Critical Hydraulic Gradient Research Articles

Prediction of Backward Erosion, Pipe Formation and Induced Failure Using a Multi‐Physics SPH Computational Framework

ABSTRACTSeepage‐induced backward erosion is a complex and significant issue in geotechnical engineering that threatens the stability of infrastructure. Numerical prediction of the full development of backward erosion, pipe formation and induced failure remains challenging. For the first time, this study addresses this issue by modifying a recently developed five‐phase smoothed particle hydrodynamics (SPH) erosion framework. Full development of backward erosion was subsequently analysed in a rigid flume test and a field‐scale backward erosion‐induced levee failure test. The seepage and erosion analysis provided results consistent with experimental data, including pore water pressure evolution, pipe length and water flux at the exit, demonstrating the good performance of the proposed numerical approach. Key factors influencing backward erosion, such as anisotropic flow and critical hydraulic gradient, are also investigated through a parametric study conducted with the rigid flume test. The results provide a better understanding of the mechanism of backward erosion, pipe formation and the induced post‐failure process.

Sediment size effects on non-Darcy flow: insights from Izbash equation and Forchheimer inertial coefficient analysis

AbstractThe transition from Darcy to non-Darcy flow regimes was investigated using column experiments. This revealed key relationships between sediment characteristics and critical thresholds for the onset of the non-Darcy flow regime, as well as inertial flow parameters. An exponential dependence of the critical Reynolds number (Rec) on hydraulic conductivity (K) and a linear dependence on sediment size (d50) was found. The analysis revealed a potentially universal relationship between the critical hydraulic gradient (Ic) and K, with a power-law exponent of –3/2, consistent with previous investigations. Additionally, Ic was found to be inversely proportional to the power law of the square root of d50. Novel relationships are derived for estimating the Izbash equation inertial exponent (n) and the Forchheimer inertial coefficient ($$\beta$$ β ) based on sediment characteristics. The exponent (n) was found to decrease with d50 and increase with K, following power-law relationships. A new equation is proposed, capable of predicting $$\beta$$ β with slightly improved accuracy, outperforming numerous and previously proposed empirical equations. Additionally, these data validate the works of Ergun and Irmay as an alternative for $$\beta$$ β estimation using porosity and sediment size. As the attainment of statistical significance in multiparameter curve fitting can be trivial, it has led to the proliferation of empirical equations for estimating β. This study highlights the limitations of existing empirical methods in determining β and emphasizes the necessity for a universal approach to predict this critical parameter, which will facilitate broader adoption of the Forchheimer equation.

Durability Analysis of Concrete Cutoff Wall of Earth-Rock Dams Considering Seepage and Dissolution Coupling Effect

In this paper, a novel numerical model for characterizing the seepage and dissolution coupling effect on the durability of anti-seepage walls of earth-rock dams is proposed. The model considers the influence of hydraulic gradient-driven seepage on the non-equilibrium decomposition of the calcium dissolution in concrete, as well as the effects of seepage dissolution on pore structure, permeability, and diffusivity. The reasonableness of the model is validated by experimental and literature data, which is then applied to analyze the deterioration and failure processes of a concrete cutoff wall of an earth-rock dam in Zhejiang Province, China. On this basis, the seepage dissolution durability control indices of anti-seepage walls are identified. The findings demonstrate that the suggested method accurately explains the calcium leaching process in concrete. Under the seepage and dissolution coupling effect, calcium in the wall continuously decomposes and precipitates, leading to varying degrees of increases in structural performance parameters, which weaken the seepage control performance of the walls and consequently result in an increase in seepage discharge and hydraulic gradient. By proposing the critical hydraulic gradient as a criterion, the service life of the wall is projected to be 42.8 years. Additionally, the upstream hydraulic head, the initial permeability coefficient, and the calcium hydroxide (CH) content are three crucial indices affecting the durability of walls, and these indices should be reasonably controlled during the engineering design, construction, and operational phases.

