An Experimental Approach to Investigating Quasi-Saturation Using Darcy’s Law

Abstract

Recent trends in abnormal weather patterns leading to sudden and localized heavy rainfall have resulted in an increased frequency of surface landslides. As a result, there is a pressing need for improved prediction and early warning systems. This research focuses on understanding soil behavior under quasi-saturation and elucidating its relationship with pore water pressure and the hydraulic head. In the present study, a mathematical model is formulated to characterize the complex dynamics of quasi-saturation based on established principles. The model demonstrates the correlation between volumetric water content and pore water pressure, considering the influence of the hydraulic gradient. Through two comprehensive model tests, empirical data are generated that highlight the intricate factors influencing quasi-saturation. The findings of this study emphasize the complex interrelationship between volumetric water content, pore water pressure, and the hydraulic head. It is worth noting that under quasi-saturation conditions, the volumetric water content may stabilize over time, indicating an equilibrium between water flow rates driven by the hydraulic gradient and pore water pressure. The present study offers new insights into soil moisture dynamics and lays the foundation for advancements in landslide prediction and mitigation strategies.

1. Introduction

In recent years, abrupt and localized heavy rain has frequently accompanied aberrant global weather patterns, usually causing surface landslides. The authors of previous studies have attempted to predict landslide failure time by correlating the steady strain rate and the creep rupture time, providing prediction formulas and graphical solution methods [1,2]. This strategy has been tested and verified in several locations [1,3]; however, its capability to predict landslide failure is limited to within a few days, rendering it unsuitable for detecting rapid slope failures caused by excessive rainfall [4].

An alternative approach proposed by Uchimura et al. [5] involves placing an inclinometer on the slope and monitoring changes in the inclination angle, which has shown potential for forecasting slope failures during heavy rain. Monitoring and early warning systems have proven to be cost-effective countermeasures for mitigating slope disasters. Nonetheless, most warning systems rely on parameters such as rainfall intensity, accumulated rainfall, or their functions to evaluate the slope failure potential. Since these parameters are measured at a representative point in each area, and all slopes are uniformly considered using these data, the risk of failure for stable slopes may be overestimated. At the same time, it may be underestimated for unstable slopes [6].

Stability problems, particularly in deep excavations close to buildings and services, are commonly encountered in geotechnical engineering contexts, with rainfall intensity, variation, and duration being among the most influential factors [7]. In response to the practical importance of these problems, assessing the stability of excavations has received significant attention from the geotechnical community [7]. A comprehensive understanding of wall deformation and ground movement characteristics has been recognized as critical for performance-based designs in urban environments; yet, predicting deformation magnitudes remains challenging [7].

Rainfall-induced slope instability is a significant issue that causes severe damage to infrastructure each year. Hence, modeling slope instability due to rainfall has been a considerable research focus in vulnerable regions worldwide [8]. The infiltration of rainfall and subsequent rise in groundwater levels can cause the failure of unsaturated slopes due to a decrease in the level of matric suction. The shear strength of unsaturated soils can be significantly weakened by suction loss; moreover, the shear stress distribution can be affected by the degree of saturation throughout the depth of the slope [8].

However, physical models that consider soil properties in addition to rainfall data are a crucial model tool for predicting rainfall-induced shallow landslides. These models offer a thorough method for evaluating slope stability and significantly contribute to our understanding and prediction of these events [9,10,11].

Compacted fine soils are frequently used in landfills and the foundations of many structures, and because these compacted soil barriers are often unsaturated, modeling their flow and transport requires knowledge of their unsaturated hydraulic properties. The Soil–Water Characteristic Curve (SWCC) relating suction (matric or total) to the water content or degree of saturation is an integral part of any constitutive relationship for unsaturated soils [12]. The SWCC can describe soil’s water storage capacity under various suction conditions and includes crucial information about the amount of water contained in the pores at any soil suction level and the pore size distribution, corresponding to the soil’s stress state [12].

