Lab Report: Geotechnical Soil Testing

1. Introduction

This laboratory study consists of two main parts: the one-dimensional consolidation test and the investigation of permeability and flow nets.

In the first part, we conducted the one-dimensional consolidation test to determine important soil properties, including the coefficient of volume change (mv) and the coefficient of consolidation (cv), as per Terzaghi's one-dimensional consolidation theory. These parameters play a crucial role in understanding how soil consolidates under specific loads over time, which is essential for designing stable foundations.

When a structure is built on soil, its weight induces soil consolidation, leading to potential settlement.

The magnitude of settlement depends on the soil's properties and its ability to consolidate. Variations in soil type across a site can result in differential consolidation rates, posing challenges to the structural integrity of buildings. For instance, iconic structures like the Leaning Tower of Pisa and Millennium Tower in San Francisco experienced settlement-related issues due to differential consolidation.

The second part of this study focuses on permeability and flow nets. It involves two tests: the constant head permeameter test and a flow tank experiment. The constant head permeameter test aims to calculate the coefficient of permeability (k) using fine sand as the soil sample. This parameter is crucial for verifying Darcy's Law, a fundamental principle governing the flow of water through soils.

By determining the hydraulic gradient (i) between two points separated by the soil sample, we can calculate the water's velocity through the soil using the equation v = i x k. Additionally, the construction of a flow net diagram allows us to estimate the flow rate through the soil with the equation q = k nfnd HB.

Such information is invaluable when designing structures like reservoirs, dams, or conducting general construction projects that involve digging pits or trenches, helping us choose suitable drainage solutions for efficient water removal.

2. Theory

2.1 Consolidation

Consolidation refers to the phenomenon in which loading causes a fully saturated soil to decrease in volume over time. This reduction in volume is attributed to the expulsion of pore water under excess pore water pressure. Unlike compressibility, consolidation accounts for the time-dependent nature of volume change, making it intrinsically linked to soil permeability (Barnes, 2016).

Terzaghi's Theory of One-Dimensional Consolidation

The rates of soil volume change, pore pressure dissipation, and surface settlements are determined by the rate at which water is expelled from the soil, as proposed by Terzaghi's theory of one-dimensional consolidation. The head (h) responsible for water flow out of a stressed soil element can be calculated as:

h = uγ_w

Where:

u = Excess pore water pressure in the element,

γ_w = Specific unit weight of water.

In a homogenous soil with vertical pore water movement, the hydraulic gradient can be expressed as:

-Δh/Δz = -Δu/Δz (1/γ_w)

According to Darcy's Law, the average water velocity through the soil element is given by:

v = -kγ_w × Δu/Δz

Thus, the velocity gradient across the element is:

Δv/Δz = -kγ_w × (Δ²u)/(Δz²)

If the volume of the soil element changes over time, the rate of volume change per unit time can be expressed as:

dV/dt = -kγ_w × Δu/Δt × dx × dy × dz

This change in volume is equal to the variation in the void space within the element (dx × dy × dz = total element volume). Therefore, the rate of change of void space with respect to time is given by:

e(1 - e₀) × Δe/Δt × dx × dy × dz = e(1 - e₀) × Δe/Δσ' × Δσ'/Δt × dx × dy × dz = -mv × Δu/Δt × dx × dy × dz

Where:

mv = Δe(1 - e₀) / Δσ',

Δσ'/Δt = -Δu/Δt.

This equation illustrates that the rate of dissipation of excess pore water pressure (u) is equal to the rate of increase of effective stress (σ'). Equating equations (5) and (9) and dividing by the total element volume (dx × dy × dz) yields:

Δu/Δt = cv × (Δ²u)/(Δz²) = k × k × (mv × γ_w) × (Δ²u)/(Δz²)

Where:

cv = Coefficient of consolidation,

mv = Coefficient of volume change,

k = Coefficient of permeability.

Therefore, the dimensionless time factor (Tv) can be expressed as:

Tv = (cv × t₉₀) / (d × d)

This theoretical framework assumes that the soil is fully saturated and homogeneous, with incompressible solid particles and pore water. Additional assumptions include vertical flow and no secondary compression, although these assumptions may not hold true in all soil conditions, especially in real-world construction scenarios (Figure 1).

