2.1. Grey Relational Analysis
Grey relational analysis (GRA), grey incidence analysis (GIA) and grey correlation analysis are three names for the same phenomenon; first reported by Javed in 2018 [
6,
22,
25,
26]. Scholars from engineering fields are more familiar with the term “GRA” [
6] and this convention has been followed throughout the current study. GRA methods constitute one of the core areas of grey system theory (GST), which was proposed by Chinese scientist, Deng Julong, in 1982 to manage the uncertain systems containing poor information. GST belongs to the family of uncertainty theories, where other family members are fuzzy theory, interval theory, rough set theory, so on [
27,
28]. However, grey system theory, guided by its own distinct approaches, deals with uncertainty unlike other uncertainty theories [
29,
30]. GST categorizes all systems of the world into three classes; black, white and grey [
31]. A black system implies a system for which no information is available and the white system is the system for which the entire information is available. Thus, a grey system becomes a system with partially known and partially unknown information [
32]. The key strength of GST and its models is their predictions and decision making using small sample size, poor data and incomplete information [
33,
34,
35,
36]. The GRA models strive to understand the uncertain relations among the parameters associated with the grey systems. GRA is not only an important part of GST, but also the cornerstone of the grey system analysis, modeling, prediction, and decision making [
37]. GRA models have been used for many years to analyze the relationships between system factors [
33]. The underlying notion of GRA is that the closeness of a relationship between the system parameters is predicted based on the level of similarity of the geometrical pattern of the data sequences representing those parameters [
38]. This closeness is also known as proximity in the literature [
6,
39]. Here, a misconception in the literature needs to be addressed. Even though Zhang et al. [
37] argued that GRA models (e.g., Deng’s GRA and absolute GRA) “can only be applied to times series”, the influential work by Liu et al. [
3] (p. 69) maintained that a series of observations may or may not be over time. It can be observations, experiments, years, indexes, etc. The work of Javed [
6] supports this notion, so does the current study.
The concept of GRA was pioneered by Deng in the 1980s. The foundation of Deng’s GRA model rests on the estimation of a degree of proximity, grey relational grade (GRG). According to Javed [
6], GRG is “a degree of partial proximity (or, partial closeness) between two curves and is estimated by taking the average of the grey relational coefficients” at each point in a data sequence. Deng’s GRA model measures the trend similarity of system factors according to the distance between corresponding points of the sequences [
39]. Later, in 1991, Liu Sifeng proposed another degree, absolute GRG, and laid the foundation of the absolute degree grey incidence analysis model, also called the absolute GRA model [
39]. A detailed discussion on the absolute GRA model can be found in Liu et al. [
3,
20,
21]. According to Javed [
6], absolute GRG is “a degree of integral proximity between two curves represented by two data sequences and is estimated by considering the integral perspective on the proximity (closeness) between the two curves”.
Deng’s GRG and absolute GRG are mutually exclusive thus they can produce different orders/ranks for the parameters under study. However, this dilemma failed to seek the attention of scholars [
6]. For instance, according to Arce et al. [
16], grey relational grade (GRG) indicates the “degree of correlation” between the reference data sequence and the data sequences to be compared, which can also be considered the “degree of influence” of the comparison sequence on the reference sequence. In a statistical paradigm, correlation and influence have totally different meanings. Correlation simply means two factors are correlated with no information about whether they are influencing each other or not. Further, in the works of Tung and Lee [
40] and Yu et al. [
41], absolute GRG serves as a correlational measure. Javed [
6] discussed a number of studies that lead him to argue that the real purpose of [Deng’s] GRG and absolute GRG is not clear in the literature. Meanwhile, Javed et al. [
4] tried resolving this dilemma by stating “in a nutshell, Deng’s GRA model gives the measure of influence that one variable represented by a data sequence exerts on the other and absolute GRA model gives the measure of association between them.” Here, the measure of association can be loosely translated as a measure of correlation [
6]. Considering the fact that the change in the order of sequences can affect the value of GRG but not of absolute GRG, this proposal can be supported as it is hard to imagine, correlation can vary if two variables switch their positions. Acknowledging this dilemma in the literature where two different versions of GRG were being used for comparison purposes, Javed [
6] proposed the “second synthetic GRG” and demonstrated its feasibility through an application in a project management environment. To date, the associated model, the second synthetic GRA (SSGRA) model, has been successfully tested and applied in healthcare systems by Javed and colleagues [
22,
25], in construction project management environment by Sheikh et al. [
42] and in supply chain environment by Diba and Xie [
43].
Building upon the work of Liu et al. [
3], Javed and Liu [
22] argued that Deng’s GRA methodology is driven by the grey relational coefficient (GRC) of particular points while the absolute GRA model is governed by an integral (rather comprehensive) perspective. Thus, the closeness or proximity, as reported by Deng’s GRG, can be referred to as “partial proximity” (or partial closeness) and the closeness or proximity as reported by absolute GRG can be referred to as “integral proximity” (or integral closeness) [
6]. The work of Dong et al. [
17] presents a theorem that proves “the absolute degree of grey incidence satisfies the integral closeness of Deng’s grey incidence axiom, but does not satisfy the partial closeness.” This theorem justified the understanding on which the second synthetic GRG was proposed as a measure of “inclusive proximity” that synthesizes both partial proximity and integral proximity of grey relation (grey incidence). The theorem and its proof have been reproduced below.
Theorem 1. The absolute GRG satisfies the integral proximity/closeness of Deng’s grey relational axiom but does not satisfy the partial proximity/closeness [
17].
