Recent developments in design criteria for granular and geotextile filters

. Granular and geotextile filters are commonly provided in several hydrological infrastructures to limit soil erosion and allow unimpeded water seepage. The success of a filter depends on forming a bridging structure, which is governed by the grain size distribution of soil and the constriction size distribution of filter. Currently, the retention requirement is satisfied considering representative grain and opening size, whereas the hydraulic conductivity requirement is satisfied considering empirical factors for avoiding excessive clogging. In this paper, the design criteria for granular and geotextile filters are reviewed, and improved design criteria are presented. A probabilistic retention criterion is developed, considering the grain size and constriction size as random variables. The influence of filter thickness is incorporated into the criterion by considering the number of constrictions in a filtration path. A hydraulic conductivity criterion is developed theoretically based on governing flow equations and the expected partial clogging of geotextiles. The limit states for the developed criteria are evaluated based on the wide range of experimental data. The developed design criteria are applicable to granular and nonwoven geotextiles, which offers an improvement in design compared to the existing criteria in practice.


Introduction
Seepage through the porous earth structure might cause instability due to the migration of soil grains. Filters are provided to control the excessive erosion while allowing the unimpeded seepage of water. In Civil Engineering, the filters are classified as granular and geotextile filters. Filters are required to satisfy two conflicting requirements of retention and hydraulic conductivity. The retention and hydraulic conductivity criteria are required to be satisfied such that erosion is limited and the flow is unimpeded, respectively.
The filtration phenomenon is primarily governed by Grain Size Distribution (GSD) of soil and Constriction Size Distribution (CSD) of a filter. The filter works on the principle of self-filtration, as shown in Fig. 1, where the coarser grains are retained by the filter, and the coarser grains retain the finer grains by forming a stable skeleton structure, also commonly known as bridging structure. The grain and constriction sizes belong to the category of random variables of aleatory uncertainty. Consequently, the filters must be designed considering the probability that a random-sized grain infiltrates into random constriction size.
However, in practice, the retention requirement is satisfied considering the representative grain size and opening size of filter. This paper aims to present improved design criteria for retention and hydraulic conductivity requirement for granular and geotextile filters. The specific objectives are: (1) To review the existing design criteria for granular and geotextile filters, (2) To develop a probabilistic assessment criterion for the retention requirement, and (3) To develop a hydraulic conductivity criterion based on the governing flow equations and expected partial clogging of filter.

Granular Filters
Granular filters are required to satisfy three requirements of retention, hydraulic conductivity, and internal stability. Internal stability refers to the ability of a coarser fraction of soil to prevent the loss of fine fraction caused by seepage flow. The details of the requirements are given below.

Retention criterion
Terzaghi [1] provided a rational criterion for retaining soil with uniformly graded granular filters. The retention criterion (Eq. 1) states that 15 [3,4,5] recommended different coefficients and grain sizes for nonuniform filter gradation. The development of the CSD estimation has improved the empirical retention criteria. Indraratna et al. [6] proposed a retention criterion based on the controlling constriction size. Srivastava and Sivakumar Babu [7] presented analytical expressions for the safety of retention and hydraulic conductivity. Nguyen et al. [8] improved Indraratna and Vafai's [9] analytical solution based on energy conservation to solve the Navier Stoke equation.

Hydraulic conductivity criterion
Terzaghi [1] provided the hydraulic conductivity criterion (Eq. 2), which states that the hydraulic conductivity of granular filters should be approximately 25 times more than the hydraulic conductivity of soil; hydraulic conductivity is directly proportional to the 2 15 d . 15 15 The hydraulic conductivity requirement includes no excess pore pressure development at the soil-filter interface, and the flow rate should be greater than the flow rate in the soil without a filter. Giroud [2] has reported that excess pore pressure is not developed if the condition given in Eq.

Geotextile filter
Geotextile filters are required to satisfy three requirements: retention, hydraulic conductivity, and clogging. The details of each requirement are given below.

