The motivation for this paper is the practical problem of interpreting bi-exponential rate-of-rise curves that are observed during many field piezometric tests. In the authors' previous study, a laboratory W-tube system of water flow through two samples of sand was introduced with an adequate mathematical model. The desired bi-exponential character of water flow was obtained by placing two different samples of sand in separate but connected columns. In the present paper, a so-called inverse problem is solved. The optimization procedure is applied in order to jointly estimate a pair of hydraulic conductivity values based on experimentally recorded bi-exponential rate-of-rise curves. The obtained values of hydraulic conductivities are presented and compared to the values determined from independent constant-head permeability tests conducted for the analyzed sands. The results of this identification procedure varied in accuracy. The mean percent errors between the hydraulic conductivity values measured jointly and independently for the analyzed experimental series were in the range of 8.8 to 37.2%. The discussion presents the restrictions of this interpretational method and suggests further modeling plans.

## INTRODUCTION

Piezometric permeability tests are usually interpreted by assuming homogeneity of water-bearing formations, however many of these formations are characterized by the presence of different systems of void space (Shapiro & Hsieh 1998; Wu *et al.* 2005; Altunkaynak 2007; Alexander *et al.* 2011). In such cases, qualitative deviations in the rise of the water level can be observed (Wang *et al.* 2004; Marciniak *et al.* 2010, 2013). The research described here is motivated by some pressure-induced permeability tests that result in recording bi-exponential rate-of-rise curves which clearly differ from the usual exponential rise of the water level. The bi-exponential curves may indicate the existence of two distinct pathways for water flow within a water-bearing formation (Szymkiewicz 2008; Balogun *et al.* 2009).

In the authors' earlier paper (Marciniak *et al.* 2013), a laboratory model of an inhomogeneous formation, called W-tube system, was considered. Numerous experiments were conducted in this system for pairs of sand samples representing a wide range of permeability values, which resulted in recording bi-exponential curves that were successively interpreted by applying an adequate mathematical model.

This paper aims to present the procedure of a joint estimation of hydraulic conductivities for two sand samples in the W-tube system, whenever a bi-exponential curve is observed. This so-called inverse problem is solved using optimization techniques along with the results of tests and the model for W-tube response. The obtained values of hydraulic conductivity are presented and compared to the values determined from constant-head permeability tests conducted for the analyzed sands. The discussion presents possible sources of errors, restrictions of the considered interpretation method, and suggestions for further modeling.

## MOTIVATION

*et al.*1965, 1967; Bouwer & Rice 1976; Bredehoeft & Papadopulos 1980; Cardiff

*et al.*2011; Chen

*et al.*2012). Nearly 12% of these tests resulted in obtaining bi-exponential rate-of-rise curves (Marciniak

*et al.*2013). Any attempts to find an exponent that would characterize the curves have proved to be unsuccessful (Figure 1).

Some piezometers, where bi-exponential movement was observed, are screened in Late Cretaceous limestone rubble. The filter zone of others is located within layered formations consisting of different Quaternary sands. These types of lithology suggest the possible presence of dual-permeability flow conditions within the screened formations that may be responsible for the deviations of the exponential rise of the water level.

The authors of this paper introduced a triple-columned system (Marciniak *et al.* 2013) that allows observing bi-exponential water level movement in laboratory conditions along with an adequate mathematical model of water flow. Owing to its shape, the system was called a ‘W-tube’ and, to some degree, is considered to re-enact the hydraulic properties of field piezometric tests in dual-permeability formations. The desired character of water flow was obtained by placing two different samples of sand in separate but connected columns (Wolny 2012; Marciniak *et al.* 2013). The laboratory nature of the presented research was dictated by the possibility of selecting numerous sets of hydraulic conductivity values for testing. Moreover, laboratory tests also allowed verifying the values of conductivities derived from the bi-exponential rate-of-rise curve with the values obtained for the analyzed sand samples using the constant-head method. The verification of research results would be severely restricted in the field.

The successful solution of the inverse problem, i.e., determining the hydraulic conductivities of two sands in the W-tube based on the experimentally observed bi-exponential rate-of-rise curve, can serve as an idea to elaborate interpretation algorithms based on permeability tests such as the PARAMEX, slug or boil test. Such algorithms could be implemented in field conditions, whenever a bi-exponential water level rise is observed. This would increase the reliability of piezometric permeability tests in heterogeneous formations.

## THE PRESSURE-INDUCED PERMEABILITY FIELD TEST

The pressure-induced permeability test is conducted in the following manner. First, the upper part of a piezometer is sealed off with a sealing device. Simultaneously, a probe is placed inside the piezometer to measure water level fluctuations. During the test, air is pumped into the sealed piezometer to decrease the water level. Next, pumping is stopped and the water level stabilizes. Afterwards, the air valve of the sealing device is opened and the air inside the piezometer decompresses, leading to the free rise of the water level to its initial state. The rate at which water returns to its initial level determines the hydraulic conductivity of the analyzed aquifer.

The mathematical model of water level rise within the piezometer characterizes two possible cases: damped oscillations or aperiodic movement (Krauss 1974, 1977). By using the model in the identification algorithms of the PARAMEX test, it is possible to calculate the hydraulic conductivity of an aquifer (Marciniak 2012).

*h*(

*t*), can be described by the following exponential function: where

*h*

_{0}is the initial water level (cm),

*λ*< 0 is an exponent dependent on the hydraulic conductivity (1/s).

