Abstract
Hydraulic conductivity measurement through a fixed wall permeameter is a common practice to obtain the fluid transmissibility characteristics of soil matrix; however, sidewall leakage due to rigid wall effect may significantly influence the observed values for coarse-grained soils. In this study, the boundary flow error was identified through characterizing the geometrical properties of voids adjacent to the sidewall, and a parameter known as the boundary void ratio () was proposed to account for this effect. The findings suggest that a fixed wall cell containing coarse soils would unavoidably generate extra voids at the interface between soil grains and inner rigid wall, contributing to a larger at the wall than void ratio within the soil bed; the measured hydraulic conductivity is increased primarily due to the apparatus-induced error. A two-dimensional geometric model was then established to estimate the value for uniformly sized coarse soils confined by a rigid permeameter wall, based on which a method was obtained for eliminating the boundary flow error from a fixed wall cell. The mathematical method was finally validated against experimental data from existing literature. It can be concluded that the boundary condition at sidewall featuring unwanted gaps lead to overestimation of the coefficient of permeability; however, the proposed correction method could adequately eliminate the boundary flow error for uniformly sized coarse-grained soils tested within a rigid wall cell.
1. Introduction
Soil mass made up of solid particles with necessary voids can allow water to pass through [1, 2]. Hydraulic conductivity (also referred to as the permeability coefficient) defined by Darcy’s law is an essential parameter for identifying the permeability of soils under the laminar flow region. Currently, there are two typical types of permeameter used for measuring the hydraulic conductivity in the laboratory, namely, fixed wall permeameter [3] and flexible-wall permeameter [4]. Fixed wall cell mainly consists of a rigid cylindrical tube and offers the benefits, such as simplicity, low cost, and the convenience for specimen preparation. At the same time, the shortcoming is obvious since the wall effect for particles packing adjacent to the wall might lead to potential sidewall leakage and flow maldistribution and an exaggerated permeability of laboratory test as well; i.e., the hydraulic conductivity value for finite bed is greater than that for infinite bed [5, 6]. A flexible-wall device comprises a triaxial cell, in which the bed is enclosed by a flexible membrane and subjected to controlled confining pressure. It can effectively eliminate boundary flow issue, and the measured data are close to the real hydraulic conductivity of unconfined bed. Thus, more researchers were focused on employing flexible-wall permeameter for hydraulic conductivity measurements [7–9]. However, complicated operations and high cost set limits for a widespread use [10].
Concerning a fixed wall cell, the boundary effect has long been investigated. Franzini [11] demonstrated that the porosity in the vicinity of sidewall is greater than within the bed packed with grains because the grains close to the wall cannot be packed as tightly as those inside soil bed. The bed can be divided into two zones, namely, the outer shell near the wall and the core whose porosity is free from the boundary effect.
Meanwhile, Roblee et al. [12] and Ridgway and Tarbuck [13] conducted several elaborate measurements of the porosity on uniformly sized grains contained in a rigid cylindrical tube, through pouring the wax into the bed to fix them in position. By analyzing variation law of porosity with the distance away from sidewall, Cohen and Metzner [14] suggested using a triregional model to partition the cross section of confined bed, namely, a wall region, a transition region, and a bulk region, each with a different porosity. Moreover, based on the capillary model from Carman [15], the flow distribution in each region was considered, and they concluded that the boundary influence on water flow could be ignored when the ratio of tube diameter to grains diameter is greater than approximately 30 for Newtonian fluids and 50 for non-Newtonian fluids. de Klerk [16] established a mathematical model with an exponential decay function for describing the porosity variation in the radial direction and predicted the average porosity of confined bed.
