Exploring the Effects of Fissures on Hydraulic Parameters in Subsurface Flows from the Perspective of Energy Changes

Abstract

Reynolds number (Re), pore water pressure (P), and water flow shear force (τ) are primary indicators reflecting the characteristics of subsurface flow. Exploring the calculation of these parameters will facilitate the understanding of the hydrodynamic characteristics in different subsurface flows and quantify their differences. Hence, we conducted a study to monitor soil water content, matrix potential, and pore water pressure in two typical soil profiles (with and without fissures). The distribution of Re, P, and τ in both matrix flow (MF) and preferential flow (PF) were calculated with an improved calculation method, focusing on their energy changes. Results showed that these hydrologic parameters are quite different between MF and PF. Re values in MF remained below 0.1, indicating lower water flow velocities, while the Re values ranged from 0.8 to 2 in PF, indicating higher flow velocities. The P values in PF was tens to hundreds of times higher than that in MF, which is mainly due to the rapid accumulation and leakage of water within soil fissures. Additionally, the larger hydraulic radius and gradient in PF also resulted in higher τ values in PF (2~6 N m⁻²) than in MF (0~1.5 N m⁻²). In PF, the pressure potential was the significant factor for τ, while τ in MF was dominated by the matrix potential and varies with the magnitude of the matrix potential gradient. This study suggests that Re, P, and τ could be considered as the major indexes to reflect dynamic characteristics of subsurface flow.

1. Introduction

Subsurface flow refers to water movement within the soil, either laterally along soil horizons or vertically through percolation. The factors influencing subsurface flow include soil physical properties, soil horizon stratification, and water supply conditions [1,2]. Additionally, features like macropores, animal burrows, and plant roots can facilitate the formation of subsurface flow [2]. Compared to surface runoff, subsurface flow is generally slower and can persist for longer durations, spanning days, weeks, or even longer [3]. The dynamics of subsurface flow play a crucial role in understanding its behavior and characteristics. Understanding subsurface flow dynamics allows researchers to calculate important parameters, such as the Reynolds number and water flow shear force, for different flow patterns. By identifying the key factors that influence these forces, researchers can better understand the mechanisms driving subsurface flow. Meanwhile, these parameters provide valuable information about the flow regime and its impact on the surrounding environment, such as contributing to the prevention of soil erosion induced by subsurface flow.

Matrix flow (MF) and preferential flow (PF) are two typical types of subsurface flow [4,5]. When MF occurs, soil moisture moves slowly in the capillary pores of soil, when PF occurs, soil moisture infiltrates rapidly directly through the large pores in the soil, resulting in a faster water flow than MF [4,5]. Hence, MF follows Darcy’s law, in which the flow rate is positively correlated with the pressure difference and the permeability of the soil [6]. However, the soil macropores enable rapid water movement, where PF occurs, and its flow characteristics do not follow Darcy’s law [7]. Both MF and PF are influenced by gravitational potential, matrix potential, and pore water pressure potentials [8]. In MF, the difference in matrix potential among adjacent soils drives subsurface flow, and the water always flows from the high-potential region to the low-potential region [9]. Nevertheless, in PF, water movement through macropores is not affected by matrix potentials when the macropores are saturated.

Pore water pressure is another driving force for subsurface flow, pushing soil water flow from areas of high pore water pressure to low, and this process is often accompanied by rapid and dramatic changes in pore water pressure [10]. Furthermore, changes in soil matrix potential and pore water pressure not only drive subsurface flow but also impact the physical properties and turbulent structure of water flow [11]. These changes result in alterations in energy loss and hydraulic characteristics. Therefore, understanding the dynamics of subsurface flow and its different flow patterns is essential for effective management and protection of groundwater resources.

Soil water potential, pore water pressure, water velocity, the Reynolds number (Re), and water flow shear force (τ) are primary parameters used to characterize the hydraulic characteristics of water flow. Among these, soil water potential and pore water pressure can be directly measured while the other three need to be calculated. The calculation of the water flow rate is straightforward, but Re and τ are slightly more complicated. Re is a dimensionless parameter that represents the ratio of inertial force to viscous force in fluid flow [12]. In 1883, Reynolds suggested that the flow pattern of a fluid is related to its flow velocity, the diameter of the tube, and the viscosity and density of the fluid (ρ) [12]. Thus, the Re of the subsurface flow can be calculated as Re = vd/4υ, where v (m s⁻¹) is the linear velocity of the water in the soil, υ (N s⁻¹ m⁻²) is the viscosity coefficient of water at a certain temperature, and d (m) is the diameter of the soil pores. When Re is small, the flow is mainly influenced by viscous forces, resulting in a stable flow field [13]. Conversely, when Re is large, inertial forces dominate, leading to an unstable flow field with turbulent flow [14]. In the case of water flowing vertically downward in a soil pipe, the flow is affected by matrix potential tension in addition to pressure, friction, inertial, and viscous forces [15]. The influence of the inertial force is negligible in this scenario, and the flow is primarily determined by the tension generated by matrix potential and pressure, with gravity acting as a constant term [16]. The soil water flow state and the friction and viscous forces vary with different Re values [17]. While friction and viscous forces cannot be directly observed, their resultant force can be numerically represented by τ, according to the Law of Conservation of Energy and the Principle of Balance of Forces.

