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In research using heat tracing technology to investigate the lateral hyporheic exchange in the shallow geological body of the riparian zone, the accurate estimation of temperature changes can provide a scientific basis for quantifying the process of lateral hyporheic exchange. To improve the accuracy of estimating temperature changes in the riparian zone, a hydrothermal coupling model considering parameter heterogeneity was established based on existing models of the relationship between thermal conductivity and saturation. The model was verified by temperature data from laboratory experiments, and the effect of the thermal conductivity prediction models was compared with that of the partial differential equation (PDE) modeling approach. The results show that the established hydrothermal coupling model can effectively characterize the temperature changes observed in a generalized laboratory model of the riparian zone, and the model simulation effects vary with the equivalent thermal conductivity models. In addition, several thermal conductivity empirical models are suggested for further application. The model parameter sensitivity analysis indicated that the hydraulic conductivity

The riparian zone is an important transitional area for the exchange of matter, energy, and information between the river ecological system and the land ecological system [

Heat transport constantly occurs during the lateral hyporheic exchange process. Therefore, observations of the temporal and spatial temperature changes in the riparian zone can be used quantify the hyporheic exchange flux and even help us understand the processes of lateral hyporheic exchange occurring in the riparian zone [

Many studies have used temperature data to analyze hydrothermal exchange in the riparian zone [

In the history of soil thermal conductivity model development, many scholars have made many contributions. Some of researches obtained formulas by analyzing and fitting test data (i.e., empirical models), and some of them established models through theoretical analysis to predict the thermal conductivity of the soil (i.e., theoretical models) [

Summary of several representative thermal conductivity prediction models.

Model types | Model formula | Physical meaning of the parameters |
---|---|---|

Johansen model [ | ^{3}°C); ^{3}°C); _{r} is the degree of saturation ^{3} | |

Campbell model [ | ||

Côté and Konrad model [ | _{e} and _{r}; the suggested value is 4.6 for crushed granite, 3.55 for sand, and 1.9 for silt and clay; | |

Lu model [ | ||

Ren model [ | _{eq} (_{d} is the dry volume weight of soil in KN/m^{3}, and _{d} = _{b}·_{sand} and _{silt} are the mass fractions of sand and silt particles, respectively, %; _{om} is the mass ratio of organic matter, % |

On the basis of our previous research works [

The Richards equation based on conservation of mass and Darcy’s law is used for the numerical simulation of water movement in saturated-unsaturated zones [^{2}; ^{3}, which is assumed to be constant; _{m} is the negative reciprocal of the slope of the soil water characteristic curve (SWCC), m^{−1}; _{e} is the relative saturation of the unsaturated zone; _{s} and _{r} (_{s} is the elastic water storage rate, Pa; ∇ is the Laplace operator; and _{m} is the source term of the seepage field, kg/(m^{3}·s).

The SWCC of the unsaturated zone is described by the VG model [^{−1}; _{r} is the residual water content, m^{3}/m^{3}; _{s} is the saturated water content, m^{3}/m^{3}; and _{p} is the pressure head (_{p} = ^{3}),

Equations (_{e}), volumetric water content (_{p}), the water capacity (_{m}), and the relative hydraulic conductivity of the unsaturated zone (_{r} (

Referring to the report of Healy and Ronan [^{3}/m^{3}, which is equal to the porosity in the saturated zone; ^{3}°C); _{s} is the heat capacity of soil, J/(m^{3}°C); _{eq} (^{3}°C); _{H} is the hydrodynamic dispersion coefficient, m^{2}/s; _{s} is the source term of the temperature field, W/m^{3}._{T} and _{L} are transverse dispersion and longitudinal dispersion, respectively, _{ij} is a kriging constant equal to 1 when

In the hydrothermal coupling problem involving the riparian zone, the interaction between the land and water fields includes two components: (1) water movement will cause a change in temperature in the riparian zone and (2) a lateral temperature change in the riparian zone will cause a change in the physical state of the water flow. These two processes are highly coupled. In this paper, COMSOL software (COMSOL Multiphysics 5.2a, COMSOL Inc., Stockholm, Sweden) was used to solve this strong coupling problem. Notably, COMSOL can be used to solve linear and nonlinear problems, as well as time-dependent steady-state and transient problems. The material properties and boundary conditions in the model can be defined as constants or functions. In addition, the software contains predefined modules (such as fluid flow, heat conduction, and structural mechanics) and a self-defined mathematical module, which makes it particularly suitable for solving complex physics-related coupling problems [

In this paper, the temperature field model is a highly nonlinear partial differential equation (PDE), and there is no fixed calculation module to determine the coupling effect between the temperature (_{eq} (_{a} is the mass coefficient, kg/(m·s°C); _{a} is the damping coefficient, J/(m^{3}°C); ^{2}°C); ^{2}; ^{2}°C); a is the absorption coefficient, W/(m^{3}°C); and ^{3}. These coefficients are defined by the users according to the governing equations.

