分布热源作用下裂隙岩体渗流-传热的拉氏变换-格林函数半解析计算方法

以裂隙岩体高放射性核废物地下处置库性能评估为目标,提出了分布热源作用下单裂隙岩体渗流-传热的简化概念模型、控制微分方程和拉氏变换-格林函数半解析法,为进一步采用半解析法计算分布热源作用下多裂隙岩体的渗流-传热问题奠定了基础。针对单裂隙岩体的渗流-传热问题,建立考虑岩石内热源和二维热传导的控制微分方程,利用拉氏变换域微分方程的基本解建立格林函数积分方程,采用解析法处理其中的奇点,通过数值积分和拉氏数值逆变换求解,计算任意时刻裂隙水和岩石的温度分布。通过算例,与基于岩石一维热传导假定的解析解进行了对比,并计算分析了分布热源作用下单裂隙岩体的渗流-传热特征及其对裂隙开度、岩石热传导系数和热流集度的敏感度。算例表明,(1)就裂隙水温度而言,由于考虑了岩石的二维热传导,拉氏变换-格林函数半解析解小于基于岩石一维热传导假定的解析解;(2)裂隙水温度和岩石温度对裂隙开度和热流集度的敏感度较大,对岩石热传导系数的敏感度较小。

A Laplace transform and Green function method for calculation of water flow and heat transfer in fractured rocks

For performance assessment of nuclear waste repositories in fractured rocks, a simplified conceptual model and mathematical formulation is proposed; and a Laplace transform and Green function semi-analytical method is developed for calculation of flow and heat transfer in single-fracture rock with distributed heat source. Taking into account of distributed heat source and 2D thermal conduction in single-fracture rock of infinite extent, the mathematical model is formulated and solved by using a Green function approach, in which a fundamental solution of the governing differential equations after Laplace transform is employed. The singularities in the integral equation are handled through analytical integration; and a numerical procedure is developed to solve the transient temperature distributions in fracture water and rock matrix. Numerical examples are provided for illustration of the proposed method with comparison of an analytical solution based on 1D rock thermal conduction, as well as for features of flow and heat transfer in single-fracture rock and the sensitivities to fracture aperture, rock thermal conductivity and heat source intensity. The calculations indicate: (1) The temperature of water in the fracture calculated from using the semi-analytical method is lower than that calculated the analytical solution, due to the fact that the former method takes account 2D thermal conduction in the rock matrix, whereas the latter assumes 1D conduction. (2) Temperatures in fracture water and the rock matrix are more sensitive, in a relative sense, to the fracture aperture and the heat source intensity than to the thermal conductivity of the rock matrix. The proposed model and solution method may serve, among other possible applications, as a foundation for semi-analytical calculation of flow and heat transfer in multiple fracture rocks with distributed heat sources.