The τ of subsurface flow is the force generated by the flow of water moving in the direction of the hydraulic gradient [18], which is mainly frictional and viscous force. These forces acting in the direction of the hydraulic gradient will scour the soil, disrupt the soil structure, and result in soil particles transporting out of the slope body along with the water flow [19,20]. According to the Law of Conservation of Energy, in uniform pipeline flow, the work performed by the water flow overcomes frictional resistance (f), also known as head loss (hf), caused by τ [21]. The formula for shear stress of thin-layer water flow under ideal conditions is given as τ = ρ_wgRJ, where τ represents water flow shear force, ρ_w denotes the density of water, g is the gravitational acceleration constant, R represents the hydraulic radius, and J signifies the hydraulic gradient [22]. The hydraulic gradient is crucial for calculating τ and is the biggest difference between calculating the τ of the subsurface flow and the pipeline flow [13].

Currently, limited research has been conducted on the calculation and application of Re and τ in subsurface flow. We argue that it is possible to draw on the principles and methods of pipeline flow hydraulics and improve the corresponding formulas for use in calculating the hydraulic factors of subsurface flow [23]. The Reynolds equation is the most typical calculation for Re, and τ is usually calculated with the water head loss equation which is followed by energy conservation. We attempt to apply the basic rules of Re and τ of pipeline flow to calculate τ in subsurface flow. Thus, the objectives of this study are the following: (1) to calculate Re for two typical subsurface flow patterns (matrix flow and preferential flow) and examine the effect of pore conditions on the size and distribution of Re and (2) to investigate the variations in τ under matrix flow and preferential flow, and explore the key factors affecting τ under these two subsurface flow patterns. This knowledge can contribute to the development of effective management and mitigation strategies for subsurface flow-related problems, ultimately leading to improved water resource management and decreased environmental degradation risk.

2. Materials and Methods

2.1. The Calculation of Hydraulic Factors

2.1.1. Reynolds Number (Re)

For the two typical subsurface flows, the preferential flow (PF) and the matrix flow (MF), the water flow velocity and Re can be expressed as the following equations:

υ = as/t

(1)

Re = vd/4υ

(2)

where v (m s⁻¹) represents the water flow linear velocity in the soil, while υ (N s⁻¹ m⁻²) is the viscosity coefficient of water at a specific temperature (in this study, it is 0.8 × 10⁻⁶ m² s⁻¹ at an air temperature of 30 °C). d (m) is the soil-equivalent pore diameter at a given matrix suction. The variables s (m) and t (s) denote the length of the water flow infiltration path (the distance between different soil layers) and the time taken for rainwater to infiltrate into different soil layers, respectively. The parameter a represents the curvature of the soil water infiltration channel (the ratio of the actual length of the path through which water flows to the distance between soil layers). According to the Brooks–Corey equation [24], a was valued as 2 in the calculation for MF, while it is the distance between different soil layers for PF and valued as 1 in this study.

The most important aspect of the calculation process was the determination of the infiltration time and the infiltration rate. The average infiltration rate is computed by considering the length of the water flow infiltration path and the time taken by rainwater to infiltrate into different soil layers. When calculating Re for MF, the starting time point selected is when rainwater begins to maintain a stable and uniform infiltration between the two soil layers. Conversely, for PF, the starting time point is when PF begins to occur between the two soil layers during the infiltration process. The equivalent pore diameter is determined by the matrix suction, which is calculated using the soil water content when the rainwater reaches a specific soil layer. The relationship between soil water content and matrix suction is obtained from the soil water characteristic curve (SWCC).