The methods for predicting the SWCC of a particular soil can be classified into five groups [12]:

• Equations of the appropriate type for the SWCC, in which experimental data are fitted to a straightforward mathematical equation [13,14,15].
• Regression analysis followed by a curve fitting process is typically needed to relate water contents at various suction levels to particular soil parameters, such as D10 (sieve size for 10% passing) and porosity [16,17].
• Regression analysis correlates an analytical equation’s parameters with fundamental soil characteristics, such as dry density and grain size distribution [18,19].
• The distribution of the water content and corresponding pore pressure are related in the physical–empirical modeling of SWCC, which transforms the grain size distribution into a pore size distribution [20,21,22,23].
• Numerous civil engineering disciplines have used artificial intelligence (AI) techniques, including neural networks, genetic programming, and other machine learning techniques [24,25,26]. This category includes SWCC prediction utilizing artificial intelligence [27,28,29].

Based on these observations, Johari [30] described a stochastic framework to consider the inherent ambiguity of site-specific soil characteristics and the unsaturated condition, and it was used to assess the reliability indices of individual failure modes in unsaturated soldier-piled excavations. It has been demonstrated that considering the unsaturated state allows researchers to achieve the objective of reliability analysis by raising the mean value of the factor of safety (FS) and lowering the associated standard deviation. Additionally, it was determined that the uncertainty of soil properties significantly impacted the global safety factor of the excavation [30]. Based on the hydrological reactions of model slopes, researchers have also developed forecast methods for rainfall-induced slope failure [31,32]. Through small-scale and actual-world slope-model studies, the potential of tensiometers for slope failure prediction was investigated [33].

2. Research Methodology

The state of the slope takes due to precipitation infiltration is referred to as a quasi-saturated state since the saturated state is not fully achieved due to trapped air [4].

The soil sample temporarily reaches equilibrium at a specific volumetric water content without being completely saturated. The volumetric water content increases further due to the subsequent rise in the saturation zone because of capillary saturation. The volumetric water content at equilibrium in the column experiment corresponds to the initial quasi-saturated volumetric water content in the model slope when the sprinkling water investigation was conducted under the same physical property conditions [4].

Unsaturated hydraulic conductivity is theoretically balanced with the sprinkling water intensity in the column and model slope tests when the intensity is constant unless the intensity exceeds the hydraulic conductivity [4].

Given this rationale, Koizumi et al. [34] defined the initial quasi-saturated volumetric water content as the condition where infiltration and drainage are balanced. They also proposed that slope health monitoring during rainfall is possible using the initial quasi-saturated volumetric water content as a regulation criterion, as deformation only occurs if this threshold is exceeded [34].

Previous studies on quasi-saturation have primarily concentrated on its behavior after attaining steady-state conditions. However, a limited amount of research explores the underlying mechanisms responsible for the quasi-saturation state and its relationship with other soil properties, such as pore water pressure and hydraulic head. This study addresses this knowledge gap by investigating quasi-saturation formation and its connection to these soil properties.

Ultimately, the findings of this study will enhance the understanding of quasi-saturation behavior and its relationship with pore water pressure and hydraulic head. These findings lay the groundwork for developing more accurate models of soil moisture dynamics in landslides induced by heavy rainfall and early warning systems. Consequently, the study combines Terzaghi’s consolidation theory and Darcy’s law to derive an equation that characterizes quasi-saturation behavior. This approach enables the examination of intricate interactions between soil moisture, pore water pressure, and the hydraulic head, offering insights into the behavior of unsaturated soils under various environmental conditions.