The temperature field model (equation (_{eq} (_{H}∇_{s}) in equation (_{a} = 0, _{a} = _{s}, _{eq} (_{H}, _{eq} (_{eq} (_{eq} (

The experiment was performed in a 3-D sand tank with a length of 60 cm, a width of 20 cm, and a height of 80 cm. The upstream and downstream sections of the sand tank were separated into an upstream water inlet tank and a downstream water outlet tank with a width of 10 cm by a plexiglass plate. The upstream water inlet tank had overflow ports of 30 cm and 50 cm to maintain a stable infiltration head, and an outlet was arranged at a height of 5 cm of the downstream water outlet tank. A schematic diagram of the entire test device structure is shown in Figure ^{3}, and the particle size distribution of the studied sands is shown in Figure

Layout of the laboratory experiment sand tank (adapted from Ren et al. [

Particle size distribution of the studied sands.

Based on the established hydrothermal coupling model and observed data from laboratory experiments, the finite element solution of the hydrothermal coupling model was obtained using a combination of the groundwater flow module and PDE module in COMSOL software. A mapping grid was used for the computational domain, and the grid size was 1.5 cm. The whole computational domain included 2160 elements and 2255 grid nodes. The computational domain and mesh layout are shown in Figure

(a) Computational domain and (b) finite element mesh (mesh layout) (

For the initial condition of the seepage field, it is assumed that the pressure head is 0 m and the initial condition of the temperature field is set to 20°C according to the initial temperature of sand in the experiment.

According to the laboratory experiment and related literature, the calculation parameters of the hydrothermal coupling model are given in Table _{s}, _{s}, _{r}, and _{b} were obtained from the laboratory experiment, and the values of parameters _{s} and _{clay}, _{sand}, _{silt}, and _{om} were obtained from Carsel and Parrish [_{s}, _{clay}, _{sand}, _{silt}, and _{om}) are also considered similar and can provide a reference for the findings of this study. The thermal conductivity of water is set to 0.58 W/(m°C), and the density of water is 1000 kg/m^{3}.

Parameters of the hydrothermal coupling model.

Parameter | Units | Value |
---|---|---|

_{s} | m/s | 8.25 × 10^{−5} |

_{s} | m^{3}/m^{3} | 0.342 |

_{r} | m^{3}/m^{3} | 0.023 |

m^{−1} | 10.5 | |

Β | — | 2.68 |

— | 0.43 | |

_{b} | kg/m^{3} | 1560 |

_{s} | J/(m^{3}°C) | 1100000 |

J/(m^{3}°C) | 4200000 | |

% | 35 | |

_{sand} | % | 93.00 |

_{clay} | % | 2.00 |

_{silt} | % | 5.00 |

_{om} | % | 0.00 |

To quantitatively evaluate the simulation effect of the hydrothermal coupling model, the root-mean-squared error (RMSE), coefficient of determination (^{2}), and relative error (Re) are used as the model evaluation indices [^{2} varied from −1.17 to 1.00, and small ^{2} values were observed at monitoring points T20, T21, T23, T24, T25, T26, T27, and T28. These monitoring points were all located near the upper surface of the sand tank, which implies that the consistency between the simulated temperature and the actual temperature is relatively poor when the established model is used to estimate the temperature of the shallow riparian zone. However, the RMSE values at these monitoring points were relatively small, which indicates that the deviations between the observed and simulated temperatures are small. The Re value ranged from 0.46–10.89% with an average value of 3.58%, and large values were only occurred at monitoring points T4 and T10. Therefore, the established hydrothermal coupling model can reasonably reflect the lateral temperature changes in the riparian zone during the water infiltration.

Comparison of the observed and simulated temperature data at equal distances from the temperature source. (a) A1. (b) A2. (c) A3. (d) A4. (e) A5.

Statistical evaluation indices at different monitoring points.

