Introduction

Changes in the pore pressure within a landslide body, particularly at the sliding surface, must be precisely evaluated to effectively assess and mitigate the hazards that arise when a slow-moving landslide is reactivated by rainfall or snowmelt. Quantitative modeling of pore-pressure fluctuations at a specific sliding depth allows the critical amount and intensity of rain/melt-water supply to be evaluated, from which the behavior of a reactivated landslide can be predicted (e.g., Iverson and Major 1987; Matsuura et al. 2003). In pore-pressure modeling, the major difficulty lies in the nonlinearity of the response, which is affected by the overlying vadose zone that buffers the infiltrated water, enabling it to percolate as unsaturated flows. The function of unsaturated flow is one of the most significant unpredictable factors because it depends strongly on the preceding water supply from rainfall or snowmelt and spatial heterogeneity in the hydraulic properties of the subsurface materials (Kool et al. 1987; Bogaard and Greco 2016).

Significant efforts have been made to model and simulate groundwater percolation and pressure propagation through the vadose zone. The Richards equation (Richards 1931) provides the physical basis for water transport in porous media, and the closed-form equations of water retention (van Genuchten 1980; Kosugi 1994) make it numerically possible to solve subsurface water seepage. Based on the Richards equation, hydrological processes in various types of natural and artificial slopes have been simulated with the finite element approach (e.g., Šimůnek et al. 2008, 2016). The distribution of relevant software has accelerated practical applications in a wide range of fields, including environmental hydrology, engineering geology, and in disaster mitigation planning (e.g., Talebi et al. 2008; Essig et al. 2009; Lanni et al. 2012; Lu et al. 2012; Yi and Fan 2016).

However, in cases where the initial and boundary conditions are uncertain, which is typical in natural systems, such deductive numerical computations frequently fail to ensure the accuracy of their outputs. For natural hillslopes, a deterministic model cannot be free from inherent parameter uncertainties derived from probabilistic changes in the secular hydrological conditions. Even when the subsurface materials or structures are relatively uniform, the stochastic aspects of the wetness and thickness of the vadose zone prevent the deterministic calculation of the pore pressure in the saturated zone (Vogel et al. 2000; Shao et al. 2015). Modeling that focuses on spatial heterogeneity is also challenging because the unknowability of actual subsurface hydrogeological structures precludes the deterministic computation of complex groundwater behavior through preferential flow paths (e.g., Malet et al. 2005; van Schaik 2009; Krzeminska et al. 2012; Shao et al. 2015).

Recent approaches invoking artificial intelligence to simulate transient hydrological responses and associated slope destabilization deserve significant attention in landslide hazard assessment. Machine/deep learning has enabled high-precision reproduction of observed pore-pressure waveforms (Mustafa et al. 2012; Babangida et al. 2016; Wang et al. 2020; Wei et al. 2021; Zhang et al. 2022). However, the use of such technology requires extensive hydrological datasets to serve as a training database. Such data are not always available in the target area, especially for the groundwater responses near the sliding surface. Furthermore, since the model treats the hydrological processes as a “black box” in the computation, the algorithm does not facilitate physical explanations for the characteristic responses in the observed data.

An approach that preserves the physical meaning of the parameters would provide practical advantages in explaining pore-pressure responses when combined with an empirical strategy for estimating the parameter values. A semiempirical approach can ensure certain accuracy for the outputs since the parameters are determined from the observed data regression. Even with empirically determined parameters, a physically based model that can explain pore-pressure responses will provide insights into the subsurface hydrological processes and controlling factors for the reactivation of dormant landslides.

For cases in hillslopes with a thin regolith cover and a high ground water table, a linear pressure diffusion model can be appropriate for reproducing the pore-pressure responses. The model regards pressure diffusion as the major process causing subsurface pore-pressure fluctuations (Iverson and Major 1987; Reid 1994; Iverson 2000), presuming saturated or nearly saturated hydrological conditions. Given an initial pressure condition, diffusion modeling yields an analytical output that mimics a quick response in pore pressure on a short time scale, which is commonly observed by field monitoring (Matsuura 2000; Rahardjo et al. 2005; Matsushi and Matsukura 2007; Schulz et al. 2009; Berti and Simoni 2010, 2011; Osawa et al. 2018). Several previous studies have demonstrated the model characteristics (e.g., D’Odorico et al. 2005) and the reproductivity of the actual pore-pressure fluctuations (Crosta and Frattini 2003; Berti and Simoni 2010; Zieher et al. 2017).