Tests that resulted in observing bi-exponential water movement were examined thoroughly. Initially, piezometer damage, i.e., the presence of faulty sand packs around the piezometers and leaky isolating clay seals, was taken into consideration. Such types of damage have been observed during pressure-induced permeability tests. However, in such cases, the first stage of water level rise is rapid and short, and the next stage is exponential. In order to find hydraulic connections resulting in the bi-exponential behavior of the water level, several laboratory systems of water flow have been investigated by Marciniak *et al.* (2013). Only the W-tube model of water flow presented in this paper allowed such behavior to be observed. This system of water flow indicates that a weak hydraulic connection between the two subdomains of a dual-permeability medium is required to observe curves deviating from the exponential rise.

## THE LABORATORY W-TUBE MODEL OF A PIEZOMETRIC TEST

The preparation of a test requires filling all columns with water to the same level. Afterwards, the water level in the central column is displaced down and the water in the outer columns is displaced up by pumping air into the central column using an air compressor. The water level stabilizes over time. A test is initialized at the same moment when the air valve located at the top of the central column is opened, causing water to return to the initial level due to its filtration through the samples in the outer columns. Since the W-tube is equipped with two valves, V1 and V2, it is also possible to perform experiments for singles samples of sand. Closing one of the valves lead to a U-tube system with a possibility of generating exponential curves typical for a homogeneous water-bearing medium.

*S*, are equal.

*S*

_{1}and

*S*

_{2}are the cross-sections of the samples of sand,

*L*

_{1}and

*L*

_{2}denote the heights of these samples,

*k*

_{1}and

*k*

_{2}are their hydraulic conductivities. Initially, the level of water in the central column is lower than in the adjacent columns and serves as the reference point for the measurements of the water level

*h*

_{r}in the central column and the

*h*

_{1}and

*h*

_{2}levels in the outer columns. The proposed mathematical model neglects the inertial forces related to the motion of water in the columns and in the pore space of the samples. The interaction forces between the fluid and the water and the porous material are approximated according to the linear Darcy's law. Taking into consideration the hydrostatic forces caused by the water columns as well as the gravitational forces and the interaction forces between the water and the porous material, the following balance equations for the pore water can be written (Rehbinder 1992; Marciniak

*et al.*2013): where

*ρ*is the density of water,

*g*is the acceleration due to gravity, and the derivative

*dh*/

_{i}*dt*denotes the rate at which the water level changes. This change is equal to the filtration velocities in the appropriate samples.

*dt*can be given by the following equation: After conducting several mathematical calculations described by the authors in a previous paper (Marciniak

*et al.*2013), the following formula was obtained: where the constant

*B*is derived from the following equation: and

*λ*

_{1,2}equals: Furthermore, the constants

*A*

_{1}and

*A*

_{2}are related to the geometric parameters of the analyzed samples of sand:

*S*

_{1},

*S*

_{2},

*L*

_{1},

*L*

_{2}, and to their hydraulic conductivities:

*k*

_{1}and

*k*

_{2}: By solving Equation (4), it is possible to determine the bi-exponential response of the water level in the central column as a function of time. The same approach can also be used to model water flow in a U-tube system, when one of the valves is closed. All of the other assumptions are identical. It should be underlined that the W-tube model re-enacts the results of field piezometric tests conducted in heterogeneous formations, i.e., the rate-of-rise curves, rather than the exact character of water flow occurring during such tests. However, there are many similarities between the field and laboratory setups, which will allow modifying the W-tube model in the future to fully describe the field situation.

## MATERIALS AND METHODS

^{−5}to 10

^{−3}m/s. Their conductivity values were determined empirically from the grain size distributions and experimentally in a permeameter using the constant-head method (Table 1). Moreover, minor adjustments to the W-tube apparatus made it possible to perform additional constant-head permeability measurements for all samples placed in the small cylindrical chambers. Since testing required filling the W-tube with the same type of sand more than once, the hydraulic conductivity of every analyzed sample was known. The results of hydraulic conductivity measurements performed in the W-tube are shown in Figure 3.

. | . | Hydraulic conductivity k [m/s]. | |||
---|---|---|---|---|---|

No. . | Sand type . | From the Hazen equation . | From the USBSC equation . | Based on the constant-head test in a permeameter . | Based on the constant-head test in the U-tube . |

I | Very coarse sand | 7.81 × 10^{−3} | 3.77 × 10^{−3} | 1.71 × 10^{−3} | (2.24 ± 0.25) × 10^{−3} |

II | Gravelly sand | 3.38 × 10^{−4} | 4.90 × 10^{−4} | 3.04 × 10^{−4} | (1.66 ± 0.23) × 10^{−4} |

III | Fine sand | 5.10 × 10^{−5} | 2.74 × 10^{−5} | 5.35 × 10^{−5} | (4.99 ± 0.96) × 10^{−5} |

IV | Silty sand | n/a* | 1.08 × 10^{−5} | 1.81 × 10^{−5} | (1.42 ± 1.21) × 10^{−5} |

. | . | Hydraulic conductivity k [m/s]. | |||
---|---|---|---|---|---|

No. . | Sand type . | From the Hazen equation . | From the USBSC equation . | Based on the constant-head test in a permeameter . | Based on the constant-head test in the U-tube . |

I | Very coarse sand | 7.81 × 10^{−3} | 3.77 × 10^{−3} | 1.71 × 10^{−3} | (2.24 ± 0.25) × 10^{−3} |

II | Gravelly sand | 3.38 × 10^{−4} | 4.90 × 10^{−4} | 3.04 × 10^{−4} | (1.66 ± 0.23) × 10^{−4} |

III | Fine sand | 5.10 × 10^{−5} | 2.74 × 10^{−5} | 5.35 × 10^{−5} | (4.99 ± 0.96) × 10^{−5} |

IV | Silty sand | n/a* | 1.08 × 10^{−5} | 1.81 × 10^{−5} | (1.42 ± 1.21) × 10^{−5} |

*The following condition is not satisfied: 0.1 < d_{10} < 3.