Segregation test—a standardisable test for suffusion assessment of granular soils

Suffusion is arguably the most complicated type of internal erosion. Although there are several popular assessment methods, the most realiable assessment is possibly still laboratory testing. However, there is not a standardised test for suffusion yet as different laboratories use different equipment and test configurations. Hence, a reliable comparison of outcomes across laboratories may not be able to achieve yet. This paper presents a new and simple, but very effective, way to test the susceptibility of soil to internal erosion using a novel segregation test. The test employs standard equipment which can be easily found in any geotechnical laboratory. There are some common characteristics of internal erosion and transport segregation, where fine particles are transported through the pore constrictions formed by the soil’s primary fabric. In segregation, particles are transported by gravitational/mechanical force to the bottom of the soil mass. Meanwhile, they are washed out of the soil mass by hydraulic force in internal erosion. Laboratory testing for internal erosion often requires specific equipment and a long duration. Meanwhile, segregation test could be undertaken with standardised sieving tower, which is available in any geotechnical laboratory. The approach was verified with an acrylic setup and some 3D-printed details. Later, the tests of 25 mixtures were undertaken with standard sieving sets. The correlation of laboratory results shows good agreement and prompts the common application of the new approach. The new test may not be able to completely replace the conventional suffusion test yet as it overlooks the critical hydraulic gradients at this stage, but it can be very useful if the research focuses on only the erodible mass and susceptibility to suffusion. In addition, it is a standardisable test with no specific requirements on equipment. The new approach may also be a starting point to study other types of internal erosion.

Hydraulic breakthrough of clay smears due to technical and natural actions

We outlined earlier in this journal by means of finite element simulations how patterns of normal faults arise by a synsedimentary tectonic extension, and how clay smears evolve therein. In the present paper, we show how hydraulic breakthroughs of clay smears can arise so that water, gas and mud rise in faults. Mechanical properties of sand and clay are introduced first for the stable range and then for rupture and internal erosion. Our numerical simulations for the evolution of normal faults and clay smears are discussed in light of critical phenomena. Water assembled in an open-cast mine about 20 years ago as the critical hydraulic gradient in a clay smear dropped to the actual one due to the rapid excavation-induced deformation. The latter led to a critical point under an excavation the slope of which was parallel to a nearby normal fault. Clay smears can also break by earthquakes so that the critical hydraulic gradient drops to the actual one caused by methane with an excess pressure. This can lead to hydraulic breakthroughs and cold eruptions at outcrops of faults.

Hydraulic heave in granular soils under hypergravity conditions

A series of discrete element method (DEM) simulations have been conducted to investigate the initiation and development of hydraulic heave for granular soils under hypergravity. During the development of hydraulic heave, local failure can occur when the stress-reduction factor (α) is less than 0.8, in which the particles are reorganized by the fluid and subsequently regain stability as particles settle down. However, if α is greater than 0.8, global failure occurs, causing the mobilization of the entire soil. The triggering mechanism of the hydraulic heave is the force equilibrium between the contact force and critical fluid force, which scales up proportionally with g-level. The non-linear variation of critical hydraulic gradient (icr) with g-levels is attributed to the non-linear relationship between fluid force and hydraulic gradient. Finally, an analytical model along with a fitting formula, is proposed to predict this non-linear relationship between icr and g-level.

Overburden failure and water–sand mixture outburst conditions of weakly consolidated overlying strata in Dananhu No.7 coal mine

This study presents a case of weakly consolidated strata developed in Dananhu No.7 coal mine. Using a combination of numerical simulation, field measurement comparison, and the critical hydraulic gradient criterion, we investigate the overburden failure and the risk possibility of water–sand mixture inrush during excavation. The following are the principal findings: (1) Weakly consolidated rocks have poor physical characteristics, particularly when they are mudded and disintegrated after encountering water, which may become a favorable source of water–sand inrush; (2) The water-conducting zone develops to a height of 160.5 m with a crack-mining ratio of 15.29 times, extending upward to Toutunhe Formation aquifer. The predictions are consistent with measurements in adjacent mines with similar geological conditions; (3) Cracks without larger subsidence are developed at the front edge of the mining direction, and some parallel stepped cracks behind the goaf could be easily observed. Ground subsidence along the goaf center finally displays a symmetrically wide-gentle U shape; (4) The critical hydraulic gradient of Toutunhe Formation aquifer, aquifer above 3# coal seam, and aquifer of 3#–7# coal seam in Xishanyao Formation is 1.314, 1.351, and 1.380, the actual value is 0.692, 2.089, and 7.418 accordingly. It is inferred water–sand mixture outburst will not occur in Toutunhe Formation aquifer, while the potential risk exists in the aquifers of Xishanyao Formation. Through drainage and depressurization projects, a water–sand mixture outburst accident does not occur during excavation. This study reveals the overburden failure characteristics and the initiation mechanism of water–sand inrush in weakly cemented strata, as well as the internal relationship between them, which provides new research ideas for safe operation in other mining areas with similar geological conditions. The research work has certain practical guiding significance.

