Technical note: The beneficial role of a natural permeable layer in slope stabilization by drainage trenches

Slope stabilization through drainage trenches is a classic approach in geotechnical engineering. Considering the low hydraulic conductivity of the soils in which this measure is usually adopted, a major constraint to the use of trenches is the time required to obtain a significant pore pressure decrease, here called “time lag”. In fact, especially when the slope safety factor is small, the use of drainage trenches may be a risky approach due to the probability that slope deformations will damage the system well before it will become fully operative. However, this paper shows that the presence of persistent permeable natural soil layers can provide a significant benefit by increasing drainage efficiency and reducing time lag. As a matter of fact, any permeable layer that is intercepted by trenches may operate as part of the global hydraulic system, reducing the drainage paths. A simplified approach to designing a drainage system that accounts for the presence of a persistent permeable layer is proposed. This approach, which can exploit solutions available in the literature for parallel drainage trenches, has been validated by numerical analyses.


Introduction
The stabilization of deep landslides in clay is one of the greatest challenges to engineers due to the high cost and the 20 unreliability of many structural solutions. Often, the only available approach is by deep drainage, which can lead to some shear strength increase through a generalized pore pressure decrease. Available solutions (Hutchinson, 1977;Bromhead, 1984;Stanic, 1985;Desideri et al., 1997;Pun and Urciuoli, 2008;Urciuoli and Pirone, 2013) concern the case of deep parallel trenches (and of deeper drainage panels as well), which is dealt with also in this paper, and the case of tubular drains in a homogeneous soil. 25 Considering the fine-grained nature of the soil, a major constraint to slope stabilization by draining trenches is the long time required to obtain a significant pore pressure decrease (time lag). Especially when the slope is characterised by a small safety factor or is subjected to slow movements (Urciuoli, 1998), the use of draining trenches is in fact problematic due to the probability that slope deformations will damage the system well before it will become fully operative thus vanishing its potential effectiveness. However, as higher is the depth of trenches (or of drainage panels) as higher the probability that 30 these intercept even thin soil layers of higher hydraulic conductivity at an intermediate depth between the ground surface and the slip surface. This would be a lucky chance since the incorporation of such layers in the drainage system may play a highly beneficial role on both the time to attain the final steady-state condition, and the system efficiency.
The scope of this paper is just examining the influence on the drainage system, of a pre-existing permeable soil layer parallel to the ground surface. 35

The basic model
The solutions presented below are based on the following assumptions: the groundwater flow is two-dimensional; each soil layer is homogeneous, isotropic and is characterized by a linear elastic constitutive law; total stresses are constant during the consolidation process (this allows to uncouple the analysis of the hydraulic and 40 of the mechanical soil response).
The technical literature reports solutions concerning the case of parallel draining trenches and of tubular drains in 45 homogeneous soils, which are generally presented in the form of dimensionless design charts, providing the average efficiency, ̄( , Γ), along the slip surface Γ: (2) In Eq. (2), (0, Γ) is the initial pore pressure on the slip surface, , and ̄( , Γ) is the average pore pressure at time t modified by the draining elements; (0, Γ)is generally assumed to be hydrostatic. During the consolidation phase, pore 50 pressures decrease towards the minimum steady-state value ̄(∞, Γ), which is attained at time t→∞ when the efficiency ̄( ∞, Γ) reaches the highest value.
The available solutions for parallel trenches, featured by a thickness H0 and a width b, consider the soil volume between the two axes of symmetry, which respectively coincide with the middle of a trench and the centreline between two adjacent trenches (Fig. 1a). This volume is delimited by the ground surface and by an impermeable bottom surface located at the 55 distance H from the ground surface. The ground and bottom surfaces are both horizontal: the slope angle is indeed assumed to play a negligible role on the hydraulic process (Aloi et al., 2019). The slip surfaceis a horizontal plane as well, located at depth D. In this paper it is assumed to be coincident with the base of trenches (D=H0).
A key hypothesis, which strongly affects the solution, is the presence of a permanent film of water at the ground surface (Burghignoli and Desideri, 1987;D'Acunto and Urciuoli, 2006;D'Acunto et al., 2007;D'Acunto and Urciuoli, 2010). 60 However, due to local formation of water ponding and saturation of vertical cracks in the ground, often this is not far from the truth, at least during the wet season. Based on this assumption, the pore pressure decrease is uniquely due to rotation of the flow lines towards the drainage trenches. Pore pressures in the zone between parallel trenches are then at any time less than hydrostatic (Fig. 1b). In contrast, beyond the bottom of trenches, the upward direction of the flow lines leads to a pore pressure distribution higher than hydrostatic. It is just for this reason that the drains should always reach a depth close to the 65 slip surface.

Time of consolidation
As outlined above, the presence of one or more persistent permeable layers in the soil body to be stabilized (a not unlikely situation in deep clay deposits to be stabilised with draining panels) may play a highly beneficial role on time lag and 75 effectiveness of the drainage system. The influence of a layer parallel to the ground surface, here indicated as the "draining layer d", featured by a thickness Hd as in Fig. 1c, has been investigated by FEM analyses using the code SEEP® (GEO-SLOPE Int. Ltd., 2012). The cases examined in this paper are indicated in Tab. 1; the results of the analyses have been elaborated in a dimensionless form. 80 The value of t90 in Eq.