The cylindrical chambers of the apparatus were filled with samples of sand either to the height of 25 or 50 cm (in order to double the amount of curves obtained when the chambers are filled to one height). To analyze the dual-permeability of all sand combinations, six experimental series were carried out, with two sand samples tested in each series: I_II, I_III, I_IV, II_III, II_IV, and III_IV.

A total of 24 rate-of-rise curves were recorded for single samples of sand during U-tube tests (with one valve of the apparatus closed), and another 24 curves were recorded for various combinations of sample type and height during the W-tube tests.

*k*were compared with the values derived from constant-head measurements (in accordance with Darcy's law)

_{W}*k*performed in the W-tube apparatus for each sand sample by calculating the absolute and percent errors (Equations (8) and (9), respectively): Furthermore, theoretical rate-of-rise curves were generated for the numerically obtained

_{D}*k*values. To assess their agreement with the experimental curves, a root mean square error (RMSE) was calculated for every experimentally recorded curve and the corresponding results of mathematical modelling: where

_{W}*s*is the water level recorded during an experiment,

_{D}*s*is the water level calculated from the mathematical model based on numerically obtained hydraulic conductivities

_{W}*k*,

_{W}*n*is the number of measurements.

It should be noted that the agreement of the experimental curves and the ones generated using the mathematical model for *k _{D}* values from constant-head tests was discussed in the authors' previous paper (Marciniak

*et al.*2013).

## RESULTS

*k*and the values calculated based on the mathematical model using numerical optimization techniques

_{D}*k*are presented in Tables 2–7. The values of absolute and percent errors between

_{W}*k*and

_{D}*k*are given in the above-mentioned tables. Moreover, examples of rate-of-rise curves are depicted in Figures 4–9 for every experimental series. In each figure three curves are presented: (1) the experimentally recorded rate-of-rise curve for an experiment with two sand types fully filling the chambers of the apparatus; (2) the corresponding curve calculated using the mathematical model for hydraulic conductivities derived from numerical optimization (

_{W}*k*values); and (3) the corresponding curve calculated using the mathematical model for hydraulic conductivities obtained from constant-head tests in the apparatus (

_{W}*k*values). A logarithmic timescale was used for each figure. Every set of curves

_{D}*h*=

*h*(

*t*) was normalized using the following equation (Equation (11)): Furthermore, a breakdown of hydraulic conductivity identification results (

*k*and

_{D}*k*values) for all experimental series is shown in Figure 10.

_{W}Experiment . | Sand sample . | Darcy test k [m/s]
. _{D} | W-tube test k [m/s]
. _{W} | Absolute error Δk [m/s]
. | Percent error δ [%]
. _{k%} |
---|---|---|---|---|---|

I25 | I | 3.31 × 10^{−3} | 2.87 × 10^{−3} | 4.40 × 10^{−4} | 13.3 |

I50 | I | 3.31 × 10^{−3} | 2.96 × 10^{−3} | 3.50 × 10^{−4} | 10.6 |

II25 | II | 2.88 × 10^{−4} | 3.00 × 10^{−4} | 1.20 × 10^{−5} | 4.2 |

II50 | II | 2.88 × 10^{−4} | 2.33 × 10^{−4} | 5.50 × 10^{−5} | 19.1 |

I25_II25 | I | 3.31 × 10^{−3} | 3.42 × 10^{−3} | 1.10 × 10^{−4} | 3.3 |

II | 2.88 × 10^{−4} | 2.77 × 10^{−4} | 1.10 × 10^{−5} | 3.8 | |

I25_II50 | I | 3.31 × 10^{−3} | 3.60 × 10^{−3} | 2.90 × 10^{−4} | 8.8 |

II | 2.88 × 10^{−4} | 2.43 × 10^{−4} | 4.50 × 10^{−5} | 15.6 | |

I50_II25 | I | 3.31 × 10^{−3} | 3.42 × 10^{−3} | 1.10 × 10^{−4} | 3.3 |

II | 2.88 × 10^{−4} | 3.03 × 10^{−4} | 1.50 × 10^{−5} | 5.2 | |

I50_II50 | I | 3.31 × 10^{−3} | 3.52 × 10^{−3} | 2.10 × 10^{−4} | 6.3 |

II | 2.88 × 10^{−4} | 2.52 × 10^{−4} | 3.60 × 10^{−5} | 12.5 | |

| | | average | 1.40 × 10 ^{−4} | 8.8 |

Experiment . | Sand sample . | Darcy test k [m/s]
. _{D} | W-tube test k [m/s]
. _{W} | Absolute error Δk [m/s]
. | Percent error δ [%]
. _{k%} |
---|---|---|---|---|---|