Effects of Clay Content on Non-Linear Seepage Behaviors in the Sand–Clay Porous Media Based on Low-Field Nuclear Magnetic Resonance

Clay is widely encountered in nature and directly influences seepage behaviors, exerting a crucial impact on engineering applications. Under low hydraulic gradients, seepage behaviors have been observed to deviate from Darcy’s law, displaying a non-linear trend. However, the impacts of clay content on non-linear seepage behavior and its pore-scale mechanisms to date remain unclear. In this study, constant-head seepage experiments were conducted in sand–clay porous media under various hydraulic gradients. Low-field nuclear magnetic resonance (LF-NMR) technology was utilized to monitor the bound-water and free-water contents of sand–clay porous media under different seepage states. The results show a threshold hydraulic gradient (i0) below which there is no flow, and a critical hydraulic gradient (icr) below which the relationship between the hydraulic gradient (i) and seepage velocity (v) is non-linear. Both hydraulic gradients increased with clay content. Moreover, the transformation between bound water and free water was observed during the seepage-state evolution (no flow to pre-Darcy or pre-Darcy to Darcy). As the hydraulic gradient reached the i0, the pore water pressure gradually overcame the adsorption force of the bound-water film, reducing the thickness of the bound-water film, and causing non-linear seepage behavior. When i0 < i < icr, the enlarging hydraulic gradient triggers the thinning of bound water and enhances the fluidity of pore water. Moreover, the increasing clay content augments the bound-water content required for the seepage state’s change.

Effect of movability of water on the low-velocity pre-Darcy flow in clay soil

Water seepage in soil is a fundamental problem involving various scientific and engineering fields. According to the literature, low-velocity water seepage in low-permeability porous media, such as clay, does not follow Darcy's law, also known as pre-Darcy flow. The formation of immovable water due to water adsorption on the pore wall is believed to be responsible for the formation of pre-Darcy flow. However, this view lacks direct solid evidence. To investigate the pre-Darcy water flow in clay, head permeability experiments are conducted on six clay samples with different densities. The results indicate that water seepage in clay at low hydraulic gradients does not follow Darcy's law. A clear nonlinear relationship between flow velocity and hydraulic gradient is observed. Water flow in clay can be divided into the pre-Darcy flow and Darcy flow regions by the critical hydraulic gradient, which is 10–12 for the Albic soil with dry density between 1.3 g/cm3 and 1.8 g/cm3. According to the disjoining pressure theory, immovable water due to water adsorption on the pore wall is the primary reason for water flow deviating from Darcy's law in clay. The results indicate that the percentage of movable water ranges from 39.7% to 59.3% for the six samples at a hydraulic gradient of 1. As the hydraulic gradient increases, the percentage of moveable water also increases. Additionally, there is a strong correlation between the percentage of movable water and the variation in hydraulic conductivity with the hydraulic gradient. Furthermore, a quantitative relationship between the percentage of movable water and the hydraulic conductivity has been established. The results of this study suggest that water adsorption on the pore wall not only affects the water movability, but is also closely related to the pre-Darcy flow phenomenon in clay.