More recently, Allen et al. [17], Yadav [18], and Larsson et al. [19] reached an agreement on the effect of permeameter boundary on water flow primarily displayed in two contradictory aspects: (1) the increased porosity near the wall would reduce the resistance to flow; therefore, sidewall leakage occurs; (2) the presence of shearing stress due to a larger surface area at the wall increases flow resistance, inducing reduction in hydraulic conductivity of confined bed. Kango et al. [20] carried out a series of permeability tests for natural soils with different permeameters to particle size ratios (i.e., , where is the permeameter diameter and is the particle diameter) and found that the influence of increasing porosity is more pronounced than surface are at the wall for , and thus the hydraulic conductivity of confined bed (k) would be greater than unconfined conditions (). However, there would be a reversal phenomenon for ranging between 40 and 120, and the wall effect is negligible provided that .
In summary, small-scale fixed wall permeameter does not require excessive soil samples and high water flow from permeability apparatus that make it difficult to conduct in the laboratory [21]. However, the related boundary flow issue in a permeability test shall be carefully dealt with. In this paper, the underlying mechanism of the wall effect was firstly elucidated by analyzing boundary void characteristics in the confined bed. Second, within the 2D theoretical framework, a geometric model was proposed to estimate the boundary void ratio of coarse-grained soils adjacent to the wall. Finally, based on the relationship between void ratio and hydraulic conductivity, a correction method was developed for hydraulic conductivity measured in a fixed wall cell and verified against test data from the existing literature. It should be emphasized that this is only a simplistic approach for uniformly sized coarse-grained soils. Further investigation is required to extend its application to soils having different grain dimensions.
2. Boundary Void Characterization
Several techniques (SEM, MIP, and X-ray computed microtomography) have been developed to reveal the inner structure of compacted soils. There are not only intraparticle voids but also interparticles, and the voids present a wide range of dimensions, varying with different conditions such as loading and compaction [22]. They jointly contribute to the real complexity of the inner structure, which cannot be fully accounted for in this study. Some simplifications have been made before characterizing boundary voids of coarse-grained soils. Figure 1 shows a plane view of composition characteristics of voids near the sidewall within a fixed wall cell, where soil voids are of two general types: typical void among the matrix particles, , and boundary void adjacent to rigid wall, . In order to further distinguish and , a few hypothetical matrix particles, which have the same packing characteristics as the existing matrix particles, are assigned along with the boundary voids, as specified in Figure 1. Thus, consists of two parts: refers to the volume of voids encircled by the hypothetical matrix particle and existing matrix particles next to it, equaling to ; Vp denotes the overlap fraction of the hypothetical matrix particle plus .
As demonstrated in Figure 1, it is also reasonable to assume that consists of and , where represents the volume associated with the existing matrix particles surrounding and refers to a volume formed by the hypothetical matrix particle next to it. This implies that some extra voids would exist around the surface of the wall when compared to those in the interior of the bed since the soil solids are not essentially contacting the rigid wall. The extra void (i.e., ) caused by sidewall can be expressed as follows:
As shown in Figure 1, in equation (1) is associated with the grain size as well as the compaction level of the bed and is independent of permeameter diameter. in equation (1), however, is connected to all the factors mentioned above. Besides, a larger permeameter diameter D leads to a smaller volume of hypothetical matrix particle that is placed next to boundary particles, as well as a reduction in .
Hence, it can be concluded that can also be treated as the sum of and , which are related to the existing matrix particles surrounding and the permeameter wall, respectively. Referring to the definition of void ratio from soil mechanics, boundary void ratio (BVR, ) at the inner wall for confined bed can be accordingly defined as follows:where is the partial volume of the existing matrix particles near the wall, which contributes to the void (i.e., the shaded area in Figure 1) and e denotes the void ratio of the soil bed.
The presence of extra void, , would produce a larger than e, increase the seepage velocity in the vicinity of wall, and ultimately induce sidewall leakage. It should be noted that the measured hydraulic conductivity would be overestimated when is ignored.
