A new approach of accounting for impedance effects in Gardner’s method of determining the hydraulic conductivity of unsaturated soils

. Based on tests carried out on a specific device allowing to determine the water retention and transport properties of granular media at low suctions, an alternative approach to Kunze and Kirkham’s method of accounting for the impedance effects due to the high air entry value ceramic disk when using Gardner’s method is proposed. Impedance effects are accounted for by proposing analytical solutions to the equations governing water transfers occurring within the specimen and the ceramic disk. By using some experimental data obtained on a volcanic granular substrate used for urban green roofs, the method is successfully compared to Kunze and Kikham’s graphical method. Its advantages are to be simpler of use and not operator dependent. A detailed examination of the performance of our method compared to those of Gardner and Kunze and Kirkham is carried out based on experimental data, that confirm its validity.


Introduction
Gardner's method [1] was the first analytical method of calculating the hydraulic conductivity of unsaturated porous media based on the measurement of transient outflow under suction step in the pressure plate apparatus. Gardner's method assumes both the linearity of the water retention curve (WRC) and a constant diffusivity over the suction step. However, Gardner's method doesn't account for the impedance effects of the plate (made up of a saturated ceramic porous disk with high air entry value) that may have a significantly lower hydraulic conductivity than that of the saturated specimen. Miller & Elrick [2] were the first to consider the impedance effect, while based on their analytical solution Kunze & Kirkham [3] developed a well-known graphical method. This method is nowadays rarely used, since it has been replaced by numerical back analysis methods (e.g. [4,5]) that deal with impedance issue through the simulation of the water flow in two-layered media (specimen -disk), by numerically solving Richards equation [6].
In order to avoid some level of subjectivity in Kunze & Kirkham's method, or tedious numerical computation related to the selection of the hydraulic properties model in numerical back analysis method, a new and simpler approach of accounting for the impedance effects in Gardner's method has been developed. The advantage of the new approach compared to Gardner's and Kunze & Kirkham's method is validated based on the comparison with experimental data for three different materials.

Theory
The method presented in this work originates from an experimental investigation carried out by [7] based on the device represented in Fig. 1, that schematically illustrates the hanging column apparatus used for simultaneous determining the WRC and the hydraulic conductivity function (HCF). It consists of a metal cell in which the specimen is placed on a saturated HAEV ceramic disk. A suction step is applied by moving down a mobile system in which the constant suction is controlled by the level of the top of the inner tube. The water extracted due to the suction step overflows in the outer tube where the change in water level is monitored by means of a high precision differential pressure gauge ( [7]).

Gardner's method
If the hydraulic conductivity of the ceramic disk is enough higher than that of the specimen, there is no impedance effects and the constant imposed suction increment ǻh i is immediately transferred through the disk to the specimen bottom (ǻh k (z=0, t) = ǻh i = const.) -see dotted lines in Fig. 1.
Since the suction step at bottom (¨h k (z,t)/z)| z=0 governs the outflow from the specimen through Richards equation, Gardner proposed to express ǻh k (z, t) based on an analogy with Terzaghi-Fröhlich consolidation equation [8]: Where V is the total volume of water [L 3 ] extracted from the specimen during the suction step. Finally, the hydraulic conductivity where D(h k ) is adjusted to obtain the best possible agreement between Equation (2) and experimental data, while C(h k ) is computed based on the measured values as ¨ș/¨h i = V /(AH s¨hi ).

Kunze & Kirkham's method
The governing equation in this case is modified equation presented in [2]: where λ n is the nth solution of the equation , while a is the ratio between the impedance of the ceramic disk and that of the specimen.

Kunze & Kirkham's solution is graphically presented through various curves showing the changes in
The various curves correspond to various values of parameter a. Experimental data are presented in the form V(t) / V versus t, and they are shifted along the axis Ȝ 1 2 D(h k )t/H s 2 in order to find the best fitting theoretical curve that defines the value of a. Based on the chosen value of a, the corresponding value of Ȝ 1 2 is adopted from the table presented in [3], while a reference time t RP is graphically determined

A new method accounting for impedance effects
As an alternative to existing methods of accounting for impedance effects, it is proposed to first apply Darcy's law to the saturated porous disk of thickness ǻz d , of saturated hydraulic conductivity K d and of cross sectional area A, like in [4]. One obtains the following expression of the changes in the increment of suction at the specimen bottom: where ǻV [L 3 ] is the extracted water volume during the time interval ǻt. The sooner ǻh k (z = 0, t) reaches ǻh i , the less significant the impedance effect is, and vice versa.
Suction profiles when accounting for impedance effects are calculated using Equation (5) (solid lines in Fig. 1). Compared to Gardner's method (Equation 1) that does not account for impedance effects (dotted lines in Fig. 1), in this case ǻh k (z=0, t) gradually approaches ǻh i -the longer the delay, the stronger the resistance of the ceramic disk. Note that larger N s secures smoother curves obtained using Equations (5) and (6), where the adequate value of this parameter can be obtained based on the sensitivity analysis. Since the computation is not time consuming, N s = 1000 was adopted as the value large enough for all cases presented.

Experimental validation
The validity of the method was established by considering the experimental data obtained on three quite different materials, provided by [7] on a coarse granular material, and [5] on both a poorly graded sand and an undisturbed silty clay.

