# Investigation of the Models of Flow through Fractured Rock Masses Based on Borehole Data

Authors: Long Tan; Wei Xiang; Jin Luo; Qingbing Liu; Xu Zuo
Journal: Advances in Civil Engineering
 (2020)

---

## Abstract

When only limited borehole data are available, making optimum use of the existing data is crucial for performing a preliminary assessment of the investigated site. In this paper, the relationships between the borehole data and the permeability coefficient were first analyzed. These relationships were then used to establish a model for estimating the permeability coefficient of rock mass that takes into account the influence from the confining pressure on the seepage flow. The proposed model can reduce the number of hydraulic tests which are time consuming and very costly and allow the determination of change in the permeability coefficient throughout the borehole. The flow model could assist in providing important references for selecting an appropriate permeability coefficient in hydrogeological simulation and in evaluating the condition of large cracks developed in boreholes. In general, the seepage flow model developed in this study will contribute to the design practice of a tunnel project constructed in fractured rock masses.


---

## Body

## 1. Introduction

The success of underground water-sealed oil storage projects depends on two major factors: the stability of the surrounding rock and the condition of the water seal. Among them, the water seal condition is considerably affected by the underground seepage field. A commonly used method of establishing the underground seepage field model during construction is obtaining the rock permeability coefficient of a specific area using in situ tests in the borehole [1, 2].As a rock type of low permeability, natural granite rock mass comprises discrete blocks of intact rock and discontinuities, such as factures, fissures, and joints [3]. Its permeability is mainly controlled by the development state of discontinuities. Therefore, evaluating the groundwater seepage characteristics in granite areas is essential for assessing the development level of fissures within underground rock masses.During the construction process, and in particular, the prefeasibility stage when there is only limited borehole data available, it is often difficult to evaluate the seepage field of the site with a small amount of data [4]. Attempts to obtain more borehole data by the hydraulic pressure test will require a considerable amount of time and cost, and therefore exploring the relationship between available borehole data and fracture seepage and then applying it to estimate the permeability coefficient can effectively reduce the number of hydraulic pressure tests and greatly save construction costs.The research of fracture seepage often starts from a series of simplified models. The goal of researchers is to deduce the general formula from various models. Scholars from various countries have done a lot of research on fracture seepage. Many scholars consider the influence of the rough fracture surface from different aspects and introduce the roughness influence coefficient into the cubic law to analyze seepage characteristics of the rough fracture [3, 5–8]. With the rapid development of numerical simulation technology, the establishment of a three-dimensional or two-dimensional model of the fracture surface to describe the surface morphology characteristics of the fracture surface and its application in seepage calculation has become a popular method of studying fracture seepage [9–12]. The theory of the lattice Boltzmann method is usually used to establish a numerical model to simulate the seepage of cracks in rough rock mass. This became one of the effective numerical research tools in solving the seepage problem of cracks in rough rock mass [13–17].The research methods mentioned above are generally too sophisticated for engineers to use in practice. A convenient method is now put forward in this article, which is straightforward and accurate enough for engineering application. In this study, we established the relationship between the parameters of the structural plane in the fractured rock mass and the permeability coefficient of the fractured rock mass by analyzing several crack indices such as aperture and inclination angle.The studied site is located in a specific region in Yantai, Shandong, as shown in Figure1. It is geographically located in the Shandong Peninsula of China, separated from Dalian by the sea and connected to Shanghai and Qingdao by land routes. The proposed underground water-sealed oil storage project is located in a hilly area with a tectonically denudated terrain. The middle region is covered by convex hills, which possess deep cut slopes revealing bare rocks at local gullies. The surrounding terrain is a hilly alluvial plain with a low altitude and a gentle slope. The mound is composed of coarse-medium grained porphyritic monzogranite and biotite monzogranite. The bedrocks at the hilltop and the slope are mostly exposed directly to ambient weather with certain local regions covered by a thin layer of residual soil.Figure 1
Geographic schematic map of the selected site of the water-sealed storage caverns in the Yantai city of Shandong province in China.According to on-site geological mapping and geological drilling, the regional exposed strata are rocks formed at the Quaternary period during the Cenozoic era and a large area of intrusive rocks. The intrusive rocks identified in the target site are primarily Luliang, Sinian, and early Yanshanian intrusive rocks. In addition, some diorite-like porphyry, lamprophyre, and granite porphyry rocks also invaded the Luliang and Yanshanian magmatic rocks in the vein shape.