2.1. Quasi-Saturation State Equation

The phenomenon of quasi-saturation refers to the situation where soil is not fully saturated with water. It has reached a state of partial saturation where the water content has temporarily stabilized. In this case, soil hydraulic conductivity will likely be reduced. In other words, hydraulic conductivity (k) is not a constant in this scenario, but rather a function of the volumetric water content (θ), k = k(θ), which can be determined using unsaturated permeability tests and the Soil–Water Characteristic Curve (SWCC). Consequently, soil hydraulic conductivity will likely be reduced in this case; this can affect the infiltration rate of rainfall and potentially trigger landslides.

Soil consolidation is a very complex process and has been studied by researchers [35,36]. The current commonly used consolidation theory was initially developed by TERZAGHI [37] and marked the birth of modern soil mechanics [38].

Equation (1) describes the change in pore water pressure as a function of time and depth during consolidation.

$$\frac{\partial \mathrm{u}}{\partial \mathrm{t}}=\mathrm{k} \ast \frac{{\partial }^2\mathrm{u}}{\partial {\mathrm{z}}^{2 }}$$

(1)

where: ∂u/∂t—the rate of change in pore water pressure (u) with respect to time (t); k(θ)—hydraulic conductivity, which is a function of the volumetric water content (θ); representing the ability of soil to transmit water; ∂²h/∂z²—the rate of change in pore water pressure gradient with respect to depth (z).

Next, Darcy’s law Equation (2) (q) presents the flux or flow rate. It indicates how quickly water is moving through the soil and is typically expressed in units of length per time (meters per second or centimeters per day).

$$\mathrm{q}=-\mathrm{k}\left(\mathsf{\theta }\right)\ast \frac{\partial \mathrm{h}}{\partial \mathrm{z}}$$

(2)

Darcy’s law relates flow rate (q) to the soil’s hydraulic conductivity (k) and gradient.

The equation’s negative sign denotes that water flows from the area of the higher hydraulic head to the location of the lower hydraulic head (i.e., water flows downhill).

The hydraulic head is a measure of energy per unit water weight given by Equation (3).

$$\mathrm{h}=\frac{\mathrm{u}}{\mathsf{\rho }\mathrm{g}}+\mathrm{z}$$

(3)

where: ρ is the density of water, g is the acceleration due to gravity, and z is the elevation.

By differentiating the relationship between hydraulic head and pore water pressure (u) concerning z, we obtain Equation (4):

$$\frac{\partial \mathrm{h}}{\partial \mathrm{z}}=\left(\frac{1}{\mathsf{\rho }\mathrm{g}}\right)\ast \frac{\partial \mathrm{u}}{\partial \mathrm{z}}$$

(4)

By differentiating it again with respect to z, we obtain Equation (5):

$$\frac{{\partial }^2\mathrm{h}}{\partial {\mathrm{z}}^2}=\left(\frac{1}{\mathsf{\rho }\mathrm{g}}\right)\ast \frac{{\partial }^2\mathrm{u}}{\partial {\mathrm{z}}^2}$$

(5)

First, it is important to understand what θ represents. θ is the ratio of the volume of water to the total volume of the soil sample. Mathematically, it can be defined as θ = Vw/Vs, where θ represents the volumetric water content, Vw represents the volume of water, and Vs represents the total volume of the soil sample.

The continuity equation needs to be considered to analyze the change in volumetric water content. The continuity equation relates the change in volumetric water content and the water flow rate (q), as shown in Equation (6):

$$\frac{\partial \mathsf{\theta }}{\partial \mathrm{t}}=-\frac{\partial \mathrm{q}}{\partial {\mathrm{z}}^{ }}$$

(6)

Now, the expression for q from Darcy’s law is substituted into continuity Equations (7) and (8):

$$\frac{\partial \mathsf{\theta }}{\partial \mathrm{t}}=-\frac{\mathrm{d}\left(-\mathrm{k}\left(\mathsf{\theta }\right)\ast \frac{\partial \mathrm{h}}{\partial \mathrm{z}}\right)}{\partial \mathrm{z}}$$