2.1.2. Water Flow Shear Force (τ)

According to the Law of Conservation of Energy, the energy balance equation of the water flow movement process in a pipeline flow (Figure 1) can be described as follows [25]:

$$h_m+{\displaystyle \frac{P_m}{{\rho }_{w}g}}+{\displaystyle \frac{a_m{v_m}^2}{2g}}=h_n+{\displaystyle \frac{P_n}{{ \rho }_{w}g}}+{\displaystyle \frac{a_n{v}_n^{ 2}}{2g}}+h_f$$

(3)

Since there is a uniform flow, acceleration is zero and thus v_m = v_n. Equation (3) can be simplified as follows:

$${ h}_f=\left(h_m-h_n\right)+\left({\displaystyle \frac{P_m}{{\rho }_{w}g}}-{\displaystyle \frac{P_n}{{\rho }_{w}g}}\right)$$

(4)

The equation for all external forces in the direction of motion is as follow:

$${ P}_{m}A-P_{n}A+{\rho }_{w}gAl\ast cos\theta =\tau \chi l$$

(5)

where

$$cos\theta ={\displaystyle \frac{h_m-h_n}{l}}$$

(6)

$$R={\displaystyle \frac{A}{\chi }}={\displaystyle \frac{d}{4}}$$

(7)

$$J={\displaystyle \frac{h_f}{l}}$$

(8)

Per the above equations, τ is calculated as follows:

$$\tau ={\rho }_{w}g{\displaystyle \frac{d}{4}}{\displaystyle \frac{h_f}{l}}$$

(9)

where h is the potential energy, α (≈1) is the kinetic energy correction factor, P (Pa) is the dynamic water pressure at both ends, A (m²) is the area of the water flow section, l (m) is the linear distance of the water flow path, and χ (m) is the wetted perimeter.

For subsurface flow where water moves vertically downwards in a soil pipe, the water flow shear force of subsurface flow can be expressed as in Figure 2. Due to the influence of the matrix potential and gravity, based on the Law of Conservation of Energy, the energy balance of the water flow movement process follows Equations (10) and (11).

$$h_m+{\displaystyle \frac{P_m}{{\rho }_{w}g}}+{\displaystyle \frac{{\alpha }_m{v}_m^{ 2}}{2g}}+{\phi }_m=h_n+{\displaystyle \frac{P_n}{{\rho }_{w}g}}+{\displaystyle \frac{{\alpha }_n{v}_n^{ 2}}{2g}}+{\phi }_n+h_s$$

(10)

$$h_s=\left(h_m-h_n\right)+\left({\displaystyle \frac{P_m}{{\rho }_{w}g}}-{\displaystyle \frac{P_n}{{\rho }_{w}g}}\right)+\left({\phi }_m-{\phi }_n\right)+{\displaystyle \frac{{\alpha }_m{v}_m^{ 2}-{\alpha }_n{v}_n^{ 2}}{2g}}$$

(11)

In PF (Figure 3), flow velocity changes and the acceleration is (α_nv_n² − α_mv_m²)/2l. The energy required for acceleration is negligible and can be disregarded during calculations. The soil surrounding the water pathway is fully saturated, resulting in a zero-matrix potential. The equilibrium of all external forces in the direction of motion follows Equation (12).

$$P_{m}A-P_{n}A+{\rho }_{w}gAl-{\rho }_{w}gA{\displaystyle \frac{{\alpha }_m{v}_m^{ 2}-a_n{v}_n^{ 2}}{2l}}=\tau \chi l$$

(12)

At this time, τ is calculated as follows:

$$\tau ={\rho }_{w}g{\displaystyle \frac{d}{4}}{\displaystyle \frac{h_s}{l}}$$

(13)

For MF, the hydraulic gradient is as follows:

$$J={\displaystyle \frac{\partial \psi }{\partial l}}$$

(14)

In this case, τ is calculated as follows:

$$\tau ={\rho }_{w}g{\displaystyle \frac{d}{4}} {\displaystyle \frac{\partial \psi }{\partial l}}$$

(15)

According to Darcy’s law, i is derived as follows:

$$i=k{\displaystyle \frac{d\psi }{dy}}$$

(16)

The calculation of τ for MF is further simplified as follows:

$$\tau ={\displaystyle \frac{i}{k}}{\displaystyle \frac{d}{4}}{\rho }_{w}g$$

(17)

where ψ (m) is the soil water potential, k (m s⁻¹) is the saturated hydraulic conductivity, and i (m s⁻¹) is the water flow flux.

2.2. Field Validation

2.2.1. Study Location Description

The study area is situated in Tongcheng County (29.33° N, 113.77° E), Hubei Province, China. It exhibits a typical subtropical monsoon climate characterized by an annual rainfall of 1512.8 mm and an average annual temperature of 16.1 °C (China Meteorological Administration). The soil in this region predominantly originates from granite parent material, which has undergone significant weathering due to consistent rainfall and high temperatures. Consequently, the soil in the study area is loose and deep, ranging from 10 to 20 m in depth. According to the U.S. Soil Taxonomy System, it falls under the classification of Ultisols.

