Preliminary discussion on the principle of minimum energy consumption rate controlling hierarchical groundwater flow systems
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摘要: 20世纪60年代初期，Tóth基于定水头上边界条件推导出解析解，得出多级次地下水流系统，是水文地质学里程碑式的突破，成功地解决了一系列理论和实际问题。但Tóth解析解存在的缺陷也长期沿袭：单纯重视数学模拟而忽视物理机制；将地形控制地下水位看成是普适性规律；忽视给定水头上边界数学模拟的失真。这些缺陷，尤其是忽视物理机制探究，不仅妨碍Tóth理论自身发展，而且导致地下水流系统理论尚未被国际水文地质界普遍接受。参照河流动力学中应用的最小能耗率原理，类比提出地下水流最小能耗率的表达式。基于已有的通量上边界地下水流模式数值模拟结果，进一步探究物理机制，归纳得出地下水流系统遵循最小能耗率原理的结论。Abstract: In the early 1960s, Tóth obtained hierarchical groundwater flow systems by using analytical solution based on given-head upper boundary, which is a milestone breakthrough in hydrogeology and successfully solved a series of theoretical and practical problems.However, the defects of Tóth's analytical solution have been followed for a long time such as focusing solely on mathematical simulation and ignoring the physical mechanism; taking terrain control of water table as a universal law; and ignoring the distortion of the mathematical simulation based on given-head upper boundary.These shortcomings, especially the lack of the physical mechanisms exploration, not only hindered the development of Tótian theory itself, but also made the theory difficult to be understood, so that the theory has not being widely applied yet by the international hydrogeological community.This paper proposes an expression for the minimum energy consumption rate of groundwater flow referring to the principle of minimum energy consumption rate applied in river dynamics.Based on the exited results of "numerical simulation of groundwater flow patterns using flux as upper boundary", the physical mechanism is further explored, and it is concluded that groundwater flow follows the principle of minimum energy consumption rate.
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图 1 模型A1：固定渗透系数K、改变入渗强度ε，不同流场平均水力梯度Ric下的地下水流模式(据文献[5]修改)
Figure 1. Groundwater flow patterns for different infiltration intensities and fixed hydraulic conductivity for Model A1
图 2 模型A2：固定入渗强度ε、改变渗透系数K，不同流场平均水力梯度Ric下的地下水流模式(据文献[5]修改)
Figure 2. Groundwater flow patterns for different hydraulic conductivity and fixed infiltration intensity for Model A2
图 3 模型B：改变盆地深度得出的地下水流系统模式
a.Rld=20; b.Rld=10; c.Rld=5
Figure 3. Groundwater flow patterns under different basin depths for Model B
表 1 模型A(Rld=10)参数系列与模拟结果(据文献[5]修改)
Table 1. Parameters and simulated results of Model A(Rld=10)
A1(K不变，K=0.1 m/d) A2(ε不变，ε=0.49 m/d) 模拟得出的地下水流模式 亚型 ε/(mm·d-1) Ric 亚型 K/(m·d-1) Ric A1-a 1.97 0.0197 A2-a 0.025 0.0197 单一局部系统(L) A1-b 0.98 0.0098 A2-b 0.050 0.0098 局部-中间嵌套系统(LI) A1-c 0.49 0.0049 A2-c 0.100 0.0049 局部-中间-区域嵌套系统(LIR) A1-d 0.25 0.0025 A2-d 0.200 0.0025 局部-区域嵌套系统(LR) A1-e 0.08 0.0008 A2-e 0.600 0.0008 单一区域系统(R) 注：R1d.盆地长深比；ε.入渗强度；Ric.水力梯度；K.渗透系数 
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表 2 模型B设定与模拟结果(改变Rld)
Table 2. Setup and simulated results of Model B(variable Rld)
亚型 Rld 潜在势汇位置(x, z) 模拟得出的地下水流模式 S1, S2, S3 B-a 20 (0, 10);(200, 13);(400, 17) 单一局部系统(L) B-b 10 (0, 35);(200, 38);(400, 42) 局部-中间-区域嵌套系统(LIR) B-c 5 (0, 85);(200, 88);(400, 92) 局部-区域嵌套系统(LR)
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