I25 | I | 3.31 × 10^{−3} | 2.87 × 10^{−3} | 4.40 × 10^{−4} | 13.3 |

I50 | I | 3.31 × 10^{−3} | 2.96 × 10^{−3} | 3.50 × 10^{−4} | 10.6 |

II25 | II | 2.88 × 10^{−4} | 3.00 × 10^{−4} | 1.20 × 10^{−5} | 4.2 |

II50 | II | 2.88 × 10^{−4} | 2.33 × 10^{−4} | 5.50 × 10^{−5} | 19.1 |

I25_II25 | I | 3.31 × 10^{−3} | 3.42 × 10^{−3} | 1.10 × 10^{−4} | 3.3 |

II | 2.88 × 10^{−4} | 2.77 × 10^{−4} | 1.10 × 10^{−5} | 3.8 | |

I25_II50 | I | 3.31 × 10^{−3} | 3.60 × 10^{−3} | 2.90 × 10^{−4} | 8.8 |

II | 2.88 × 10^{−4} | 2.43 × 10^{−4} | 4.50 × 10^{−5} | 15.6 | |

I50_II25 | I | 3.31 × 10^{−3} | 3.42 × 10^{−3} | 1.10 × 10^{−4} | 3.3 |

II | 2.88 × 10^{−4} | 3.03 × 10^{−4} | 1.50 × 10^{−5} | 5.2 | |

I50_II50 | I | 3.31 × 10^{−3} | 3.52 × 10^{−3} | 2.10 × 10^{−4} | 6.3 |

II | 2.88 × 10^{−4} | 2.52 × 10^{−4} | 3.60 × 10^{−5} | 12.5 | |

| | | average | 1.40 × 10 ^{−4} | 8.8 |

Experiment . | Sand sample . | Darcy test k [m/s]
. _{D} | W-tube test k [m/s]
. _{W} | Absolute error Δk [m/s]
. | Percent error δ [%]
. _{k%} |
---|---|---|---|---|---|

I25 | I | 3.48 × 10^{−3} | 3.11 × 10^{−3} | 3.70 × 10^{−4} | 10.6 |

I50 | I | 3.48 × 10^{−3} | 3.46 × 10^{−3} | 2.00 × 10^{−5} | 0.6 |

III25 | III | 7.71 × 10^{−5} | 1.11 × 10^{−4} | 3.39 × 10^{−5} | 44.0 |

III50 | III | 7.71 × 10^{−5} | 8.60 × 10^{−5} | 8.90 × 10^{−6} | 11.5 |

I25_III25 | I | 3.48 × 10^{−3} | 3.59 × 10^{−3} | 1.10 × 10^{−4} | 3.2 |

III | 7.71 × 10^{−5} | 1.12 × 10^{−4} | 3.49 × 10^{−5} | 45.3 | |

I25_III50 | I | 3.48 × 10^{−3} | 3.74 × 10^{−3} | 2.60 × 10^{−4} | 7.5 |

III | 7.71 × 10^{−5} | 7.91 × 10^{−5} | 2.00 × 10^{−6} | 2.6 | |

I50_III25 | I | 3.48 × 10^{−3} | 3.85 × 10^{−3} | 3.70 × 10^{−4} | 10.6 |

III | 7.71 × 10^{−5} | 1.15 × 10^{−4} | 3.79 × 10^{−5} | 49.2 | |

I50_III50 | I | 3.48 × 10^{−3} | 3.97 × 10^{−3} | 4.90 × 10^{−4} | 14.1 |

III | 7.71 × 10^{−5} | 8.63 × 10^{−5} | 9.20 × 10^{−6} | 11.9 | |

| | | average | 1.46 × 10 ^{−4} | 17.6 |

Experiment . | Sand sample . | Darcy test k [m/s]
. _{D} | W-tube test k [m/s]
. _{W} | Absolute error Δk [m/s]
. | Percent error δ [%]
. _{k%} |
---|---|---|---|---|---|

I25 | I | 3.48 × 10^{−3} | 3.11 × 10^{−3} | 3.70 × 10^{−4} | 10.6 |

I50 | I | 3.48 × 10^{−3} | 3.46 × 10^{−3} | 2.00 × 10^{−5} | 0.6 |

III25 | III | 7.71 × 10^{−5} | 1.11 × 10^{−4} | 3.39 × 10^{−5} | 44.0 |

III50 | III | 7.71 × 10^{−5} | 8.60 × 10^{−5} | 8.90 × 10^{−6} | 11.5 |

I25_III25 | I | 3.48 × 10^{−3} | 3.59 × 10^{−3} | 1.10 × 10^{−4} | 3.2 |

III | 7.71 × 10^{−5} | 1.12 × 10^{−4} | 3.49 × 10^{−5} | 45.3 | |

I25_III50 | I | 3.48 × 10^{−3} | 3.74 × 10^{−3} | 2.60 × 10^{−4} | 7.5 |

III | 7.71 × 10^{−5} | 7.91 × 10^{−5} | 2.00 × 10^{−6} | 2.6 | |

I50_III25 | I | 3.48 × 10^{−3} | 3.85 × 10^{−3} | 3.70 × 10^{−4} | 10.6 |

III | 7.71 × 10^{−5} | 1.15 × 10^{−4} | 3.79 × 10^{−5} | 49.2 | |

I50_III50 | I | 3.48 × 10^{−3} | 3.97 × 10^{−3} | 4.90 × 10^{−4} | 14.1 |

III | 7.71 × 10^{−5} | 8.63 × 10^{−5} | 9.20 × 10^{−6} | 11.9 | |

| | | average | 1.46 × 10 ^{−4} | 17.6 |

Experiment . | Sand sample . | Darcy test k [m/s]
. _{D} | W-tube test k [m/s]
. _{W} | Absolute error Δk [m/s]
. | Percent error δ [%]
. _{k%} |
---|---|---|---|---|---|