The study site is an active Benggang, characterized by vertical collapse due to water and gravity [26]. It consists of a catchment slope, gully head (Benggang wall), avalanche pile, flow channel, and alluvial fan (Figure 4). Our study focused on a representative Benggang in an active area where three collapses occurred during observation. The Benggang wall is steep, with a height difference of 5.2 m and a slope angle of over 75°. Soil layering varies with slope position: an upper shallow surface soil layer (layer A), a middle and lower red clay layer (layer B) exposed by water erosion, and even a sandy layer (layer BC) in some areas. The depth of 20–40 cm is consistently red clay, followed by a 40–80 cm transition layer, and a 60–80 cm sandy soil layer on the Benggang wall. The detritus layer (layer C) is deep and not visible in the soil profile.

2.2.2. Monitoring of Soil Water Content, Pore Water Pressure, and Matrix Suction

To investigate the changes in soil water dynamics during rainfall infiltration, two soil profiles were selected on the Benggang wall (Figure 4). These profiles were formed naturally as a result of the Benggang collapse. Profile A represents a homogeneous soil profile, while profile B exhibits visible fissures on the exposed side of the Benggang (Figure 5).

At specific depths of 20, 40, 60, and 80 cm on both profiles, we embedded soil water sensors (SMEC 300 SM/EC/Temp Sensor, Spectrum Technologies, Inc., Chicago, IL, USA), pore pressure sensors (Navigation Technology Company, Xi’an, China), and micro-tensiometers (Umwelt-Geräte-Technik GmbH, Berlin, Germany). These sensors were positioned consistently between the profiles. A 2 min resolution was used for the monitoring of soil water content, pore water pressure, and soil matrix potential. In the cracked profile (profile B), the fissure widths at the sensor positions were recorded as 2.11 mm, 2.34 mm, 1.98 mm, and 2.02 mm.

Table 1. Soil physical properties of four soil layers at two profiles.

Profile	Soil Layer (cm)	Soil Particle Size Distribution (%)			Soil Porosity (%)				Bulk Density (g cm−3)
		Sand *	Silt	Clay	>1 mm	1~0.5 mm	0.5~0.2 mm	<0.2 mm
A	0~20	43.66	32.19	24.15	1.52	1.16	1.76	38.59	1.58
	20~40	43.33	32.38	24.29	1.51	3.6	1.52	36.77	1.54
	40~60	44.2	35.87	19.93	1.26	3.36	0.93	34.2	1.55
	60~80	44.65	35.58	19.77	1.3	2.98	0.74	34.6	1.54
B	0~20	47.67	32.69	19.64	1.6	3.06	1.11	35.72	1.55
	20~40	50.59	31.89	17.52	2.95	6.27	3.27	28.61	1.56
	40~60	56.97	29.07	13.96	5.45	5.9	1.01	29.94	1.53
	60~80	58.94	26.9	14.16	9.72	10.17	0.54	22.17	1.52

Notes: * The size of sand, silt, and clay were 2~0.05 mm, 0.05~0.002 mm, and less than 0.002 mm, respectively (according to USAD).

provides an overview of the soil physical properties for the four soil layers present in profiles A and B.

2.2.3. Measurements of Soil Macropores

According to the saturated volumetric soil water content and residual soil water content (drying method) of the different soil layers in the two profiles, and the soil water characteristic curve (SWCC), the soil macropores distribution was obtained. Among them, the soil water characteristic curve is drawn using RETC software 6.02. The soil water content under a series of matrix suction (under the suction of 15 cm-H₂O, 9 cm-H₂O, and 3 cm-H₂O) was measured with a micro tensiometer, and then the obtained soil water characteristics data were input into the RETC computer program (American Salinity Laboratory, 4500 Glenwood Drive, Riverside, CA 92051, USA). In RETC, the VG model [27] is used to fit and draw the SWCC. According to Brooks’ research, the equivalent pore diameter is calculated as d = 3/h, where h (cm-H₂O) is the height of the water applied to the matrix suction, d (mm) is the diameter of the soil pore equivalent to the suction [28]. Different matrix suctions represent different soil pore diameters. Finally, according to the change in soil water content under different suction conditions, the specific distribution of soil macropores with different diameters is calculated.

2.3. Data Statistics and Analysis

All statistical analyses were performed with the software Origin 2021 (Origin Lab Corporation, Northampton, MA, USA). An alpha of 0.05% was used to identify significant results.

