Evaluating the Hydraulic Conductivity of Dense Nonaqueous Phase Liquid in a Single Fracture of Rock-like Material

Abstract

To investigate the seepage characteristics of dense nonaqueous phase liquids (DNAPLs) in rock fractures, two types of NAPLs (paint and creosote) were used in triaxial permeability tests conducted on single-fracture samples. The hydraulic conductivity of rock fractures with different apertures, confining pressures, and fluid properties was measured, and the influence of various physical factors on transmissivity was explored. The results demonstrated the following: (1) Fracture aperture and fluid viscosity are the main factors influencing transmissivity; (2) The widely used cubic law fails to effectively predict the transmissivity of high-viscosity liquids in a fracture, and the influence of liquid viscosity is considerably higher than that predicted by the cubic law; and (3) This study proposed a transmissivity prediction model of DNAPLs in a rock fracture based on multivariate regression analysis. The proposed model provides more accurate prediction results than those predicted by the cubic law, and is applicable to fracture apertures ranging from 5 × 10⁻⁴ to 2.5 × 10⁻³ m as well as to every kind of fluid used in this study.

1. Introduction

Among the pollutants in soil and groundwater, organic liquids that are insoluble or slightly soluble in water are called nonaqueous phase liquids (NAPLs). These pollutants are commonly present as a result of leakage accidents, including those involving oil storage tanks, oil refineries, industrial parks, and other facilities. NAPLs can be divided into two categories based on their densities: light NAPLs (LNAPLs), which are lighter than water, and dense NAPLs (DNAPLs), which are heavier than water. LNAPLs include gasoline and heating oil. DNAPLs include chlorinated solvents (dichloroethylene, trichloroethylene, and tetrachloroethylene), creosote, coal tar, and polychlorinated biphenyls (PCBs). DNAPLs can infiltrate into the ground and migrate rapidly. These DNAPLs exist in the pores of fractured rock and affect groundwater for a long time [1]. They can penetrate the fractured rock mass, migrate to a certain depth below the water level and then slowly dissolve into the flowing underground water. To remediate DNAPLs in fractured rock, some have authors added chemical oxidants (such as permanganate and persulfate) to the fractured rock [2,3,4,5]. Therefore, the release of DNAPLs from ground pollution sources can lead to long-term pollution in both unsaturated and saturated areas underground. Simultaneously, the area affected by DNAPL leakage will continue to expand. Since the 1980s, the impact of DNAPLs on rock formations and groundwater pollutants has gradually attracted the attention of environmental engineers. In Taiwan, by the end of 2020, the underground pollution of NAPLs was controlled at 188 sites, covering a total area of 6.5 million square meters [6]. These pollutants are present as a result of leakage accidents, or oil storage tanks, oil refineries, industrial parks, and other facilities. The NAPL pollution of an oil refinery plant located on the carbonate rocks is particularly noted.

The distribution range of DNAPLs in the rock strata is estimated based on sample drilling and the permeability theory. Pankow and Cherry [7] proposed that, when DNAPLs are released from the ground, they will move both vertically and horizontally underground. Some DNAPLs will remain at the bottom of the overburden, whereas the rest will move to a significant depth within the fractured rock mass. Therefore, the absence of a DNAPL pool in the overlying soil does not mean that DNAPLs do not exist in the fractured rock. Chlorinated solvents, which are DNAPLs, have a density significantly higher than that of water and a relatively low viscosity. According to field investigations, it is likely that chlorinated solvents have migrated to a significant depth of the fracture network and will no longer move. However, DNAPLs such as creosote and coal tar are characterized by low density and high viscosity. Even though a leakage may have occurred decades ago, the migration of DNAPLs in the fracture network may still be in progress.