Retention criterion
Inspired by Terzaghi's [1] retention criterion for granular filters (Eq. 1), the retention criteria for geotextile filters are commonly given by Eq. (5). Wilson-Fahmy [10] and Moraci [11] provided an extensive summary of various proposed geotextile design criteria. CFEM [12] provides retention criteria based on the experimental investigation by Lafleur [13]. Giroud [2] developed a rational retention criterion similar to Eq. (5) with a correction factor for soil with Cu (coefficient of uniformity) and density into consideration.
95 y Od   (5) where 95 O is the opening size of filter, y d is grain size corresponding to y percent passing, and  is the retention ratio.

Hydraulic conductivity criterion
The seepage through geotextile filters is unimpeded if the cross-plane saturated hydraulic conductivity of geotextile ( gt s k ) is greater than the saturated hydraulic conductivity of soil (

Clogging criterion
In a bridging structure formation, coarser grains are retained with a partial restriction on the fine grains. Consequently, the migration of fines leads to a partial clogging of geotextile. The extent of clogging depends on the GSD of soil, CSD, and geotextile thickness [18]. The common practice to evaluate clogging potential involves laboratory testing based on in-situ conditions. Numerous test methods have been proposed to evaluate clogging potentials, such as the long-term flow test, hydraulic conductivity ratio test, and the Gradient Ratio (GR) test. Holtz et al. [15] recommended Eq. (7) to be satisfied for less critical or severe conditions and the GR test for critical or extreme conditions. 95 15

Improved Design Criteria
The improved design criteria of retention and hydraulic conductivity for granular and geotextile filters are given below.

IMPROVED RETENTION CRITERION
Consider uniform-size (single-sized) grains infiltrating into uniform-size constrictions similar to sieving; the filter is effective if the constriction size is smaller than the grain size, whereas ineffective if the constriction size is larger than the grain size. This analogy is extended to non-uniform size grains by considering a safety margin limit state function for retention requirement as Eq. (8).
Here,  ( ) number of times 0 r g p N  = (9) To limit the excessive washout of soil fines, the infiltrated fines must be trapped to a possible extent within the filter. For a typical soil-filter system, the r p decreases with each constriction layer along the filtration path as the number of retained grains increases. Therefore, the r p at the filter thickness must be considered for assessing the soil-filter system.

Hydraulic conductivity criterion for granular filters
For testing soil-filter, a downward flow is simulated in a one-dimensional flow test set up by applying a constant pressure head at the inlet greater than at the exit of the soil-filter system, which is considered the most adverse condition in filtration. The equivalent hydraulic conductivity ( sf s k ) for the soil-filter system is given by Eq. (10).
According to Darcy's law, the flow rate ( s Q ) through a cylindrical soil column with constant pressure head at boundaries, as shown in Fig. 1, is evaluated using Eq. (11).   Figure 2 shows the variation of hydraulic and pressure heads as a function of depth for the three soil-filter systems. The hydraulic gradient in soil ( s i ) and hydraulic gradient in filter ( f i ) in a soil-filter system are defined as Eqs. (12) and (13) (12) isf e f f f P P t i t −+ = (13) where, s h  is the hydraulic head loss in soil and isf P is the pressure head at soil-filter interface.