In this paper, we attempted to validate this semiempirical approach via supplemental modification to a model proposed by Iverson (2000). In our modeling, the buffering effects of the unsaturated overburden were empirically incorporated as functional parameters that vary with the wetness and thickness of the vadose zone. Over 2 years, we monitored pore pressures at several locations within a dormant landslide underlain by soft sedimentary rocks in a high-snowfall area of central Japan. The strong seasonality of the surface and subsurface hydrological processes in this region (Osawa et al. 2018) allowed exploration of the applicability of this approach for reproducing the pore-pressure response to diverse water input patterns, such as episodic rainstorms in summer and daily snowmelt in winter. We fitted the model outputs to the observed data and formulated the hydraulic diffusivity and conductivity coefficients, which represent the buffering function of the vadose zone. The obtained formulations to determine the parameters were tested regarding their applicability for reproducing the observed pressure fluctuations at several sites in the same landslide.

Study area

We focused on a typical dormant slide, the Busuno landslide (Fig. 1), located in a mountainous area in Niigata Prefecture, central Japan. The bedrock in this area consists of soft Neogene sedimentary rocks and includes massive black mudstone with thin sandstone layers and fine-grained dacitic tuff (Takeuchi and Kato 1994). The attitude of the bedding observed at outcrops and boreholes indicates a NE–SW strike that dips 20–40° to the southeast (Fig. 1). The shallow subsurface materials comprise heavily weathered tuff covered by < 1 m of organic gravelly colluvium. The colluvial soil is sensitive to the release of pore water under low negative pressure conditions, while the tuffaceous residual clay has a higher ability to retain water (Osawa et al. 2018).

Fig. 1
figure 1

Topographic map showing a longitudinal profile of the study area on the Busuno landslide, Niigata Prefecture, Japan. The sliding depth of the landslide was investigated during a borehole survey (Osawa et al. 2018)

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The studied hillslope is an east-facing, elongated sliding body 370 m in length and 50–70 m in width, with an elevation range of 550–650 m and a slope of 10–15° (Fig. 1). The shallow subsurface materials across the scale of the entire slope are strongly deformed and fractured by past landslide activity to a depth of a few meters above the sliding surface. The sliding surface formed within a weathered mudstone layer at 3–7 m depth and is indicated by clear striations in a thin layer of weathered tuff in some cases. The landslide had historically moved over the sliding surface as four distinct blocks: upper, middle, lower, and terminal. The middle and lower blocks moved actively at a rate of up to 2 m year−1 in the 1990s and 2000s (Matsuura et al. 2003; Okamoto et al. 2008). However, the landslide has not moved since 2011, when it experienced a slight movement caused by a MW 6.4 earthquake, for which the epicenter was located 15 km west of the study site (Okamoto et al. 2015).

The climate in this area is characterized by monsoonal seasonal precipitation with substantial amounts of snowfall. More than half (1500–2000 mm) of the total 3000 mm of annual precipitation in this area is snowfall, which occurs when large volumes of water vapor are drawn up from the warm Sea of Japan and transported landward by air masses. The ground surface is covered by thick snowpack from December to May (Figs. 2A, B and 3), with maximum depths of 3–5 m in early March. Warm air is occasionally transported from the high-pressure Pacific Ocean toward the low-pressure Sea of Japan during foehn events in early spring, causing rapid snowmelt. In summer, seasonal frontal activity and occasional typhoons provide intense rainfall events. The landslide surface is covered by dense vegetation (Fig. 2D), mainly herbaceous species, from May to October (Osawa et al. 2017).

Fig. 2
figure 2

Photographs of the meteorological station and pore-pressure monitoring sites in winter (A, B) and summer (C, D), with stochastic diagrams of meteorological (E) and pore-pressure (F) observations. The surface of the landslide body is covered by herbaceous vegetation in summer with a height of up to 3 m and snowpack in winter with a thickness of up to 5.5 m (see Fig. 1 for the locations and directions of the cameras)

