccaccini et al.

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1995; Matikas et al. 1997; Maitra and Phani, 1994) or that the power-law

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relationship is converted to a linear model when f ≡ 1 (Lam et al. 1994). In the

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latter case, φc is considered to be the initial powder porosity. There have been some

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studies that state that the theory behind Equation (12) can be justified by

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percolation theory for the behavior of Young’s and shear moduli of porous

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material (Kovacik, 1999). Porous materials (including rocks) are generally

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prepared from powders or particles of different sizes and shapes. During the

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consolidation process, a range of porosities can be achieved based on tectonic

119

parameters, such as temperature, external pressure or time. As particles bump into

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each other, the compaction process starts and creation and growth of the necks

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between particles leads to a decrease in porosity. Subsequently, pore channel

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closure results in the elimination of the pores. According to percolation theory,

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there is a critical volume fraction ‘nc,’ called the percolation threshold, at which a

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continual solid phase is formed that spans the whole system. At and above this

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threshold, there is a relationship between the volume fractions of the solid material

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and other geometrical, physical and mechanical properties, as shown by Equation

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(13):

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Π ∝ (n − n c )f

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for n ≥ n c

(13)

where Π is the property under investigation (it could be any parameter, such as E,

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Vp, Vs), ‘n’ is the volume fraction of the solid material, and ‘f’ is the critical

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exponential that is predicted for the investigated property. Another result of

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percolation theory is that the value of the critical exponential (f) does not depend

133

on the structure or geometrical properties of the system, but only depends on the

134

dimension of the problem (Stauffer and Aharony, 1992). Moreover, the value of

135

the percolation threshold (nc) significantly depends on the structure. Therefore,

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experimental values are reported in the range of 0.06 - 60 vol% of the three-

137

dimensional structure (Schueler et al. 1997; Balberg and Binenbaum, 1987). If

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porosity is used instead of the volume fraction of the powder (as it is known that φ

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= 1 - n), another form of percolation theory is obtained:

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Π ∝ (ϕ c − ϕ )f

for ϕ ≤ ϕc

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Applying the boundary conditions of Π = Π0 at φ = 0 leads to Equation (15):

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ϕ −ϕ   Π = Π 0  c   ϕc 

(14)

f

for ϕ ≤ ϕ c

(15)

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This is identical to Equation (12). Percolation theory predicts that for Young’s

144

modulus in a 3D structure, ‘f = 2.1’ (Sahimi, 1994).

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Kovacik (1999) calculated the value of ‘f’ for different sintered materials and

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obtained various values depending on the shape of the powders. Table 2 shows the

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results of his studies.

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Table 2

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The values of ‘f’ in different sintered materials (Kovacik, 1999) Sintered material

Porosity Range

E0 (GPa)

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Th2O powder (2µm)

0 – 0.33

263.9 ± 3.4

1.22 ± 0.10

Th2O powder (4µm)

0 – 0.39

264.2 ± 3.9

1.36 ± 0.17

Si3N4 doped with CeO2

0 – 0.38

284.1 ± 4.3

1.13 ± 0.12

Sintered Iron

0 – 0.28

213.2 ± 3.1

1.19 ± 0.28

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α-Al2O3 Powder

0 – 0.39

f

399.6 ± 7.7

1.25 ± 0.13

In the present study, the same approach was followed to predict static Young’s

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modulus from dynamic Young’s modulus. We also assumed that φc = 1 and that

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the new proposed model for Young’s modulus of rock was:

154

E = E 0 (1 − ϕ )f

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To generate a more simple equation by applying mathematical series, the general

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form of Equation (16) is nearly equal to the following form (see also Appendix I):

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E ≈ E 0 [1 − fϕ ]

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(16)

(17)

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Considering (static or dynamic) Young’s modulus as a function of E0 and porosity

159

(as in Equation 17) and bearing in mind the general form of the conversion of static

160

Young’s modulus to dynamic Young’s modulus (Equation (1)), the new proposed

161

equation is Equation (18):

162

 E s = aE d + b  ⇒  E s = a' [E d0 (1 − fϕ )] + b' ( ) ϕ E E 1 f = − 0   

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where ‘a'’ and ‘b'’ are constant values. This semi-empirical approach was

164

employed for the above-mentioned dataset that contained different rock types. The

165

values of the constants and correlation coefficients are shown in Table 3. In this

166

Table, the adjusted R2 (R2 adjusted for the number of independent variables in the

167

model) and predicted R2 (that reflects how well the model will predict future data)

168

were presented to adjust R2 for unrepresentative statistical values. It should be

169

clarified that the R2, adjusted R2, and predicted R2 values shown in Table 3 are in

170

reasonable agreement with each other. Adjusted R2 is especially important in this

171

research because if unnecessary variables are included, R2 can be misleadingly

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high. For all data sets, adjusted R2 is close to R2, which implies that both variables

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Ed and ϕ significantly contribute to the prediction equation.