The leakage and inflow of DNAPLs into rock fractures may occur both above and below the groundwater table [8,9,10]. Capillary pressure at the entrance of the fracture is directly proportional to surface tension and inversely proportional to the fracture aperture [11]. Therefore, DNAPLs will infiltrate preferentially into fractures with large apertures. Therefore, in rock formations, where the seepage flow rate is primarily controlled by fractures, the orientation and distribution of fractures determine the main direction of DNAPL migration. Once DNAPLs enter the fracture network, it is most likely that downward or lateral leakage will continue until the intact rock is reached. Based on the principle of capillary pressure, Lerner et al. [12] estimated the fracture aperture required to stop DNAPL leakage; that is, the maximum hole diameter at the corresponding depth that stagnates DNAPL leakage. Johns et al. [13] measured fracture apertures inside granite cores using computed tomography and found them to range from 0.076 mm to 3.397 mm. Thus, actual apertures of rock fractures are considerably greater than that which is needed to stop DNAPL infiltration.

Understanding the hydraulic characteristics of a single fracture is the first step in evaluating the hydraulic conductivity of a fractured network in a rock mass. Conventionally, the fracture was simplified and considered to be formed by two smooth parallel plates, which provided the basis for the development of the cubic law. The flow in the fracture can be described using the Navier–Stokes equation. Based on the following assumptions and the smooth parallel-plate model, Equation (1) was derived [14,15].

(1)

The flow rate is fixed.

(2)

When the flow rate is small, the flow in the smooth parallel-plate fracture is laminar.

(3)

The flow rate variation along the direction perpendicular to the parallel plate is negligible.

$$Q=\frac{g{a}^3}{12vC}J$$

(1)

where Q is the average flow rate, J is the hydraulic gradient, a is the fracture aperture, g is the gravitational constant, v is the kinematic viscosity, and C is the roughness reduction factor.

Since the actual fracture surface of rocks is rough, it is not appropriate to directly apply the original cubic law to the fractures; thus, the cubic law should be modified. In Equation (1), the variable C is used to indicate that the flow in the fracture is affected by surface roughness, which is a hypothesis for modifying the cubic law based on the parallel-plate model. The value of C will significantly impact the seepage flow rate. Based on experimental data, Louis [16] proposed that the value of C is related to the fracture aperture a and surface roughness ∆, as shown in Equation (2).

$$C=1+8.8{(\frac{\mathrm{\Delta }}{2a})}^{1.5}$$

(2)

Most of the above theories are based on water or low-viscosity fluids. It is unclear whether they can be applied to the transport behavior of DNAPLs in rock fractures. Experiments were conducted in this study to explore the transport characteristics of NAPLs in rock fractures. To simulate the behavior of NAPLs in rock fractures, artificial materials were used to prepare test specimens, and different fracture apertures were considered. Different confining pressures were applied to the specimen in a triaxial test to simulate the earth pressure exerted by overburdens. The experimental results revealed the relationship between the flow rate of seepage and the fracture aperture, which serves as the basis upon which the flow rate can be transformed into transmissivity. This transmissivity can be used as a basis to evaluate the potential impact scope of a DNAPL site on a rock stratum in the future. It can also be used to determine whether the cubic law is applicable to this situation.

2. Experimental Methods

2.1. Test Sample Preparation

In this study, gypsum and epoxy resin were used to cast two kinds of rock-like materials. There are two main advantages of using artificial materials. First, artificial materials can be easily made into various shapes and sizes with different fracture widths. Secondly, gypsum samples can simulate the real state of rock pores, whereas epoxy resin samples have no pores and can be used to study the influence of fracture apertures on the seepage flow rate. In addition, as gypsum is made of cemented particles, and epoxy resin is a polymer, the experimental results can be used for comparison. These two kinds of samples are introduced below, and the finished samples are shown in Figure 1.

(1)

Epoxy resin test sample

The size of the test sample was prepared in accordance with the International Society for Rock Mechanics [17], and the length-to-diameter ratio was greater than or equal to 2.5. A transparent filled-type epoxy resin was used in the study. To prepare epoxy resin samples, the weight ratio of agent A (base) to agent B (hardener) was 3:1. Agent A and agent B with predetermined weights were thoroughly mixed with a stir rod. Care was taken during mixing to avoid creating excessive air bubbles. After mixing, the liquid was poured into a mold and left for approximately one day. After solidification, the mold could be removed to complete the sample preparation. The epoxy resin samples were 5 cm in diameter and 11.5 cm in length. Stainless steel plates of different sizes were inserted into the samples to simulate six different fracture apertures: 2.5 mm, 2 mm, 1.5 mm, 1 mm, 0.8 mm, and 0.5 mm, according to the investigation data from an oil refinery plant located on the carbonate rocks in Taiwan.