Figures 2a and 2b
, which report some results concerning the horizontal plane located at depth D = H0, suggest quite a rapid attainment of ̅ = 90%, which is a crucial issue of the design. For significant values of trench spacing in the practice (i.e. 90 s/H0<3), the following considerations may be drawn: i) for H1/H0 = 0.75 and Kd/K = 100 (Fig. 2a), the dimensionless time T90 ranges between one half and one third of the value that would be obtained in the absence of the draining layer; ii) for Kd/K= 1000 (Fig. 2b), T90 significantly decreases with depth of the layer d (for H1/H0= 0.75, it drops to about 20% of the value obtainable in homogenous soils).

Steady state condition 100
The presence of a permeable layer allows to obtain higher values of ̅ (∞, ), and sooner than in homogeneous soils. Some significant data are provided: i) in Fig. 2c, showing the steady-state efficiency for H1/H0= 0.75 reported as a function of the ratio Kd/K and of trench spacing; as shown, as higher is the hydraulic conductivity of the draining layer as higher the efficiency (as an example, for Kd/K=1000 and s/H0 = 3 it practically doubles); a major effect of layer d is in fact diversion 105 of a significant part of water coming from the ground surface towards the trench thus strongly reducing water flow towards the slip surface; ii) in Fig. 2d, showing the efficiency for Kd/K=1000 reported as a function of depth of layer d and trench spacing; the figure shows that it increases as the dimensionless distance, H1/H0, increases; the effect of layer d is a strong pore pressure reduction at depth H1; as a consequence, pore pressure decrease, due to the action of layer 110 d, increases with its depth; iii) in Fig. 2e, showing the non-dimensional pore pressure distribution, ud, along the lower boundary of the draining layer d plotted as a function of trench spacing for H1/H0= 0.75 and Kd/K=1000; near the trench boundary, the pressure head is less than Hd, hence, a free water surface forms in the layer d (here water can move towards the trench only below this surface where pore pressures are positive). 115

3.2.1A simplified approach to predict the steady-state condition
In the following, a simplified model for the optimization of the design is briefly described. A very efficient working condition is achieved if, at the centreline between two adjacent trenches, the atmospheric pressure is attained at the uppermost point of layer d.
The first step in the design of the drainage system is just creating the conditions for full layer activation. This is obtained 120 when the spacing, s, of trenches is equal to the value sd, according to the following expression: The values of sd/H0 in Eq. (4) have been obtained from the results of the numerical integration of Eq. (1). These have been reported in Figure 2f, which shows the dependency of sd/H0 on Kd/K and H1/H0, having fixed Hd/H0.
In case of full activation of layer d, the response of the entire draining system may be analysed by a simplified approach. 125 Since the fluid pressure at the uppermost boundary of the layer d is equal to the atmospheric pressure (or to a small suction especially near the trench boundary), a water film may be fictitiously assumed at the same depth (Fig. 1d). This obviously leads to a generalized pore pressure decrease in the lowermost soil. In the following, any parameter referred to this fictitious condition will be indicated with the apex*.    The values of ̅ * (∞, ) and ̅ * (∞, ) may be obtained from the well known dimensionless solutions for the case of parallel trenches in homogeneous soil, as a function of spacing (see to the simplified scheme in Fig. 1d). The steady-state efficiency thus 140 ̄ * (∞, ) = * (0, )(1 − * (∞, )) = ( − 1 )(1 − * (∞, )).

Conclusions and final considerations 150
The scope of this paper is to demonstrate that the presence of soil layers of higher permeability, a not unlikely condition in some deep landslides in clay, may be exploited to improve the efficiency of systems of drainage trenches for slope stabilization. Once established the depth of trenches, which should reach the slip surface, the selection of a proper spacing may create a hydraulic system in which such layers can work as additional drains. The problem has been examined for the case that a unique permeable layer is present at an elevation higher than the bottom of trenches. 155 The results of numerical analyses show that it significantly speeds up the consolidation process triggered by drainages, leading also to a higher steady efficiency of the system. However, as mentioned in the Introduction, in many practical cases the critical aspect of the design concerns the time requested to achieve an adequate effective stress and safety factor increase.
In these cases, trench spacing should be established looking essentially at the T90 value.
If pore pressures in the draining layer do not exceed the atmospheric pressure, a hydraulic disconnection forms between the 160 two parts of the landslide body respectively located above and below the layer. In such a way, the water film which is normally assumed at the ground surface ideally moves to the depth of the draining layer. This simple consideration allows to employ the design charts available for the design of drainage trenches in homogeneous soils in the equivalent scheme characterised by groundwater level located at the depth of the draining layer, in order to calculate the final system efficiency.
It is worth to mention that the hydraulic continuity of layer d is a fundamental condition for the design. Considering the 165 variability and the unpredictability of many natural situations, proper investigations to check the validity of such an assumption are then warmly recommended. In particular, the adoption of such a stabilization measure should be always managed through the "observational method", i.e. by monitoring the system response in order to i) check the validity of the design and ii) to adopt proper modifications to it due to unexpected or neglected factors. The installation of piezometers is an obvious measure to check in real-time the efficiency of the drainage system (especially during the critical rainy season). The 170 piezometers should be installed both in proximity of the slip surface (near and far from the trenches) and, if possible depending on thickness, in the permeable layer. This will allow to verify the full activation of the permeable layer. dimensional time corresponding to ̅ (t, ) = 90%