Monitoring points | RMSE (°C) | ^{2} | Re (%) |
---|---|---|---|

T1 | 0.88 | 0.82 | 6.07 |

T2 | 1.49 | 0.64 | 8.11 |

T3 | 0.68 | 0.96 | 4.77 |

T4 | 1.87 | 0.74 | 10.88 |

T7 | 1.17 | 0.86 | 5.75 |

T8 | 0.21 | 1.00 | 1.28 |

T9 | 1.02 | 0.92 | 5.40 |

T10 | 2.01 | 0.47 | 10.49 |

T11 | 0.57 | 0.96 | 3.80 |

T13 | 0.49 | 0.98 | 2.51 |

T15 | 0.83 | 0.82 | 3.83 |

T16 | 0.61 | 0.92 | 2.72 |

T18 | 0.59 | 0.90 | 2.40 |

T19 | 0.18 | 0.98 | 0.72 |

T20 | 0.52 | 0.41 | 1.85 |

T21 | 0.38 | −0.52 | 1.29 |

T23 | 0.32 | −2.38 | 1.12 |

T24 | 0.16 | 0.14 | 0.62 |

T25 | 0.18 | −0.63 | 0.72 |

T26 | 0.32 | −0.61 | 1.41 |

T27 | 0.35 | −1.03 | 1.57 |

T28 | 0.31 | −0.75 | 1.35 |

The simulation effects of different models vary with the monitoring points. Therefore, it is necessary to comprehensively evaluate the effect of each equivalent thermal conductivity model. In this section, the simulated and observed temperature data collected at all the monitoring points in the sand tank are considered for global comparison, and the results are shown in Figure

Global comparison of the observed and simulated temperature data for different equivalent thermal conductivity models.

To quantitatively evaluate the simulation effects of the hydrothermal coupling model with different equivalent thermal conductivity models from a global perspective, the statistical results of the simulated and observed temperature values at 22 monitoring points in the sand tank for different thermal conductivity models are given in Table ^{2}, and Re values for the Ren model, Lu model, and Côté and Konrad model were 0.86°C, 0.95, and 3.57%, respectively, and each evaluation index reflected a satisfactory result. The Lu model is a thermal conductivity model that has been widely used in recent years [

Statistical results of evaluation indices of different equivalent thermal conductivity models.

Equivalent thermal conductivity models | Evaluation induce | ||
---|---|---|---|

RMSE (°C) | ^{2} | Re (%) | |

Johansen model | 0.87 | 0.92 | 3.63 |

Campbell model | 0.92 | 0.90 | 3.92 |

Côté and Konrad model | 0.86 | 0.95 | 3.58 |

Lu model | 0.86 | 0.95 | 3.58 |

Ren model | 0.86 | 0.95 | 3.58 |

Figure

Comparison of the observed and simulated mean temperature changes in the sand tank for different equivalent thermal conductivity models.

From the above analysis results, the simulation effects of the hydrothermal coupling model vary with the model of equivalent thermal conductivity used. Therefore, the results suggest that it is necessary to select a reasonable equivalent thermal conductivity model when using a hydrothermal coupling model to analyze the hydrothermal dynamics in the riparian zone. The traditional methods of coupled hydrothermal modeling are generally based on the default module of heat transfer in porous media and focus on the internal heat transfer of objects, which limits the selection of an equivalent thermal conductivity prediction model. However, the PDE module is used in this paper to model the heat transfer equation, which makes the selection of the equivalent thermal conductivity model more flexible. These results can provide a reference for the promotion and reasonable selection of an equivalent thermal conductivity model to simulate temperature variations in saturated-unsaturated sandy soils.

The established hydrothermal coupling model contains many parameters and requires extensive model calibration. Therefore, the Morris method was selected to analyze the global sensitivity of the parameters in the hydrothermal coupling model with the Ren model to the temperature output. The specific calculation principle of the Morris method can be found in [

The model parameters selected in this paper include: hydraulic conductivity (_{s}), saturated water content (_{s}), residual water content (_{r}), porosity (_{s}). In the numerical calculation process, each parameter was evaluated based on a 10% increase, and the calculation time was 500 minutes. The difference between the temperature field and the rated temperature field due to changes in the parameters was determined for each finite element node, and the average change was used to represent the change in temperature. According to the calculation principle of the Morris method, the combination of parameters was input into the hydrothermal coupling model, and then, the sensitivity of the parameters and the corresponding interactions was obtained by solving the model in turn. Table

Parameter combinations and temperature fluctuations based on the morris sensitivity analysis method.