Fig. 2 Variation of hydraulic head and pressure head along the depth of a soil-filter system
The conservation of mass states that the water flux (flow rate per unit area) in the soil and the filter are the same. Using Darcy's law, water flux through the system ( sf v ) can be expressed as Eq. (14). (14) It can be seen from Fig. 2 that the surplus pressure heads are not developed for a soilfilter with (14) is rewritten as Eq. (15) by substituting (15) represents the pressure head requirement of filter, and it is similar to the pressure head criterion proposed by Giroud [2]. fs s s s k k i  (15) The flow rate requirement is established by equating the flow rate in the soil without a filter (Fig. 1) and the flow rate in a soil-filter system (Fig. 2). The filter will be deemed acceptable if the flow rate in the soil-filter system is at least equal to the flow rate in soil. Referring to Fig. 2, the hydraulic gradient in soil-filter system ( sf i ) is defined as Eq. (16). (16) where sf h  is the hydraulic head loss in soil-filter system. Substituting Eqs. (10), (12), and (16) in Darcy's law (Eq. 11) give the flow rate in soil and the soil-filter system as Eqs. (17) and (18) Substituting Eqs. (17) and (18) (15) is provided by Kalore and Sivakumar Babu [16]. The developed criterion is independent of the thickness of filter, indicating the applicability of the criterion to granular and geotextile filters. For granular filters with internally unstable soils, it is observed that the zone of influence is marginal compared to its thickness. Therefore, it is rational to consider the hydraulic conductivity of soil and granular filter without clogging, and the hydraulic conductivity requirements for the granular filter are satisfied by Eq. (15) with internally stable and unstable soils. For nonwoven geotextile filters, the thickness indirectly influences the hydraulic conductivity depending on the mass per unit area. Also, the applicability of the developed criterion is limited to internally stable soils and filters. Geotextile filters are highly susceptible to clogging due to their limited thickness compared to granular filters. The clogging significantly affects the flow through geotextile and is needed to be considered in the design criterion.

Hydraulic conductivity criterion for geotextile filters
The applicability of the developed criterion (Eq. 15) can be extended to internally unstable soils if the hydraulic conductivities of soil and geotextile filters are examined after the formation of the bridging structure. The zone of soil where the significant movement of soil grains is observed to be relatively thin and less than 10 mm for the formation of bridging structure or filter cake formation. Therefore, considering the relative depth of soil (>100mm) compared to the zone of influence, it is rational to consider the hydraulic conductivity of the original soil in the design criterion. For geotextiles, the extent of clogging is a function of retention efficiency and hydraulic properties. Therefore, the hydraulic conductivity criterion considering the partial clogging of geotextile and original soil is given by Eq. (20).  [17,18,19].

Estimation of limit state of retention criterion for granular filters
The probability of ineffective retention ( r p ) is defined as the fraction of soil grains the filter might fail to retain. The r p is a filter performance indicator regarding the formation of  [15] criterion is inefficient in demarcating effective and ineffective soil-geotextile systems. Note that the limits developed are applicable for hydraulic gradients less than 10.

Summary and conclusions
i.
This paper presents improved filter design criteria for retention and hydraulic conductivity for granular and geotextile filters. The developed design criteria are based on the complete GSD of soil and the CSD of filter, an important consideration, and improvement over the empirical criteria in published literature. Therefore, the influence of GSD shape is inherently captured in the design. Also, the framework is responsive to filter thickness which was not considered in the existing criteria. ii.
The limit states for the probabilistic assessment for retention requirements are estimated based on the experimental data. The probabilistic assessment considers the most influencing factors, such as GSD, CSD, RD, and filter thickness. The comparison between the probabilistic assessment and other widely-used models has demonstrated the refinement in soil-filter system assessment. iii.
For the hydraulic conductivity requirement of fs s s s k k i  , an analytical criterion is developed considering the governing flow equations for soil-filter system. iv.
The hydraulic conductivity and clogging criteria ensure common requirements of unimpeded flow for effective performance. These requirements are satisfied in the developed approach by considering a criterion based on the hydraulic conductivity of clogged geotextile and soil in contact ( gc s ss kk >1). The hydraulic conductivity of partially clogged geotextile is estimated based on the semi-empirical model. The model has been calibrated to predict the GR test results and depends on the considered test results dataset. The comparison between the proposed and widely used criteria has shown an improvement in the soil-geotextile system assessment. v.
The developed criteria provide a primary means for assessing and designing filters. Experimental or advanced computation-based analysis is suggested for critical structures.