Prediction of thermal conductivity of rock through physico-mechanical properties

Abstract

The transfer of energy between two adjacent parts of rock mainly depends on its thermal conductivity. Present study supports the use of artificial neural network (ANN) and adaptive neuro fuzzy inference system (ANFIS) in the study of thermal conductivity along with other intrinsic properties of rock due to its increasing importance in many areas of rock engineering, agronomy and geo environmental engineering field. In recent years, considerable effort has been made to develop techniques to determine these properties. Comparative analysis is made to analyze the capabilities among six different models of ANN and ANFIS. ANN models are based on feedforward backpropagation network with training functions resilient backpropagation (RP), one step secant (OSS) and Powell–Beale restarts (CGB) and radial basis with training functions generalized regression neural network (GRNN) and more efficient design radial basis network (NEWRB). A data set of 136 has been used for training different models and 15 were used for testing purposes. A statistical analysis is made to show the consistency among them. ANFIS is proved to be the best among all the networks tried in this case with average absolute percentage error of 0.03% and regression coefficient of 1, whereas best performance shown by the FFBP (RP) with average absolute error of 2.26%. Thermal conductivity is predicted using P-wave velocity, porosity, bulk density, uniaxial compressive strength of rock as input parameters.

Introduction

Thermal conductivity of rock is studied along with different physico-mechanical properties due to its increasing importance in the geotechnical engineering, geothermal engineering, nuclear disposal, etc. Its study is important because it is main characteristic of energy transfer phenomenon. Energy transfer arising from the temperature difference between the adjacent parts of the body is called heat conduction. The amount of heat to be transferred through any body depends upon a number of factors, such as the particle shape, porosity, temperature range, solid constituents, moisture content, uniaxial and/or triaxial pressure exerted on the rock, etc. [1], [2], [3], [4], [5]. It widely influences the energy transfer between adjacent rocks in underground mines and in insulation of the building by providing an energy efficient solution. Energy saving is the important part of any national energy strategy and its conservation for underdeveloped countries with inadequate resources is even more important [5]. Thermal insulation material is most important part of any thermal insulation system and a lower system thermal conductivity can be achieved by combination of more thermal insulation material. Such systems are generally characterized by an effective thermal conductivity. The thermal conductivity is determined by the measurements of temperature gradient in the rock and heat input [6].

In general, the thermal conductivity can be calculated using Fourier's Law as given below:

$$\frac{\mathrm{d}Q}{\mathrm{d}t}=-\mathit{kA}\frac{\mathrm{d}T}{\mathrm{d}x}[18]\text{,}$$

$$\frac{\mathrm{d}Q}{\mathrm{d}t}=-(k_0+\lambda P^n)A\frac{\mathrm{d}T}{\mathrm{d}x}[18]\text{,}$$

$$Q=-(k_0+{\lambda }_{\mathit{cp}}{P}^{n_{\mathit{cp}}})C_p^{{\beta }_{\mathit{cp}}}\mathit{At}\frac{\mathrm{d}T}{\mathrm{d}x}[18]\text{,}$$

where dQ/dt is the time rate of heat transfer; A the area of the body whose heat is to be transferred; dT/dx the temperature gradient, k the thermal conductivity of the material; k₀ the thermal conductivity of the rock under normal conditions, W/(m °C); λ and n are the rock-dependent constant parameters in uniaxial compression; P is the applied uniaxial pressure, Mpa; ${\lambda }_{\mathit{cp}}$, n_cp and β_cp are the rock-dependent constant parameters in triaxial compression, C_p is the confining pressure, MPa. Transport and thermodynamic properties which is referred as thermo-physical properties affect the thermal conductivity of rocks. Properties like k the thermal conductivity of the material comes under transport properties while in thermodynamic analysis properties like density and specific heat are commonly used.

Considering the physical mechanism associated with conduction in general, the thermal conductivity of a solid is larger than that of a liquid, and that of liquid is larger than gas.

Knowing a greater usability of thermal property in the civil engineering, a lot of work has been carried out to find thermal conductivity of rock and to predict by a definite simple model for the judgment the thermal property of rocks. Progress has been made in recent years in the ability to predict the thermal conductivity, but the state of the art is deficient in many ways. On the basis of detailed investigation, a viable approach for the prediction of thermal conductivity is necessary. An artificial intelligence (AI) comes in handy to fulfill this problem. Artificial neural networks (ANNs) and adaptive neuro fuzzy inference system (ANFIS) are used in the present study. ANN can be viewed as an interesting class of statistical pattern recognition algorithm, which provides explicit facilities for modeling non-linear and non-Gaussian statistical regularities and proves to be strong tool to prepare an equivalent model by virtue of its capabilities of function approximation and classification. The potential of modeling the material behavior using ANN was first proposed by Ghaboussi et al. [7]. ANNs have been used to solve a wide variety of application in Geomechanics and Rock Engineering [8]. In the last few years, fuzzy inference systems (FISs) began to be used in the areas of rock mechanics and engineering geology [9], [10], [11], [12]. According to Setnes et al. [13], an interesting and perhaps the most attractive characteristics of fuzzy models compared with other conventional methods commonly used in geosciences. Such as in statistics, they are able to describe complex and non-linear multivariable problems in a transparent way. Fuzzy logic is a logical system, which is an extension of multivalued logic. Fuzzy propositions are statements that posses fuzzy variables. The concept of a fuzzy set is the basis of a fuzzy logic. A fuzzy set is a set without a crisp, clearly defined boundary. ANFIS and ANN can be viewed as strong tools in the statistical pattern recognition algorithm and to prepare an equivalent model by virtue of their capabilities of function approximation and classification. Data sets for the analysis are taken from literature [14] based on the laboratory experiments.