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Hydraulic modeling of water flow in the thick vadose zone under precipitation

Hydraulic modeling of water flow in the thick vadose zone under precipitation Data from field monitoring and artificial rainfall experiments suggest that the thick vadose zone can be divided into two sub-zones based on soil water variation, namely the active and steady zones. The soil water content of the top active zone (2–5 m depth) is sensitive to precipitation and evaporation and dominated by transient water flow. Soil water content of the underlying steady zone remains constant over time and there is a steady flow under the force of gravity. However, since the transition from transient flow to steady flow is difficult to observe in nature, the physical mechanism of this transition remains poorly understood. This study establishes a hydraulic model to visually demon- strate water flow in the entire vadose zone under multiple infiltration events. The model comprises of a series verti- cally aligned water tanks, each with a small outlet at the bottom, and each representing a soil unit. The water level in a tank represents the water content and the related permeability of the soil unit. The results of an experiment con- ducted with the model clearly show that transient flow in the upper active zone will transfer to steady flow. A zoomed out data with an annual rainfall record at a site in the central Chinese Loess Plateau is applied in the model to simulate the water content and the flow state of the vertical profile, and the results are in accordance with in-situ monitoring data. The outcomes of this study suggest that although water content in the steady zone remains unchanged, there is a constant steady flow seeping downward through the zone, acting as a typical source of groundwater recharge in the loess region. Keywords: Vadose zone, Hydraulic model, Transient flow, Steady flow, Unsaturated loess Introduction of water seepage along the joints and cracks in the loess The lower loess layer is naturally higher than a nearby profile. The argument for piston flow was due to many riverbed due to its origin through wind mobilization of field observations and artificial rainfall experiments had sand and the entire Loess Plateau is composed of a series shown that the depth of the wetting front in loess was of independent hydrogeological units bounded by river limited, generally around 2 m and no more than 4 m (Li valleys (Xu et  al. 2020; Wang et  al. 2017). Under natural et  al. 2016; Tu et  al. 2009; Xu et  al. 2011). Studies based condition, rainfall is typically the only source of ground- on long-term monitoring in the Chinese Plateau by Hou water recharge in the Chinese Loess Plateau. However, et  al. (2017, 2019, 2020) and Zhang et  al. (2014) found the mode for rainwater infiltration through the thick that the depth at which soil moisture was most sensitive vadose zone between preferential flow and piston flow to rainfall and evaporation-transpiration was only 2.0 m, remains contentious (Li et  al. 2017; Wang et  al. 2018). whereas water content below 2.0  m remained constant This disagreement stems from the intuitive observation over time. Figure  1 shows the results of up-to-date data. The monitored data for the Bet Dagan and Rio Galeria basins similarly demonstrated that the response of soil water content to precipitation events began to weaken *Correspondence: dcdgx08@chd.edu.cn Department of Geological Engineering, Chang’an University, at a depth exceeding 0.3  m, regardless of whether the Xi’an 710054, Shaanxi, China response was measured during the dry season or rainy Full list of author information is available at the end of the article © The Author(s) 2022. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. Wang et al. Geoenvironmental Disasters (2022) 9:7 Page 2 of 9 Fig. 1 Conceptual diagram representing two regions of the unsaturated zone season, and water content remained basically unchanged rendered to flow along the joints, pores, and holes, at a depth exceeding 1  m (Russo and Fiori 2008). The referred to as “preferential flow”. However, to date, field upper wetting zone is commonly referred to as the “active observations and measurements have not supported this zone”, whereas the underlying zone with an unchanged assumption. Wu et  al. (2011) conducted in-situ infiltra - water content is referred to as the “steady zone”. Past sci- tion tests across fractures in loess ground and did not entific understanding had been that piston flow occurred identify a hydraulic connection between flow in wide fis - in the active zone, whereas there was no flow in the sures and groundwater. A study by Xu and Zhao (1993) steady zone (Gazis and Feng 2004). on rainfall seepage in an excavated profile with fissures On the other hand, deep vertical joints, larger root suggested that fissures and macro-pores did not play a pores and animal holes were widely spread in loess slopes key role in water diversion. The vague understanding (Derbyshire 2011; Zhang and Liu 2010; Zhuang et  al. of preferential flow can be attributed to confusion over 2017), which motivated for the theory that rainwater the condition of saturation and unsaturation. It is easy W ang et al. Geoenvironmental Disasters (2022) 9:7 Page 3 of 9 to demonstrate that preferential flow occurs in saturated water flow from transient flow behavior in the active soil. But in unsaturated soils, the macro spaces, such zone to the steady flow behavior in the steady zone under joints, large pores and holes, have zero potential while multiple infiltration events. Furthermore, a differential the micro pores have potential. The smaller pore typically solution for Richards equation is provided to simulate the has a lower potential. Consequently, water will fill the flow in the vadose zone under a magnified rainfall series. minor pores first, then expand the bigger ones in order. The result demonstrates the behaviors of water flow in During the heavy rain period, a saturated zone can tem- the vadose zone as mentioned above. porarily form in shallow ground, with preferential flows forming along some macro paths. However, the prefer- The hydraulic model ential flow in macro pores diminishes immediately when Figure  2 is a schematic conceptual representation of the saturated flow changes to unsaturated flow. hydraulic model. The model is composed of a set of open Loess is a typical unsaturated soil which is sensitive to water tanks with a small outlet at the bottom of each moisture (Li et al. 2015). Preferential flow occurs tempo - tank. In the model, a tank represents a soil unit, the tank rarily on the ground surface during rain events. Uniform itself represents the soil skeleton, and the water in the unsaturated seepage, namely piston flow, is the dominant tank represents water in a soil unit. The water-saturated mode of flow in the thick loess vadose zone. tank represents a saturated soil unit. A set of vertically However, it was difficult to observe piston flow in both aligned tanks represents a soil column. active and steady zones (Cheng et  al. 2018; Basara and Water supplied