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Geothermal gradient is one of the most important parameters for geothermal exploration and exploitation. Mud is generally used as a drilling fluid in geothermal wells. According to energy conservation law, when the drilling penetrates through an aquifuge or aquitard, its temperature can be used to establish the mathematical analysis model for estimating the geothermal gradient. The recorded mud temperatures in some practical drilling cases in southwest China have been applied to study the influence radius of geotherm that is conducted by surrounding rock into the derived formula. The fitting calculation results show that the absolute error is generally very low, being less than 10%, which indicates that the proposed formula can be used to effectively predict the geothermal gradient. The calculation indicates that it is only when the rock conductive influencing radius is very small (around 0.25m) that it is possible to have a reasonable solution.
Drilling fluid of mud, Geothermal gradient, Temperature, Energy conservation law, Rock conductive influencing radius.
Geothermal resource is one of the newest sources of energy. It has recently gained considerable attention due to its renewable and clean characteristics. For the sake of energy conservation and to address the energy shortage, the exploitation and utilization of geothermal resources is becoming increasingly important, which leads to the reasonable utilization of geothermal resources to be an important strategic action that can alleviate the problems of resources shortages and environmental degradation [14].
Since geothermal energy has great potential for development and broad prospects for use, the realization of geothermal resources for sustainable development and utilization is a critical issue for the world's geothermal industry [514]. Scholars all over the world have done much research on the geothermal resources and hot springs, including studies on heat flow, the thermal state of the rock, the thermal structure of the Earth’s crust or upper mantle, the geothermal effects and climate change, mining geotherm, and other geothermal resources. [1519]. Strata temperature, which is difficult to measure directly, is a key parameter in the research and development of underground hot water, the thermal storage to divide the drilling depth in the geothermal system, and the evaluation of the potential of geothermal resources. Currently, the geothermal temperature scale approach is an economical and effective means of providing this parameter. This approach has been widely used in estimating reservoir temperatures, including cationic, silica, gas chemistry, isotope, and so on [1930].
The drilling fluid of mud, which can interfere with the temperature field near the borehole during the drilling process, is generally used in exploration and development of geothermal wells. When the drilling time is long enough to make the temperature balance with the surrounding rock, the surface temperature can reflect the actual status of the stratum temperature. Some related work has been done by Lachenbruch and Hrewer (1959), Albright (1976), Barelli and Palama (1981), and so on [3137].
Geothermal gradient is one of the most important parameters for geothermal exploration and exploitation. In the drilling process, the effect of exchanging heat between the drilling fluid, which mainly consists of drilling mud, and the strata that surround the borehole will result in a distinct temperature increase in the circulated outflow fluid. If drilling penetrates through aquifuge or aquitard with a certain thickness, the drilling fluid will have almost no leakage, and the heat attracted from the surrounded strata will be mainly supplied by conduction. For a relatively constant fluid flowrate Q_{l}, the geothermal gradient can be calculated by energy conservation, and the reservoir temperature can be estimated resulting in a more reasonable drilling program being put forward.
The physical model of calculation is shown in Fig.1. When a borehole is drilled at a depth of H_{w}, for the given temperature of the inflow (T_{0}) and outflow drilling fluid (T_{n}), if the depth is divided into n parts evenly, the thickness of each part is:
z=H_{w}/n(1)
Figure 1. Model of drilling fluid temperature prediction of the reservoir temperature
Fig. 1 shows that the rising temperature in each part is obviously caused by the heat absorbed from the preceding part of the surrounding rock:
$\left\{\begin{array}{l}{C_{l}\left(T_{1}T_{0}\right) \rho_{l} Q_{l}=Q_{r 1}} \\ {C_{l}\left(T_{2}T_{1}\right) \rho_{l} Q_{l}=Q_{r 2}} \\ {C_{l}\left(T_{3}T_{2}\right) \rho_{l} Q_{l}=Q_{r 3}} \\ {\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots} \\ {C_{l}\left(T_{n}T_{n1}\right) \rho_{l} Q_{l}=Q_{m}}\end{array}\right. $(2)
where C_{l}——specific heat of the drilling fluid, KJ/(Kg·^{0}C); $\rho_{l}$——density of the drilling fluid, Kg /m^{3}; Q_{l}——constant fluid flowrate, m^{3}/s; Q_{r1}, Q_{r2}, Q_{r3}…Q_{rn}——conductive heat flow from surrounding strata in different depth (upwards from bottom of borehole) (KJ/s);
T_{0}, T_{1}, T_{2}, ……T_{n1}——temperature of the drilling fluid corresponding to the divided depth (^{0}C).
According to the conductive equation, the heat flow that comes into the borehole from radial surrounding strata for a given divided depth can be described as:
$Q_{r i}=K_{r}(2 \pi r \Delta z) \frac{d T}{d r}, i=1,2 \cdot 3 \ldots \ldots n$(3)
where K_{r}——surrounded rock conductivity, W /(m×k); r——the maximum distance that will cause heat flow from the surrounded strata while drilling, m; $\frac{d T}{d r}$——horizontal radial geothermal gradient at a given depth,^{0}C /m; 2$\pi r \Delta z$——the heat exchanging area between the drilling fluid and the surrounding strata at a given depth, m^{2}.