(2)

Gypsum test sample

To prepare gypsum samples, the weight ratio of gypsum to water was 2.5:1. The samples were 5 cm in diameter and 11.5 cm in length. Before sample preparation, the inside of the mold and the outside of the stainless-steel plates were uniformly coated with a thin layer of Vaseline to facilitate subsequent mold removal. Thereafter, the mold was assembled and placed on a horizontal workbench. Furthermore, the stainless-steel plate was fixed in the center to simulate the corresponding fracture aperture. The required amount of gypsum was mixed with water, and then the gypsum slurry was poured into the mold for further mixing. After the sample solidified, the stainless-steel plate was carefully pulled out to avoid joint surface roughness (to allow the assumption of smooth joint surfaces). The test sample was then placed in the oven for approximately three to five days. The test sample was weighed every day until its weight was stable. Stainless-steel plates of different sizes were inserted into the samples to simulate six different fracture apertures: 2.5 mm, 2 mm, 1.5 mm, 1 mm, 0.8 mm, and 0.5 mm.

Figure 1. Cylinder specimen with a fracture for this study: (a) epoxy resin specimen; and (b) gypsum specimen.

Figure 1. Cylinder specimen with a fracture for this study: (a) epoxy resin specimen; and (b) gypsum specimen.

2.2. Test Fluid

To simulate the transport behavior of DNAPLs, creosote was selected as the test liquid in this study. Creosotes have high viscosity and hydrophobicity, and their density is greater than that of water. For comparison, black oil paint with the same characteristics was selected for the experiment due to its similarities to DNAPLs. In addition, water was used as the control group. The black paint and creosote used in the experiment are shown in Figure 2. The physical properties of the selected liquids were analyzed first. The kinematic viscosity (v), which has an effect on the seepage flow rate, was calculated based on the viscosity and liquid density using Equation (3).

$$v=\frac{\eta }{\rho }$$

(3)

Table 1 shows the experimental results of the physical properties of the three liquids.

Table 1. Physical properties of the test fluids.

Type	Water	Creosote	Paint
ρ (kg/m3)	997.08	1161.83	1377.78
Viscosity (Pa·s)	8.94 × 10−4	1.469	1.044
Kinematic viscosity (m2/s)	8.961 × 10−7	1.264 × 10−3	7.577 × 10−4

Table 1. Physical properties of the test fluids.

Type	Water	Creosote	Paint
ρ (kg/m3)	997.08	1161.83	1377.78
Viscosity (Pa·s)	8.94 × 10−4	1.469	1.044
Kinematic viscosity (m2/s)	8.961 × 10−7	1.264 × 10−3	7.577 × 10−4

Figure 2. Liquid for the permeability test: (a) paint; and (b) creosote.

Figure 2. Liquid for the permeability test: (a) paint; and (b) creosote.

2.3. Test Procedure

The triaxial permeability test was selected to simulate field overburden conditions and to measure the seepage flow rate of different kinds of liquids in samples with different apertures. Furthermore, during the triaxial tests, different overburden depths were simulated. The experimental equipment in this study included air pressure and water pressure control systems and triaxial test equipment. The detailed experimental equipment diagram is shown in Figure 3. The air compressor in the air pressure control system provided the pressure required for the test. The upper limit of the pressure was 7.0 MPa. The manual pressure regulation valve was used to provide the required confining pressure of the sample.

In addition, the manual pressure regulation valve was used to apply pressure to the deaired bucket and then through a plastic pipe; the back pressure was applied to the inside of the test sample. The pressure gauge showed the confining pressure and back pressure magnitudes. In this study, only the confining pressure was applied to the sample without back pressure. The triaxial test equipment was primarily connected with the lower part of the triaxial chamber through the stainless-steel inner support. The main body of the triaxial chamber was a hollow cylinder, which was made of transparent acrylic. Therefore, the variations in the sample during the test can be easily observed. The maximum air pressure was 1 MPa. The base conformed to the sample size with a diameter of 50 mm. The permeability test procedures for the three fluids were as described below.