Figure 1: Effect of Load Duration

Coefficient of volume change, mv = Δe(1 - e₀) / (Δσ') (1/Δσ')

Dimensionless time factor, Tv = (cv × t₉₀) / (d × d) = 0.848

2.2 Permeability

The flow of water through soil occurs when there is a difference in hydraulic head between two points. The total head (h_T) at a specific location can be defined using Bernoulli's equation in relation to a datum:

h_T = h_Z + h_B + (v² / 2g)

Where:

h_T = Total head,

h_Z = Position head,

h_B = Piezometric head,

v² / 2g = Velocity head (usually negligible).

Darcy's Law, a fundamental principle, states that the discharge velocity (v) of water is directly proportional to the hydraulic gradient (i), and can be expressed as:

Q_A = v = k_i

Where:

k = Darcy coefficient of permeability (m/s),

Q = Flow rate (m³/s),

A = Cross-sectional area of flow (m²).

Darcy's Law is applicable to most soil types, especially in cases of laminar flow, where it accurately describes water flow. However, in soils with larger pores, turbulent flow may occur, deviating from the ideal behavior described by Darcy's Law.

Flow Nets

Flow nets are graphical representations used to visualize water flow through soils, particularly when water flows beneath or around structures due to hydraulic head differences. A flow net consists of two main types of lines: flow lines and equipotential lines.

Flow lines depict the path of water flow through the soil. While there can be numerous potential flow lines, only a few need to be drawn for estimation purposes, as flow nets are approximations. The channels between flow lines carry an equal flow quantity (∆Q), leading to the calculation of the total seepage flow.

Equipotential lines represent levels of equal total head. As water flows through the soil, it loses energy, resulting in a drop in head. Each equipotential line signifies an equal drop in head, making it possible to calculate the change in head between adjacent lines based on the total change in head and the number of potential drops.

Flow net construction follows specific guidelines, as outlined by Barnes (2016):

• Flow lines and equipotential lines intersect at right angles.
• Sections formed by flow lines and equipotential lines are roughly square (a ≈ b).
• Impermeable boundaries are represented by flowlines.
• Permeable boundaries and water bodies are represented by equipotential lines.

From Darcy's Law and with reference to Figure 2:

∆Q = Aki = -ak∆h_B

Since each equipotential line represents an equal potential drop:

∆h = H/N_d

Since each section of the flow net is roughly square (a ≈ b):

∆Q = -kH/N_d

To calculate the total flow for all channels with unit width (w):

Q = Σ_(c=1)^N_f [-kH/N_d] = -kN_fN_dH

Where:

N_f = Number of flow channels,

N_d = Number of equipotential drops,

H = Total drop in head (hydraulic head difference).

This calculation assumes a homogenous soil with a constant coefficient of permeability (k) throughout the sample.

3. Experimental Procedure

3.1 One-Dimensional Consolidation Test

The one-dimensional consolidation test involved the following steps:

1. The saturated soil sample was loaded into the steel mould within the consolidation cell, following the guidelines specified in BS 1377-5:1990.
2. The initial height of the soil sample was measured to calculate the change in height.
3. The consolidation cell was placed into the consolidation loading apparatus, as shown in Figure 1.
4. An initial loading of 20N was applied from the loading hanger, which was then increased by a factor of 10 to reach a load of 200N applied to the sample due to mechanical leverage from the loading arm.
5. The setup was left undisturbed until the dial gauge settled, signifying that the sample had reached equilibrium.
6. At equilibrium, the final dial gauge reading was taken and multiplied by 0.002 to convert it to millimeters. This value was then subtracted from the initial sample height to determine the equilibrium height.
7. An additional 20N weight was added to the loading hanger, increasing the hanger load to 40N and the load on the sample to 400N.
8. Upon adding the load, a timer was started, and dial gauge readings were recorded at specified time intervals. The timings were in minutes, as this scale is used to plot the graph for finding the time for 90% consolidation (t₉₀).
9. When the dial gauge readings plateaued, the final dial gauge reading for the 400N load was taken, converted to millimeters, and subtracted from the initial height to find the equilibrium height.

