#### 2.1. Mathematical Analysis of Erodibility Parameters in Persence of Crude Oil

Wilson [

26] developed a non-linear model to estimate erosion rates, based on stabilizing and removing forces and associated moment lengths for particle displacement. The forces that act to remove soil particles in the presence of crude oil are presented in

Figure 1. These forces include the weight of the soil particle

$({w}_{s})$, drag force

$({F}_{d})$, lift forces

$({F}_{L})$, the contact forces between adjacent soil particles

$({F}_{c1},{F}_{c2},\dots ,{F}_{cn})$, and the contact forces between adjacent soil particles and adjacent oil particles

$({F}_{o1},{F}_{o2},\dots ,{F}_{on})$. Particle detachment takes place if the resisting moment is less than the driving moment [

26]. In this study, the Wilson model was modified to include the influence of oil contamination, based on the original framework developed by Wilson [

26]. The detachment of soil particles is expected to occur if the drag force is higher than the cohesive force and weight, relative to the moment around point

A, and can be defined with the introduction of the terms

${M}_{c}$ and

${M}_{co}$ as the following Equations (1)–(4):

in which

${M}_{c}$ refers to the sum of moments of the frictional and cohesive forces,

${M}_{co}$ is the sum of moments exerted by the frictional and cohesive forces on oil particles,

n_{c} is the number of contact areas for soil particles,

n_{co} is the number of contact areas for oil particles,

F_{ci} is the contact forces between adjacent soil particles,

${F}_{coi}$ is the contact forces between adjacent soil and oil particles,

${\sigma}_{ci}$ is the soil particle to particle stress,

${\sigma}_{coi}$ is the oil particle to soil particle stress,

${a}_{i}$ is the contact area,

${\overline{l}}_{i}$ is the moment length for each contact force,

$\alpha $ is the angle slope of channel,

${k}_{v}$ is the volume constant of a spherical particle,

${\rho}_{w}$ and

${\rho}_{s}$ are water and soil particle densities, respectively, and

d is the equivalent diameter of a soil particle.

Chepil [

31] and Wilson [

26] assumed a proportional correlation between

$({F}_{d})$ and

$({F}_{L})$ (i.e.,

${K}_{L}/{K}_{f}={F}_{L}/{F}_{d}$), in which

K_{f} is the proportion of the projected area of the

${F}_{d}$ and

${F}_{L}$ forces, and

K_{L} is the proportion of drag and lift coefficients along with that of the velocities [

26]. Consequently, Equation (1) can be modified as the following Equations (5)–(8):

in which

K_{ls} is a dimensionless parameter dependent on particle size, its orientation within the slope, and the bed;

$S\text{}(=\mathrm{tan}\alpha )$ is channel slope;

f_{c} is a dimensionless parameter based on soil cohesion; and

f_{co} is a dimensionless parameter based on soil and oil cohesions. The values of

f_{co} were derived and calibrated from tests on contaminated soils.

Adapting the mathematical approach proposed by Wilson [

26] and Al-Madhhachi et al. [

24,

25], the time-averaged net force,

F_{n}, acting in the direction of movement of the oil particles, was modified as the following Equation (9):

in which

K_{t} is a factor of the cumulating instantaneous fluid forces,

${\overline{F}}_{d}$ is the time-averaged drag force,

μ_{f} is the coefficient of friction,

μ_{o} is the oil coefficient of cohesion between soil and oil particles as proposed in this study,

${w}_{o}=g({\rho}_{w}-{\rho}_{o}){k}_{v}{{d}_{o}}^{3}$ is the oil particle submerged weight,

d_{o} is the oil particle diameter, and

ρ_{o} is oil density. This study proposed that the oil particle diameter is equivalent to the soil particle diameter (

d_{o} =

d).