3. Mathematical Formulation
Figure 2 illustrates an overview of uniformly sized disks packing in a 2D problem. The regular triangle packing (Figure 2(a)) and the square packing (Figure 2(c)) can produce the densest and loosest states, respectively. Therefore, the density of other rhombus packing falls within this range (Figure 2(b)). By connecting the centers of adjacent four disks, e.g., in Figure 2, and defining the packing angle as (π/3 ≤ α ≤ π/2), the plane void ratio (PVR) can be expressed as follows:
(a)
(b)
(c)
As α increases, the void size in a packing unit and would monotonously increase, leading to a reduction in density. In particular, if α equals to π/3 or π/2, the minimum and the maximum can be thus obtained, respectively [23].
Owing to numerous factors affecting the boundary void, some necessary simplifications are made before establishing a theoretical model. In this paper, the soil mass in a permeameter cell are assumed to be composed of uniform circular disks in a plane with packing angle α as an indication of packing density. Through replacing partial disks of a packing unit with the permeameter wall, a simplified geometric model is developed to estimate , as shown in Figure 3.
(a)
(b)
(c)
(d)
In this model, the inner wall of permeameter has a radius of , and the disks having a uniform diameter of d next to the wall represent the soil particles. Moreover, the blank area between the wall and the particles refers to the boundary void , around which are two tangent points between the wall and the particles. Within the framework of the computational model, a disk refers to soil particle, while the gap between the wall and the particles represents the boundary void. The thickness of sidewall has thus been ignored.
In the packing unit where π/3 < α ≤ π/2, three boundary-packing types are clarified for , with multiple disks removed and replaced by the wall. Thereafter, the plane boundary void ratio (PBVR, ) is proposed, which can be determined for certain boundary-packing types.
Type I: is surrounded by any two tangent particles of unit and permeameter wall. For instance, as shown in Figure 3(a), and are assumed to be replaced by the wall; Hence, the wall is tangent to the particles and , and the corresponding can be calculated as
The area A1B1C1 excluding disks in Figure 3(a) represents , which can be easily inferred:
A complete void for unit corresponds to an area of an individual particle. is surrounded by two particles in Figure 3(a), based on the adjacent complementation law of the interior angles for rhombus, and the disk corresponding to void in Figure 1 shall be
Combining Equations (2), (5), and (6), the PBVR for Type I can be obtained as follows:
Type II: is jointly surrounded by any three particles in unit and permeameter wall. In addition, the lines connecting the centers of these particles generate a triangle (e.g., when α = π/2, a right triangle will be). For instance, by replacing particle with the wall in Figure 3(c), can be calculated by
Similarly, the area standing for equals to
As can be seen in Figures 2(b) and 3(c), the particle is replaced by a wall; therefore, the disk area representing can be obtained:
To sum up, the PBVR for Type II is
Type III: is jointly surrounded by any three particles of unit and the permeameter wall. It should be noted that the lines connecting the centers of these particles create an obtuse triangle (when α = π/2, it will then be a right triangle). If so, replacing particle with the wall in Figure 3(d), will be expressed by
Following the same procedure for Type II, the area representing and the PBVR for Type III are calculated as follows:
Among all the boundary-packing types mentioned above, it is apparent that PBVRs for Type I, Type II, and Type III are in the ascending order, primarily due to the increased representative area of .
To be more specific, the particle contacts the particle O3 when α = π/3 (Figure 2(a)). More importantly, it also serves as a boundary between Type II and Type I in Figure 3. Hence, there are two general boundary-packing types for the particles adjacent to the wall, namely, Type I and Type III. As shown in Figure 2(a), it is three particles that surround a single pore of unit , so that a pore corresponds to half of a particular area that is different from Figures 2(b) and 2(c). Because the wall replaces one particle and combines the other two particles to form such a boundary void in Type I (Figure 3(b)), based on the sum of the interior angles of a triangle, the disk area of is given as follows:
Substituting equations (5) and (15) into equation (2) yields
Moreover, another boundary-packing type (i.e., Type III) is the same as that in Figure 3(d); then, PBVR for Type III remains unchanged when compared to , indicating that it can be calculated by equation (14).