3. Results

3.1. The Reynolds Number under Different Subsurface Flows

Table 2. The equivalent pore diameter (the equivalent pore diameter for PF refers to the width of the fracture of the soil layer where the PF occurs), infiltration rate, and the Reynolds number due to pore pressure when MF and PF occurred during the initial stage of infiltration at each profile and each soil layer.

Profile	Rainfall Amount	Rainfall Intensity	Soil Layer	Initial Water Content	Equivalent Pore Diameter (PF/MF)	Infiltration Rate (PF/MF)	Reynolds Number (MF)	Reynolds Number (PF)
	mm	mm h−1	cm	m3 m−3	mm	mm s−1
A	60, artificial rainfall	10	0~20	0.222	0.20
			20~40	0.336	0.32	0.033	0.033
			40~60	0.267	0.29	0.043	0.029
			60~80	0.278	0.31	0.013	0.014
	60, artificial rainfall	10	0~20	0.254	0.25
			20~40	0.354	0.34	0.046	0.049
			40~60	0.301	0.34	0.018	0.021
			60~80	0.302	0.33	0.016	0.016
	34, natural rainfall	10	0~20	0.243	0.23
			20~40	0.373	0.35	0.083	0.091
			40~60	0.291	0.32	0.033	0.034
			60~80	0.319	0.34	0.015	0.016
B	30, artificial rainfall	10	0~20	0.284	2.11
			20~40	0.294	2.34	1.334		0.975
			40~60	0.196	1.98	1.334		0.825
			60~80	0.187	2.02	2.667		1.684
	30, artificial rainfall	10	0~20	0.215	2.11
			20~40	0.267	2.34	1.334		0.975
			40~60	0.174	1.98	2.667		1.652
			60~80	0.266	2.02	2.667		1.684
	27, natural rainfall	10	0~20	0.381	2.11
			20~40	0.373	2.34	1.334		0.975
			40~60	0.344	1.98	1.334		0.825
			60~80	0.283	2.02	2.667		1.684

presents the relevant calculation parameters and Re values during the initial stage of infiltration for three rainfall events, considering pore pressure. When subsurface flow is driven by matrix flow (MF), Re values are generally small. Among the nine scenarios (three soil layers × three rainfall events), profile A consistently showed Re values for MF below 0.1. The highest Re value (0.091) was observed in the 20–40 cm soil layer during the second artificial rainfall. Re values decrease from the surface soil to deeper layers, with the lowest values appearing at 60–80 cm. The velocity of water flow in MF is also very low, usually below 0.1 mm s⁻¹, and there is a time lag of over an hour between the soil water content probes in different layers (Figure 6). This suggests that the small Re values during MF can be attributed to the extremely slow infiltration velocity of subsurface flow.

In profile B, subsurface flow occurs as preferential flow (PF) in fissures, and the velocity of water flow in PF is much higher than that in the soil matrix (Figure 7). When PF drives subsurface flow, the Re values due to pore pressure are tens-of-times greater than that in MF, which ranged from 0.8 to 2. However, due to the limitations of the soil layer thickness (20 cm) and monitoring frequency (every 5 min), the maximum Re value during PF cannot be measured. Under the same rainfall intensity, the infiltration velocity of rainwater in PF is significantly higher than that in MF, generally exceeding 1 mm s⁻¹. This fast infiltration velocity likely leads to higher Re values. Additionally, in profile B, the deeper soil layer is predominantly sandy with numerous macropores (Table 1). Consequently, Re values tend to increase from the surface to deeper layers in most rainfall events (Table 2).

3.2. The Distribution of Soil Pore Pressure under Different Subsurface Flows

Pore water pressures were compared between profiles A and B in three rainfall events to examine the impact of soil water content on pore water pressure distribution in MF and PF (Figure 8). High soil water content promotes pore water pressure. During MF, when water content is below 0.3 m³ m⁻³, pore water pressures in different soil layers are close to 0 kPa. Nevertheless, pore water pressures become polarized when soil water content exceeds 0.3 m³ m⁻³. However, certain soil water contents correspond to multiple pore water pressure values, especially at the 20 cm and 40 cm soil layers, where maximum and minimum values of 2 kPa and −2 kPa appear, respectively, in the 40 cm soil layer. Due to the thickness of the soil layer and its depth, positive pore water pressure is more likely, particularly in the deeper layer (such as 80 cm) (Figure 8a).