(7)

$$\frac{\partial \mathsf{\theta }}{\partial \mathrm{t}}=\frac{\mathrm{d}\left(\mathrm{k}\left(\mathsf{\theta }\right)\ast \frac{\partial \mathrm{h}}{\partial \mathrm{z}}\right)}{\mathrm{dz}}=\mathrm{k}\left(\mathsf{\theta }\right)\ast \frac{{\partial }^2\mathrm{h}}{\partial {\mathrm{z}}^2}+\frac{\mathrm{dk}\left(\mathsf{\theta }\right)}{\mathrm{dz}}\ast \frac{\partial \mathrm{h}}{\partial \mathrm{z}}$$

(8)

To evaluate dk(θ)/dz in the equation, conducting SWCC (Soil–Water Characteristic Curve) and unsaturated permeability tests on the soil is necessary.

To establish a connection between the equation and Darcy’s law, ∂²h/∂z² with an expression of ∂²u/∂z² is substituted. By differentiating the relationship between the hydraulic head (h) and pore water pressure (u) with respect to z, we can derive Equation (9):

$$\frac{\partial \mathsf{\theta }}{\partial \mathrm{t}}=\mathrm{k}\ast \left(\frac{1}{\mathsf{\rho }\mathrm{g}}\right)\ast \frac{{\partial }^2\mathrm{u}}{\partial {\mathrm{z}}^2}$$

(9)

At this point, it is not just the amount of water in the soil that can fluctuate substantially, but also how that water is distributed and moves. Specifically, the second derivative of the hydraulic head with respect to depth (∂^2 h/∂z^2) will be zero, leading to a linear distribution of the hydraulic head, represented by h = a*z + b, where ‘a’ and ‘b’ are constants that depend on the specific conditions of the soil and the surrounding environment, respectively. Equation (9) demonstrates an intriguing phenomenon. When soil reaches the quasi-saturation stage, the volumetric water content (θ) ceases to exhibit drastic changes. The rate at which it changes concerning time (∂θ/∂t) approaches zero, as deduced from the consolidation equation and the derivatives of Darcy’s law. At this point, it is not only the amount of water in the soil that can fluctuate substantially, but also how that water is distributed and moves.

Essentially, this indicates a state of equilibrium between two opposing forces in the soil: the hydraulic gradient-induced flow and the consolidation-induced flow. It is as if these two forces find a balance, slowing down the change in pore water pressure (u) over time (t). As a result, the flows no longer compete, but coincide, establishing a quasi-steady state within the soil matrix.

Essentially, Equation (9) captures a snapshot of the soil’s behavior in quasi-saturation, connecting the rate of the change in volumetric water content (∂θ/∂t), pore water pressure (u), and the distribution of the hydraulic head (h). This mathematical representation provides a deeper understanding and quantification of quasi-saturation events and the linear distribution of hydraulic heads, enhancing our comprehension of this complex soil behavior.

Soil exhibits a homogeneous hydraulic head and pore water pressure at this quasi-saturation stage. The volumetric water content, however, exhibits minute oscillations throughout time and is not static. This is the soil’s response to the ongoing interplay between its inherent hydraulic characteristics and external influences.

However, the quasi-saturation stage is not a permanent state. It is inherently transient and susceptible to external disruptions. Changes in the surrounding environment can unsettle this delicate balance, such as increased water infiltration due to rainfall, evaporation due to heat, or changes in the soil’s mechanical properties. These influences might push the soil out of quasi-saturation and initiate a new hydraulic adjustment and consolidation cycle.

In essence, Equation (9) encapsulates a snapshot of the soil’s state in quasi-saturation, tying together the rate of the change in volumetric water content (∂θ/∂t), pore water pressure (u), and the hydraulic head (h).

Moreover, this equation highlights the criticality of balance in the soil’s hydraulic behavior. In the quasi-saturation stage, what matters is not the absolute amounts, but the rates at which these parameters change. The equilibrium between water flow rates resulting from the hydraulic gradient and those induced by consolidation signifies a stage where changes in the volumetric water content become minimal.