Numerical simulation of the critical hydraulic gradient of granular soils at seepage failure by discrete element method and computational fluid dynamics

Seepage failure is a common problem in engineering, and the calculation and analysis of critical hydraulic gradient are of great significance for the safety and protection of engineering. Based on the principle of discrete element method and computational fluid dynamics, the fluid–solid coupled models were established to study the critical hydraulic gradient and particle loss rate of granular soils at seepage failure. The evolution of seepage failure was divided into four stages: seepage development stage, local damage stage, volume expansion stage and overall damage stage. The validity of numerical simulation was demonstrated by comparing the critical hydraulic gradient obtained by numerical simulation and by Terzaghi’s formula. According to the fabric damage and flow velocity variation of the models at seepage failure, the influences of model size and particle size on the critical hydraulic gradient and particle loss rate were analyzed. The results indicate that critical hydraulic gradient and particle loss rate were not sensitive to changes in model size. A wide particle size distribution range resulted in large critical hydraulic gradient and small particle loss rate at seepage failure. The discrete element numerical simulation can not only be used to determine the critical hydraulic gradient of geotechnical and hydraulic engineering, but also offer a visual portrayal of the evolution of seepage failure, serving as an important complement to comprehend the intricate microscopic mechanisms underlying soil seepage failure.

Study on Voids and Seepage Characteristics within Rock Fracture after Shear Dislocation Viewing from CT Test and Numerical Modeling

In rock mass engineering, stress balance changes often cause the relative slip of fractures along a wall surface, impacting the seepage behavior of fluid in the fractures. Using computer tomography (CT) scanning, spatial models of fractures with dislocations ranging from 0 to 10 mm were created to explore the relationship between changes in fracture dislocation and changes in fluid flow behavior, respectively. The spatial fractal dimension of cavity distribution within the fractures was calculated using a thin-plate filling approach to characterize the complexity of the fracture cavity distribution. The fluid flow within the dislocation fractures was then simulated using COMSOL, and the effect of cavity alterations in the form of dislocation on the fluid seepage behavior was analyzed using the spatial fractal. The results show that the values of mechanical aperture after dislocation of the fracture obtained by a CT test are normally distributed, the distribution range of mechanical aperture gradually widens with an increase in the dislocation distance, and the average mechanical aperture increases on a logarithmic curve. The relative spatial fractal dimension decreases gradually with an increase in dislocation distance, and the interconnected pathways within the fracture decrease; in addition, it is observed that the change in the relative spatial fractal dimension is closely correlated with the change in the mean mechanical aperture. Numerical simulations of dislocation fracture seepage found that the permeability increases nonlinearly with increasing dislocation distance. When the dislocation distance reaches 5 mm, nonlinear behaviors such as eddy currents occur, and the influence range of eddy currents gradually expands with the increase in dislocation distance under the influence of the boundary. Moreover, the inertia coefficient B in the Forchheimer equation and the critical hydraulic gradient Jc, which can describe the nonlinear seepage characteristics, show a power function decreasing trend with increasing dislocation distance, and the fluid in the fracture is more likely to produce nonlinear flow.

Internal instability of cohesionless soils in upward and downward flow: an experimental verification of theory

Rigid-wall permeameter testing of two potentially unstable gradations established that the critical hydraulic gradient to trigger instability is smaller for upward flow than for downward flow. The experimental finding is explained with reference to the Skempton-Brogan stress reduction factor α in the finer fraction content of the soil. Theory uses the α-factor to define a hydro-mechanical envelope in gradient-stress space that is independent of flow direction, and the experimental results are found to be in good agreement with the theory. Testing with upward flow is recommended to determine the value of α for an internally unstable gradation because, in contrast to downward flow, there is no requirement for an outflow boundary.