Data of [7]
In the apparatus presented in [7] (Fig. 1), a 70 mm diameter and 24 mm height specimen is placed on a Δz d = 5 mm thick ceramic porous disk with an air entry value of 50 kPa, and a saturated hydraulic conductivity K d = 4.02 x 10 -8 m/s. Water exchanges are monitored by using an outer tube (15 mm diameter) that surrounds the thin inner tube (inner diameter 5 mm, outer diameter 8 mm).

Data of [5]
Wayllace and Lu in [5] developed a transient water release and imbibition (TWRI) method for determining the WRC and HCF of two materials along both the drying and wetting paths. In this device, they imposed, through the axis translation method, two suction increments to drain water from the soil specimen, followed by a suction decrease, allowing for subsequent water imbibition. The TWRI apparatus consisted of: • a flow cell accommodating a soil specimen of 60.7 mm diameter placed on 300 kPa HAEV ceramic disk (saturated hydraulic conductivity K d = 2.5x10 -9 m/s, thickness ǻz d = 3.2 mm), • a pressure regulator connected to cell top, • a water jar placed on a weight scale connected to the cell bottom to collect the drained outflow (more details in [5]).

Validation of the method
Gardner's, Kunze & Kirkham's and our method are now compared with the transient outflow data of the three materials presented. The data sets for each of them are related to the first suction step, during which the impedance effects are the most significant.

Coarse volcanic substrate ([7])
By   Fitting our experimental data following Kunze and Kirkham's method provided Ȝ 1 2 = 0.097 and t RP = 2750 s, with a = 10, giving D(h k ) = 2.17x10 -6 m 2 /s and finally K(h k ) = 1.82x10 -6 m/s. However, the choice of the adequate theoretical curve (value of a) is somewhat operator-dependent, especially for higher values of a (> 0.5), where some of the curves are almost overlapping. A wrong choice of parameter a can lead to significantly different values of K(h k ), up to one or two orders of magnitude.
The best fit between Equation (6) and experimental data is obtained for D(h k ) = 1.2x10 -6 m 2 /s, which finally gives K(h k ) = 1.04 x 10 -6 m/s. The Figure shows excellent agreement between experimental data and both Kunze & Kirkham and our method. Also, both methods confirm the occurrence of impedance effects, since K(h k ) > K d in both cases. Unsurprisingly, the extracted volume estimated by Gardner's method (D(h k ) = 7.5x10 -8 m 2 /s) for times smaller than 1 h is higher than those measured and calculated with the two other methods.

Poorly graded sand and Undisturbed silty clay ([5])
In Fig. 3 are presented the same kind of data as in Fig. 2, for the poorly graded sand. In this case H s = 2.67 cm, ǻh i = 0.2 m and V = 4.66 x 10 -6 m 3 , thus giving C(h k ) = 0.302 m -1 . The best agreement between measurements and Equations (2) and (6) is obtained for D(h k ) = 1.0x10 -8 m 2 /s (K(h k ) = 3.0x10 -9 m/s) and D(h k ) = 1.8x10 -8 m 2 /s (K(h k ) = 5.4x10 -9 m/s), respectively. In case of Kunze & Kirkham's method, the best fitting theoretical curve is a = 0.389 (Ȝ 1 2 = 1.323) with t RP = 23700 s, which finally gives K(h k ) = 6.9x10 -9 m/s. For the undisturbed silty clay (Fig. 4), H s = 2.41 cm, ǻh i = 0.2 m and V = 1 x 10 -6 m 3 , thus giving C(h k ) = 0.07 m -1 . The value of D(h k ) is adjusted to 8x10 -8 and 5.0x10 -7 m 2 /s for Equations (2) and (6), respectively. Also, the theoretical curve at a = 0.5 (Ȝ 1 2 = 1.16) shows the best agreement with experimental data (not in small times) for t RP = 2300 s, leading to K(h k ) = 1.6x10 -9 m/s. Based on the data from both our method and Kunze & Kirkham's one, it can be concluded that the impedance effect does occur (with K(h k ) > K d for both soils), especially in case of the silty clay where the calculated value of K(h k ) is about an order of magnitude larger than K d for both methods. Unsurprisingly, Gardner's method shows significantly lower K(h k ) values, because of the perturbation caused by the low hydraulic conductivity ceramic disk.

Conclusion
The experimental data from various materials analyzed (a coarse green roof substrate, a poorly graded sand and an undisturbed silty clay) showed that the proposed simple analytical method fairly well accounts for the impedance effects of the ceramic disk. This method is believed to be more reliable than Kunze & Kirkham's graphical method, especially in the case of significant impedance effect, because it is not dependent of the difficulty in choosing the best fitting theoretical curve among the family of curves provided by Kunze & Kirkham. The proposed method, based on the analytical resolution of the water transfer equations in the different parts of the system, only requires the accurate monitoring of outflow measurements, a requirement that is typical of any method of determining the hydraulic conductivity of multiphase porous material. Compared to numerical back analyses method, our method provides the values of hydraulic conductivity without the need to assume a parametric expression for the hydraulic conductivity function. Also, this analytical method is considered simpler in the sense that it does not require the use of any numerical simulations with optimization algorithms, since the analysis of outflow data and the derivation of hydraulic conductivity value is much more straightforward.