Step 1: The test sample was placed in the rubber membrane, and it was ensured that all the liquids in the test would flow into the fracture. The sample was then placed on the base of the triaxial chamber. In contrast to the soil test, permeable stones should not be placed on the upper and lower ends of the test sample. Otherwise, the seepage flow rate will be controlled by the permeable stones rather than by the fracture itself. Therefore, it was only necessary to place an O-ring on the top and bottom of the sample and test instrument, respectively, so that the liquid could flow smoothly into the fracture of the test sample. The test sample was then fixed using the base and top cap of the triaxial chamber. Finally, the triaxial acrylic cover was put on, and the screws were locked.

Step 2: Water was injected into the triaxial chamber through the chamber pressure valve. After the chamber was filled, the chamber pressure was raised to 100 kPa, 200 kPa, and 300 kPa to simulate conditions with different overburden thicknesses.

Step 3: The valve connecting the reservoir was opened, and the elevation head of the liquid surface was kept at a constant of 1.3 m. The distilled water was first used to infill the fracture of the specimen. Then, the DNAPL liquids were adopted for the permeability test. Owing to the constant elevation head, the test fluids flowed into the instrument. Until the outflow was steady, the measuring cylinders were placed at the outlet to receive the test liquid flowing out (as shown in Figure 3). The outflow and time were measured to calculate the flow rate (m³/s) and transmissivity (m²/s).

Step 4: At the end of the test, the paint and creosote in the instrument and pipeline were removed by using turpentine or toluene while following steps 1 to 3 to prevent the liquid from blocking the experimental equipment so that the subsequent experimental data were not affected.

3. Experimental Results

3.1. Test Results of Epoxy Resin Specimens

Variations in the seepage flow rates of the test fluids in the epoxy resin samples with different fracture apertures are investigated in this section. Each fluid was tested three times under the same fracture aperture and the same confining pressure to ensure the reproducibility of the experimental results. The measured seepage flow rate (Q) was converted into transmissivity (T), which can represent the hydraulic conductivity of the fracture. The seepage in the fracture was assumed to conform to Darcy’s law. The calculation was based on Equation (4):

$$T=\frac{Q}{Jb}$$

(4)

T: transmissivity (m²/s); Q: seepage flow rate (m³/s)

J: hydraulic gradient (dimensionless); b: fracture width (m)

Figure 4 shows the transmissivity of the test fluids under different fracture apertures. When water is used as the test liquid, the difference in transmissivity is not significant. It is speculated that as the viscosity of water is quite low, the fracture aperture has a negligible influence on transmissivity. When paint is used as the test liquid (Figure 4), there is a high correlation between transmissivity and the fracture aperture, which may be caused by the high viscosity of the paint. High liquid viscosity reduces the seepage flow rate.

Permeability tests using creosote showed that its correlation between transmissivity and the fracture aperture was the highest among the three test fluids (as shown in Figure 4). During the experiment, test results were successfully obtained for samples with fracture apertures of 2.5 mm, 2.0 mm, 1.5 mm, and 1.0 mm. The fracture apertures of 0.8 mm and 0.5 mm may have been appropriate threshold values due to their high liquid viscosity. After adjusting the test equipment several times, the seepage flow rate could still not be measured. According to the four groups of measured transmissivities, creosote’s test results have the highest degree of correlation, and its viscosity is also the highest among the three liquids.

Figure 4. Transmissivity variation of epoxy resin specimens with different apertures.

Figure 4. Transmissivity variation of epoxy resin specimens with different apertures.