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(18)

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Further statistical investigation showed that the constant ‘f’ could be considered to

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be unity, as in Lam et al (1994), within an acceptable error range. Therefore, the

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final form is Equation (19):

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E s = a' E d (1 − ϕ ) + b'

(19)

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Table 3

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Statistical parameters for different rock types according to Equation (19) Rock

No. of

a'

b'

Normal

Adjusted

Predicted

Data

Type

data

R2

R2

R2

a'

b'

Price et al (1994)

Tuff

12

0.83091

-0.48656

0.8712

0.8658

0.8518

0.0001

0.0001

Martin et al

Tuff

26

2.16135

-49.14988

0.7868

0.7654

0.6234

0.0001

0.0001

38

0.90255

-2.07400

0.8621

0.8583

0.8463

0.0001

0.0001

Shale

38

0.61345

7.26414

0.7870

0.7811

0.7620

0.0001

0.0001

Shale

14

0.48898

-0.59017

0.8528

0.8405

0.8068

0.0001

0.0001

52

0.93913

-3.69662

0.8946

0.8925

0.8879

0.0001

0.0001

All Tuff Yale and

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Jamieson (1994) Lashkaripour (1996)

Al-Tahini (2003)

Sand

Morales and

Sand

Marcinew (1993)

p-value

30

0.99581

-11.10263

0.8561

0.8510

0.8362

0.0001

0.0001

9

0.89071

-5.15572

0.9451

0.9372

0.8850

0.0001

0.0001

39

0.92596

-7.48396

0.8949

0.8921

0.8831

0.0001

0.0001

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Reference

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Lime

15

0.64897

8.90570

0.9396

0.9350

0.9268

0.0001

0.0001

Author 1

Lime

17

1.04057

-7.60778

0.9514

0.9481

0.9400

0.0001

0.0001

Author 2

Lime

8

0.93596

-4.95594

0.9604

0.9538

0.9314

0.0001

0.0001

Yale and

Carbonates

22

0.66762

8.86661

0.8917

0.8863

0.8747

0.0001

0.0001

All Lime

62

0.90804

-2.07677

0.9391

0.9380

0.9311

0.0001

0.0001

All Data

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0.89561

-2.84374

0.9141

0.9136

0.9114

0.0001

0.0001

Yale and

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As can clearly be concluded from Table 3, in Equation (19), a' is often a coefficient

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that is less than unity and b' is generally a negative value depending on the range

184

and normal distribution of the input values. All of the p-values presented in Table 3

185

state that there is only a 0.01% chance that this model could occur due to noise.

186

Considering the optimization of the constant parameters, a general model for all

187

rock types is described by Equation (20), with a 16.1% average error (estimated

188

static Young’s modulus with ± 4.5 GPa accuracy) and 4.14 standard error:

189

E s = 0.88Ed (1 − ϕ ) − 3.7

190

Figure 2 illustrates the scatter of the predicted Es values against the observed

191

values based on Equation (20) for all data on a 45° symmetry line. If the predicted

192

values match the observed values, all of the points will lie on the 45° line. The

193

fairly symmetric spread of points along the 45° line in this figure visually confirms

194

the verification of the prediction.

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(20)

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Figure 2: Estimated Es versus Measured Es using Equation (20)

In the next step we observed that constant b' in Equation (20), which is a negative

198

value, improved the prediction when it was related to porosity. Therefore, we

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further improved and optimized Equation (20) into Equation (21), and R2 increased

200

to 0.92, the average error decreased to 15.2 and the standard error decreased from

201

4.14 to 3.89:

202

E s = E d (1 − ϕ ) − 3lnφ

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Conclusion

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Comprehensive statistical and theoretical studies regarding the relationship

206

between static and dynamic Young's moduli for different rock types, including

207

sandstone, shale, limestone, and tuff, revealed that porosity plays an important

208

role. We borrowed the concept of effective Young’s modulus to propose the

209

impact of porosity on the stiffness of rock material. Therefore, a new approach for

210

estimating static Young’s modulus from dynamic Young’s modulus was

211

suggested, while previous general forms of the prediction equations were also

212

considered, simultaneously. Accordingly, using dynamic Young's modulus and

213

porosity, static Young's modulus can be estimated with an acceptable error range

214

of 15% and correlation coefficient of 0.9. This is a significant improvement for

215

predicting the static properties of rock material from petrophysical data in deep

216

underground explorations.

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Exhibition of the 10.2118/24689-MS

Society

of

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Engineers,

Washington

DC,

DOI:

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Yale, D. P., Jamieson, W. H., 1994, Static and Dynamic Rock Mechanical Properties in the Hugoton and Panoma Fields, Kansas, SPE 27939, SPE Mid-Continent Gas Symposium, Texas, DOI: 10.2118/27939-MS

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Zisman, W. A., 1933, A comparison of the statically and seismologically determined elastic constants of rock, Proceedings of National Academy of Science of USA., Vol. 19, No. 7, pp. 680– 686. http://www.jstor.org/stable/86139.

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Appendix I:

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Let g(x) be a function and ‘a’ some point from the interior of its domain. Assume

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that g(x) has derivatives of all orders at ‘a.’ Then, we define its Taylor series at ‘a’

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by the following formula:

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T(x) =

∞

∑

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g (k) (a ) (x − a )k k!

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(A-1)

In the case of a = 0, the series simplifies to the McLaurin series form: T(x) =

∞

∑ k =0

' " "' g (k) (0 ) k (x ) = 1 + g (0) x + g (0) x 2 + g (0) x 3 + ... k! 1! 2! 3!

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In Equation (5), the function form can be converted as follows:

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g(x) = (1 − ϕ )f

⇒ g(x) = 1 +

f (− 1) − f (f − 1) 2 ϕ+ ϕ + ... 1! 2!

(A-2)

(A-3)

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0 < ϕ < 1,

φ2 and higher exponential forms are negligible in

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As porosity is

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comparison with the first order of φ, the final equation is:

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E ≈ E 0 [1 − fϕ ]

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(A-4)

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Highlights: - Empirical equation based on two input parameters (dynamic Young's modulus, porosity)

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- Theoretical backgrounds of porous media were used to explain the meaning of equation

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- We successfully added data from different authors to our own data