On the premise of that α is a constant, with the various numbers and locations of particles within a certain unit replaced by a sidewall, the number of combinations of the residual particles to form the boundary void should be 8 (referring to Table 1). Assuming that all the combinations follow the principle of random events, the probability of boundary-packing types, namely, , , and corresponding to Types I, II, and III, respectively, have been tabulated in Table 1, including such particular conditions as π/3 < α ≤ π/2 and α = π/3. When π/3 < α ≤ π/2, there are four particle combinations for Type I-1, two for Type II, and two for Type III; when α = π/3, there are six particles combinations in total for Type I-2 and just two for Type III.
4. Correction Method
As discussed above, if extra void contributing to potential sidewall leakage is known, the error of measured hydraulic conductivity through fixed wall permeameter can be corrected accordingly. Based on a plane-geometric model for calculating PBVR, the main computation procedure to correct the hydraulic conductivity of uniformly sized coarse-grained soils is given as follows.
Step 1. determining the average grain size of the material tested. The disks diameter (d) in the PBVR geometric model is regarded as the geometric-equivalent diameter of poorly graded coarse soils obtained from standard sieve analysis. The applicability of this correction method in well graded soils has not been fully discussed and illuminated.
Step 2. determining the correction factor for PBVR. Following the assumption that the uniform disks with α = π/3, π/2 corresponds to the densest or the loosest state of the soil sample, respectively. Thus, the soil bed in the permeameter with any given compaction level can be reproduced by the plane packing of uniform disks in Figure 2 with a specific packing angle α ranging from π/3 to π/2.
Since the soil particles are simplified as mono-sized 2D disks in the model development, it cannot fully reflect the real packing characteristics under 3D conditions, under the effect of grain shape, spatial dimension, and the variability between the overlapping 2D disks. Assuming that the compaction levels for the uniformly sized disks in a plane and compacted soil bed are controlled as the same, the difference between PVR () and the real void ratio should be accordingly modified, given bywhere ξ is the correction factor for PVR.
For the densest scenario, the corresponding correction factor ξmin is calculated through comparing to the minimum void ratio emin as follows:Likewise, the correction factor ξmax for the loosest scenario shall bewhere emax denotes the maximum void ratio.
Therefore, the correction factor ξ for the confined bed with any given compaction level corresponding to the void ratio e could be obtained through linear interpolation between ξmin and ξmax, and thus,
Step 3. determining the packing angle. It can be seen from equation (17) that e is dependent on . Also, is positively related to α in equation (3). Hence, in order to reflect the compaction level of the confined bed in plane whose void ratio is e, α for uniform disks packing in plane is expressed from equation (21) by considering equations (3) and (17):
Step 4. calculating the area ratio of wall region and PBVR. Employing the measurement of water flow rate Qt at the permeameter outlet and hydraulic gradient J across the test length of the soil bed for certain period, the observed value of hydraulic conductivity for confined bed in a fixed wall cell, i.e., kt, could be obtained by Darcy’s law:where At is the clearance cross-sectional area of permeameter, At = πD2/4.
It is reasonable to assume that Qt is divided into two parts, i.e., Qc flows through the interior region of the bed and flows through the wall region, consistent with the assumption of [11]. If the water flow near the wall still obeys Darcy’s law and has the same hydraulic gradient as the interior one, equation (22) can be transformed intowhere kc is the hydraulic conductivity for the interior region, which equals to the hydraulic conductivity without confinement; is the hydraulic conductivity for the wall region; and Ac and represent the cross section areas of the interior region and the wall region, respectively. In addition, .
As illustrated in equation (23), only when the area ratio of wall region λ (where ) is small enough that it can be ignored. In other words, the interior region is much larger than the wall region; kt would approximate the unconfined value, kc. Owing to the empirical equation proposed by Liu [24, 25], which revealed a third-order nonlinear relationship between porosity and hydraulic conductivity of cohesionless soil shown in equation (24), a relationship between kt and kc thus is deduced in equation (25) by the combination of equations (23) and (24):where n is the porosity of soils and β refers to the coefficient for the influence of particle graduation.According to the geometric model in a plane used for calculating PBVR, the area of the wall region in Type I, Type II, and Type III, namely, , , and , respectively, are inferred as follows:where , , and can be calculated in Table 1; the expression of SA1B1C1, SA2B2C2J2, and SA3B3C3J3 can be found from equations (5), (9), and (13), respectively.