3.2. Test Results of Gypsum Samples

Figure 5 shows the transmissivity of the three test fluids flowing through samples made of gypsum with different fracture apertures. When water is used as the test liquid, the seepage flow rate increases as the fracture aperture increases. Gypsum is a porous material; therefore, the impact of fracture roughness should be considered. The experimental results demonstrate that roughness reduces transmissivity. When paint is used as the test liquid, its transmissivity and the fracture aperture maintain a high degree of correlation. As shown in Figure 5, transmissivity increases as the fracture aperture becomes larger, and the test results also reflect the influence of roughness. The surface roughness of the aperture was further measured by atomic force microscopy (AFM). Figure 6 shows the variations in the surface roughness of the apertures. The average roughness of epoxy resin and gypsum are 45.88 nm and 134.20 nm, respectively. For the same aperture, higher roughness reduces transmissivity.

3.3. Influence of Confining Pressure

To simulate earth pressure caused by overburdens, the confining pressure was determined to be 100 kPa, 200 kPa, and 300 kPa, which corresponded to overburden depths of approximately 5 to 15 m. The experimental results of different fluids are discussed below.

Choosing water as the test fluid, Figure 7 shows the variations in water transmissivity under different confining pressures. The influence of confining pressure on water transmissivity was insignificant. Due to the high bulk modulus of epoxy resin specimens in this study (4.1 GPa), the aperture deformation induced by the confining pressure is quite small. For example, when a maximum confining pressure of 300 kPa was applied to the specimen, the axial compressive strain was 1.27 × 10⁻⁵, and deformation was calculated to be 1.2 × 10⁻³ mm. Therefore, the initial fracture aperture was considerably larger than the size of the deformation. When different confining pressures were applied, the water transmissivity was almost the same. The influence of confining pressure on the overall test is minor, which is true for each fracture aperture. When paint was selected as the test fluid, Figure 8 shows transmissivity variations under different confining pressures. Similar to the results obtained using water, paint transmissivity at each confining pressure was quite stable and was almost unaffected.

4. Prediction Model of Transmissivity

Transmissivity prediction according to the cubic law.

The widely used cubic law was adopted to predict test results, as shown in Figure 9a. The predictions underestimate transmissivity when the aperture is less than 1.0 × 10⁻³ m and apparently overestimate transmissivity when the aperture is greater than 1.0 × 10⁻³ m. Romm [18] indicated that the fracture aperture should be less than 2 × 10⁻⁴ mm to ensure that the flow in the fracture is laminar and conforms to the cubic law. However, the actual apertures of rock fractures are usually larger than 2 × 10⁻⁴ mm, especially in carbonate rocks. According to He et al. [19], the cubic law often leads to significant errors in the estimated flow conductivity. Various methods have been proposed in the literature to modify the cubic law to consider the deviation caused by the non-ideality of the fracture’s geometric properties [20,21,22]. He et al. [19] summarized dozens of modified cubic law-based models from the literature, by which correction factors related to the fracture aperture, surface roughness, and flow tortuosity are considered. In addition, DNAPLs exhibit high viscosity and may not fulfill the cubic law assumptions. Therefore, this study proposed a transmissivity prediction model based on multivariate regression analysis.

4.1. Multivariate Regression Analysis of Factors Influencing Transmissivity

Regarding the permeability results, the multivariate regression analysis proposed by Jeng et al. [23] was used to discuss the relevance of the relationship between different influencing factors and transmissivity. First, it is assumed that the object, experimental result, or phenomenon to be discussed is a multivariate function with multiple influencing factors, as expressed in the functional Equation (5), where A is a constant, and x₁, x₂, … x_n denote the influencing factors.

$$T=A{\mathit{f}}_1\left({\mathit{x}}_1\right) {\mathit{f}}_2\left({\mathit{x}}_2\right) {\mathit{f}}_3\left({\mathit{x}}_3\right)\dots {\mathit{f}}_{\mathit{n}}\left({\mathit{x}}_{\mathit{n}}\right)$$

(5)

Based on the above assumption, function regression is conducted for the factors one by one. The most correlative influencing factor (maximum correlation coefficient, r²) and its function can be determined from the results of regression. The most influential factors with the maximum correlation coefficient and function obtained from normalization are called the first-class factor x and function f₁(x), respectively. In addition, the expression of the regression function must conform to the rationality of the analysis to ensure the accuracy of subsequent multivariate regression analyses. When the first-class function f₁(x₁) to transmissivity is normalized, function regression is performed for the composition data of various test results using the ratio T/f₁(x₁) one by one to determine the second-class influencing factor f₂(x₂). Subsequently, the ratio T/f₂(x₂) is calculated and regressed again to determine the corrected first-class function f₁(x₁). The iteration process continues until the function satisfies the related requirement (correlation coefficient r² is stable). Similar procedures are performed to determine the other influencing factors and an empirical regression equation is obtained.