If the variable is separated and the integral transformation is used on equation (3) [3839], the heat flow occurring at depth i can be derived as:
$Q_{r i}=\frac{2 \pi r_{w} K_{r}\left(T_{r i}T_{i1}\right)}{\ln \frac{R_{i}}{r_{w}}} \Delta z, \quad i=1.2 .3 \ldots \ldots n$(4)
where T_{ri}——rock temperature of the divided depth i, ^{0}C; r_{w}——borehole radius, m; R_{i}——rock conductive influencing radius of the divided depth i, m;
Combining equation (4) in equation (2) yields the following:
$1000 C_{l}\left(T_{i}T_{i1}\right) \rho_{l} Q_{l}=Q_{r i}=\frac{2 \pi r_{w} K_{r}\left(T_{r i}T_{i1}\right)}{\ln \frac{R_{i}}{r_{w}}} \Delta z$(5)
Or:
$\frac{1000 C_{l} \rho_{l} Q_{l}}{2 \pi r_{w} K_{r}} \ln \frac{R_{i}}{r_{w}} \frac{T_{i}T_{i1}}{\Delta z}=T_{r i}T_{i1}$(6)
Equation (6) is high nonlinear and remains hard to solve. However, when the drilling depths are divided into enough sections evenly, the thickness of each section is very small. The heat transfer in the upper and lower sections of the surrounding rock is approximately equal in scope, and the $\frac{R_{i}}{r_{w}}$ could be considered as a constant.
In this case, the following applies:
$b=\frac{1000 C_{l} \rho_{l} Q_{l}}{2 \pi r_{w} K_{r}} \ln \frac{R_{i}}{r_{w}}$(7)
This parameter has length dimension. Thus, the equation (7) can be simplified as:
$b \frac{T_{i}T_{i1}}{\Delta z}=T_{r i}T_{i1}$(8)
Take the temperature of drilling fluid with the depth as function T(z) (Coordinate of Z is positive downwards from the ground), and the surrounding rock temperature function as Tr(z), then equation (8) can be expressed as a differential equation:
$b \frac{d T}{d z}=T_{r}T$(9)
Considering that geothermal growth is linear in the aquifuge, the gradient is fixed, as expressed by the following:
$G(t)=\frac{\left(T_{r i}T_{r i+1}\right) n}{H_{w}} i=1,2,3 \ldots \ldots n1$(10)
Or:
$T_{r}=T_{s}+G(t) z$(11)
G(t) is simplified as G. Combining equation (11) and equation (9) yields the following:
$b \frac{d T}{d z}=T_{s}+G zT$(12)
Its general solution is:
$T=b G+\left(T_{s}+G z\right)C_{0} e^{z / b}$(13)
where C_{0 }is an undetermined constant. Making use of the surface temperature condition:
$T(z=0)=T_{n}=b G+T_{s}C_{0}$(14)
Hence,
$C_{0}=b G+T_{s}T_{n}$(15)
Then, we have:
$T(z)=b G+\left(T_{s}+G z\right)\left(T_{s}+b GT_{n}\right) e^{z / b}$(16)
To determine the geothermal gradient G, the bottom hole temperature T_{0} is inserted into equation (16), yielding the following:
$T\left(z=H_{w}\right)=T_{0}=b G+\left(T_{s}+G H_{w}\right)\left(T_{s}+b GT_{n}\right) e^{H_{w} / b}$(17)
Namely, equation (17)can be changed to:
$G=\frac{T_{s}+\left(T_{n}T_{s}\right) \exp \left(H_{w} / b\right)T_{0}}{b \exp \left(H_{w} / b\right)\left(b+H_{w}\right)}$(18)
Integral constants can also be obtained by the bottom hole temperature T_{0}.
Note that we define:
$C_{0}=\frac{b G+T_{s}+G H_{w}T_{0}}{\exp \left(H_{w} / b\right)}$(19)
Yielding:
$T(z)=b G+\left(T_{s}+G z\right)\left(T_{s}+b G+G H_{w}T_{0}\right) e^{\left(zH_{w}\right) / b}$(20)
Moreover, the above equation can be rewritten as:
$T(z=0)=b G+T_{s}\left(T_{s}+b G+G H_{w}T_{0}\right) e^{H_{w} / b}$(21)
The equation (16) and equation (20) are equivalent.