I25 | I | 3.30 × 10^{−3} | 3.46 × 10^{−3} | 1.60 × 10^{−4} | 4.8 |

I50 | I | 3.30 × 10^{−3} | 3.45 × 10^{−3} | 1.5 × 10^{−4} | 4.5 |

IV25 | IV | 1.90 × 10^{−5} | 1.50 × 10^{−5} | 4.00 × 10^{−6} | 21.1 |

IV50 | IV | 1.90 × 10^{−5} | 1.85 × 10^{−5} | 5.00 × 10^{−7} | 2.6 |

I25_IV25 | I | 3.30 × 10^{−3} | 4.75 × 10^{−3} | 1.45 × 10^{−3} | 43.9 |

IV | 1.90 × 10^{−5} | 2.11 × 10^{−5} | 2.10 × 10^{−6} | 11.1 | |

I25_IV50 | I | 3.30 × 10^{−3} | 4.52 × 10^{−3} | 1.22 × 10^{−3} | 37.0 |

IV | 1.90 × 10^{−5} | 2.29 × 10^{−5} | 3.90 × 10^{−6} | 20.5 | |

I50_IV25 | I | 3.30 × 10^{−3} | 4.00 × 10^{−3} | 7.00 × 10^{−4} | 21.2 |

IV | 1.90 × 10^{−5} | 2.00 × 10^{−5} | 1.00 × 10^{−6} | 5.3 | |

I50_IV50 | I | 3.30 × 10^{−3} | 4.53 × 10^{−3} | 1.23 × 10^{−3} | 37.3 |

IV | 1.90 × 10^{−5} | 2.04 × 10^{−5} | 1.40 × 10^{−6} | 7.4 | |

| | | average | 4.11 × 10 ^{−4} | 18.1 |

Experiment . | Sand sample . | Darcy test k [m/s]
. _{D} | W-tube test k [m/s]
. _{W} | Absolute error Δk [m/s]
. | Percent error δ [%]
. _{k%} |
---|---|---|---|---|---|

I25 | I | 3.30 × 10^{−3} | 3.46 × 10^{−3} | 1.60 × 10^{−4} | 4.8 |

I50 | I | 3.30 × 10^{−3} | 3.45 × 10^{−3} | 1.5 × 10^{−4} | 4.5 |

IV25 | IV | 1.90 × 10^{−5} | 1.50 × 10^{−5} | 4.00 × 10^{−6} | 21.1 |

IV50 | IV | 1.90 × 10^{−5} | 1.85 × 10^{−5} | 5.00 × 10^{−7} | 2.6 |

I25_IV25 | I | 3.30 × 10^{−3} | 4.75 × 10^{−3} | 1.45 × 10^{−3} | 43.9 |

IV | 1.90 × 10^{−5} | 2.11 × 10^{−5} | 2.10 × 10^{−6} | 11.1 | |

I25_IV50 | I | 3.30 × 10^{−3} | 4.52 × 10^{−3} | 1.22 × 10^{−3} | 37.0 |

IV | 1.90 × 10^{−5} | 2.29 × 10^{−5} | 3.90 × 10^{−6} | 20.5 | |

I50_IV25 | I | 3.30 × 10^{−3} | 4.00 × 10^{−3} | 7.00 × 10^{−4} | 21.2 |

IV | 1.90 × 10^{−5} | 2.00 × 10^{−5} | 1.00 × 10^{−6} | 5.3 | |

I50_IV50 | I | 3.30 × 10^{−3} | 4.53 × 10^{−3} | 1.23 × 10^{−3} | 37.3 |

IV | 1.90 × 10^{−5} | 2.04 × 10^{−5} | 1.40 × 10^{−6} | 7.4 | |

| | | average | 4.11 × 10 ^{−4} | 18.1 |

Experiment . | Sand sample . | Darcy test k [m/s]
. _{D} | W-tube test k [m/s]
. _{W} | Absolute error Δk [m/s]
. | Percent error δ [%]
. _{k%} |
---|---|---|---|---|---|

II25 | II | 2.12 × 10^{−4} | 2.16 × 10^{−4} | 4.00 × 10^{−6} | 1.9 |

II50 | II | 2.12 × 10^{−4} | 2.03 × 10^{−4} | 9.00 × 10^{−6} | 4.2 |

III25 | III | 6.21 × 10^{−5} | 1.11 × 10^{−4} | 4.89 × 10^{−5} | 78.7 |

III50 | III | 6.21 × 10^{−5} | 7.97 × 10^{−5} | 1.76 × 10^{−5} | 28.3 |

II25_III25 | II | 2.12 × 10^{−4} | 2.77 × 10^{−4} | 6.50 × 10^{−5} | 30.7 |

III | 6.21 × 10^{−5} | 1.15 × 10^{−4} | 5.29 × 10^{−5} | 85.2 | |

II25_III50 | II | 2.12 × 10^{−4} | 2.28 × 10^{−4} | 1.60 × 10^{−5} | 7.5 |

III | 6.21 × 10^{−5} | 7.97 × 10^{−5} | 1.76 × 10^{−5} | 28.3 | |

II50_III25 | II | 2.12 × 10^{−4} | 1.59 × 10^{−4} | 5.30 × 10^{−5} | 25.0 |

III | 6.21 × 10^{−5} | 1.37 × 10^{−4} | 7.49 × 10^{−5} | 120.6 | |

II50_III50 | II | 2.12 × 10^{−4} | 2.02 × 10^{−4} | 1.00 × 10^{−5} | 4.7 |

III | 6.21 × 10^{−5} | 8.17 × 10^{−5} | 1.96 × 10^{−5} | 31.6 | |

| | | average | 3.24 × 10 ^{−5} | 37.2 |

Experiment . | Sand sample . | Darcy test k [m/s]
. _{D} | W-tube test k [m/s]
. _{W} | Absolute error Δk [m/s]
. | Percent error δ [%]
. _{k%} |
---|---|---|---|---|---|