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Monitoring methods

Meteorological variables were measured for 2 years on flat ground close to the Busuno landslide (Figs. 1 and 2). The amount of water that reached the ground (meltwater and/or rainwater, hereafter referred to as MR) was monitored using two sets of 4-m2 lysimeters that were devoid of soil and tipping-bucket gauges with a resolution of 0.125 mm (Fig. 2A, C, E). We chose this type of tipping bucket because it was sufficiently large to catch all of the MR that percolated through a layered snowpack of average thickness (Matsuura et al. 2013). Systematic errors in the measured meltwater input values caused by local topographic effects during early snowmelt periods were corrected to estimate spatially representative snowmelt rates. We adopted the procedure proposed by Matsuura et al. (2013) and used an empirical function for the MR capture ratio based on the lysimeter measurements, which was dependent on decreasing snow depths and snow water equivalents. The snow depth and snow water equivalent were measured using a laser-type sensor and a pressure pillow snow gauge, respectively (Fig. 2C, E). The data from all of the equipment were recorded in 10-min intervals via a multichannel logger.

Fluctuations in pore pressure in the landslide body were monitored using bedrock bore holes in the middle and lower blocks. Two bedrock borings were drilled at each location of B1 in the center of the middle block and B2 at a side slope of the lower block (Figs. 1 and 2). Pressure sensors at the B1 site were installed at depths of 2.00 m below the land surface, 5.20 m within the weathered mudstone and immediately above the sliding surface at 5.26 m below the surface (Fig. 2F; Osawa et al. 2018). For the B2 site, pressure sensors were installed at 1.36 m depth and 2.85 m depth within the heavily weathered mudstone. The sliding surface in the B2 borehole was defined at 3.62 m below the surface (Fig. 2F; Okamoto et al. 2015). Each pressure sensor was packed into 20 cm of permeable silica sand and held between impermeable bentonite layers to ensure that the pore-pressure fluctuations at the specified depths were measured directly (Fig. 2F). These materials made a continuous line with the bare borehole wall. The remaining upper section of the borehole was filled with local soil material. The data were recorded at 5-min intervals via a data logger (Fig. 2F).

Monitoring results

The MR supply at ground level was concentrated during the periods of intense rainfall events in summer to autumn and daily snowmelt in early spring. The water inputs during summer (July to September) were characterized by intense rainfall that lasted for a few hours (10–20 mm h−1), while medium-intensity rainfall that spanned entire days (50–100 mm day−1) dominated in autumn (September to December) (Fig. 3). The land surface was covered by a thick snowpack during winter (December to March). The snow reached maximum depths of 5.1 and 2.5 m during the two winters of the monitoring period. The snow melted continuously from the bottom of the snowpack at a rate of 0.5–2 mm day−1 during the mid-winter period and melted rapidly, at rates of up to 130 mm day−1, during occasional foehn events (Osawa et al. 2018). Water inputs from the circadian snowmelt and rainwater that fell on the snowpack ranged from 20 to 120 mm day−1 from late winter to early spring.

Fig. 3
figure 3

Seasonal variations in the pore pressure in response to meltwater and/or rainwater inputs

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The response of pore pressure to the MR input in the landslide body varied with season. During summer months without any snow cover, large pore-pressure peaks were observed sequentially in response to intermittent rainfall. Drying between the rainfall events depresses the pore pressure significantly in midsummer. In winter, less sensitive fluctuations were observed in the subsurface pore pressure under a thick snowpack. The pore pressure rises to its base level and oscillates finely as large amounts of meltwater are supplied onto the ground surface in early spring.

The seasonal response patterns in pore pressure were similar for the monitoring points at differing depths (2.00 m and 5.20 m) at site B1 and even at another location (1.36 m depth at site B2) ca. 100 m from B1 (Figs. 1 and 3). However, the pore pressure recorded at 2.85 m depth in B2 showed a different trend in its fluctuation characteristics. The pore pressure at this point fluctuated irregularly, showing unclear correspondence to the individual MR inputs (Fig. 3). Sharp responses were seldom observed to both intense rainfall in summer and meltwater supply in early spring. Such responses seem to result from local pressure dissipation at the gauging point, set in heavily weathered mudstone damaged by fracturing via landslide displacement (Fig. 2F).

The detailed fluctuations in pore pressure in summer and winter were represented by the data obtained at a depth of 5.20 m at the B2 site. The typical response waveform was characterized by a sharp increase and tailed decay, depending on the individual pattern of MR input, as shown in Fig. 4. During a snow-free period, the pore pressure increased rapidly in response to intense rainfall with a short lag time (Fig. 4A). For sequential rainfall events during the rainy season, pore pressure fluctuated in a sawtooth pattern with a systematic base-level rise (Fig. 4B). The pore pressure in early spring responded rhythmically to diurnal meltwater inputs over varying base levels depending on the stage of snowmelt (Fig. 4C, D).