In PF, as soil water content increases, pore water pressure in each soil layer gradually transitions to positive pressure and continues to rise (Figure 8b). Generally, pore water pressure throughout the entire profile is positive in PF, except for the 20 cm soil layer where soil water content is below 0.3 m³ m⁻³ and an isolated area in the 40 cm soil layer. In deeper soil layers, pore water pressures are positive during infiltration, except a minimum value of −1 kPa observed at low soil water content in the 20 cm soil layer. At the 80 cm soil layer, pore water pressure increases linearly with water content, reaching a maximum value of 1.5 kPa. In PF, the rapid pooling of soil water may be the reason for the linear increase in pore water pressure.

3.3. The Water Flow Shear Force under Different Subsurface Flows

Table 3. The water flow shear force when MF and PF occurred during the initial stage of infiltration at each profile and each soil layer.

Profile	Soil Layer (cm)	Pore Water Pressure (KPa)	Soil Matrix Potential (MF) (KPa)	Water Flow Shear Force (N m−2)	Water Flow Shear Force (N m−2)
					via Pore Pressure	via Gravity	via Matrix Potential
A, artificial rainfall	0–20	0.016	−2.3
	20–40	0.014	−2.1	0.547	0.001	0.591	−0.045
	40–60	0.011	−5.2	1.579	0.001	0.564	1.014
	60–80	0.052	−4.6	0.343	−0.012	0.588	−0.233
A, artificial rainfall	0–20	0.346	−2.2
	20–40	0.008	−1.6	0.461	0.164	0.711	−0.414
	40–60	0.011	−3.1	1.216	−0.001	0.711	0.506
	60–80	0.076	−3.0	0.648	−0.024	0.709	−0.039
A, natural rainfall	0–20	0.014	−2.1
	20–40	−0.279	−1.2	0.593	0.126	0.761	−0.294
	40–60	−0.292	−4.2	1.729	0.004	0.662	1.063
	60–80	0.077	−2.8	0.076	−0.143	0.760	−0.543
B, artificial rainfall	0–20	0.842
	20–40	1.0536		4.439	−1.335	5.737
	40–60	0.161		6.902	2.047	4.855
	60–80	0.277		4.576	−0.384	4.960
B, artificial rainfall	0–20	−0.561
	20–40	0.257		2.406	−3.332	5.737
	40–60	0.341		3.623	−1.232	4.855
	60–80	−0.357		6.949	1.989	4.960
B, natural rainfall	0–20	−0.059
	20–40	0.231		5.621	−0.116	5.737
	40–60	0.115		5.121	0.266	4.855
	60–80	0.091		5.046	−0.086	4.960

presents the distribution of shear stress (τ) in profiles A and B at the initial stage of rainwater infiltration across three rainfall events. In MF, τ values are generally below 2 N m⁻², with the maximum value (1.729 N m⁻²) occurring in the 40~60 cm soil layer. Noteworthy, the τ generated via matrix potential is tens to hundreds of times higher than that via pore water pressure, with the magnitude of τ determined by the matrix potential gradient. In PF, τ is approximately ten times higher than in MF. The value of τ in PF is typically above 2 N m⁻², with the maximum value (6.949 N m⁻²) occurring in the 60~80 cm soil layer. Moreover, the τ value generated via gravity is fixed and only related to the vertical height difference and soil pore diameter. However, the pore water pressure in the fissure of PF is tens to hundreds of times higher than that in MF, and it is the key factor determining τ in PF.

Due to the long equilibrium time and lag in the determination of soil matrix potentials, we calculated τ for MF according to Darcy’s law.

Table 4. The water flow shear force calculated via Darcy’s law when MF occurred during the initial stage of infiltration at each profile and each soil layer.

Profile	Soil Layer (cm)	Equivalent Pore Diameter (MF) (mm)	Saturated Hydraulic Conductivity (cm d−1)	Water Flow Flux (mm s−1)	Water Flow Shear Force Due to Pore Pressure (N m−2)
B, artificial rainfall	0–20	0.3	0.163
	20–40	0.32	0.353	0.033	0.907
	40–60	0.23	0.349	0.043	0.252
	60–80	0.24	0.577	0.013	0.147
B, artificial rainfall	0–20	0.21	0.163
	20–40	0.34	0.353	0.046	0.631
	40–60	0.29	0.349	0.018	0.766
	60–80	0.29	0.577	0.016	1.419
B, natural rainfall	0–20	0.21	0.163
	20–40	0.35	0.353	0.083	1.606
	40–60	0.27	0.349	0.033	0.465
	60–80	0.31	0.577	0.015	0.132

shows that the calculated τ for MF ranges from 0 to 2 N m⁻², with a maximum value of 1.606 N m⁻². The water flow flux at this time is 0.083 mm s⁻¹. This result is consistent with the calculation based on the Law of Energy Conservation. However, the size of τ calculated with the two methods differs in the same soil layer. When using Darcy’s law, water flow flux is the determining factor for τ. In our three rainfall events, water flow flux gradually decreased from the surface downward, while saturated hydraulic conductivity increased. Two of the rainfall events indicated that the Re for MF decreased from the surface downward.