Table 1. The bestfit list for predicting geothermal gradient through the actual drilling mud
Arguments Hole location 
T_{0} 
T_{s} 
T_{n} 
ρ_{l}(Kg /m^{3}) 
Q_{l} (l/s) 
C_{l} KJ/（Kg·^{0}C） 
r_{w} 
H_{w} 
R_{i} 
K_{r} W /(m×k) 
CalculationG(t) 
Actual measurementG(t) 
Relative error (%) 
(^{0}C) 

(m) 
(^{0}C /m) 

Yunnan elderly activity center 
23 
17 
31 
1190 
3.00 
4.25 
0.088 
1382 
0.25 
2.4 
0.030 
0.028 
7.14 
Yunnan Tuodong sports center 
22 
17 
31 
1150 
2.50 
4.25 
0.08 
1285 
0.25 
2.4 
0.034 
0.033 
3.03 
Yunnan Yiliang city coal mining administration 
20 
17 
23 
1200 
2.50 
4.12 
0.108 
420 
0.23 
2.1 
0.051 
0.050 
2.00 
Kunming Lin Yuan real estate company 
23 
17 
30 
1100 
2.50 
4.25 
0.108 
801 
0.25 
2.8 
0.049 
0.045 
8.89 
Yunnan Xuanwei hot well 
20 
18 
26 
1070 
2.50 
4.25 
0.108 
1311 
0.26 
2.1 
0.019 
0.018 
5.56 
Yunnan Xundian County Beidaying pastures 
22 
17 
24 
1200 
1.88 
4.12 
0.108 
692 
0.25 
2.8 
0.028 
0.023 
21.74 
Guiyang Wudang District xiangzhigou scenic area 
19 
17 
24 
1200 
2.50 
4.12 
0.076 
1150 
0.24 
1.8 
0.021 
0.021 
0 
Kunming Haigeng sports training base 
23 
17 
30 
1150 
2.00 
4.26 
0.108 
720 
0.25 
2.5 
0.055 
0.051 
7.20 
The mud temperatures in some actual drilling cases in southwest China were recorded to study the influence radius of geotherm that is conducted by surrounding rock. By using Excel spreadsheets, the related parameters from these boreholes have been input into the corresponding column, and comprehensive parameters (b) have been calculated according to equation (7). For each borehole, when different values for the influence radius of the surrounding rock’s thermal conductivity in each well were given，the corresponding geothermal gradient G_{i }was calculated based on equation (19). By comparing the geothermal gradient value that was calculated with the actual measured temperature in each borehole, the best fitting data (Table 1) were determined. Since the friction heat generated during the drilling process is ignored in the formula derivation of the geothermal gradient, all the calculated values of geothermal gradient appear higher than the actual ones. Furthermore, the simulation is available only when the rock conductive influencing radius is very small (about 0.25m). This indicates that to form a stable heat flow field, the role of the surrounding rock in supplying heat for drilling mud is not essential. The simulation results show that the absolute error rate is generally low in the mass, and the relative error rate is less than 10%.
Setting 0.25m of the surrounding rock conductive influencing radius as the empirical value, we can predict the computation in the process of several drilling construction sites in Yunnan and Guizhou Provinces, in southwest China. Comparing the calculated value with the actual detected data of the final hole, the relative error rate is less than 10% (Table 2), which proves the derived formula is very practical.
Table 2. Comparison of the predicting geothermal gradient through several drilling mud sites in Yunnan and Guizhou Provinces with the actual measured value
Arguments Hole location 
T_{0} 
T_{s} 
T_{n} 
ρ_{l}(Kg /m^{3}) 
Q_{l} (l/s) 
C_{l} KJ/（Kg·^{0}C） 
r_{w} (m) 
H_{w} (m) 
K_{r} W /(m×k) 
Calculation G(t) 
Actual measurement G(t) 
Relative error(%) 
(^{0}C) 


(^{0}C /m) 

Guizhou Pingba County Xujiadu 
22 
17 
21 
1200 
2.00 
4.12 
0.108 
400 
1.3 
0.0103 
0.0105 
1.91 
Yunnan Dali city haidong hot well 
20 
17 
21 
1180 
1.67 
4.26 
0.076 
760 
2.4 
0.0145 
0.0135 
6.89 
Kunming New Asia sports city 
21 
17 
25 
1100 
3.00 
4.425 
0.108 
773 
2.5 
0.031 
0.030 
3.33 
Yunnan Luoping county people’s hospital 
20 
17 
21 
1180 
2.00 
4.26 
0.076 
860 
1.8 
0.0129 
0.0124 
3.88 
Yunnan Yuxi cigarette factories filter rods factory 
19 
17 
23 
1180 
2.00 
4.12 
0.108 
660 
1.9 
0.0305 
0.0279 
8.52 
(1) In the process of geothermal exploration, the drilling fluid of mud will extract heat from the surrounding rock when penetrating through aquifuge, which will inevitably result in a significant temperature increase in the outflow drilling fluid. Since the heat thoroughly transfers by conduction, Fourier law and energy conservation law can be employed to calculate the strata temperature.
(2) Once the parameters of the drilling mud fluid and the rock thermal conductivity are measured, the general derived formula $G=\frac{T_{s}+\left(T_{n}T_{s}\right) \exp \left(H_{w} / b\right)T_{0}}{b \exp \left(H_{w} / b\right)\left(b+H_{w}\right)}$ can be used to estimate the geothermal gradient in the aquifuge.
(3) The fitting calculation results of the thermal influence radius, which is conducted and formed by surrounding rock, show that the relative error rate is less than 10% in some practical drilling cases in southwest China.
(4) Only when the rock conductive influencing radius is very small, it is possible to have a reasonable solution, which indicates that the role of the surrounding rock in supplying heat for drilling mud is not essential to form a stable state. The suggested value is around 0.25m empirically.
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