Fig. 4
figure 4

Examples of the observed (gray) and simulated (red) transient pore-pressure fluctuations in response to meltwater or rainwater inputs. The short-term responses are extracted from periods S1, S2, W1, and W2 (see Fig. 3 for the periods). Gray shading indicates the analytical interval of the one-dimensional linear diffusion model and the Nash–Sutcliffe efficiency coefficient NS

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Data analysis and discussion

Pressure diffusion model and strategy for application

A one-dimensional linear diffusion model proposed by Iverson (2000) was adopted in this study to analyze the short-term pore-pressure responses near the sliding surface to the MR input. The model has an advantage in reproducing transient pore-pressure fluctuations with only two parameters (hydraulic conductivity and hydraulic diffusivity). This model is derived from the conditional approximation of the Richards equation (Richards 1931) and is given as

$$\frac{\partial \psi }{\partial t}={D}_{0}{\mathrm{cos}}^{2}\alpha \frac{{\partial }^{2}\psi }{\partial {Z}^{2}},$$
(1)

where \(\psi\) is pore-water pressure (kPa), t is time elapsed (s), \({D}_{0}\) is constant hydraulic diffusivity of subsurface materials (m2 s–1), \(\alpha\) is local slope angle (°), and \(Z\) is vertical depth (m).

A solution to Eq. (1) can be obtained given the initial pore-water pressure (\({\psi }_{ini}\)) and boundary conditions as described in Iverson (2000):

$$\psi \left(t\right)={\psi }_{ini}+Z\frac{{I}_{z}}{{K}_{z}}\left\{\begin{array}{ll}\left[R\left({t}^{*}\right)\right]&\left({t}^{*}\le {T}^{*}\right)\\ \left[R\left({t}^{*}\right)-R\left({t}^{*}-{T}^{*}\right)\right]&\left({t}^{*}> {T}^{*}\right)\end{array}\right.,$$
(2)

where \({{K}_{z}}\) is constant hydraulic conductivity at the surface (m s–1), \({I}_{z}\) is the rate of water supply to the ground (i.e., MR intensity in this study), and t* and T* are the dimensionless time and rainfall duration, respectively, which are defined as time normalized by the slope-normal pressure diffusion timescale, \({Z}^{2}/(4{D}_{0}\;{\mathrm{cos}}^{2}\;\alpha )\), (where \({t}^{*}=4{D}_{0}{\mathrm{cos}}^{2}\alpha t/{Z}^{2}\) and \({T}^{*}=4{D}_{0}\;{\mathrm{cos}}^{2}\;\alpha T/{Z}^{2}\)). \(R\left({t}^{*}\right)\) is a response function, which is expressed by

$$R\left({t}^{*}\right)=\sqrt{\frac{{t}^{*}}{\pi }}\mathrm{exp}\left(-\frac{1}{{t}^{*}}\right)-\mathrm{erfc}\left(\sqrt{\frac{1}{{t}^{*}}}\right),$$
(3)

where erfc is the complementary error function (\(\mathrm{erfc}\left(x\right) =\frac{2}{\sqrt{\pi }}{\int }_{\!x}^{\infty }{\text{e}}^{{-\mu }^{2}}\mathrm{d}\mu\)). The time-series fluctuations of the pore-water pressure can be obtained by summing the calculated response function for discrete water inputs, as described in Godt et al. (2008).

The linear-diffusion model was deductively derived from the assumption that \({{K}_{z}}\) and \({D}_{0}\) are constant under full saturation of the subsurface materials. However, this assumption is unsuitable for natural hillslopes owing to an existing vadose zone over the saturated zone (Handwerger et al. 2013; Shao et al. 2016; Finnegan et al. 2021). In general, the hydraulic properties of unsaturated soils, such as conductivity (i.e., net flow velocity of water-molecule percolation) and diffusivity (i.e., celerity in propagation of pore-pressure diffusion waves), vary in response to changes in moisture content. Under dry conditions, wetting-front migration bears the pore-pressure responses in the vadose zone that result from the large gap in hydraulic properties between the zones before and behind the percolating edge. In contrast, pressure diffusion dominates in the case of infiltration into wet media with the condition that full saturation is unnecessary, but sufficient moisture to enclose bubbles in the pores, which function as a buffering fluid in the pressure transmission must be present. Under this condition, the degree of saturation should affect the celerity in the diffusive pressure-wave propagation through the unsaturated but pressurized transmission zone. This invention enables us to strategically apply this exquisite model to reproduce the observed pore pressure in natural hillslopes.