II25 | II | 2.12 × 10^{−4} | 2.16 × 10^{−4} | 4.00 × 10^{−6} | 1.9 |

II50 | II | 2.12 × 10^{−4} | 2.03 × 10^{−4} | 9.00 × 10^{−6} | 4.2 |

III25 | III | 6.21 × 10^{−5} | 1.11 × 10^{−4} | 4.89 × 10^{−5} | 78.7 |

III50 | III | 6.21 × 10^{−5} | 7.97 × 10^{−5} | 1.76 × 10^{−5} | 28.3 |

II25_III25 | II | 2.12 × 10^{−4} | 2.77 × 10^{−4} | 6.50 × 10^{−5} | 30.7 |

III | 6.21 × 10^{−5} | 1.15 × 10^{−4} | 5.29 × 10^{−5} | 85.2 | |

II25_III50 | II | 2.12 × 10^{−4} | 2.28 × 10^{−4} | 1.60 × 10^{−5} | 7.5 |

III | 6.21 × 10^{−5} | 7.97 × 10^{−5} | 1.76 × 10^{−5} | 28.3 | |

II50_III25 | II | 2.12 × 10^{−4} | 1.59 × 10^{−4} | 5.30 × 10^{−5} | 25.0 |

III | 6.21 × 10^{−5} | 1.37 × 10^{−4} | 7.49 × 10^{−5} | 120.6 | |

II50_III50 | II | 2.12 × 10^{−4} | 2.02 × 10^{−4} | 1.00 × 10^{−5} | 4.7 |

III | 6.21 × 10^{−5} | 8.17 × 10^{−5} | 1.96 × 10^{−5} | 31.6 | |

| | | average | 3.24 × 10 ^{−5} | 37.2 |

Experiment . | Sand sample . | Darcy test k [m/s]
. _{D} | W-tube test k [m/s]
. _{W} | Absolute error Δk [m/s]
. | Percent error δ [%]
. _{k%} |
---|---|---|---|---|---|

II25 | II | 2.38 × 10^{−4} | 1.95 × 10^{−4} | 4.30 × 10^{−5} | 18.1 |

II50 | II | 2.38 × 10^{−4} | 1.91 × 10^{−4} | 4.70 × 10^{−5} | 19.7 |

IV25 | IV | 1.90 × 10^{−5} | 1.50 × 10^{−5} | 4.00 × 10^{−6} | 21.1 |

IV50 | IV | 1.90 × 10^{−5} | 1.85 × 10^{−5} | 5.00 × 10^{−7} | 2.6 |

II25_IV25 | II | 2.38 × 10^{−4} | 2.22 × 10^{−4} | 1.60 × 10^{−5} | 6.7 |

IV | 1.90 × 10^{−5} | 1.80 × 10^{−5} | 1.00 × 10^{−6} | 5.3 | |

II25_IV50 | II | 2.38 × 10^{−4} | 2.48 × 10^{−4} | 1.00 × 10^{−5} | 4.2 |

IV | 1.90 × 10^{−5} | 2.37 × 10^{−5} | 4.70 × 10^{−6} | 24.7 | |

II50_IV25 | II | 2.38 × 10^{−4} | 2.09 × 10^{−4} | 2.90 × 10^{−5} | 12.2 |

IV | 1.90 × 10^{−5} | 1.69 × 10^{−5} | 2.10 × 10^{−6} | 11.1 | |

II50_IV50 | II | 2.38 × 10^{−4} | 2.14 × 10^{−4} | 2.40 × 10^{−5} | 10.1 |

IV | 1.90 × 10^{−5} | 2.52 × 10^{−5} | 6.20 × 10^{−6} | 32.6 | |

| | | average | 1.56 × 10 ^{−5} | 14.0 |

Experiment . | Sand sample . | Darcy test k [m/s]
. _{D} | W-tube test k [m/s]
. _{W} | Absolute error Δk [m/s]
. | Percent error δ [%]
. _{k%} |
---|---|---|---|---|---|

II25 | II | 2.38 × 10^{−4} | 1.95 × 10^{−4} | 4.30 × 10^{−5} | 18.1 |

II50 | II | 2.38 × 10^{−4} | 1.91 × 10^{−4} | 4.70 × 10^{−5} | 19.7 |

IV25 | IV | 1.90 × 10^{−5} | 1.50 × 10^{−5} | 4.00 × 10^{−6} | 21.1 |

IV50 | IV | 1.90 × 10^{−5} | 1.85 × 10^{−5} | 5.00 × 10^{−7} | 2.6 |

II25_IV25 | II | 2.38 × 10^{−4} | 2.22 × 10^{−4} | 1.60 × 10^{−5} | 6.7 |

IV | 1.90 × 10^{−5} | 1.80 × 10^{−5} | 1.00 × 10^{−6} | 5.3 | |

II25_IV50 | II | 2.38 × 10^{−4} | 2.48 × 10^{−4} | 1.00 × 10^{−5} | 4.2 |

IV | 1.90 × 10^{−5} | 2.37 × 10^{−5} | 4.70 × 10^{−6} | 24.7 | |

II50_IV25 | II | 2.38 × 10^{−4} | 2.09 × 10^{−4} | 2.90 × 10^{−5} | 12.2 |

IV | 1.90 × 10^{−5} | 1.69 × 10^{−5} | 2.10 × 10^{−6} | 11.1 | |

II50_IV50 | II | 2.38 × 10^{−4} | 2.14 × 10^{−4} | 2.40 × 10^{−5} | 10.1 |

IV | 1.90 × 10^{−5} | 2.52 × 10^{−5} | 6.20 × 10^{−6} | 32.6 | |

| | | average | 1.56 × 10 ^{−5} | 14.0 |

Experiment . | Sand sample . | Darcy test k [m/s]
. _{D} | W-tube test k [m/s]
. _{W} | Absolute error Δk [m/s]
. | Percent error δ [%]
. _{k%} |
---|---|---|---|---|---|