3.2 Constant Head Permeameter Test

The constant head permeameter test followed these steps:

1. The cylindrical chamber (as depicted in Figure 2) was filled with the prepared soil in accordance with BS 1377-5:1990. Measurements of the chamber diameter and the distance between the two standpipe outlets were taken.
2. The pump was activated, and the control valve was opened to ensure full saturation of the soil.
3. In the first stage, the outlet tap was fully opened, and water was allowed to flow into an empty container until the water levels in the standpipes settled. Both head 1 and head 2 were recorded once settled. Then, the container was replaced with a measuring cylinder, and a stopwatch was started.
4. The time taken for 1000cm³ of water to exit the system was recorded, after which the tap was closed.
5. In the second stage, the tap was opened slightly less, and the standpipes were allowed to settle. The time was recorded for 500cm³ of water to pass.
6. For the final stage, the tap was opened even less, and the standpipes were allowed to settle again. The time was recorded for 250cm³ of water to flow through.
7. Velocity was calculated using the formula v=qAt, where q represents the flow volume, A is the cross-sectional area of the chamber, and t is the time to collect the flow volume.
8. The hydraulic gradient (i) was calculated using i=(h₂-h₁)/l, where h₂ and h₁ are the heads in the standpipes (with h₂ greater than h₁), and l represents the distance between the standpipe outlets.

3.3 Flow Tank Test

The flow tank test involved the following steps:

1. The tank, as illustrated in Figure 3, was filled with the selected soil sample so that the soil level was between the water inlet/outlet and the bottom of the sheet pile.
2. The pump was then activated, allowing the water levels on either side of the sheet pile to equilibrate.
3. Measurements were taken for the head of water on either side of the sheet pile, the height of the soil on both sides, the width of the container, and the length (B) between the glass sides. These measurements were essential for calculating the flow rate using a flow net.
4. A measuring cylinder was placed under the outlet pipe, and a stopwatch was started.
5. The time required to fill 600cm³ of water was recorded, and this process was repeated three times to allow for the calculation of a mean flow rate (q_measured).

4. Results

4.1 One-Dimensional Consolidation Test

The results of the one-dimensional consolidation test are summarized in Table 2:

Hanger load (N)	Load on sample (N)	Applied pressure (kN/m2)	Final dial gauge (-)	Piston displacement (mm)	Equilibrium height (mm)
0	0	0	0	0	-
20	200	25.460	622.500	1.245	17.760
40	400	50.930	857.000	1.714	17.290

Calculation of mv:

m_v = (H₁-H₂)H₁ × 1/(P₂-P₁) = (17.76-17.29)17.76 × 1/(50.93-25.46) = 0.039 m²/kN

Finding t₉₀:

t₉₀ = 2.8 min¹²

Finding coefficient of consolidation Cv:

t₉₀ = 2.8 min¹² = 470 s

d = ((H₁ - H₂)²)² = ((17.16 - 17.29)²)² = 8.76

C_v = (T_v × d) / t₉₀ = (0.848 × 8.76) / 470 = 0.0176 mm²/s = 0.56 m²/year

4.2 Constant Head Permeameter Test

The results of the constant head permeameter test are presented in Table 6:

Duration of water flow, t (s)	Flow volume, q (cm3)	Potential head 1, h1 (cm)	Potential head 2, h2 (cm)	Velocity, v (cm/s)	Hydraulic gradient, i
110	1000	15.20	47.70	0.22	3.25
82	500	44.50	65.60	0.15	2.11
90	250	73.20	83.00	0.07	0.98

Calculation of velocity and hydraulic gradient:

v = qAt, i = (h₂ - h₁) / l

4.3 Flow Tank Test

The results of the flow tank test are detailed in Table 8:

Measurement number	Duration of water flow (s)	Flow volume (cm3)	Flow rate (cm3/s)
1	87	600	6.9
2	87	600	6.9
3	85	600	7.06

Calculation of flow rate based on the flownet:

q_calculated = k * n_f * n_d * H * B

n_f = number of flow channels = 5, n_d = number of potential drops = 10

H = potential drop = 12.5 cm, k = coefficient of permeability = 0.07 cm/s, B = length of tank = 20 cm