The Criteria for Transition of Fluid to Nonlinear Flow for Fractured Rocks: The Role of Fracture Intersection and Aperture

Understanding the fluid pattern is of special significance for estimating the hydraulic conductivity of fractured rock masses. The nonlinearity of fluid flow in discrete fracture networks (DFNs) originates from inertial effects and is enhanced by complex geometric topologies, which produces additional viscous friction and is subject to inertia effects, consequently transitioning the fluid to the nonlinear flow regime. Therefore, it is important to obtain the critical conditions for the transition of a fluid from laminar to turbulent flow. To investigate the role of fracture aperture and fracture intersection on the onset of the transition of a fluid to nonlinear flow in fractured rocks, the fluid dynamic computation was performed by solving Navier–Stokes (N–S) equations in DFN models. The results show that the flow flux initially linearly correlates with the hydraulic gradient (J) and the permeability of DFNs initially remains constant. As the hydraulic gradient increases, the flow flux presents a strong nonlinear relationship with the hydraulic gradient, and the permeability decreases dramatically. In particular, significant inertial effects appear earlier with a large fracture aperture or a dense fracture intersection. A critical hydraulic gradient (Jc) is proposed to judge the onset of nonlinear flow. The mathematical expression of Jc and Forchheimer coefficients A and B involving the fracture aperture and fracture intersection density is established through a multiple regression algorithm. Finally, the reliability of the predictive model was verified by comparing the results of the prediction and fluid dynamic computation of a series of DFN models with well-known geometric distributions. The consistency of the fitted equations and a correlation coefficient greater than 0.9 between them indicate that the predictive model proposed in this study is reliable.

Critical Hydraulic Gradient of Piping Erosion Under Free Flow and Seepage Flow Coupling Model

Critical Hydraulic Gradient of Piping Erosion Under Free Flow and Seepage Flow Coupling Model

Seepage safety evaluation of high earth-rockfill dams considering spatial variability of hydraulic parameters via subset simulation

Seepage failure of high earth-rockfill dams have devastating consequences, and its safety analysis is of great importance in the design phase. However, the highest safety standards in high dams pose a challenge to the calculated efficiency of safety assessment. To address this problem, a random seepage safety assessment method which can consider the spatial variability of hydraulic parameters is proposed. Firstly, the spatial variability of the hydraulic parameters in the impervious materials is simulated using the covariance matrix decomposition under the framework of the non-intrusive finite element method. Then, according to the safety index of critical hydraulic gradient, the seepage stability formula is established to evaluate the safety state. Finally, considering that the highest safety standards in high dams make the probability of seepage hazard small (low failure probability event), it may require more than 10,000 deterministic calculations to complete the assessment process. Therefore, an efficient method based on subset simulation is proposed to optimize the computational efficiency. After the seepage safety evaluation of a 315 m high dam, the results show that the hydraulic conductivity of the core wall has the greatest influence on the seepage safety of the dam, and even under the influence of strong parameter spatial variability, its seepage failure probability is lower than 10-3. Compared with Monte Carlo simulation, the proposed method requires only 1/11 of the computation time at the 10-3 failure level. In addition, the results of grey relational analysis show that the coefficient of variation and correlation distance have obvious effects on seepage safety, and the autocorrelation distance has a greater effect.

A resolved CFD–DEM investigation into the onset of suffusion: effect of confining pressure and stress anisotropy

AbstractThe susceptibility of a granular soil to suffusion is strongly dependent on its grain size distribution (GSD) and the mechanical and hydraulic conditions it is subjected to. This study investigates the onset of suffusion considering the effect of confining pressure and stress anisotropy using a fully resolved computational fluid dynamics and discrete element method (CFD–DEM). Three benchmarks, including the sedimentations of single and two adjacent spheres and the classic one‐dimensional (1D) consolidation are performed to demonstrate the capability of this method for high‐fidelity particle‐fluid simulations. A modified hydraulic criterion for the onset of suffusion considering stress anisotropy is presented. The microstructural changes of soil specimens before and during global suffusion are inspected, with emphasis on the evolutions of particle kinetic energy and displacements, force chain networks, and stress anisotropy. We found that the critical hydraulic gradient is negatively correlated with the confining pressure and the degree of stress anisotropy. Fine particles in the soil matrix are locally detached at small hydraulic gradients before the apparent global suffusion, as manifested by the variation of particle kinetic energy and coordination numbers. The roles of different contact types on force transmission and stress anisotropy in eroded specimens are also examined.