III25 | III | 8.09 × 10^{−5} | 1.05 × 10^{−4} | 2.41 × 10^{−5} | 29.8 |

III50 | III | 8.09 × 10^{−5} | 8.41 × 10^{−5} | 3.20 × 10^{−6} | 4.0 |

IV25 | IV | 1.90 × 10^{−5} | 1.50 × 10^{−5} | 4.00 × 10^{−6} | 21.1 |

IV50 | IV | 1.90 × 10^{−5} | 1.85 × 10^{−5} | 5.00 × 10^{−7} | 2.6 |

III25_IV25 | III | 8.09 × 10^{−5} | 1.14 × 10^{−4} | 3.31 × 10^{−5} | 40.9 |

IV | 1.90 × 10^{−5} | 1.78 × 10^{−5} | 1.20 × 10^{−6} | 6.3 | |

III25_IV50 | III | 8.09 × 10^{−5} | 8.66 × 10^{−5} | 5.70 × 10^{−6} | 7.0 |

IV | 1.90 × 10^{−5} | 2.42 × 10^{−5} | 5.20 × 10^{−6} | 27.4 | |

III50_IV25 | III | 8.09 × 10^{−5} | 4.41 × 10^{−5} | 3.68 × 10^{−5} | 45.5 |

IV | 1.90 × 10^{−5} | 3.69 × 10^{−5} | 1.79 × 10^{−5} | 94.2 | |

III50_IV50 | III | 8.09 × 10^{−5} | 9.00 × 10^{−5} | 9.10 × 10^{−6} | 11.2 |

IV | 1.90 × 10^{−5} | 2.74 × 10^{−5} | 8.40 × 10^{−6} | 44.2 | |

| | | average | 1.24 × 10 ^{−5} | 27.9 |

Experiment . | Sand sample . | Darcy test k [m/s]
. _{D} | W-tube test k [m/s]
. _{W} | Absolute error Δk [m/s]
. | Percent error δ [%]
. _{k%} |
---|---|---|---|---|---|

III25 | III | 8.09 × 10^{−5} | 1.05 × 10^{−4} | 2.41 × 10^{−5} | 29.8 |

III50 | III | 8.09 × 10^{−5} | 8.41 × 10^{−5} | 3.20 × 10^{−6} | 4.0 |

IV25 | IV | 1.90 × 10^{−5} | 1.50 × 10^{−5} | 4.00 × 10^{−6} | 21.1 |

IV50 | IV | 1.90 × 10^{−5} | 1.85 × 10^{−5} | 5.00 × 10^{−7} | 2.6 |

III25_IV25 | III | 8.09 × 10^{−5} | 1.14 × 10^{−4} | 3.31 × 10^{−5} | 40.9 |

IV | 1.90 × 10^{−5} | 1.78 × 10^{−5} | 1.20 × 10^{−6} | 6.3 | |

III25_IV50 | III | 8.09 × 10^{−5} | 8.66 × 10^{−5} | 5.70 × 10^{−6} | 7.0 |

IV | 1.90 × 10^{−5} | 2.42 × 10^{−5} | 5.20 × 10^{−6} | 27.4 | |

III50_IV25 | III | 8.09 × 10^{−5} | 4.41 × 10^{−5} | 3.68 × 10^{−5} | 45.5 |

IV | 1.90 × 10^{−5} | 3.69 × 10^{−5} | 1.79 × 10^{−5} | 94.2 | |

III50_IV50 | III | 8.09 × 10^{−5} | 9.00 × 10^{−5} | 9.10 × 10^{−6} | 11.2 |

IV | 1.90 × 10^{−5} | 2.74 × 10^{−5} | 8.40 × 10^{−6} | 44.2 | |

| | | average | 1.24 × 10 ^{−5} | 27.9 |

The calculated RMSE values for the experimental curves and the ones derived from the mathematical model based on numerically obtained hydraulic conductivities *k _{W}* are shown in Table 8.

No. . | Experiment . | RMSE [cm] . | No. . | Experiment . | RMSE [cm] . | No. . | Experiment . | RMSE [cm] . |
---|---|---|---|---|---|---|---|---|

Series I_II | Series I_IV | Series II_IV | ||||||

1. | I25 | 1.124 | 1. | I25 | 1.455 | 1. | II25 | 0.400 |

2. | I50 | 1.141 | 2. | I50 | 0.574 | 2. | II50 | 0.411 |

3. | II25 | 0.396 | 3. | IV25* | 0.380 | 3. | IV25* | 0.380 |

4. | II50 | 0.515 | 4. | IV50* | 0.307 | 4. | IV50* | 0.307 |

5. | I25_II25 | 0.691 | 5. | I25_IV25 | 0.781 | 5. | II25_IV25 | 0.467 |

6. | I25_II50 | 0.777 | 6. | I25_IV50 | 0.691 | 6. | II25_IV50 | 0.740 |

7. | I50_II25 | 0.460 | 7. | I50_IV25 | 0.588 | 7. | II50_IV25 | 0.272 |

8. | I50_II50 | 0.558 | 8. | I50_IV50 | 0.785 | 8. | II50_IV50 | 0.335 |

Series I_III | Series II_III | Series III_IV | ||||||

1. | I25 | 1.057 | 1. | II25 | 0.428 | 1. | III25 | 0.194 |

2. | I50 | 0.578 | 2. | II50 | 0.791 | 2. | III50 | 0.464 |

3. | III25 | 0.458 | 3. | III25 | 0.745 | 3. | IV25* | 0.380 |

4. | III50 | 1.336 | 4. | III50 | 0.549 | 4. | IV50* | 0.307 |

5. | I25_III25 | 0.722 | 5. | II25_III25 | 0.299 | 5. | III25_IV25 | 0.387 |

6. | I25_III50 | 0.635 | 6. | II25_III50 | 0.551 | 6. | III25_IV50 | 0.325 |

7. | I50_III25 | 0.577 | 7. | II50_III25 | 0.448 | 7. | III50_IV25 | 0.538 |

8. | I50_III50 | 0.730 | 8. | II50_III50 | 0.352 | 8. | III50_IV50 | 0.692 |

No. . | Experiment . | RMSE [cm] . | No. . | Experiment . | RMSE [cm] . | No. . | Experiment . | RMSE [cm] . |
---|---|---|---|---|---|---|---|---|