to the top tank will successively flow to Crawford 2010), and this difficulty may have contributed the tanks below. All tanks have the same cross-section to misunderstanding regarding the existence of flow in area and height. The water level relative to the height of the steady zone. The Richards equation described water the tank corresponds to the water content of the soil unit, flow in unsaturated porous media and can help explain and the velocity of the change in water level of each tank this phenomenon (Richards 1931). The use of the Rich - corresponds to the permeability of the soil unit. The suc - ards equation readily shows that the active zone has tran- cessive flow of water from the top tank to those below sient flow, whereas the steady zone has steady flow. In reflects the properties of the one-dimensional infiltration a natural profile, intermittent rainfall infiltration could process in the vadose zone. The water levels of the tanks form transient flow in the shallow active zone, which should be controlled without overflow to simulate unsat - transitioned to steady flow in the underlying steady zone urated flow. (Hou et al. 2017, 2020). Following steady flow, there was a constant flow of water downward to recharge groundwa - Experiment with the hydraulic model ter (Huang et al. 2017, 2020). As shown in Fig.  3, the experiment was conducted using The present study uses a hydraulic model by Zhang nine glass tanks vertically fixed onto a wooden board at et  al. (2019) to visually demonstrate the transition of 10 mm intervals. Each tank had a diameter and height of Fig. 2 Representation of a soil column using the hydraulic model in the current study Wang et al. Geoenvironmental Disasters (2022) 9:7 Page 4 of 9 Fig. 3 Water levels in the tanks of the hydraulic model over time under an intermittent supply 36 mm and 180 mm, respectively. A water level scale was The water level changes of different amplitudes in marked onto the wall of each tank. A water supply bottle some tanks imply that both the water content and per- connected to a faucet was installed above the tanks. As meability of the soil zone represented by each tank are an initial condition, all tanks had a water level of 30 mm. variable, which reflect the behavior of the active zone The boundary condition of the experiment was an inter - with transient flow. Meanwhile, the ones with the con- mittent water supply to the top tank C0 for simulating stant water level reflect the behavior of the steady zone rainfall events, as shown in the first row in Fig.  3. As the with a steady flow. The overall profile of the tanks water supply was initiated, the water level of each tank shows a change in the water level from an active zone was recorded every 30  s. In total, 3,000  s of data were to a steady zone, which is associated with the transi- recorded. tion from transient flow to steady flow. The graphs on the left-hand side of Fig.  3 show water Since rainfall events are permanent features in nature, levels of the tanks over time. The water level in C1 fluctu - the final state in the experiment corresponds to the nat - ated widely as a direct response to water supply C0. The ural state. The variations in water content in the experi - variation in the water level of C2 was more gentle than ment are consistent with in-situ observations in the soil that in C1. An examination of the variation in water level profile. The experiment conducted in the present study of the tanks from C1 to C8 showed that variation in water directly demonstrates that there is water flow in both level progressively decreased from the upper to lower the transient and steady zones. However, this flow is tanks. The water level in the last 2 tanks (C7 and C8) sta - difficult to observe or monitor in a natural soil profile. bilized at 10 mm after 2000 s. Since the water level of C8 was constant, it could be inferred that any tanks below C8 would also have constant water levels. W ang et al. Geoenvironmental Disasters (2022) 9:7 Page 5 of 9 Numerical solution for the hydraulic model t = kΔt (k = 1, 2, …, m), the water level h can be calcu- 1 1 1 2 2 2 The present study conducts a theoretical numerical solu - lated in the order of h , h , …, h ; h , h , …, h ; … 1 2 n 1 2 n m m m tion for the model to demonstrate the reasons for the h , h , …, h . Figure  4 demonstrates the calculation 1 2 n ability of the hydraulic model to represent the behavior of steps. unsaturated flow. The following differential expression is The outlet radii of the tanks are set as two cases to deduced from the model (Zhang et al. 2019): simulate the flow behaviors in uniform and layered soils, respectively, as shown in Table 1. In case 1, all tanks have �q � �H the same outlet radius r , which represents a uniform soil = k(H) − 1 (1) �t �z �z column with the same hydraulic conductivity function among the soil units. In case 2, each tank has a different In Eq.  (1), q is the ratio of water quantity in the ith outlet radius r , which represents a layered soil column tank to the total volume, z is the height of the tank, H is with different hydraulic conductivity functions among the water head in the tank, and k(H) is equivalent to the the soil units. The initial condition of the water content is hydraulic conductivity function, which is expressed as also set as two states. Under state 1, the water levels of all follows: the tanks are set to zero, whereas under state 2, the water levels are set according to the values listed in Table  1. r �z k(H) = 2gh (2) 2 The daily rainfall data recorded at the monitoring site in R h + �z the Central Chinese Loess Plateau, from March 2018 to In Eq.  (2), h is the water level of the ith tank, r is the March 2019, are used as reference data for the boundary i i outlet radius of the ith tank, R is the radius of the tank, condition. The present study applies daily precipitation and each tank has the same radius. data, but the time interval of one day is zoomed to 1 s in The Richards equation (Richards 1931) for vertical one- the simulation. The long-term effect of water flow in the dimensional flow in unsaturated soils is: vadose zone is achieved by repeatedly applying one year of daily precipitation over 5 years. Since the time interval ∂θ ∂ dH of the experiment is set to 1  s, the total duration of the = k(H) − 1 (3) ∂t ∂z dz experiment is 5 cycles of 365 s. In Eq.  (3), z is the vertical coordinate (downward is positive), θ is the volumetric water content of soil, H is Analysis of the results the pressure head, and k(H) is the hydraulic conductivity The simulated results are summarized in Fig.  5. The function. dashed lines indicate the case under which the ini- Comparing Eqs. (1) and (3) shows their similarity in tial water levels of all tanks are zero, whereas the solid form and physical context. The left-hand sides of both lines indicate the case under which the tanks have dif- equations represent the rate of variation in water content ferent initial water levels. The green lines show the to duration, whereas the right-hand sides of both equa- case under which all tanks have the same outlet radius tions, ΔH/Δz or ∂H/∂z, correspond to the hydraulic gra- dient. k(H) is the function of the water level h as shown in Eq.  (2) and is correlated to the water content h /z or pressure head (H = − h ) of the tank. Therefore, k(H) is equivalent to the hydraulic conductivity function, which is also a function of the water content and pressure head. Substituting Eq.  (2) and q = h /z, H = − h into Eq.  (1), i i i the first-order differential equation of Eq.  (1) for time t and coordinate z can be written as Eq. (4): 2g t t−1 2 t−1 2 t−1 h = h + r h − r h t (4) i i i−1 i−1 i i Equation (4) can be used to calculate the water level of each tank from the upper-most tank downward. The calculation begins from t = 1. In this case, h (i = 1, 2, …, n) for the right-hand side of Eq. (4) are given as ini- tial conditions, R and r are fixed, and g is 9800  mm/s . Fig. 4 Flow chart demonstrating the process used to derive the numerical solution for the model Given the boundary condition h at the time intervals 0 Wang et al. Geoenvironmental Disasters (2022) 9:7 Page 6 of 9 Table 1 The properties of the model Tank C C C C C C C C C 0 1 2 3 4 5 6 7 8 Initial condition h (mm) State 1 Rainfall 0 0 0 0 0 0 0 0 State 2 Rainfall 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Outlet radius r (mm) Case 1 1.80 1.70 1.70 1.70 1.70 1.70 1.70 1.70 1.70 Case 2 1.80 1.80 1.80 1.75 1.75 1.70 1.70 1.65 1.60 Boundary condition h (mm) Daily rainfall; t = kΔt, 0 k Δt = 1 s, (k = 1, 2, …, m) Fig. 5 Change in water level in the system under two cases of initial tank water level state of each tank is independent of the initial condition. (r = 1.70 mm), whereas the red lines show the case under Regardless of identical (zero) or different initial water lev - which each tank has a different outlet radius. els, the final water levels of the upper tanks will eventu - Comparison of the results for an identical initial con- ally reach a state of periodic fluctuation, whereas those dition (state 1, zero water level, dashed line) with those of the lower tanks tend to stabilize. The dashed line and generated under different initial conditions (state 2, dif - corresponding solid line eventually converge. However, ferent water levels, solid line) shows that the final flow W ang et al. Geoenvironmental Disasters (2022) 9:7 Page 7 of 9 hysteresis can be observed on the curves under which the fluctuation in the water level of C1, with a short time lower tanks require more time to reach the final state. lag relative to C0. The water levels in C2 and C3 show Table  1 shows the conditions under which all tanks continuous and smooth fluctuation with a time lag rela - have the same outlet radius (case 1, green lines) and tive to the tanks above, whereas C4 to C6 show similar under which each tank has a different outlet radius (case trends in water level variation. The amplitudes of water 2, red lines). And the simulated results in Fig.  5 show level in tanks C1 to C6 rapidly decrease from the top that the tanks with larger outlet radii (case 2, C1 to C4, tank to the bottom tank. The water levels of C7 and C8 red lines) have lower water levels, whereas those with eventually reach a steady state. As mentioned above, smaller outlet radii (case 1, C1 to C4, green lines) have the final water levels of the tanks are independent of higher levels. The same regularity in water level appears initial conditions. Based on the final state, the model in the C7 and C8. Both C7 and C8 have higher water lev- column can be divided into two zones, namely the els, whereas they have a smaller and larger outlet radius active zone and the steady zone. under case 2 (red lines) and case 1 (green lines), respec- The water levels in the active zone are influenced by tively. Tanks with an identical outlet radius (in both cases, the boundary condition, whereas those in the steady C5 and C6) show similar water levels. The green and red zone are constant with a steady water flow. The flow line for C5 and C6 are close and finally converged. The velocity of each tank in the steady zone is equal to the simulated results reflect the flow behavior in unsaturated average boundary recharge. As demonstrated by the soils. A soil layer with good hydraulic conductivity and monitoring results shown in Fig.  1, the flow behaviors low water retaining capacity is represented by the tanks simulated by the model are similar to those observed in with a larger outlet radius, whereas the soil layer with low natural soil layers. The hydraulic model provides strong hydraulic conductivity and high water retaining capacity evidence for the existence of steady flow, which acts as is represented by tanks with a small outlet radius. a typical source of groundwater recharge in the loess Figure 6 shows the soil water retention curves (SWRC) region. of the silt and sand, which clearly illustrate the differ - ence in the ability of each of the two soils to retain water. Under the same water content (e.g., 17.5%), the soil suc- Discussion tion of silt (40 kPa) would far exceed that of sand (2 kPa), Figure 7(a) shows the in-situ monitoring results of water representing a much lower permeability of the silt. content of the soil profile in the central Chinese Loess The variations in the water levels of the tanks tend to plateau (Hou et al. 2020). Figure 7(b) shows the simulated decrease from the upper to lower tanks, regardless of results using the monitored precipitation data. The simu - their initial condition. The water level in C0 represents lated water content of the soil profile is consistent with the influence of rainfall events. There is a continuous the observed water content. The mechanism of the simi - larity between the results of the physical model and the natural situation is discussed below. To solve Eq.  (3), two key soil constitutive functions should be provided in advance, namely the SWRC and hydraulic conductivity function (HCF) (Fredlund 2006; Lu and Likos 2006). These functions can use the empiri - cal equations, such as the commonly used SWRC and HCF equations provided by van Genuchten (1980) and Fredlund et  al. (1994), and the parameters should be fit - ted using laboratory test data. The equivalent SWRC and HCF for the hydraulic model can be deduced based on the principle of the model. Supposing that the equivalent SWRC is expressed as equivalent matric suction against the degree of saturation and that HCF is expressed as the equivalent hydraulic conductivity against the degree of saturation, then the water level height of the ith tank is h , and its degree (S ) of saturation is: S = (5) Fig. 6 Representative soil water retention curves (SWRC) of sand and silt from China The equivalent matric suction (ψ ) is: i Wang et al. Geoenvironmental Disasters (2022) 9:7 Page 8 of 9 Fig. 7 Comparison of soil water content and calculated value using rainfall data ψ = �z − h i i (6) Therefore, the equivalent SWRC of the tank is ψ = (1 − S )�z i i (7) From Eqs. (2) and (5), the equivalent HCF of the tank can also be easily obtained: k(S ) = 2gS �z i i (8) S + 1 Suppose Δz is unity, then Eqs. (7) and (8) can be simpli- Fig. 8 Equivalent matric suction and hydraulic conductivity function fied