algebraic equations of the Laplace transformed temperatures of water in the fractures are formed by dividing the integrals into elemental ones; in particular, the fracture faces are discretized into rectangular elements, over which the integrations are carried out either analytically for singular integrals when the base point is of the repository of high-level nuclear waste due to the radioactive decay of nuclides may affect the hydrological, mechanical, and chemical properties of the host rocks and hence the performances of the for water-flow facilitated heat transfer in fractured rocks, for example, Lauwerier  presented an
INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS
Int. J. Numer. Anal. Meth. Geomech. 2014; 38:1149–1171
Published online 23 January 2014 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nag.2252analytical solution for heat advection and conduction in single-fracture rocks under the action of a pair of parallel injection-extraction wells, Čermak and Jetel  presented an analytical solution for heat transfer in a homogeneous aquifer with geothermal gradient, Heuer et al.  employed anrepository in terms of confinement, retardation, and dilution of the radioactive nuclides for protection of the environments. Therefore, heat transfer in fractured rocks with saturated water flow and distributed heat source must be analyzed in repository design and performance assessments.
There have been a number of publications in the aspects of analytical or semi-analytical approachesinvolved or numerically for regular integrals when otherwise. The solutions of the algebraic equations are inverted numerically to obtain the real-time temperatures of water in the fractures, which may be employed to calculate the temperatures at prescribed locations of the rock matrix. Three example calculations are presented to illustrate the workability of the developed approach. The calculations found that water flux in the fractures may decrease the rate of temperature rise in regions close to the distributed heat source and increase the rate of temperature rise in regions downstream away from the distributed heat source and that the temperature distribution and evolvement in a sparsely fractured rock mass may be significantly influenced by water flow exchange at intersection of fractures. Copyright © 2014 John Wiley & Sons, Ltd.
Received 28 March 2013; Revised 22 September 2013; Accepted 9 December 2013
KEY WORDS: sparsely fractured rocks; water flow; distributed heat source; heat transfer; singular integration 1. INTRODUCTION
The processes of heat transfer in fractured rockswith saturated water flowmay need to be studied in several types of engineering fields, such as radioactive waste disposal, carbon dioxide sequestration, and thermal energy exploration in deep geological environments. In particular, the rising temperatures in the host rocksA semi-analytical modeling approach for three-dimensional heat transfer in sparsely fractured rocks with water flow and distributed heat source
Zhang Yong and Xiang Yan-yong*,†
School of Civil Engineering, Beijing Jiaotong University, Beijing100044, China
A semi-analytical approach is developed for modeling 3D heat transfer in sparsely fractured rocks with prescribed water flow and heat source. The governing differential equations are formulated, and the corresponding integral equations over the fracture faces and the distributed heat source are established in the Laplace transformed domain using the Green function method with local systems of coordinates. The*Correspondence to: Xiang Yan-yong, School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China. †E-mail: email@example.com
Copyright © 2014 John Wiley & Sons, Ltd. analytical model of water flow and heat transfer in a rock column with a fracture to study the process of heat dispersion in hot and dry rocks, Cheng et al.  developed a semi-analytical integral equation approach to take into account two-dimensional heat conduction in the rock matrix, Xiang and Guo  and Xiang and Zhang  employed similar approaches but with incorporation of distributed heat source and with multiple sparse fractures, respectively, and finally, in particular relevance to the present study, Ghassemi et al.  developed a semi-analytical model to take into account 3D heat conduction in the rock matrix,
In China, geological explorations have revealed that the major potential site for the first high-level nuclear waste repository in the country is composed of large saturated granite rock bodies with sparsely distributed fractures (average fracture spacing on the order of meters). In order to study the relevant radioactive waste-induced heat transfer process, a semi-analytical model is proposed for 3D advective–conductive heat transfer in sparsely fractured water-saturated rocks with prescribed water flow and heat source, and an integral equation solution scheme via discrete Laplace transform and numerical integration is developed. The approach draws on Ghassemi et al.  and Zhang and Xiang  for 3D heat transfer in water-saturated rocks involving one single fracture and a distributed heat source and on Xiang and Zhang  for two-dimensional heat transfer in water-saturated rocks involving multiple sparse fractures and distributed heat source. Three example calculations, involving regular or irregular fracture networks and water flow interchange at intersection of fractures, are presented to illustrate the developed approach and the relevant features of 3D heat transfer in sparsely specific heat and prescribed steady unit-width flux of water in fractures k = 1 ~K, respectively.
Assume small fracture apertures and instantaneous thermodynamic equilibrium at the fracture walls 1150 Z. YONG AND X. YAN-YONGbetween the rock matrix and water in the fractures, and negligible thermal diffusion, dispersion and
Figure 1. 3D conceptual model for heat transfer in sparsely fractured rocks.fractured rocks with prescribed water flow and heat source. 2. CONCEPTUAL MODEL AND GOVERNING EQUATIONS