Group | Parameter combination | Temperature fluctuation (%) |
---|---|---|

1 | _{s} | −0.40 |

2 | _{s} | −0.12 |

3 | _{r} | 0.03 |

4 | 0.11 | |

5 | 0.29 | |

6 | −0.35 | |

7 | _{s} | 0.34 |

8 | _{s}, _{s} | −0.53 |

9 | _{s}, _{r} | −0.17 |

10 | _{r}, | 0.12 |

11 | 0.40 | |

12 | 0.01 | |

13 | _{s} | 0.01 |

14 | _{s}, _{s}, _{r} | −0.56 |

15 | _{s}, _{r}, | −0.06 |

16 | _{r}, | 0.47 |

17 | 0.13 | |

18 | _{s} | 0.40 |

19 | _{s}, _{s}, _{r}, | −0.45 |

20 | _{s}, _{r}, | 0.29 |

21 | _{r}, | 0.20 |

22 | _{s} | 0.47 |

23 | _{s}, _{s}, _{r}, | −0.03 |

24 | _{s}, _{r}, | 0.05 |

25 | _{r}, _{s} | 0.53 |

26 | _{s}, _{s}, _{r}, | −0.32 |

27 | _{s}, _{r}, _{s} | 0.37 |

28 | _{s}, _{s}, _{r}, _{s} | 0.00 |

To facilitate the analysis, the temperature fluctuation results for each parameter combination in Table _{s}, _{s}, and _{s}, _{r}. When multiple parameters change (Group 8–28), groups 8, 11, 14, 16, 18, 19, 22, and 25 are the parameter combinations that cause the largest temperature fluctuations, and all of these parameter combinations include _{s}, _{s}, or _{s}, _{s}, and

Temperature fluctuations for different parameter combinations based on sensitivity analysis.

In addition, Table

Although the sensitivity of hydrothermal coupling model parameters to the calculation output results has been assessed by some scholars in previous studies [_{s}) and VG model parameter (_{s} was the most important and sensitive parameter. However, their results were based on a single-factor sensitivity analysis and did not consider the influence of multiple factors on the results. Ren et al. [_{s}) and VG model parameter (_{s}) and VG model parameter (_{s}) and VG model parameter _{s}) and VG model parameter (_{s}) influence the heat transport process.

Based on the seepage theory of unsaturated soil and the principle of energy conservation, a coupled solution to the PDE of the temperature and seepage fields is achieved through COMSOL software and the appropriate equivalent thermal conductivity model. Combined with temperature series data from laboratory experiments, the validity of the model was verified, and the results showed that the established model can effectively reflect the dynamic temperature changes in the soil based on the results of a sand box test.

A reasonable selection of the equivalent thermal conductivity model will improve the effect of hydrothermal coupling simulations. The temperature data observed in the laboratory test were compared with the simulated temperatures based on each equivalent thermal conductivity model, and the Ren model, Lu model, and Côté and Konrad model are recommended for future hydrothermal coupling analysis in saturated-unsaturated soils. However, the experimental material used in this paper was sandy soil; therefore, the results are only applicable to the sandy soil. The suitability for other soil types must be further confirmed by applying the proposed modeling approach in subsequent studies.

The results of sensitivity analysis revealed that the hydraulic conductivity (_{s}), VG model parameters (_{s}) greatly affect the temperature output of the hydrothermal coupling model. Therefore, in the application of the model, the observation accuracy of these parameters should be improved; thus, the numerical model that yields simulation results similar to the measured values can be obtained, even if the secondary factors are not fully considered or ignored.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

W. Z. and J. R. designed the framework and wrote the manuscript. W. Z. and G. C. collected the data. W. Z. and Z. S. were mainly responsible for the simulation analysis. W. Z. and L. X. verified the results of the model. J. R. and Z. S. provided funding support.

This research was funded by the National Natural Science Foundation of China (Grant nos. U1765205 and 51679194), the Natural Science Foundation of Shaanxi Province (Grant no. 2020JM-448), the Planning Project of Science and Technology of Water Resources of Shaanxi (Grant no. 2019slkj-12), the Belt and Road Special Foundation of the State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering (Grant no. 2019490711), and the Priority Academic Program Development of Jiangsu Higher Education Institutions (YS11001).