to: against degree of saturation of the model ψ = 1 − S i i (9) it can visually show the two-dimensional profile of water r 1 k(s ) = 2gS i i (10) flow in the vadose zone and can help to clarify the argu - S + 1 ment between piston flow and preferential flow. Figure  8 shows the curves of Eqs. (9) and (10). Equa- tion  (9) is a linear function in which the equivalent mat- Conclusions ric suction linearly increases with decreasing saturation, The paper uses a hydraulic model to simulate the flow similar to the trend in SWRC variation for real soil. How- behavior in the vadose zone. The mathematical expres - ever, it is difficult to fit a linear SWRC function for real sion of the hydraulic model is deduced and is similar to soil. In addition, Eq. (10) shows that the trend of HCF is the Richards governing equation. The experiment and similar to that of real soil. However, the HCF cannot be mathematical solutions demonstrate that under a con- applied in real soils. The advantage of the model is that dition of intermittent precipitation events, the model W ang et al. Geoenvironmental Disasters (2022) 9:7 Page 9 of 9 Hou X, Vanapalli SK, Li T (2017) Water infiltration characteristics in loess associ- column can be divided into two zones, namely active ated with irrigation activities and its influence on the slope stability in zone and steady zone. The water content in the active heifangtai loess highland, china. Eng Geol 234:27–37 zone changes the variation of the boundary conditions Hou X, Li T, Vanapalli SK, Xi Y (2019) Water percolation in a thick unsaturated loess layer considering the ground-atmosphere interaction. Hydrol (e.g., precipitation, evaporation in natural conditions), Process 33(5):794–802 whereas it keeps constant in the steady zone. The results Hou X, Vanapalli SK, Li T (2020) Water flow in unsaturated soils subjected to of the model are consistent with the soil water content multiple infiltration events. Can Geotech J 57(3):366–376 Huang T, Pang Z, Liu J, Ma J, Gates John (2017) Groundwater recharge mecha- profile observed in the field. The model can overcome the nism in an integrated tableland of the Loess Plateau, northern China: difficulty of observing unsaturated flow in nature by visu - insights from environmental tracers. Hydrogeol J 25(7):2049–2065 ally demonstrating the two-dimensional profile of water Huang T, Ma B, Pang Z, Li Z, Li Z, Long Y (2020) How does precipitation recharge groundwater in loess aquifers? Evidence from multiple environ- flow in both the active and steady zones, and particularly mental tracers. J Hydrol 583:124532 the transition from transient water flow to steady flow. Li P, Zhang X, Shi H (2015) Investigation for the initiation of a loess landslide based on the unsaturated permeability and strength theory. Geoenviron Acknowledgements Disast 2(1):24 The author would like to thank the National Natural Science Foundation of Li P, Li T, Vanapalli SK (2016) Influence of environmental factors on the wetting China for its funding and trust. front depth: a case study in the Loess Plateau. Eng Geol 214:1–10 Li Z, Chen X, Liu W, Si B (2017) Determination of groundwater recharge Authors’ contributions mechanism in the deep loessial unsaturated zone by environmental trac- Yu Wang was involved in the design and operation of experiment, data collec- ers. Sci Total Environ 586:827–835 tion and analysis, simulations and writing the manuscript. And all other author Lu N, Likos WJ (2006) Suction stress characteristic curve for unsaturated soil. J contributed in operation of experiment, data analysis, simulations and writing Geotech Geoenviron Eng 132(2):131–142 the paper. All authors read and approved the final manuscript. Richards AL (1931) Capillary conduction of liquids through porous mediums. Physics 1(5):318–333 Funding Russo D, Fiori A (2008). Equivalent vadose zone steady state flow: an assess- The study was supported by the National Natural Science Foundation of China ment of its capability to predict transport in a realistic combined vadose (Grant Nos. 41772278, 41790442, 41807242). The above financial support is zone–groundwater flow system. Water Resour Res 44(9) gratefully acknowledged. Tu XB, Kwong AKL, Dai FC, Tham LG, Min H (2009) Field monitoring of rainfall infiltration in a loess slope and analysis of failure mechanism of rainfall- Availability of data and materials induced landslides. Eng Geol 105(1–2):134–150 Some or all data, models, or code that support the findings of this study are Van Genuchten MT (1980) A closed-form equation for predicting the hydraulic available from the corresponding author upon reasonable request. conductivity of unsaturated soils. 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J Changchun Univ Earth Sci 23(03):326– Observation and Research Station for the Chinese Loess Plateau, Ministry 329 (in Chinese) of Education, Lanzhou 745399, Gansu, China. Key Laboratory of Shale Gas Xu L, Dai FC, Tham LG, Tu XB, Min H, Zhou YF, Xu K (2011) Field testing of irriga- and Geoengineering, Institute of Geology and Geophysics, Innovation Acad- tion effects on the stability of a cliff edge in loess, North-West China. Eng emy for Earth Science, CAS, 10029 Beijing, China. Geol 120(1–4):10–17 Xu P, Zhan Q, Qian H, Yang F, Zheng L (2020) Investigating the mechanism Received: 12 May 2021 Accepted: 14 February 2022 of ph effect on saturated permeability of remolded loess. Eng Geol 284(6):105978 Zhang M, Liu J (2010) Controlling factors of loess landslides in western China. Environ Earth Sci 59(8):1671–1680 Zhang C, Li T, Li P (2014) Rainfall infiltration in Chinese loess by in situ observa- References tion. 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J Geotech Geoenviron Eng 132(3):286–321 Publisher’s Note Fredlund DG, Xing A, Huang S (1994) Predicting the permeability function for Springer Nature remains neutral with regard to jurisdictional claims in pub- unsaturated soils using the soil-water characteristic curve. Can Geotech lished maps and institutional affiliations. J 31(4):533–546 Gazis C, Feng X (2004) A stable isotope study of soil water: evidence for mixing and preferential flow paths. Geoderma 119(1–2):97–111 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Geoenvironmental Disasters Springer Journals

Hydraulic modeling of water flow in the thick vadose zone under precipitation

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Abstract