2.2. Test Material

To verify the suggested Equation (9) and contrast it with actual model test results, the authors conducted two different types of model testing

Table 1. Properties of model tests.

Test Type	Relative Density	Soil Type	Rainfall Intensity (mm/h)	Measurement
Multi-layer model test	60	Edosaki	20	VWC
One-dimensional Column test	60	Edosaki	20	VWC 1, PWP 2

¹ Volumetric water content sensor. ² Pore water pressure sensors.

. These tests mainly examined the volumetric water content and pore water pressure behavior, including multi-layer shear tests and column tests. To evaluate the precision and application of Equation (9) in forecasting quasi-saturation behaviors, the authors would like to report the findings from these tests in the current work.

The plateau extending from the southern part of the Ibaraki Prefecture to the northern section of the Chiba Prefecture is formed by Edosaki sand, a sandy soil layer of the marine Narita layer deposited during the Quaternary period. As mountain sand, it is frequently studied. Figure 1 demonstrates the grain size distribution of the material used for model tests.

Figure 2 displays the results of compaction testing, indicating that the optimal moisture levels for soil are between 13.89% and 17.17%. The soil attains its maximum dry density within this range of approximately 1.76 g/cm³. These findings adhere to the guidelines in ASTM D698-70 [39] for soil sample compaction curves.

3. Model Tests

3.1. Multi-Layer Model Test

The authors of this study conducted a laboratory experiment using a multi-layer shear model test designed to represent a slope and provide insights into the quasi-saturation phenomenon. The primary objective of the experiment was to investigate the volumetric water content in each of the five layers, each 5 cm high, during a simulated rainfall event. The apparatus used for this model test measured 5 cm in height and 60 cm in width (Figure 3), providing a controlled environment for the analysis of soil moisture dynamics.

The study’s findings included a lab experiment using a multi-layer shear model test designed to resemble a slope. This test arrangement gave important information about the quasi-saturation phenomena, which was the focus of this study.

Figure 3 shows a typical slope during a rainstorm, illustrating the circumstances resulting in the soil’s quasi-saturation state.

The ‘soil box’ was then hypothetically laid horizontally, providing a conceptual link to laboratory model test. Finally, the model was prepared with the required density, mirroring the complexity of natural conditions, with five layers measuring 5 cm high, to examine the variability in volumetric water content during a simulated rainfall event.

The apparatus employed for this model test measures 5 cm in height and 60 cm in width, as depicted in Figure 4, with open drainage at the bottom. This controlled environment scrutinizes the intricacies of soil moisture dynamics under rainfall conditions similar to those in the real world within a manageable laboratory setting.

Volumetric water content sensors must be carefully positioned in each layer for the experiment. These sensors captured the changes in soil moisture during simulated rainfall at 20 mm per hour. Fifteen-minute raindrop applications were spaced fifteen min apart with a fifteen-minute pauses. Nine cycles were completed, and the precipitation was applied constantly until the model sheared.

To fully understand the complexity of the quasi-saturation phenomenon, the emphasis now primarily hinges on volumetric water content data. Previous tests with this setup have repeatedly shown that the volumetric water content remains stable during rainfall-induced infiltration from the top layer.

The current work highlights investigating soil conditions and their importance in real-world circumstances, such as a slope prone to rainfall, even though it does not directly address slope stability. This comprehension, in turn, establishes the framework for indirect, but essential, contributions to thorough slope stability evaluations in future studies.

Although shear displacement data were also collected during the experiment, due to the primary objective and limitations of the current study, only volumetric water content was considered in the analysis.

This comprehensive analysis of the multi-layer shear model test results will provide valuable insights into quasi-saturation phenomena in various soil layers and contribute to a better understanding of the factors affecting slope stability under heavy rainfall conditions.