Figure 6 and Figure 7 provide insights into the continuous changes in τ during rainfall for MF and PF, respectively. These values were calculated using Darcy’s law and the Law of Energy Conservation. In MF, τ remains below 0.5 N m⁻² throughout the rainfall process. The influence of τ becomes evident only when the soil water content begins to change, and the response of τ in a soil layer is positively correlated with the change in water content. However, the water content level does not determine the magnitude of τ between different soil layers. In the case of PF, τ undergoes significant changes during rainfall and when MF occurs. Notably, τ is particularly high at the beginning of infiltration. In Figure 8, changes in τ often occur when there are substantial changes in soil water content or pore water pressure. However, an increase in soil water content does not necessarily lead to an increase in τ. When there is a steady increase or decrease in soil water content, τ remains relatively stable. The same holds for changes in τ caused by pore water pressure, where a sharp change in pore water pressure leads to changes in τ depending on the difference in pore water pressure at both ends.

4. Discussion

4.1. The Distribution of Re in Different Profiles

Re is a crucial parameter in analyzing subsurface flow dynamics, with its value depending on the type of flow. Understanding the distribution of Re values helps to comprehend the behavior of subsurface flow, which is essential for various applications like groundwater management, material transport, and energy conversion. In this study, we focused on quantifying Re in two drastically different subsurface flows.

When MF dominated the subsurface flow, Re is small and the maximum value is below 0.1. During the infiltration process, MF follows Darcy’s law, and the soil pore diameter used to transport MF is small, both of which result in a slow flow velocity of rainwater infiltration [29] and a lower Re. Meanwhile, Re with dynamic pressure gradually decreases with an increase in profile depth, which is mainly caused by the gradual decrease in water velocity in a profile [30]. Additionally, the number of macropores gradually decreases in the deeper profile. Since macropores can promote the movement of water flow, Re with MF in the upper part of a profile, with more macropores, also will be larger.

Conversely, for PF, the water movement does not follow Darcy’s law, but bypasses the soil matrix and moves downward in a disorderly and inhomogeneous manner through the cracks [31]. Due to the larger width of fissures, water flow velocity is faster, and the Re for PF is much larger than MF. We found that when PF occurs, the Re for PF is nearly always greater than 1. In addition, during downward infiltration with PF, the soil pipe wall is saturated and not affected by the matrix potential [32]. In this situation, gravitational potential is constantly converted into kinetic energy and the velocity of water flow in the soil is constantly accelerating under the action of gravity. As the profile deepens, the Re for PF will continue to increase.

4.2. The Effects of Flow Regimes on Pore Water Pressure

The distribution of pore water pressure during subsurface flow is intricately linked to the continuous change in soil water content in different soil layers [8]. As soil water flows, it aggregates or leaks, resulting in varying water pressure states in the soil pores [33]. This interplay between soil water content and pore water pressure in different subsurface flows is complex and fascinating. When MF dominates subsurface flow, the pore water pressure rapidly polarizes as the water content increases. In different soil layers, the uniform distribution of soil pores and soil texture results in a different hydraulic conductivity, which plays a significant role in determining the variation in pore water pressure [34]. When the hydraulic conductivity of the lower soil layer is greater than that of the corresponding upper layer, a water scarcity zone is easy to form between adjacent soil layers, resulting in cavities and negative pore water pressure in the soil body. Conversely, when the lower layer’s hydraulic conductivity is lower, rainwater will easily accumulate between adjacent soil layers, resulting in a stagnant layer and positive pore water pressure [35]. Noteworthy, with the high soil water condition, the soil body is more prone to produce stagnant layers or cavities, which makes the change in positive and negative pore water pressure especially obvious in this stage [36]. Meanwhile, during rainfall events, the hydraulic conductivity of each soil layer will also change with the variation in soil water content. This suggests that the changes in soil water content within any of the two adjacent soil layers are accompanied by a change in hydraulic conductivity. Our results show that in general, the pore water pressure can vary even with constant soil water content, and different stages of rainfall can result in different pore water pressures for the same water content.