Data from field monitoring and artificial rainfall experiments suggest that the thick vadose zone can be divided into two sub-zones based on soil water variation, namely the active and steady zones. The soil water content of the top active zone (2–5 m depth) is sensitive to precipitation and evaporation and dominated by transient water flow. Soil water content of the underlying steady zone remains constant over time and there is a steady flow under the force of gravity. However, since the transition from transient flow to steady flow is difficult to observe in nature, the physical mechanism of this transition remains poorly understood. This study establishes a hydraulic model to visually demon- strate water flow in the entire vadose zone under multiple infiltration events. The model comprises of a series verti- cally aligned water tanks, each with a small outlet at the bottom, and each representing a soil unit. The water level in a tank represents the water content and the related permeability of the soil unit. The results of an experiment con- ducted with the model clearly show that transient flow in the upper active zone will transfer to steady flow. A zoomed out data with an annual rainfall record at a site in the central Chinese Loess Plateau is applied in the model to simulate the water content and the flow state of the vertical profile, and the results are in accordance with in-situ monitoring data. The outcomes of this study suggest that although water content in the steady zone remains unchanged, there is a constant steady flow seeping downward through the zone, acting as a typical source of groundwater recharge in the loess region. Keywords: Vadose zone, Hydraulic model, Transient flow, Steady flow, Unsaturated loess Introduction of water seepage along the joints and cracks in the loess The lower loess layer is naturally higher than a nearby profile. The argument for piston flow was due to many riverbed due to its origin through wind mobilization of field observations and artificial rainfall experiments had sand and the entire Loess Plateau is composed of a series shown that the depth of the wetting front in loess was of independent hydrogeological units bounded by river limited, generally around 2 m and no more than 4 m (Li valleys (Xu et  al. 2020; Wang et  al. 2017). Under natural et  al. 2016; Tu et  al. 2009; Xu et  al. 2011). Studies based condition, rainfall is typically the only source of ground- on long-term monitoring in the Chinese Plateau by Hou water recharge in the Chinese Loess Plateau. However, et  al. (2017, 2019, 2020) and Zhang et  al. (2014) found the mode for rainwater infiltration through the thick that the depth at which soil moisture was most sensitive vadose zone between preferential flow and piston flow to rainfall and evaporation-transpiration was only 2.0 m, remains contentious (Li et  al. 2017; Wang et  al. 2018). whereas water content below 2.0  m remained constant This disagreement stems from the intuitive observation over time. Figure  1 shows the results of up-to-date data. The monitored data for the Bet Dagan and Rio Galeria basins similarly demonstrated that the response of soil water content to precipitation events began to weaken *Correspondence: dcdgx08@chd.edu.cn Department of Geological Engineering, Chang’an University, at a depth exceeding 0.3  m, regardless of whether the Xi’an 710054, Shaanxi, China response was measured during the dry season or rainy Full list of author information is available at the end of the article © The Author(s) 2022. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. Wang et al. Geoenvironmental Disasters (2022) 9:7 Page 2 of 9 Fig. 1 Conceptual diagram representing two regions of the unsaturated zone season, and water content remained basically unchanged rendered to flow along the joints, pores, and holes, at a depth exceeding 1  m (Russo and Fiori 2008). The referred to as “preferential flow”. However, to date, field upper wetting zone is commonly referred to as the “active observations and measurements have not supported this zone”, whereas the underlying zone with an unchanged assumption. Wu et  al. (2011) conducted in-situ infiltra - water content is referred to as the “steady zone”. Past sci- tion tests across fractures in loess ground and did not entific understanding had been that piston flow occurred identify a hydraulic connection between flow in wide fis - in the active zone, whereas there was no flow in the sures and groundwater. A study by Xu and Zhao (1993) steady zone (Gazis and Feng 2004). on rainfall seepage in an excavated profile with fissures On the other hand, deep vertical joints, larger root suggested that fissures and macro-pores did not play a pores and animal holes were widely spread in loess slopes key role in water diversion. The vague understanding (Derbyshire 2011; Zhang and Liu 2010; Zhuang et  al. of preferential flow can be attributed to confusion over 2017), which motivated for the theory that rainwater the condition of saturation and unsaturation. It is easy W ang et al. Geoenvironmental Disasters (2022) 9:7 Page 3 of 9 to demonstrate that preferential flow occurs in saturated water flow from transient flow behavior in the active soil. But in unsaturated soils, the macro spaces, such zone to the steady flow behavior in the steady zone under joints, large pores and holes, have zero potential while multiple infiltration events. Furthermore, a differential the micro pores have potential. The smaller pore typically solution for Richards equation is provided to simulate the has a lower potential. Consequently, water will fill the flow in the vadose zone under a magnified rainfall series. minor pores first, then expand the bigger ones in order. The result demonstrates the behaviors of water flow in During the heavy rain period, a saturated zone can tem- the vadose zone as mentioned above. porarily form in shallow ground, with preferential flows forming along some macro paths. However, the prefer- The hydraulic model ential flow in macro pores diminishes immediately when Figure  2 is a schematic conceptual representation of the saturated flow changes to unsaturated flow. hydraulic model. The model is composed of a set of open Loess is a typical unsaturated soil which is sensitive to water tanks with a small outlet at the bottom of each moisture (Li et al. 2015). Preferential flow occurs tempo - tank. In the model, a tank represents a soil unit, the tank rarily on the ground surface during rain events. Uniform itself represents the soil skeleton, and the water in the unsaturated seepage, namely piston flow, is the dominant tank represents water in a soil unit. The water-saturated mode of flow in the thick loess vadose zone. tank represents a saturated soil unit. A set of vertically However, it was difficult to observe piston flow in both aligned tanks represents a soil column. active and steady zones (Cheng et  al. 2018; Basara and Water supplied to the top tank will successively flow to Crawford 2010), and this difficulty may have contributed the tanks below. All tanks have the same cross-section to misunderstanding regarding the existence of flow in area and height. The water level relative to the height of the steady zone. The Richards equation described water the tank corresponds to the water content of the soil unit, flow in unsaturated porous media and can help explain and the velocity of the change in water level of each tank this phenomenon (Richards 1931). The use of the Rich - corresponds to the permeability of the soil unit. The suc - ards equation readily shows that the active zone has tran- cessive flow of water from the top tank to those below sient flow, whereas the steady zone has steady flow. In reflects the properties of the one-dimensional infiltration a natural profile, intermittent rainfall infiltration could process in the vadose zone. The water levels of the tanks