∴ q_calculated = 0.07 * 5 * 10 * 12.5 * 20 = 8.75 cm³/s

q_calculated = 8.75 cm³/s, q_measured = 6.95 cm³/s

% difference = [(8.75 - 6.95) / 6.95] * 100 = 25.90%

5. Discussion

In the one-dimensional consolidation test, values for the coefficient of volume change (mv) and consolidation were obtained. These values are crucial for understanding how this soil type consolidates, which is vital information for foundation design. The graph illustrating the relationship between Dial Gauge Reading and time12 shows the expected trend, and no anomalous results were observed. However, to ensure accuracy and account for sample variation, it is advisable to repeat the experiment with different samples of the same soil type to calculate mean values.

In the calculation of t9012, there is a possibility of human error in drawing the tangent line on the graph. Variations in this tangent could affect the output value and, consequently, the value of cv. To improve accuracy, computer software can be used to plot and read values from the graph. Other errors may have occurred due to parallax error or misreading of the dial gauge, which could be resolved by using a digital dial gauge.

Darcy's Law states that the rate of discharge is directly proportional to the hydraulic gradient. The results of the constant head permeameter test validate this statement, as shown in the figure, which demonstrates a linear relationship between v and i, indicating that v ∝ i. The flow tank experiment would have further verified Darcy's Law by showing that qmeasured ≈ qcalculated; however, in this test, there was a 26% difference between these values. Although Darcy's Law has some assumptions, the likelihood of non-laminar flow occurring in the flow tank is low due to the very low water velocity through the soil and the small soil voids.

Errors in qmeasured could have arisen from inaccurate measurements or timing of flow volume. Since it was not possible to have the measuring vessel perfectly level while collecting the water, the scale on the cylinder may not have aligned with the expected volume of water. Additionally, the value of qcalculated relies on the coefficient of permeability, k, which was calculated from the constant head permeameter test. Any errors in this experiment would directly affect the relationship between qmeasured and qcalculated.

6. Conclusion

The primary objectives of these experiments were to determine the parameters of Terzaghi's one-dimensional consolidation theory (mv and cv) and to validate Darcy's Equation. The parameters of Terzaghi's theory were obtained from the oedometer test. The coefficient of volume change, mv, was calculated based on changes in height and pressure on the sample. A graph of Dial Gauge Reading against time12 was plotted, and the value of t9012 was derived from this graph. This value was used to calculate the coefficient of consolidation, cv. Both of these calculations rely on equations derived from Darcy's Law, contributing to the verification of this law while acknowledging its limitations.

A combination of the constant head permeameter test and the flow tank test was employed to further validate Darcy's Law. The constant head permeameter test illustrated the relationship between water velocity through soil, v, and hydraulic gradient, i. These variables were plotted on a graph and demonstrated to be directly proportional to each other, thus validating Darcy's Law, which states v = ki. However, when comparing the value of qmeasured to qcalculated in the flow tank experiment, Darcy's Law was not fully validated due to a 26% difference between the values.

Despite the potential for errors in these experiments, the results generally aligned with Darcy's Law within its limitations. The parameters for Terzaghi's theory were also successfully obtained, fulfilling the primary objectives of the experiments.

7. References

Barnes, G., 2016. Soil Mechanics: Principles and Practice. 4th ed.

BS 1377-5:1990. Standard methods of test for soils for civil engineering purposes.

Beckett, L., 2022. Sinking feeling: San Francisco’s Millennium Tower is still leaning 3in every year.

8. Appendices

Appendix A: One-dimensional consolidation test

Table 1: Testing details

Date	Soil	Initial height (mm)	Diameter (mm)	Load Multiplier (-)	Gauge factor (mm/div)
19012022	Speswhite Kaolin	19	100	10	0.002

Table 3: Consolidation data

Time	Time12 (min12)	Dial Gauge Reading	Time	Time12 (min12)	Dial Gauge Reading
0s	0.00	622.50	9m00s	3.00	823.00
10s	0.41	692.00	12m15s	3.50	834.00

Appendix B: Constant Head Permeameter test

Table 4: Constant head permeameter geometry

Distance between standpipes, l (cm)	Diameter of permeameter, d (cm)	Area of the sample, A (cm2)
10	7.30	41.85