We attempted to treat \({{K}_{z}}\) and \({D}_{0}\) as fitting parameters to characterize the hydraulic condition of the hillslope, which can be empirically determined by optimizing the values as the model output reproduces the observed pore pressure under varying seasonal conditions. The best-fit values of these variables thus represent the seasonality of the hydrological buffering effect of the vadose zone. In this study, we chose the pore-pressure record at 5.20 m depth just above the sliding surface in B1 for determining optimal parameter values via the model curve fitting, for which the data showed the most typical pressure responses to the MR inputs. Based on the optimal sets of parameters and empirical relationships between them, we examined whether the model can reproduce the data observed at another location in the same landslide.

Model curve fitting to the observed pore-pressure responses

The fitting was conducted for each MR-input episode that produced pore-pressure peaks in both occasional rainfall events in summer and circadian meltwater pulses in late winter to early spring (Figs. 3 and 4). A total of 60 events were targeted for analysis, which were configured for two categories typical for the late winter (W) or summer (S) periods: W1 (12 events from April 23 to May 5 in 2015); W2 (16 events from March 27 to April 14 in 2016); S1 (13 events from October 13 to December 7 in 2015); and S2 (19 events from July 3 to September 28 in 2016). The target of fitting in each episode started immediately prior to the onset of the MR supply and ended at the tail end of the pore pressure for the summer rains and was set between the initiation point of the meltwater supply and inflection point of the diminishing pressure curves for cases of snowmelt (Fig. 4). The other parameters were fixed constants at Z = 5.20 m and \(\alpha\) = 10°, based on the pressure monitoring depth and local slope angle, respectively.

The accuracy of the model fit was quantified using the Nash–Sutcliffe (NS) coefficient (Nash and Sutcliffe 1970):

$$NS=1-\frac{{\sum }_{i\mathop{=}1}^{n}{\left({\psi }_{i}^{obs}-{\psi }_{i}^{sim}\right)}^{2}}{{\sum }_{i\mathop{=}1}^{n}{\left({\psi }_{i}^{obs}-{\psi }^{mean}\right)}^{2}},$$
(4)

where \({\psi }_{i}^{obs}\) and \({\psi }_{i}^{sim}\) are the observed and simulated pore pressures, respectively, and \({\psi }^{mean}\) is the mean of all observed values. The NS value indicates the degree of goodness of fit, with a value of 1 representing a perfect fit. We optimized combinations of the parameters \({{K}_{z}}\) and \({D}_{0}\) by maximizing the NS coefficient because it usually became a sufficient value when greater than 0.8.

The model outputs well reproduced the observed pore-pressure fluctuations for the various water input events. Figure 4 shows typical waveform examples of the observed and simulated pore-pressure responses in the two summers and winters (see Fig. 3 for extracted periods). The model coincided with the observed values even for completely different water input patterns: short-term intense rainfall (Fig. 4A); long-term intermittent rainfall (B); circadian cycles resembling snowmelt (C); and diurnal snowmelt with increasing intensity (D). The NS coefficients range from 0.80 to 0.99 in the fitting of the model curves to the observed waveforms.

Seasonal change in the model parameters and their controlling factors

The obtained parameters \({{K}_{z}}\) and \({D}_{0}\) show a strong compensating correlation (Fig. 5). The regression yields an empirical power function (r2 = 0.95):

$${D}_{0}=1226\cdot {{K}_{z}}^{1.24},$$
(5)
Fig. 5
figure 5

Relationship between the hydraulic conductivity and hydraulic diffusivity from the hydrological model. The colors in the figure indicate the initial pore pressure. Red and blue shading illustrate the ranges in summer and winter, respectively

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Hydraulic diffusivity is generally defined as the ratio of hydraulic conductivity to specific moisture capacity (e.g., Brutsaert 2005). Hence, the resulting correlation between \({{K}_{z}}\) and \({D}_{0}\) does not contradict the nature of unsaturated porous media. Nevertheless, the physical meaning of the empirical power function between \({{K}_{z}}\) and \({D}_{0}\) is difficult to interpret rigorously. In the model formulas adopted in this study, \({{K}_{z}}\) is a factor regulating the input magnitude as the denominator of the MR intensity, while \({D}_{0}\) controls the response characteristics to the input (Eqs. 2 and 3), with which the model enables us to draw any response waveform to reproduce the pressure record. However, we cannot specify which properties or conditions of the subsurface materials constrain the functional forms and the coefficients within the empirical correlation (Eq. 5).