3.2. One-Dimensional Column Test

The authors conducted a second model test using a one-dimensional column setup to investigate the quasi-saturation phenomena further. The current experimental approach aimed to analyze the behavior of quasi-saturation and the relationship between the volumetric water content and pore water pressure in a controlled environment.

It is important to note that the current study at hand features a revised version of a one-dimensional column test conducted by previous researchers like Ohnishi et al. [40], who aimed to determine the initial quasi-saturation, and Xu Li et al. [41], who measured the wetting front. The authors simulated rainfall in the modified column test and measured the pore water pressure and volumetric water content. In contrast to earlier studies, the water used was sourced from a capillary rise, and only the volumetric water content was measured in some model tests.

The column test apparatus comprised a 1 m high and 0.1 m diameter acrylic column filled with Edosaki sand at an initial volumetric water content of 10% and a 60% relative density. In addition, volumetric water content and pore water pressure sensors were installed, as illustrated in Figure 5, at five locations within the column: 10 cm, 30 cm, 50 cm, 70 cm, and 80 cm from the top surface. Consequently, this setup allowed the authors to closely monitor the soil moisture and pressure changes during the experiment, providing valuable data on analyzing quasi-saturation behaviors.

Similar to the multi-layer shear model test, the rainfall intensity for the column test was set at 20 mm per hour. The experiment consisted of three cycles of rainfall application, with different periods for each step. The primary objective of this test was to observe the quasi-saturation behavior and its relationship with pore water pressure in the column, as illustrated in Figure 5.

The column test enabled the observation of soil behavior under controlled rainfall infiltration, allowing the measurement of changes in moisture content and pressure as the soil became increasingly saturated. Therefore, by conducting a column test, a comprehensive understanding of quasi-saturation phenomena can be obtained by comparing the results of this one-dimensional column test with those from the multi-layer shear model test.

4. Test Results and Discussion

Figure 6 demonstrates the test findings using the multi-shear model. The graph illustrates the volumetric water content variation across the nine applied rainfall cycles. The x-axis represents the 300 min timeline and rain, depicted by the blue columns, while the y-axis represents the volumetric water content and time in minutes. As water infiltrates, the volumetric water content increases, resulting in quasi-saturation, which numerous researchers have observed.

Figure 6 shows the volumetric water content responses at intervals of 5 cm (10 cm, 15 cm, 20 cm, and 25 cm) from the model’s top. The data show that the first increase in the volumetric water content occurs at a depth of 15 cm from the top, reaching a value of about 0.30 m³. Next, the layer at 5 cm from the top experiences an increase in the volumetric water content; then, the layers at 10 cm and 20 cm from the top experience an increase in the volumetric water content. However, the bottom layer (25 cm) continues to grow until it reaches 0.35 m³, at which point, the model shears.

The pattern depicted in Figure 6 provides insights into the phenomenon of quasi-saturation and the infiltration process in unsaturated soils. The initial increase in volumetric water content shows a uniform infiltration pattern. The progressive increase in volumetric water content across all strata, which eventually reaches a quasi-saturated state, emphasizes the impact of rainfall penetration on soil moisture dynamics and slope stability.

More precise models of soil moisture dynamics can be created by combining the knowledge from this work, making it easier to put sustainable practices into place for landslide prevention and early warning systems.

Figure 7 shows the results of the pore water pressure (a) and volumetric water content (b) at a depth of 10 cm from the surface of the column over time. The x-axis represents time in minutes, while the y-axis represents the pore water pressure (a) in kPa and the volumetric water content (b) as a fraction of the total volume of the sample. The graph indicates that pore water pressure and volumetric water content increase over time during rainfall cycles, which suggests water infiltration into the soil column.

Equation (9), which describes the relationship between the change in volumetric water content, time, and the gradient of pore water pressure concerning depth, can be used to explain the behavior of pore water pressure and volumetric water content in Figure 7.