Regression analysis was conducted on the experimental results, and a model was proposed accordingly, as shown in Equation (6):

$$T=4.4\times {\left(\frac{\rho }{{\rho }_w}\right)}^{2.3}\times \frac{v_w}{{\left(\frac{\eta }{{\eta }_w}\right)}^{0.85}}\times {(\frac{a}{a_0})}^{\frac{\eta }{2000{\eta }_w}+0.05}\times \phi$$

(6)

$T$: transmissivity (m²/s) ρ: the liquid unit weight (kg/m³)

${\rho }_w$: the unit weight of water at 25 °C (kg/m³)

$\eta$: the liquid dynamic viscosity (Pa·s)

${\eta }_w$: the dynamic viscosity of water at 25 °C (Pa·s)

$a$: fracture aperture (m)

$a_0$: the threshold of the fracture aperture (2.5 × 10⁻³ m in this study)

$\phi$: roughness factor (dimensionless) $\phi =1+{\left(\frac{\mathrm{\Delta }}{a}\right)}^{1.5}$ (Zhang [24]). where Δ is the average roughness (m).

According to the above equation, the factors of unit weight, dynamic viscosity, and fracture aperture can be normalized. Based on the case with a water temperature of 25 °C, the impact of each factor on transmissivity was derived. At present, the research results are applicable to fracture apertures ranging from 5 × 10⁻⁴ to 2.5 × 10⁻³ m as well as to kinematic viscosities ranging from the minimum to maximum values of the fluids used in this study.

4.2. Experimental Data Verification

In Figure 9, the calculation results using the above equation (with the input of liquid properties and fracture apertures) are compared with the experimental results. Figure 9a shows the comparison with the experimental results of epoxy resin samples. The predicted values and the test results are highly correlated for the three test fluids used in the experiment. The result is quite satisfactory. The predicted results and experimental results for creosote are slightly different; however, they are still on the same order of magnitude. Figure 9b shows the comparison of the calculated values with the experimental results of the gypsum samples. The influence of gypsum is expressed using a certain value of roughness. An influencing factor of roughness is introduced into the equation for verification. The result is quite satisfactory and highly correlated. If roughness is to be considered in future studies, its value can be estimated and modified based on prior studies. The equation that includes the roughness value can be used to predict the seepage results. Compared with the cubic law, as shown in Figure 9a, the proposed model provides more reasonable predictions.

5. Conclusions

In this study, rock fracture permeability tests were conducted on samples made of two kinds of materials (epoxy resin and gypsum). Furthermore, different fracture apertures and two kinds of DNAPLs were used to explore the influence of the physical properties of fluids on transmissivity. Based on a regression analysis of the experimental results, a seepage model for nonaqueous phase fluids in a rock fracture was proposed. The research results demonstrated the following:

• Fracture aperture has the greatest influence on transmissivity;
• The impact of liquid viscosity is considerably greater than that predicted by the cubic law. As the fracture aperture varies, the effect of viscosity also changes significantly, whereas the cubic law only predicts changes in low viscosity;
• The proposed prediction model is more accurate than conventional cubic law predictions. Prior studies found that the cubic law is applicable to fracture apertures less than 0.2 mm. However, the range of artificial fracture apertures in this test exceeded this range. Moreover, actual fracture apertures are not limited to this range. If the cubic law is to be applied to all fracture apertures, large errors will be produced. According to the prediction model, after the threshold value is exceeded, the increase in transmissivity will slow as the fracture aperture increases;
• Temperature will affect the viscosity of the liquid and consequently affect transmissivity. In the future, viscosity values corresponding to different temperatures should be considered.