Therefore, the area ratio of the wall region for three boundary-packing types (λ1, λ2, and λ3) can be accordingly expressed as follows:
Type I:Type II:Type III:where the section areas of SOA1B1, SOA2B2, and SOA3B3, which denote At in Type I, Type II, and Type III, respectively, are obtained as follows:Hence, considering the probability of boundary-packing types, i.e., P1, P2, and P3, in Table 1, eb and λ can be expressed by
Step 5. correcting hydraulic conductivity. Substituting equations (31) and (32) into equation (25) yields the correction value of hydraulic conductivity for confined bed:To verify the rationality of the proposed correction method, the constant falling head experiments conducted by Kango et al. [20] in a fixed wall cell with uniformly sized soils (e.g., sand, marble chips, and gravels) were used as a reference. The collected data of hydraulic conductivities with D/d < 40 are demonstrated in Table 2 because the permeameter boundary predominates over other factors’ influence on the permeability test results under such conditions. Unfortunately, since the values of emax and emin for these soil samples have not been provided in [20], the maximum and minimum void ratio of monodisperse sand in [26] shall be chosen to substitute them during the overall computational process, namely, emax = 0.786 and emin = 0.372. Additionally, kc in Table 2 representing the hydraulic conductivity of unconfined bed is measured with D/d greater than 120, where the wall effect is not significant concerning the permeability test results and the permeameter wall could be assumed to be infinite as well [20, 27].
The kt/kc value was greater than the allowable error (i.e., 1.25) [3], indicating that the increase in kt is mainly because of extra void Vf in Figure 1 and greater boundary void ratio eb at the wall. After adopting the above procedures to revise these permeability test results, the correction values of hydraulic conductivity are also shown in Table 2. The ratio approximates one, indicating that the corrected hydraulic conductivity of confined bed is consistent with that of unconfined bed in the proposed method.
Though the proposed method can correct hydraulic conductivity to some extent, the limitations of the model lie in that the soil particles in the confined bed are simplified as mono-sized 2D disks, which differs from the natural deposits. As a result, it is desirable to treat the soil particles as 3D balls to establish a computational model for boundary void ratio. Besides, this method can only be applied to uniformly sized soils. Given the real granulometric composition of naturally occurring soil that is composed by grains of various dimensions, further investigations need to be carried out to enhance the method’s capability. It should also be noted that things become much more complex as more than two grain diameters are involved in developing a geometric model.
5. Conclusions
The boundary effect derived from fixed wall cell was investigated by analyzing the composition characteristics of voids close to sidewall, whereby the boundary void ratio () was defined for quantifying the wall effect within confined bed. It appears that the extra void adjacent to rigid wall should depend upon the grain size, compaction level of soil bed, and permeameter diameter. It should be noted that is likely greater than the void ratio in the interior of the bed and contributes to the sidewall leakage during a permeameter test. Assuming that the soil particles of compacted bed are uniform disks packing in a plane, a geometric model for estimating was established. Based on this model and employing an empirical relationship between void ratio and hydraulic conductivity, a correction method for the measured hydraulic conductivity using a fixed wall permeameter was proposed. The experimental data from published literature verified the feasibility of this method for uniformly sized coarse-grained soils. Further developments regarding the correction method should take gradation characteristics into account and derive a generalized formula. Notwithstanding such limitations, it can be treated as a promising step towards providing reliable references for the application of a fixed wall permeameter in laboratory tests.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (grant nos. 51878560 and 41901073) and China Postdoctoral Science Foundation (grant no. 2019M663556).