The paired values, \({{K}_{z}}\) and \({D}_{0}\), tend to increase with increasing initial pore pressure, \({\psi }_{ini}\), as indicated by the colored symbols in Fig. 5. The seasonality observed here makes sense because these parameters represent the signal preservation and response speed in the conversion of the MR input to the pore-pressure increase, which is buffered by water absorption in the unsaturated part of the subsurface materials. Osawa et al. (2018) reported that the lag time between the peaks of water input and pore-pressure response at this site increased exponentially with increasing thickness and dryness of the vadose zone. The hydrological conditions of the vadose zone were controlled by both drainage and evapotranspiration, especially between the rainwater supply events in summer. In late winter to early spring, the pressure responded promptly to meltwater input under wet subsurface conditions with snow cover. The seasonal change in the parameters \({{K}_{z}}\) and \({D}_{0}\) thus reflects the signal buffering capability of the vadose zone.

The buffering effect of the vadose zone can be parametrized most simply by its apparent thickness. Here, we evaluate the apparent thickness of the vadose zone at the onset of the pressure response, which can be calculated as described in Osawa et al. (2018):

$$d={Z}_{set}-\frac{{\psi }_{ini}}{{\rho }_{w}g},$$
(6)

where d is the apparent thickness of the vadose zone (m), Zset is the installation depth of the pressure sensor (m) (5.2 m here), \({\rho}_{w}\) is the water density (1.0 kg m−3), and \(g\) is the gravitational acceleration (9.8 m s−2). This linear conversion brings no constitutive reformation of the modeling but allows refining the physical meaning of the parameters that express the seasonality in the pore-pressure responses.

The hydraulic conductivity \({{K}_{z}}\) and the hydraulic diffusivity \({D}_{0}\) as model parameters showed a clear tendency of exponential decay with increasing apparent thickness of the vadose zone d (Fig. 6). The large variation in the parameter values indicates the critical role of the thin unsaturated zone (0.2 < d < 0.7 m) in the MR–pore-pressure waveform conversion, even though it has a thickness of one order of magnitude smaller than the observed depth (Zset = 5.2 m) located in the aquifer. The contrasting parameter values between winter and summer reflect the seasonal change in hydrological conditions in this shallow buffering zone, which are characterized by a high groundwater table with a wet vadose zone in winter, whereas they become low and dry during summer (Osawa et al. 2018). Large values of \({{K}_{z}}\) and \({D}_{0}\) in winter represent a smaller water storage capacity in the wet vadose zone, resulting in effective and rapid MR–pore-pressure waveform conversion. In summer, these parameters become drastically smaller because of the significant water storage and possible heterogeneous water supply to the low groundwater table through the dry vadose zone, resulting in attenuated signal transformation.

Fig. 6
figure 6

Relationship between apparent vadose zone thickness and hydraulic conductivity (A) and hydraulic diffusivity (B) as model parameters

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A gap in the parameter values appears around the border between winter and summer at d = 0.4 m (Fig. 6), which corresponds to the boundary between colluvial and residual layers (Fig. 2F). These materials have distinct hydrological properties, which are characterized by high hydraulic conductivity (Ksat = 1 × 10−5 m s−1) and small water retentivity (field capacity: \(\theta_{\mathrm{FC}}\) = 36–42%) for the colluvial soil and by low hydraulic conductivity (Ksat = 5 × 10−7 m s−1) and large water retentivity (\(\theta_{\mathrm{FC}}\) ≈ 52%) for the residual (heavily weathered) bedrock (Osawa et al. 2018). In general, porous media with high Ksat and small \(\theta_{\mathrm{FC}}\) allow fast transportation of water molecules under a large hydraulic gradient in wet conditions, but they tend to dry out and hamper quick pressure propagation. Meanwhile, in porous media with low Ksat and large \(\theta_{\mathrm{FC}}\), water molecules move at a limited velocity, but pressure diffusion can occur quickly as long as the medium is saturated. In our results, the discontinuity in the parameter values around the colluvial–residual boundary seems more pronounced for \({D}_{0}\) than for \({{K}_{z}}\). The predominant pressure diffusion occurs below the top of the capillary fringe, which must include the vadose zone defined with the apparent thickness d here. Even when the calculated groundwater table (atmospheric equilibrium surface) becomes lower than the colluvial–residual boundary, the residual layer remains in a tension-saturated state and continues to contribute to prompt pressure diffusion. This nonlinearity in water retention and subsequent pressure diffusion behavior may appear as a stepwise change in the optimum parameter values near the hydrogeologic boundary.