Proof [
17]
. The partial proximity of Deng’s grey relational axiom pertains to the grey relational grade (GRG) as estimated in line with the distance between the corresponding points of the data sequences i.e., the smaller the distance between the reference sequence
X0 and the comparison sequence
Xi at point
k, the higher the grey relational coefficient (GRC) at point
k. From the definition of Absolute GRG, when
then
= 1, where
and
are the zero-starting images of the sequences
X0 and
Xi, respectively. Due to the positive and negative offsets of the integral operation, as long as
oscillates around
and the area of the parts of
located above
equals to that of the parts of
located beneath
,
= 1 can be satisfied. In fact, the proximity of Deng’s grey relational axiom pertains to GRC between the points of the data sequences, whereas the absolute GRG is based on the integral (rather comprehensive) perspective. Thus, the absolute GRG, which predicts similarity based on proximity, is valid in the situations where zero-starting point sequences do not cross each other or do not affect the degree of grey relation after the intersection. For further discussion on this theorem, the readers are directed to Dong et al. [
17]. □
2.2. Thermal Conductivity and Petrophysical Parameters
Thermal conductivity is an important physical property of rocks that describes how well, but not how fast, heat is conducted through a material [
44]. Thermal conductivity (TC) is a property that can be determined by different measuring methods. It can be defined as the capability of a material to transfer heat with respect to a temperature gradient. It is not only an essential criterion to evaluate the heat transfer in rocks but also has applications in the geoscientific and geotechnical field [
45]. Thermal conductivity results in the heat accumulation and determines where and how much heat flows resulting from the temperature differences in rocks [
46]. Conduction and convection are the two fundamental phenomena that determine the heat transfer in the rock formation. According to Fourier’s law, rocks with low-porous and high density possess higher conduction than convention. However, in rocks such as porous sediments are dominated by convection heat transfer. Different rocks have a different heterogeneous mineral composition, so it shows significant differences in the thermal conductivity. Different groups of rocks show the variability of thermal conductivity in the range of 2 to 4 Wm
−1K
−1 [
47]. Two very important parameters associated with a material’s thermal conductivity are porosity and density. The porosity describes the fraction of void space in the rock, where the voids may contain some fluid (e.g., air, water etc.) [
48]. The rock density is defined as the quotient of the mass and the volume of the material [
24] i.e., mass per unit volume. Permeability of a material is another property that may not directly be associated to thermal conductivity but is closely related to porosity as both permeability and porosity intrinsically depend on the microstructure of pores in porous materials [
49]. These properties frequently find their mention in the geothermic and earth sciences literature.
The relationships between thermal conductivity (TC) and the influencing parameters/properties have been studied by numerous scholars in the past. Somerton [
50] is considered as one of the pioneers to study the correlation of TC with petrophysical parameters. He found that variation in temperature significantly affects the thermal diffusivity of rocks. In later years, several scholars [
51,
52,
53,
54,
55,
56,
57] studied the effect of density, rock texture, permeability, fluid saturation, mineral composition and porosity of rocks on thermal conductivity. For instance, Barry-Macaulay et al. [
54] reported the variation in TC of soils and rocks with the variation in density. Duchkov et al. [
55] revealed a stable linear relationship between the thermal conductivity of the dry samples of Mesozoic sedimentary rocks and, porosity and permeability. Haffen et al. [
58] studied the correlation of TC and porosity maps for granite and sandstone. Their study developed the porosity maps for air and water saturated conditions. Asadi et al. [
57] and Ouali [
59] stressed the significant role of porosity in the thermal conductivity of materials. Duchkov et al. [
60] reported that TC of sedimentary rocks is determined by the composition of the mineral skeleton, porosity and permeability of the rock, as well as the type of fluid that fills the pores. Literature suggests that TC of rocks is influenced, directly or indirectly and with varying degree of extents, by various kinds of petrophysical parameters, which are highlighted in
Figure 1.
Finding an association between two continuous variables is usually performed by a correlation method. Finding a correlation is essential to reduce the range of uncertainty. It helps us to understand the causal effects, economic benefits and helps us to screen the parameter. There is a dire need to find an accurate correlation between TC and other parameters, especially relatively less explored quartzite. The literature suggests thermal conductivity is inversely and significantly related to porosity [
59,
60,
61,
62]. Duchov et al. [
60] reported this relationship to be linear while Ouali [
59] reported a harmonic average relationship. Since porosity and permeability are known to have a positive correlation [
48,
63,
64,
65] it is very plausible to assume that thermal conductivity and permeability are likely to have inverse relationship as well. This can be confirmed from Duchkov et al. [
59] who reported inverse linear relation between thermal conductivity and permeability. However, Mielke et al. [
66] failed to identify a consistent correlation between thermal conductivity and permeability. Also, since porosity and density are known to be inversely related [
46,
48,
60] it is very plausible to assume that thermal conductivity and density are likely to be positively (directly) related. This can be confirmed from the studies [
60,
61,
67] that reported a positive correlation between thermal conductivity and density. To our surprise, among all the literature reviewed by us, only Zerrouki et al. [
24], through one of their figures, reported an inverse relation between thermal conductivity and density. To confirm (or disconfirm) their results through a different approach the current study provided a promising avenue where GRA models were used to reevaluate the relationships. Based to the authentic literature, it can be emphasized that thermal conductivity is a function of density, porosity and permeability while its relation to density is positive and to porosity, and thus to permeability, is inverse. This can be represented as
or,
where, λ,
D,
and
K represent thermal conductivity, density, porosity and permeability, respectively.