Wilson [

26] assumed that the particle exchange time,

t_{e}, was a function of the exit velocity (

V_{e} =

F_{n} t_{e}/

m), in which

m is the particle mass. Incorporating the definition of

V_{e} and Equation (9), the exit velocity in the presence of crude oil can be expressed as the following Equation (10):

in which

${k}_{a}$ is the area constant of a spherical particle and

k_{r} =

k_{v}/

k_{a} is the geometrical proportion for a spherical particle. Equation (10) can be simplified by introducing the fluid factor

K_{n} = (

K_{t} K_{o}/

k_{r}) and the Shields parameter (

τ*) to give as the following Equations (11)–(13):

in which

K_{o} is the velocity JET parameter, as proposed by Al-Madhhachi et al. [

17];

C_{D} is the drag coefficient;

c_{d} is the diffusion constant of the jet;

r is the jet radius;

z_{d} is the height that the drag velocity is acting upon in the jet environment; and

C_{f} is the coefficient of friction in the jet environment. In this study, the particle exchange time was predicted by incorporating the oil coefficient as the following Equation (14):

A probability framework for turbulent forces was developed by Wilson [

26], similar to that developed by Einstein [

32] and Partheniades [

33]. Therefore, the soil erosion or soil detachment rate (

${\epsilon}_{ri}$) in presence of oil particles is defined as the following Equations (15)–(17):

in which Δ

FF_{i} is the fraction finer value for bed materials,

$P$ is the exceedance probability of drag force,

K_{e} is the exposure of the lower particle parameter (i.e., additional time to eliminate neighbouring particles),

μ_{v} is the upper limit of integration of exceedance probability distribution, and

e_{v} is the coefficient of variations. Equation (15) could be further derived following the same procedure outlined by Wilson [

26,

27] and Al-Madhhachi et al. [

24] to achieve the total detachment rate parameters of the Wilson model, including the influence of crude oil as the following Equations (18)–(21):

in which

b_{0} is the detachment parameter (g/m-s-N

^{0.5}),

b_{1} is the threshold parameter (Pa), and

${\mu}_{or}$ is the oil coefficient ratio. It should be noted that

τ* decreases while

${\mu}_{o}$ increases as the soil erosion occurs.

Wilson [

26,

27] used a calibration procedure to empirically derive some parameters included in both the cohesive (

f_{c}) and the exposure (

K_{e}) parameters. This was due to there being little information available about these parameters at the time. Similarly, in this study, the crude oil parameters (

f_{co} and

${\mu}_{or}$) were developed and calibrated using tests on contaminated soils. Both parameters

f_{co} and

${\mu}_{or}$ are functions of soil and oil particle cohesion and contamination time determined by the chemical-physical bonds between soil and oil particles. Based on experimental evidence in this study on soils that JETs were undertaken, the values of

f_{co} and

${\mu}_{or}$ range from 81 to 140 and 18 to 23, respectively.

The definitions of the parameters in the Wilson model (Equations (18)–(21)), with their values, are reported in

Table 1. In the absence of crude oil, the oil parameters can be neglected (i.e.,

${\mu}_{or}$ = 0 and

f_{co} = 0), and the developed model will match the set of equations suggested by Wilson [

26,

27]. The parameters

b_{0} and

b_{1} can be derived by employing curve-fitting techniques that reduce the errors of these functions in relation to measured erosion data obtained from JETs. Al-Madhhachi et al. [

17] developed an Microsoft Excel spread sheet to derive parameters

b_{0} and

b_{1} using the solver routine in Microsoft Excel, which utilized the generalized reduced gradient method.

The erodibility of cohesive soils, affected by crude oil contamination, can be theoretically predicted based on observed JET data without crude oil. Mini JETs were undertaken with conditions without crude oil to derive

b_{0w} and

b_{1w} (in which

b_{0w} and

b_{1w} are the Wilson model parameters without the influence of crude oil). The

b_{1w} can be converted to

b_{1} (including the crude oil term), and

b_{0w} can be converted to

b_{0} (including the crude oil term) based on the measured crude oil parameters, at any time, without conducting new JETs. The parameters,

b_{0} and

b_{1}, are mechanistically defined. Parameter

b_{1}, based on Equation (20), can be rewritten as the following Equation (22):

in which

${b}_{1w}=(\frac{\pi}{{e}_{v}\sqrt{6}})\frac{{k}_{r}({K}_{ls}+{f}_{c})}{{K}_{o}}g({\rho}_{s}-{\rho}_{w})d$ is the Wilson model parameter derived from JET data without crude oil contamination (i.e.,

f_{co} = 0). The second term in Equation (22) can be mathematically found by using the terms given in

Table 1.