Reproducibility of pressure peak values by the model

Empirical modeling of seasonal variations in parameters in the pore-pressure diffusion equation enables reproduction of observed pore-pressure peaks or prediction of the peak values based on MR inputs and subsurface hydrological conditions. The key parameters \({D}_{0}\) and \({{K}_{z}}\) covary tightly (Fig. 5) and are both highly correlated with the apparent vadose zone thickness d (Fig. 6), with a higher linearity in the relationship between \({{K}_{z}}\) and d. Regression for each correspondence and their coupling would lead to the establishment of an empirical model representing seasonality in the parameters for practical use. The hydraulic conductivity \({{K}_{z}}\) as a model parameter decayed with the apparent vadose zone thickness d (Fig. 6A), as regressed by (r2 = 0.85)

$${K}_{z}=1.1\cdot {\mathrm{e}}^{-17.26\cdot d}$$
(7)

By substituting Eq. (7) to Eq. (6), \({D}_{0}\) can also be modeled as a function of d. Thus, the pore-pressure fluctuation can be reproduced semi-empirically based on \({\psi }_{ini}\) for an arbitrary MR input by substituting the empirical functions into the theoretical model formulas (Eqs. 2 and 3).

Figure 7 compares the calculated and observed peaks of the pore-pressure responses to check the reproducibility. Although the regressive determination of the parameter values inevitably brings uncertainties in the reproduction (see residuals shown in Figs. 5 and 6A), the semiempirical model proposed here can reproduce the pore-pressure peaks with sufficient accuracy (the root mean squared error is 0.43 kPa; Fig. 7). The incorporation of this empirical function seems to impart the response seasonality in the model outputs through parameterizing the buffering effects by the vadose zone.

Fig. 7
figure 7

Comparison of the observed and calculated peak pore pressures obtained from monitoring at 5.20 m depth in B1 and the model (Eqs. 2 and 3) via empirical parameter functions (Eqs. 5 and 7). The color indicates the initial condition for each event by the apparent vadose zone thickness d

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The uncertainties contributing to the overall scattering should first come adversely from the simplicity of the modeling, which indicates the limitations of the pressure diffusion concept. The parameter d, i.e., apparent vadose zone thickness, is simply a proxy to parameterize the hydraulic connectivity from the land surface to the aquifer and is not capable of evaluating the actual function of the buffering effects of the unsaturated overburden of the hillslope. The bulk specific yield of the vadose zone depends not only on its thickness but also on the moisture depth profile. Subsurface water percolation through the unsaturated zone must be spatiotemporally inhomogeneous, recharging the aquifer via preferential flow paths originating from the existence of geostructural conduits and heterogeneity in the wetting processes. Such complexity in real hydrology cuts across the capacity of the simplified model to reproduce the pressure response.

The validity of this semiempirical approach and the parameter representativeness can be examined to determine whether the model can reproduce pore-pressure responses at another location, at least in the same landslide mass. We targeted the datasets observed at 2.00 m depth in B1 and 1.36 m depth in B2 to be reproduced by the model, at which the records seemed not to be affected by any local pressure dissipation as was at 2.85 m depth in B2. Peak pore-pressure values at these two locations were calculated forward based on \({\psi }_{ini}\) and MR input during the same periods for the parameter analysis at 5.20 m depth in B1. Z and \({\alpha}\) in the analyses for B1 and B2 were set to 2.00 m and 10° and 1.36 m and 15°, respectively.