Series I_II | Series I_IV | Series II_IV | ||||||

1. | I25 | 1.124 | 1. | I25 | 1.455 | 1. | II25 | 0.400 |

2. | I50 | 1.141 | 2. | I50 | 0.574 | 2. | II50 | 0.411 |

3. | II25 | 0.396 | 3. | IV25* | 0.380 | 3. | IV25* | 0.380 |

4. | II50 | 0.515 | 4. | IV50* | 0.307 | 4. | IV50* | 0.307 |

5. | I25_II25 | 0.691 | 5. | I25_IV25 | 0.781 | 5. | II25_IV25 | 0.467 |

6. | I25_II50 | 0.777 | 6. | I25_IV50 | 0.691 | 6. | II25_IV50 | 0.740 |

7. | I50_II25 | 0.460 | 7. | I50_IV25 | 0.588 | 7. | II50_IV25 | 0.272 |

8. | I50_II50 | 0.558 | 8. | I50_IV50 | 0.785 | 8. | II50_IV50 | 0.335 |

Series I_III | Series II_III | Series III_IV | ||||||

1. | I25 | 1.057 | 1. | II25 | 0.428 | 1. | III25 | 0.194 |

2. | I50 | 0.578 | 2. | II50 | 0.791 | 2. | III50 | 0.464 |

3. | III25 | 0.458 | 3. | III25 | 0.745 | 3. | IV25* | 0.380 |

4. | III50 | 1.336 | 4. | III50 | 0.549 | 4. | IV50* | 0.307 |

5. | I25_III25 | 0.722 | 5. | II25_III25 | 0.299 | 5. | III25_IV25 | 0.387 |

6. | I25_III50 | 0.635 | 6. | II25_III50 | 0.551 | 6. | III25_IV50 | 0.325 |

7. | I50_III25 | 0.577 | 7. | II50_III25 | 0.448 | 7. | III50_IV25 | 0.538 |

8. | I50_III50 | 0.730 | 8. | II50_III50 | 0.352 | 8. | III50_IV50 | 0.692 |

*The same IV25 and IV50 samples were used for series I_IV, II_IV, and III_IV.

## DISCUSSION AND CONCLUSIONS

The hydraulic conductivities of the analyzed sand samples identified in accordance with the mathematical model using numerical optimization techniques varied in accuracy. The lowest errors were observed for series I_II. For this series, the mean percent error between the hydraulic conductivity values measured during constant-head tests and the values identified numerically was 8.8%. This indicates a good convergence of research results. Relatively small mean percent errors were obtained for series I_III, I_IV, and II_IV – 17.6%, 18.1%, and 14.0%, respectively. The largest percent errors were obtained for series II_III, especially for the experiments with sand III filling the chamber of the W-tube to the height of 25 cm. For experiments III25, II25_III25, and II50_III25 the percent error for sand III was 78.7%, 85.2%, and 120.6%, respectively. The mean percent error for the whole experimental series was 37.2%.

The authors suggest that the changing consolidation of sand samples was one of the main causes of the observed errors. Since constant-head permeability measurements were always performed at the end of each experimental series, the analyzed samples of sand were probably more consolidated at this point in time than at the beginning of the series. Samples of sand III were the most affected by the changes in consolidation during the experimental series. Relatively high values of percent errors can be observed for the samples of this sand in every series.

In cases where there is only a small contrast of transmissivity between the two analyzed samples of sand, a high percent error was also observed. This is especially apparent in the identification results of experiment III50_IV25 – the percent error was 45.5% for sand III and 94.2% for sand IV.

Furthermore, regularity has also been observed for experiments conducted in the W-tube – percent errors for the sand with a higher value of hydraulic conductivity were usually smaller than for the sand with a lower value. This occurrence can be explained by the fact that the first part of such experiments (when most of the water level rise takes place) is determined by the flow parameters of the sand sample with a higher hydraulic conductivity. Therefore, the water flow through this sample has a greater influence on the recorded rate-of-rise curve and the accuracy of identifying the hydraulic conductivity of this sample must be greater.

All experimentally recorded rate-of-rise curves were compared to the curves calculated for the numerically obtained hydraulic conductivities using the presented mathematical model. RMSE values greater than 1 cm were observed only for three very rapid experiments with the highly permeable sand I in a U-tube configuration. In all other cases, the errors show minor discrepancies of the experimental and theoretical curves. This indicates that the hydraulic conductivity values obtained from calculations accurately reflect the hydraulic parameters of the sands used in laboratory studies.

The proposed method of determining the hydraulic conductivities of two sands in the W-tube can be used as an idea to elaborate interpretation algorithms based on permeability tests, including the pressure-induced PARAMEX test or other piezometric tests, such as the slug or boil test. The necessary algorithms could be implemented in field conditions, whenever a bi-exponential water level rise is observed, which would increase the reliability of piezometric permeability tests conducted in heterogeneous formations.

## ACKNOWLEDGEMENTS

This work was partially supported by the Polish Ministry of Science and Higher Education (NCN Grant No. 0697/B/P01/2010/39, statutory activity – M.K.).