Figure 8 shows the results of the pore water pressure (a) and volumetric water content (b) at a depth of 70 cm from the surface of the column over time. The y-axis displays the volumetric water content (b) as a percentage of the sample’s total volume, and pore water pressure (a) is measured in kPa. The x-axis represents time in minutes. The graph exhibits a behavior similar to that shown in Figure 6, with the pore water pressure and volumetric water content increasing during the rainfall cycles and leveling off towards the end.

Similarly, Figure 9 shows the results of the pore water pressure (a) and volumetric water content (b) at a depth of 80 cm from the surface of the column over time. Again, the x-axis represents time in minutes, while the y-axis represents the pore water pressure (a) in kPa and the volumetric water content (b) as a fraction of the total volume of the sample.

The graph shows a similar trend to those in Figure 6 and Figure 7, with the pore water pressure and volumetric water content increasing during the rainfall cycles and leveling off towards the end.

While the current study focuses on exploring the relationship between different properties and describing the behavior of quasi-saturation via mathematical and model tests, the authors acknowledge the importance of future investigations to validate these findings.

In conclusion, the column test results demonstrate the quasi-saturation behaviors in rainfall-triggered landslides and provide evidence for the validity of Equation (9) in describing these behaviors. The similar behaviors of the pore water pressure and volumetric water content shown in Figure 7, Figure 8 and Figure 9 during the rainfall cycles supports Equation (9).

These insights can inform future research and guide the development of sustainable solutions to address the challenges of landslides in a changing climate.

This study’s results have practical implications for geotechnical engineers and other professionals managing and mitigating landslides. Engineers can design more effective strategies to prevent and manage landslides by understanding how rainfall-triggered landslides behave when they reach quasi-saturation. Landslides can cause significant damage to infrastructure and communities, so it is essential to monitor and predict their behavior, especially in areas prone to heavy rainfall and other environmental factors that can trigger slope instability. Moreover, Equation (9) and column tests can help researchers to investigate quasi-saturation behaviors in various soils and slopes, leading to a better understanding of unsaturated soil behaviors under different conditions. This knowledge can help develop sustainable and innovative solutions to landslide challenges in a changing climate, including using advanced monitoring techniques and improving infrastructure.

5. Conclusions

This research contributes significantly to the understanding of quasi-saturation in geotechnical engineering. Firstly, while the concept of quasi-saturation is not new, this study provides a deeper exploration of its impacts, particularly on soil behavior and the susceptibility to collapse in rainfall-induced landslide scenarios.

• The present study utilizes and develops Equation (9), a mathematical model that elucidates the relationship between the volumetric water content and pore water pressure, considering the effects of a hydraulic gradient. Although this equation is well known in hydrogeology, the authors of the present research uniquely apply it to geotechnical engineering, specifically, soil quasi-saturation.
• The findings from previous research align with the observations made in this study, indicating that the volumetric water content (θ) stabilizes over time under quasi-saturation, implying equilibrium between the water flow rates due to the hydraulic gradient. Therefore, the current research explores how the stabilization of the volumetric water content relates to the dynamic interplay between the hydraulic head and pore water pressure. This study enhances our understanding of this relationship, providing a novel perspective of the existing body of knowledge.
• The research demonstrates the complexity of the relationship between the rate of change in pore water pressure (∂u/∂t) and volumetric water content (∂θ/∂t) under quasi-saturation. This complexity arises from the hydraulic characteristics of soil and external variables, which are thoroughly explored and assessed in this study.
• The empirical data generated via two model tests illustrate the intricate interplay of factors influencing quasi-saturation.

It is important to note that additional testing of the obtained equations and observations is required. Future numerical simulations and model testing will be conducted to ensure a thorough comparison and validate the accuracy of the presented equations. This additional validation process will enhance the trustworthiness and reliability of the study’s conclusions.

In summary, this study provides new insights into soil behavior under quasi-saturation, emphasizing the necessity of considering this state in geotechnical engineering practices.