# Multiple Thermal Parameter Inversion for Concrete Dams Using an Integrated Surrogate Model
## 2 Methodology
### Mathematical Model Construction for Multiple Thermal Parameters Inversion  
Calculation Theory of Temperature Field Containing Cooling Water Pipe  
Heat transfer in a concrete dam is governed by the following [30]:  
\[\frac{\partial T}{\partial t}=a\bigg{(}\frac{\partial^{2}T}{\partial x^{2}}+ \frac{\partial^{2}T}{\partial y^{2}}+\frac{\partial^{2}T}{\partial z^{2}} \bigg{)}+\frac{\partial Q}{c\rho} \tag{1}\]  
where \(T\) is the concrete temperature; \(a\) is the temperature conductivity; \(Q\) is the heat; \(c\) is the specific heat of concrete; \(\rho\) is the density of concrete; and \(t\) is the time.  
Figure 1: Methodology flowchart of this study.  
The temperature rise rate of concrete under adiabatic conditions is as follows:  
\[\frac{\partial Q}{c\rho}=\frac{\partial\theta}{\partial t} \tag{2}\]  
where \(\theta\) is the adiabatic temperature rise of the concrete.  
The governing equation is subject to the following initial conditions:  
\[T|_{t=0}=T_{0}(x,y,z) \tag{3}\]  
where \(T_{0}(x,y,z)\) is the initial distribution function of temperature.  
The boundary conditions of the external surface are as follows:  
(1) First type:  
\[T=T(t) \tag{4}\]  
(2) Second type:  
\[-\lambda\frac{\partial T}{\partial n}=f(t) \tag{5}\]  
(3) Third type:  
\[-\lambda\frac{\partial T}{\partial n}=\beta(T-T_{a}) \tag{6}\]  
where \(T(t)\) is the known temperature on the boundary; \(f(t)\) is the known heat flow on the boundary; \(n\) is the direction of the external normal to the boundary; \(\beta\) is the heat release coefficient; and \(T_{a}\) is the air temperature.  
Many serpentine cooling pipes are arranged in the concrete dams to regulate the concrete temperature, as shown in Figure 2a. In a concrete cylinder of diameter \(D\) length \(L\), as shown in Figure 2b, the inlet water temperature of the water pipe is \(T_{w}\), and the final adiabatic temperature rise is \(\theta_{0}\). Assuming that the outer surface of the cylinder is adiabatic, the average temperature of the concrete can be expressed as follows:  
\[\left\{\begin{array}{c}T(t)=T_{w}+(T_{0}-T_{w})\varnothing(t)+\theta_{0}t \psi(t)\\ \varnothing(t)=e^{-pt}\\ p=\frac{\delta ka}{D^{2}}\\ k=2.09-1.35\xi+0.32\xi^{2}\\ \xi=\lambda L/c_{w}\rho_{w}q_{w}\\ g=\frac{alt_{1}00}{ln(\frac{b}{b_{0}})+(\frac{\lambda}{\lambda_{1}})ln(\frac{ c_{0}}{b_{0}})}\\ b=0.5836\sqrt{S_{1}S_{2}}\end{array}\right. \tag{7}\]  
where \(b\) and \(c_{0}\) are the outer and inner radius of the concrete column, respectively; \(r_{0}\) is the inner radius of the cooling water pipe; \(\lambda\) and \(\lambda_{1}\) are the thermal conductivity of concrete and cooling water pipe, respectively; \(c_{w}\) is the specific heat of cooling water; \(\rho_{w}\) is the density of cooling water; \(q_{w}\) is the flow rate of cooling water; and \(S_{1}\) and \(S_{2}\) are the horizontal and vertical spacing of the cooling water pipe, respectively; \(\psi(t)\) is a function related to the adiabatic temperature rise in the concrete.  
Before constructing a mathematical model for the adiabatic temperature rise in the concrete, a functional expression is usually assumed and then fitted with experimental data. Commonly available forms are exponential, hyperbolic, double exponential, and combined exponential. The combined exponential model matches the test data better and is widely used in engineering [30]. The expressions are as follows:  
\[\theta(t)=\theta_{0}s(1-e^{-m_{1}t})+\theta_{0}(1-s)(1-e^{-m_{2}t}) \tag{8}\]  
where \(s\), \(m_{1}\), and \(m_{2}\) are the coefficients to be determined, \(m_{2}=0.1m_{1}\).  
\(\psi(t)\) can be expressed as  
\[\psi(t)= \frac{sm_{1}}{m_{1}\!-\!p}(e^{-pt}\!-\!e^{-m_{1}t})\!+\!\frac{(1-s)m_{2}} {m_{2}\!-\!p}(e^{-pt}\!-\!e^{-m_{2}t}) \tag{9}\]