Examining the criteria for contact erosion in granular soil foundations

Contact erosion, or interfacial erosion, often occurs in layered foundations composed of coarse and fine layers; it refers to the phenomenon that the particles of the fine layer are detached by the flow parallel to the interface and transported through the pore channels of the coarse layer. To visualise the initiation and progression of contact erosion at the internal interface of soil strata, transparent soil model tests were employed to simulate real soil, and a series of contact erosion tests was conducted on double-layer foundations composed of different coarse and fine layers. Based on the test results, the commonly used geometric and hydraulic criteria for contact erosion were examined. It is shown that the geometric criteria are reliable, while hydraulic criteria give diverse results. For the same fine soil layer, the critical hydraulic gradient varies greatly with the coarse soil layer, while the critical flow velocity varies in a relatively small range since flow velocity has an intrinsic relationship with the dragging force imposed on particles by seepage flow. More efforts should be invested on the criteria based on critical velocity in further studies of contact erosion.

Experimental Research on Backward Erosion Piping Progression

Internal erosion is caused by seepage body forces acting on the soil particles. One of the most dangerous modes of internal erosion at hydraulic structures is backward erosion piping, which usually initiates at the downstream end of a seepage path, e.g., at the downstream toe of the dam. The progress of backward erosion and the development of erosion pipes were tested in a newly developed laboratory device for three types of sand with grain sizes of 0/2, 0.25/2, and 0.25/1. The piezometric head along the gradually developing seepage “pipe” was observed by seventeen piezometers and seven pressure sensors. The seepage amount was measured by the volumetric method. The critical hydraulic gradient was determined and related to the soil porosity. The progression of the seepage path and relevant characteristics such as the piezometric and pressure heads and the amount of trapped sediment were observed by two synchronous cameras. Based on the analysis of the results of 42 tests, a new empirical formula for the backward erosion rate was proposed. The characteristics of lateral erosion were evaluated and compared with the available literature, which provided reasonably good agreement.

A probabilistic approach to levee reliability based on sliding, backward erosion and overflowing mechanisms: Application to an inspired Canadian case study

AbstractImproving protection against fluvial floods requires a better estimation of levee failure. We developed an assessment method of levee failure probabilities for sliding, backward erosion, and overflowing each represented by fragility curves. We tested two approaches to aggregate those fragility curves into a global fragility curve respectively using: an enveloping curve and Monte‐Carlo simulations. We implemented this approach to earthen levee reliability for several flood return periods to the Bow River in Calgary, Canada. We used limit equilibrium method to estimate the safety factor of the levee segment and Monte‐Carlo simulations to estimate sliding probabilities. We used Terzaghi's critical hydraulic gradient to estimate backward erosion failure probabilities. The estimation of overflowing probabilities required expert judgment. We discussed how the choice of the hydraulic gradient area and the consideration of a steady state or transient model impact backward erosion failure probabilities. The results showed for our study case that, even though the transient model is a closer representation of reality, the levee saturation parameter has little impact on hydraulic gradient values, by extension, on sliding and backward erosion failure probabilities. The Monte‐Carlo aggregated fragility curve is more realistic than the envelop curve of the failure mechanisms for an equivalent computation time.

Three-dimensional analytical solutions of non-Darcy seepage in front of a grouted subsea tunnel face

In this paper, the impact of non-Darcy seepage on the steady seepage field in front of a grouted subsea tunnel face is analysed. Analytical solutions are proposed by combining the three-dimensional (3D) seepage continuity equation, which is expressed in spherical coordinates, with the Hansbo seepage model. These solutions, which are also suitable for calculating the hydraulic head distribution in front of the tunnel face without a grouted zone, are verified by a series of numerical simulations. Comparisons show that this study enabled the augmentation of the classical analytical model. Finally, the influence of relevant parameters, including the nonlinear parameters in the Hansbo seepage model, the relative permeability (i.e., kr/kg), and the grouted zone thickness, are investigated. The results show that the exponential parameter m and critical hydraulic gradient i1 in the Hansbo seepage model have a more significant impact on the water inflow than the water pressure on the grouted zone. Based on the non-Darcy seepage model, the water inflow is lower and the water pressure on the grouted zone is larger than that obtained by Darcy's model.