form transient flow in the shallow active zone, which should be controlled without overflow to simulate unsat - transitioned to steady flow in the underlying steady zone urated flow. (Hou et al. 2017, 2020). Following steady flow, there was a constant flow of water downward to recharge groundwa - Experiment with the hydraulic model ter (Huang et al. 2017, 2020). As shown in Fig.  3, the experiment was conducted using The present study uses a hydraulic model by Zhang nine glass tanks vertically fixed onto a wooden board at et  al. (2019) to visually demonstrate the transition of 10 mm intervals. Each tank had a diameter and height of Fig. 2 Representation of a soil column using the hydraulic model in the current study Wang et al. Geoenvironmental Disasters (2022) 9:7 Page 4 of 9 Fig. 3 Water levels in the tanks of the hydraulic model over time under an intermittent supply 36 mm and 180 mm, respectively. A water level scale was The water level changes of different amplitudes in marked onto the wall of each tank. A water supply bottle some tanks imply that both the water content and per- connected to a faucet was installed above the tanks. As meability of the soil zone represented by each tank are an initial condition, all tanks had a water level of 30 mm. variable, which reflect the behavior of the active zone The boundary condition of the experiment was an inter - with transient flow. Meanwhile, the ones with the con- mittent water supply to the top tank C0 for simulating stant water level reflect the behavior of the steady zone rainfall events, as shown in the first row in Fig.  3. As the with a steady flow. The overall profile of the tanks water supply was initiated, the water level of each tank shows a change in the water level from an active zone was recorded every 30  s. In total, 3,000  s of data were to a steady zone, which is associated with the transi- recorded. tion from transient flow to steady flow. The graphs on the left-hand side of Fig.  3 show water Since rainfall events are permanent features in nature, levels of the tanks over time. The water level in C1 fluctu - the final state in the experiment corresponds to the nat - ated widely as a direct response to water supply C0. The ural state. The variations in water content in the experi - variation in the water level of C2 was more gentle than ment are consistent with in-situ observations in the soil that in C1. An examination of the variation in water level profile. The experiment conducted in the present study of the tanks from C1 to C8 showed that variation in water directly demonstrates that there is water flow in both level progressively decreased from the upper to lower the transient and steady zones. However, this flow is tanks. The water level in the last 2 tanks (C7 and C8) sta - difficult to observe or monitor in a natural soil profile. bilized at 10 mm after 2000 s. Since the water level of C8 was constant, it could be inferred that any tanks below C8 would also have constant water levels. W ang et al. Geoenvironmental Disasters (2022) 9:7 Page 5 of 9 Numerical solution for the hydraulic model t = kΔt (k = 1, 2, …, m), the water level h can be calcu- 1 1 1 2 2 2 The present study conducts a theoretical numerical solu - lated in the order of h , h , …, h ; h , h , …, h ; … 1 2 n 1 2 n m m m tion for the model to demonstrate the reasons for the h , h , …, h . Figure  4 demonstrates the calculation 1 2 n ability of the hydraulic model to represent the behavior of steps. unsaturated flow. The following differential expression is The outlet radii of the tanks are set as two cases to deduced from the model (Zhang et al. 2019): simulate the flow behaviors in uniform and layered soils, respectively, as shown in Table 1. In case 1, all tanks have �q � �H the same outlet radius r , which represents a uniform soil = k(H) − 1 (1) �t �z �z column with the same hydraulic conductivity function among the soil units. In case 2, each tank has a different In Eq.  (1), q is the ratio of water quantity in the ith outlet radius r , which represents a layered soil column tank to the total volume, z is the height of the tank, H is with different hydraulic conductivity functions among the water head in the tank, and k(H) is equivalent to the the soil units. The initial condition of the water content is hydraulic conductivity function, which is expressed as also set as two states. Under state 1, the water levels of all follows: the tanks are set to zero, whereas under state 2, the water levels are set according to the values listed in Table  1. r �z k(H) = 2gh (2) 2 The daily rainfall data recorded at the monitoring site in R h + �z the Central Chinese Loess Plateau, from March 2018 to In Eq.  (2), h is the water level of the ith tank, r is the March 2019, are used as reference data for the boundary i i outlet radius of the ith tank, R is the radius of the tank, condition. The present study applies daily precipitation and each tank has the same radius. data, but the time interval of one day is zoomed to 1 s in The Richards equation (Richards 1931) for vertical one- the simulation. The long-term effect of water flow in the dimensional flow in unsaturated soils is: vadose zone is achieved by repeatedly applying one year of daily precipitation over 5 years. Since the time interval ∂θ ∂ dH of the experiment is set to 1  s, the total duration of the = k(H) − 1 (3) ∂t ∂z dz experiment is 5 cycles of 365 s. In Eq.  (3), z is the vertical coordinate (downward is positive), θ is the volumetric water content of soil, H is Analysis of the results the pressure head, and k(H) is the hydraulic conductivity The simulated results are summarized in Fig.  5. The function. dashed lines indicate the case under which the ini- Comparing Eqs. (1) and (3) shows their similarity in tial water levels of all tanks are zero, whereas the solid form and physical context. The left-hand sides of both lines indicate the case under which the tanks have dif- equations represent the rate of variation in water content ferent initial water levels. The green lines show the to duration, whereas the right-hand sides of both equa- case under which all tanks have the same outlet radius tions, ΔH/Δz or ∂H/∂z, correspond to the hydraulic gra- dient. k(H) is the function of the water level h as shown in Eq.  (2) and is correlated to the water content h /z or pressure head (H = − h ) of the tank. Therefore, k(H) is equivalent to the hydraulic conductivity function, which is also a function of the water content and pressure head. Substituting Eq.  (2) and q = h /z, H = − h into Eq.  (1), i i i the first-order differential equation of Eq.  (1) for time t and coordinate z can be written as Eq. (4): 2g t t−1 2 t−1 2 t−1 h = h + r h − r h t (4) i i i−1 i−1 i i Equation (4) can be used to calculate the water level of each tank from the upper-most tank downward. The calculation begins from t = 1. In this case, h (i = 1, 2, …, n) for the right-hand side of Eq. (4) are given as ini- tial conditions, R and r are fixed, and g is 9800  mm/s . Fig. 4 Flow chart demonstrating the process used to derive the numerical solution for the model Given the boundary condition h at the time intervals 0 Wang et al. Geoenvironmental Disasters (2022) 9:7 Page 6 of 9 Table 1 The properties of the model Tank C C C C C C C C C 0 1 2 3 4 5 6 7 8 Initial condition h (mm) State 1 Rainfall 0 0 0 0 0 0 0 0 State 2 Rainfall 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Outlet radius r (mm) Case 1 1.80 1.70 1.70 1.70 1.70 1.70 1.70 1.70 1.70 Case 2 1.80 1.80 1.80 1.75 1.75 1.70 1.70 1.65 1.60 Boundary condition h (mm) Daily rainfall; t = kΔt, 0 k Δt = 1 s, (k = 1, 2, …, m) Fig. 5 Change in water level in the system under two cases of initial tank water level state of each tank is independent of the initial condition. (r = 1.70 mm), whereas the red lines show the case under Regardless of identical (zero) or different initial water lev - which each tank has a different outlet radius. els, the final water levels of the upper tanks will eventu - Comparison of the results for an identical initial con- ally reach a state of periodic fluctuation, whereas those dition (state 1, zero water level, dashed line) with those of the lower tanks tend to stabilize. The dashed line and generated under different initial conditions (state 2, dif - corresponding solid line eventually converge. However, ferent water levels, solid line) shows that the final flow W ang et al. Geoenvironmental Disasters (2022) 9:7 Page 7 of 9 hysteresis can be observed on the curves under which the fluctuation in the water level of C1, with a short time lower tanks require more time to reach the final state. lag relative to C0. The water levels in C2 and C3 show Table  1 shows the conditions under which all tanks continuous and smooth fluctuation with a time lag rela - have the same outlet radius (case 1, green lines) and tive to the tanks above, whereas C4 to C6 show similar under which each tank has a different outlet radius (case trends in water level variation. The amplitudes of water 2, red lines). And the simulated results in Fig.  5 show level in tanks C1 to C6 rapidly decrease from the top that the tanks with larger outlet radii (case 2, C1 to C4, tank to the bottom tank. The water levels of C7 and C8 red lines) have lower water levels, whereas those with eventually reach a steady state. As mentioned above, smaller outlet radii (case 1, C1 to C4, green lines) have the final water levels of the tanks are independent of higher levels. The same regularity in water level appears initial conditions. Based on the final state, the model in the C7 and C8. Both C7 and C8 have higher water lev- column can be divided into two zones, namely the els, whereas they have a smaller and larger outlet radius active zone and the steady zone. under case 2 (red lines) and case 1 (green lines), respec- The water levels in the active zone are influenced by tively. Tanks with an identical outlet radius (in both cases, the boundary condition, whereas those in the steady C5 and C6) show similar water levels. The green and red zone are constant with a steady water flow. The flow line for C5 and C6 are close and finally converged. The velocity of each tank in the steady zone is equal to the simulated results reflect the flow behavior in unsaturated average boundary recharge. As demonstrated by the soils. A soil layer with good hydraulic conductivity and monitoring results shown in Fig.  1, the flow behaviors low water retaining capacity is represented by the tanks simulated by the model are similar to those observed in with a larger outlet radius, whereas the soil layer with low natural soil layers. The hydraulic model provides strong hydraulic conductivity and high water retaining capacity evidence for the existence of steady flow, which acts as is represented by tanks with a small outlet radius. a typical source of groundwater recharge in the loess Figure 6 shows the soil water retention curves (SWRC) region. of the silt and sand, which clearly illustrate the differ - ence in the ability of each of the two soils to retain water. Under the same water content (e.g., 17.5%), the soil suc- Discussion tion of silt (40 kPa) would far exceed that of sand (2 kPa), Figure 7(a) shows the in-situ monitoring results of water representing a much lower permeability of the silt. content of the soil profile in the central Chinese Loess The variations in the water levels of the tanks tend to plateau (Hou et al. 2020). Figure 7(b) shows the simulated decrease from the upper to lower tanks, regardless of results using the monitored precipitation data. The simu - their initial condition. The water level in C0 represents lated water content of the soil profile is consistent with the influence of rainfall events. There is a continuous the observed water content. The mechanism of the simi - larity between the results of the physical model and the natural situation is discussed below. To solve Eq.  (3), two key soil constitutive functions should be provided in advance, namely the SWRC and hydraulic conductivity function (HCF) (Fredlund 2006; Lu and Likos 2006). These functions can use the empiri - cal equations, such as the commonly used SWRC and HCF equations provided by van Genuchten (1980) and Fredlund et  al. (1994), and the parameters should be fit - ted using laboratory test data. The equivalent SWRC and HCF for the hydraulic model can be deduced based on the principle of the model. Supposing that the equivalent SWRC is expressed as equivalent matric suction against the degree of saturation and that HCF is expressed as the equivalent hydraulic conductivity against the degree of saturation, then the water level height of the ith tank is h , and its degree (S ) of saturation is: S = (5) Fig. 6 Representative soil water retention curves (SWRC) of sand and silt from China The equivalent matric suction (ψ ) is: i Wang et al. Geoenvironmental Disasters (2022) 9:7 Page 8 of 9 Fig. 7 Comparison of soil water content and calculated value using rainfall data ψ = �z − h i i (6) Therefore, the equivalent SWRC of the tank is ψ = (1 − S )�z i i (7) From Eqs. (2) and (5), the equivalent HCF of the tank can also be easily obtained: k(S ) = 2gS �z i i (8) S + 1 Suppose Δz is unity, then Eqs. (7) and (8) can be simpli- Fig. 8 Equivalent matric suction and hydraulic conductivity function fied to: against degree of saturation of the model ψ = 1 − S i i (9) it can visually show the two-dimensional profile of water r 1 k(s ) = 2gS i i (10) flow in the vadose zone and can help to clarify the argu - S + 1 ment between piston flow and preferential flow. Figure  8 shows the curves of Eqs. (9) and (10). Equa- tion  (9) is a linear function in which the equivalent mat- Conclusions ric suction linearly increases with decreasing saturation, The paper uses a hydraulic model to simulate the flow similar to the trend in SWRC variation for real soil. How- behavior in the vadose zone. The mathematical expres - ever, it is difficult to fit a linear SWRC function for real sion of the hydraulic model is deduced and is similar to soil. In addition, Eq. (10) shows that the trend of HCF is the Richards governing equation. The experiment and similar to that of real soil. However, the HCF cannot be mathematical solutions demonstrate that under a con- applied