The calculated peak pore pressure generally reproduced the values observed at both the B1 and B2 sites well (Fig. 8). Residuals between calculated and observed values range within the practical level, with root mean squared errors of 0.60 kPa and 1.57 kPa for B1 and B2, respectively, for the entire data at each site. However, the model outputs overestimated the peak pore pressure under some conditions that differed by site. The overestimation became prominent in B1 under situations when the pore pressure rose regardless of the pre-event apparent vadose zone thickness d (Fig. 8A), whereas in B2, it appeared under drier conditions with larger d values (Fig. 8B). Subsurface hydrogeologic structure at individual sites may affect this occasional restraint of pore-pressure buildup.

Fig. 8
figure 8

Comparison of the observed peak pore pressures to the modeled values at 2.00 m depth at the B1 site (A) and 1.36 m at the B2 site (B). The color indicates the initial condition by the apparent vadose zone thickness d at each site

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At the B1 site, drainage through the permeable colluvial layer suppressed the pore-pressure rise beyond the colluvial–residual boundary. The observed peak pore pressure tends to be suppressed at approximately 16 kPa (Fig. 8A), which corresponds to the groundwater level meeting the colluvial–residual boundary at Z = 0.4 m (Fig. 2F). Osawa et al. (2018) reported that the overlying colluvial soil is characterized by higher permeability and lower water retention capacity than the underlying residual material, and the efficient drainage system by subsurface lateral flow over the interface leads to the cutoff of peak pore pressures. An analogous feature appeared in the results at different depths at the same site (Fig. 7) only when the groundwater table rose across the colluvial–residual boundary, i.e., for cases with d > 0.4 m of pore pressure rise beyond \(\psi\) > 48 kPa at 5.20 m depth. The overestimation would reflect difficulty in the data reproduced by the linear pore-pressure diffusion model that employed simple parameterization for the layered hydrogeological structure.

For the case at the B2 site, the locality of observation (Z = 1.36 m) near the hydrogeologic boundary (Z = 1.20 m; Fig. 2F) could affect the discordance between the model prediction and observed data. Although the model output exhibited precise reproductivity under conditions with a high groundwater table, it tended to overestimate the peak pore pressure when the event started from a thicker vadose zone (d > 0.6 m) (Fig. 8B). A larger water storage and drainage in the unsaturated colluvial layer would cause pressure dissipation, which had not been incorporated in the linear diffusion model employed here. The model seems to function practically only when the hydraulic heterogeneity exerts limited nonlinearity on the pressure diffusion processes.

These attempts at data reproduction provide significant implications for the conditional applicability of the linear diffusion model. Semiempirical modeling is valid for situations in which the subsurface pressure transfer system can be simplified because of shallow groundwater tables and high hydraulic continuity. When the bedrock has experienced significant displacement by past landslide activity, the intensively fractured body can be regarded in bulk as a porous medium. In such a case, the semiempirical approach combined with the pressure diffusion concept exerts utility to model the buffering effect of the vadose zone. Application of the model would be constrained by heterogeneous subsurface structures that produce nonlinearity in the pressure propagation. In contrast, model applicability potentially provides a clue to characterize the hydrological properties of landslides. Simultaneous monitoring and semiempirical modeling at multiple locations within a single site can help evaluate heterogeneity in the hydrogeological structure of the specific landslide and hence provide a step to progress monitoring-based landslide characterization. A better understanding of the subsurface hydrology of a landslide through model parameterization can contribute to the design of disaster mitigation strategies by using cost-effective countermeasures for groundwater drainage or pile works and to the establishment of adequate rainfall or snowmelt criteria for the evacuation of local residents.

Conclusions

This study applied field observations and modeling of the pore-pressure fluctuations in a dormant landslide underlain by soft sedimentary rocks in central Japan. The pattern of the pressure response to rainwater and/or meltwater input varied seasonally. We proposed a semiempirical model based on the pressure diffusion concept to predict transient changes in the pore pressure near the sliding surface. The model incorporates variables that parameterize the buffering effect of the vadose zone; they can be obtained empirically as inner functions. The methodological validity and the parameter representativeness of this semiempirical approach were examined by investigating the ability of the model to reproduce pore-pressure responses at different depths and locations in the same landslide mass. The optimized parameters and monitoring data are used to set the initial condition, enabling the model to accurately predict short-term fluctuations and peak values of the pore pressure in the landslide. The precision of the calculated peak pore pressure is sufficient for practical use with respect to the observed value.