In a similar fashion, the Wilson model parameter

b_{0} can also be predicted based on the observed properties of crude oil contaminated soil. The terms in Equation (19) can be mathematically defined using the values given in

Table 1, combined the range of 18 to 23 found in this study for the crude oil coefficient ratio, and

K_{e}, which can be predicted from observed JET data for soil without crude oil contamination, using the following Equation (23):

#### 2.2. Materials and Experimental Procedure

The Taji region was selected as a case study for this research. The Taji region is located about 30 km northwest of Baghdad city (

Figure 2). The study area was located between (33°35′40″–33°28′42″) N and (44°05′22″–44°18′04″) W.

Figure 2 shows the crude oil pipe lines in Iraq, including the study area. The lean clay soil used in this study was acquired from the Taji region. The physical characteristics of the soil samples that were used are listed in

Table 2 followed by their characteristics defined using the ASTM standard (ASTM, 2006). The crude oil was obtained from the Iraqi South Oil Company (Basrah, Iraq). Chemical and physical characteristics were found using its laboratory, the results of which are listed in

Table 3. Chemical analysis of crude-oil-contaminated soil was performed by the Ministry of Science and Technology, the results of which are listed in

Table 4.

The JET settings and operation followed the procedure laid out by Al-Madhhachi et al. [

16,

37]. Soil samples were first air dried and sieved through sieve number four. Then, the sieved samples were packed into small-scale (standard mold) and large-scale (in-situ soil box) setups at three different soil moisture contents: 10%, 16%, and 20%, respectively. This was to investigate the influence of crude oil contamination on deriving Wilson model parameters at two different scale setups and different soil moisture levels (

Figure 3). The small-scale setup was an ASTM standard mold with 960 cm

^{3} in volume (

Figure 3a). The large-scale setup was a soil box with 48,000 cm

^{3} in volume (

Figure 3b). A standard bulk density was achieved by packing the soil into three layers using a manual rammer packed at three previously mentioned soil moisture levels. A similar technique was accomplished for the crude oil-contaminated soil. The packed contaminated soil was covered with 3 cm of crude oil and left for one day for each scale prior to JET testing. The next day, any excess oil was removed, and the JET experiments were performed.

Al-Madhhachi and Hasan [

7] indicted that oil contamination influenced the erodibility of dry soil samples. Therefore, in this study, the influence of contamination time on contaminated soil was examined by implementing the JET device with the small-scale setup at dry soil moisture content (10%) to investigate the second objective of this study. The soil samples were covered with oil and left for 1st, 4th, and 8th days depending on the required contamination time prior to applying the JET. The procedure outlined by Khanal et al. [

5] and Al-Madhhachi and Hasan [

7] for collecting score depth vs. time was followed. A total of 48 “mini” JETs was performed for the clean and crude-oil-contaminated soils to achieve the objectives of this study. The Wilson model parameters (

b_{0} and

b_{1}) were calculated using an Excel spread sheet that was developed by Al-Madhhachi et al. [

17].

The Normalised Objective Function (

NOF), which is the ratio of the standard deviation (

STD) of differences between observed and predicted data to the overall mean (

X_{av}) of the observed data, was calculated to quantify its suitability and to examine how well the Wilson model matched the observed data from the JET. Accordingly, the

NOF is expressed as [

24,

25]

in which

O_{i} and

P_{i} are the observed and predicted data, respectively, and

N is the observation number.

The computed statistical differences in the Wilson model parameters were also investigated using the analysis of variance (ANOVA) technique between clean and crude oil contaminated soil after 1, 4, and 8 days of contamination. The median and the difference between the 25th and 75th percentiles (IQR) were described for b_{0} and b_{1}. Pairwise comparison tests were undertaken for the mechanistic detachment parameters, which revealed a significant difference compared to ANOVA with a significance level of α = 0.05.