in real soils. The advantage of the model is that dition of intermittent precipitation events, the model W ang et al. Geoenvironmental Disasters (2022) 9:7 Page 9 of 9 Hou X, Vanapalli SK, Li T (2017) Water infiltration characteristics in loess associ- column can be divided into two zones, namely active ated with irrigation activities and its influence on the slope stability in zone and steady zone. The water content in the active heifangtai loess highland, china. Eng Geol 234:27–37 zone changes the variation of the boundary conditions Hou X, Li T, Vanapalli SK, Xi Y (2019) Water percolation in a thick unsaturated loess layer considering the ground-atmosphere interaction. Hydrol (e.g., precipitation, evaporation in natural conditions), Process 33(5):794–802 whereas it keeps constant in the steady zone. The results Hou X, Vanapalli SK, Li T (2020) Water flow in unsaturated soils subjected to of the model are consistent with the soil water content multiple infiltration events. Can Geotech J 57(3):366–376 Huang T, Pang Z, Liu J, Ma J, Gates John (2017) Groundwater recharge mecha- profile observed in the field. The model can overcome the nism in an integrated tableland of the Loess Plateau, northern China: difficulty of observing unsaturated flow in nature by visu - insights from environmental tracers. Hydrogeol J 25(7):2049–2065 ally demonstrating the two-dimensional profile of water Huang T, Ma B, Pang Z, Li Z, Li Z, Long Y (2020) How does precipitation recharge groundwater in loess aquifers? Evidence from multiple environ- flow in both the active and steady zones, and particularly mental tracers. J Hydrol 583:124532 the transition from transient water flow to steady flow. Li P, Zhang X, Shi H (2015) Investigation for the initiation of a loess landslide based on the unsaturated permeability and strength theory. Geoenviron Acknowledgements Disast 2(1):24 The author would like to thank the National Natural Science Foundation of Li P, Li T, Vanapalli SK (2016) Influence of environmental factors on the wetting China for its funding and trust. front depth: a case study in the Loess Plateau. Eng Geol 214:1–10 Li Z, Chen X, Liu W, Si B (2017) Determination of groundwater recharge Authors’ contributions mechanism in the deep loessial unsaturated zone by environmental trac- Yu Wang was involved in the design and operation of experiment, data collec- ers. Sci Total Environ 586:827–835 tion and analysis, simulations and writing the manuscript. And all other author Lu N, Likos WJ (2006) Suction stress characteristic curve for unsaturated soil. J contributed in operation of experiment, data analysis, simulations and writing Geotech Geoenviron Eng 132(2):131–142 the paper. All authors read and approved the final manuscript. Richards AL (1931) Capillary conduction of liquids through porous mediums. Physics 1(5):318–333 Funding Russo D, Fiori A (2008). Equivalent vadose zone steady state flow: an assess- The study was supported by the National Natural Science Foundation of China ment of its capability to predict transport in a realistic combined vadose (Grant Nos. 41772278, 41790442, 41807242). The above financial support is zone–groundwater flow system. Water Resour Res 44(9) gratefully acknowledged. Tu XB, Kwong AKL, Dai FC, Tham LG, Min H (2009) Field monitoring of rainfall infiltration in a loess slope and analysis of failure mechanism of rainfall- Availability of data and materials induced landslides. Eng Geol 105(1–2):134–150 Some or all data, models, or code that support the findings of this study are Van Genuchten MT (1980) A closed-form equation for predicting the hydraulic available from the corresponding author upon reasonable request. conductivity of unsaturated soils. Soil Sci Soc Am J 44(5):892–898 Wang H, Li M, Zhou B, Zhou Y, Yuan Z, Chen Y (2017) Application of a hybrid model of neural networks and genetic algorithms to evaluate landslide Declarations susceptibility. Geoenviron Disast 4(1):15 Wang J, Li P, Ma Y, Li T (2018) Influence of irrigation method on the infil- Competing interests tration in loess: field study in the Loess Plateau. Desalin Water Treat The authors declare that they have no competing interests. 110(APR.):298–307 Wu C, Xu L, Dai F, Min H, Tham LG, Kwong AKL, Zhou YF (2011) Topographic Author details features and initiation of earth flows on Heifangtai loess plateau. Rock Department of Geological Engineering, Chang’an University, Xi’an 710054, Soil Mech 32(06):1767–1773 (in Chinese) Shaanxi, China. Department of Civil Engineering, Chang’an University, Xu Z, Zhao Y (1993) Research of fractural efficacy on mechanisms governing Xi’an 710054, Shaanxi, China. Water Cycle and Geological Environment water flow in unsaturated loess. J Changchun Univ Earth Sci 23(03):326– Observation and Research Station for the Chinese Loess Plateau, Ministry 329 (in Chinese) of Education, Lanzhou 745399, Gansu, China. Key Laboratory of Shale Gas Xu L, Dai FC, Tham LG, Tu XB, Min H, Zhou YF, Xu K (2011) Field testing of irriga- and Geoengineering, Institute of Geology and Geophysics, Innovation Acad- tion effects on the stability of a cliff edge in loess, North-West China. Eng emy for Earth Science, CAS, 10029 Beijing, China. Geol 120(1–4):10–17 Xu P, Zhan Q, Qian H, Yang F, Zheng L (2020) Investigating the mechanism Received: 12 May 2021 Accepted: 14 February 2022 of ph effect on saturated permeability of remolded loess. Eng Geol 284(6):105978 Zhang M, Liu J (2010) Controlling factors of loess landslides in western China. Environ Earth Sci 59(8):1671–1680 Zhang C, Li T, Li P (2014) Rainfall infiltration in Chinese loess by in situ observa- References tion. J Hydrol Eng 19(9):06014002 Basara JB, Crawford TM (2010) Improved installation procedures for deep-layer Zhang Y, Li T, Shen W, Wang Y (2019) Hydraulic model of transition of transient soil moisture measurements. J Atmos Oceanic Tech 17(6):879–884 to steady flows in the Vadose zone. J Hydrol Eng 24(12):04019052 Cheng Y, Zhan H, Yang W, Fang B (2018) Deep soil water recharge response to Zhuang J, Peng J, Wang G, Iqbal J, Wang Y, Li W, Zhu X (2017) Prediction of precipitation in mu us sandy land of china. Water Sci Eng 11(2):8 rainfall-induced shallow landslides in the Loess Plateau, Yan’an, China, Derbyshire E (2011) Geological hazards in loess terrain, with particular refer- using the TRIGRS model. Earth Surf Proc Land 42(6):915–927 ence to the loess regions of China. Earth Sci Rev 54(1–3):231–260 Fredlund DG (2006) Unsaturated soil mechanics in engineering practice. J Geotech Geoenviron Eng 132(3):286–321 Publisher’s Note Fredlund DG, Xing A, Huang S (1994) Predicting the permeability function for Springer Nature remains neutral with regard to jurisdictional claims in pub- unsaturated soils using the soil-water characteristic curve. Can Geotech lished maps and institutional affiliations. J 31(4):533–546 Gazis C, Feng X (2004) A stable isotope study of soil water: evidence for mixing and preferential flow paths. Geoderma 119(1–2):97–111

Journal

Geoenvironmental DisastersSpringer Journals

Published: Feb 28, 2022

Keywords: Vadose zone; Hydraulic model; Transient flow; Steady flow; Unsaturated loess

References