The distribution of pore water pressure in subsurface flow is influenced by the change in soil water content within different soil layers. As soil water content increases, pore water pressure transitions to positive pressure and continues to rise. In PF, the water flow bypasses the soil matrix and moves quickly through fissures, and the soil structure has little influence on water flow velocity [37]. Positive pressure is generated on the fissure wall as soil water fills the fissures [38]. However, PF can also cause blockages and dredging of fissures, leading to changes in positive and negative pore water pressures [39]. When fissures are blocked, positive pore water pressure increases, and when fissures are drained, negative pore water pressure occurs.

4.3. Factors Controlling Water Flow Shear Force

Soil water content, pore water pressure, and subsurface flow are interconnected factors that play a crucial role in various applications. τ is a function of the hydraulic radius and the hydraulic gradient, with its magnitude depending on the combination of the hydraulic radius and the hydraulic gradient [40]. While the hydraulic radius is constant, the difference in hydraulic gradient is key to controlling τ under different flow conditions. The hydraulic gradient is influenced by gravity potential, pressure potential, matrix potential, and kinetic energy of water flow [13]. Under different conditions, one or more of these energies may dominate the hydraulic gradient [41]. In addition, since soil water flow velocities are usually small, the variations in the velocity of subsurface flow have less influence on the changes in water flow energy. Therefore, τ of subsurface flow is usually calculated using a constant flow velocity [42].

When MF occurs, gravity potential and matrix potential differences become the dominant forces affecting τ [43]. As the gradient value of the gravitational potential energy is fixed, matrix potential plays the most critical factor. In our research, τ caused by matrix potential difference is tens to hundreds of times greater than τ generated by pressure potential difference when MF occurs (Table 3). The magnitude of the matrix potential difference largely controls τ in different soil layers, and it correlates positively with soil water content. Additionally, due to the continuous downward movement of water flow and its decreasing hydraulic gradient, the direction of τ is always downward [44,45]. However, since the measurement of soil matrix potential requires a long equilibrium time, it is difficult to accurately measure and also makes the calculation of τ during MF complex. It is a better choice to calculate τ using Darcy’s law; in this process, the water flow flux can be obtained by the change in soil water content per unit time.

In PF, the wall of the fissure quickly saturates, rendering the matrix potential irrelevant, and then the pressure potential becomes the key factor in controlling τ [46]. The accumulation or leakage of soil water is a key cause of changes in pressure potential and streamflow shear [47]. In our study, the water flow rate in PF is fast, and the rapid accumulation and seepage of soil water within the fissure result in a pressure potential energy that is tens to hundreds of times higher than that of MF (Table 3). Furthermore, the hydraulic radius in PF is much larger than in MF, resulting in τ in PF being tens to hundreds of times greater.

4.4. Understanding Subsurface Flow Dynamics and Energy Changes

Our results show that the changes in hydrodynamic factors driving subsurface flow can be calculated by analyzing the changes in potential and kinetic energy during different subsurface flow motions. In MF, potential energy is the key factor controlling the state of subsurface flow. In PF, the kinetic energy state of water flow and the change in pressure potential are the key factors controlling the state of subsurface flow. This provides us with a new perspective of thinking in regard to subsurface flow. Under field conditions, we can infer and predict the specific state of subsurface flow by monitoring energy changes in the soil profile. For example, some measurable factors such as soil temperature and soil pore pressure are used to determine the form of subsurface flow. These parameters provide a new way to calculate the hydraulics driving subsurface flow in different soils (e.g., rate of water flow, water flow shear force, Reynolds number, etc.). The Law of Conservation of Energy can be used to solve hydraulic equations and provide us with new parameters for the simulation of subsurface flow and the establishment of subsurface flow models. When other parameters cannot be determined due to the state of subsurface flow, energy changes can be used as a reference. Thus, in the absence of other suitable methods, justifying the movement of subsurface flow from the perspective of energy exchanges may result in new ways of thinking about subsurface flow.

5. Conclusions

In our study, we made an improvement in the calculation method for hydraulic factors in uniform pipe flow. By applying this improved method, we were able to analyze the distribution of Reynolds number (Re), pore water pressure, and τ in two profiles under two different flow regimes: preferential flow (PF) and matrix flow (MF). In the MF regime, the water flow velocity and Re values were generally below 0.1 mm s⁻¹ and 0.1, respectively. However, under the PF regime, Re values were significantly higher, ranging from 0.8 to nearly 2. We identified that high water content plays a crucial role in the generation of pore water pressure across different flow regimes. Specifically, greater water content resulted in higher pore water pressure. These findings attempt to analyze the characteristics of different subsurface flow from the perspective of energy changes and will provide some guidelines for exploring soil erosion. Moreover, this approach also holds promise for gaining a deeper understanding of the mechanisms and behavior of groundwater flow.