## 2. Fracture Seepage Model Based on Borehole Data

### 2.1. Theoretical Basis of Permeability Coefficient of Fractured Rock Mass

Poiseuille derived a theoretical formula to describe the motion of a viscous incompressible fluid in the gap between smooth parallel plates under homogeneous and constant velocity conditions; it is expressed as [18](1)u=gb212νJ,where u is the flow rate (m/s), g is the gravitational acceleration (m/s2), b is the width between smooth parallel plates (m), ν is the kinematic viscosity coefficient of water (m2/s), and J is the hydraulic gradient.Snow conducted a parallel plate fissure seepage test and suggested the popular cubic law for fractures [19, 20]:(2)q=gb312vJ,where q is the flow per unit length (m2/s).The width between two parallel plates can be treated as the crack width of the ideal fractured rock layer. When the liquid is in laminar flow, the average flow rate is given by(3)u=B212⋅γμJ,where B is the aperture of the crack (m), γ is the volumetric weight (N/m3), and μ is the dynamic viscosity of water (Pa·s).If the crack system is treated as a lot of straight cracks and all the apertures are considered equal, then the flow velocity along the line of intersection of the fracture group is given by(4)v=NB212⋅γμJ,where v is the total flow velocity of all the cracks (m/s) and N is the number of the cracks.Therefore, the permeability coefficient can be calculated by(5)K=NB212γμ,where K is the permeability coefficient (m/s).Figure2 shows the schematic of the fracture distribution in the borehole. As shown in the figure, we can always obtain the aperture and angle between the crack and the horizontal direction regardless of the shape of the fracture revealed by the borehole.Figure 2
Schematic map of fracture distribution during the borehole hydraulic pressure test.In the smooth parallel-plate model, the fracture system is treated as multiple equal-width and flat-fracture groups. In real engineering practices, however, the cracks in the fracture system revealed by drilling usually exhibit different inclination angles, a variety of aperture, and rough surfaces. Some of the cracks are even packed with filling materials. Therefore, several additional factors neglected by the smooth parallel plate model need to be considered when analyzing the permeability coefficient in engineering practices. These factors include the inclination angle, roughness, and aperture of the fracture.


### 2.2. Parameter Selection at the Structural Plane

By analyzing the results of the hydraulic pressure test performed in the borehole and the fracture development characteristics of the corresponding section, several factors affecting the permeability of the rock mass can be inferred.The results of the borehole TV (Figure3) can be used to extract basic fracture information, as shown in Figure 4.Figure 3
Picture of the borehole televiewer.Figure 4
Schematic map of the borehole information extraction and calculation method.According to the picture of the borehole televiewer, the altitude difference of the crack can be calculated. Given the known diameter of the drill hole, the inclination angle can be obtained. On the other hand, the aperture could be observed from the picture.


#### 2.2.1. Aperture of the Crack

The crack aperture revealed by the borehole is not equivalent to the hydraulic aperture of the seepage flow through the crack. However, these two parameters can be connected to each other based on Barton’s empirical equation [21]:(6)ah=am2JRC2.5,where ah is the equivalent hydraulic aperture (m), am is the mechanic aperture (m), and JRC is the roughness coefficient.Though this equation yields an approximation, the aperture of the crack can still be used to reflect the aperture of the seepage channel to a certain extent, and thus it reflects the seepage capacity of the channel.


#### 2.2.2. Inclination Angle Index at the Structural Plane

The distribution of the inclination angle at the structural plane can reflect the average direction of the inclination angle distributed in the study area. This direction can reflect the effective seepage path in the target area in the field of seepage research. There should exist a certain relationship between the distribution index of the inclination angle at the structural surface and the permeability coefficient itself.According to the established model, a change in the inclination angle will affect the length of the seepage path. Therefore, the permeability coefficient associated with this path will be reduced by cosθ owing to the variation in the crack inclination angle. Thus, cos θ is proposed as the inclination angle index at the structural plane.


#### 2.2.3. Joint Roughness Coefficient (JRC)

Barton established the straight-edge method of JRC measurement (Figure5) on the basis of the in situ rock mechanics test, by which JRC is determined by measuring the surface fluctuation amplitude of the rock mass structural plane [22].Figure 5
Barton’s straight edge method.The crack development diagram shown in boreholes should theoretically adhere to the triangular function curve, as illustrated in the following function:(7)y=Asinωx+t,where A is the amplitude, ω is the angular velocity, and t is the initial phase.After measuring the amplitude from the drill televiewer, the basic form of the crack curve can be determined using the period of the curve function which can be calculated. Then, the curve could be used as the datum line to measure the fluctuation amplitude. The amplitude will be put on Barton’s straight edge figure to obtain the final JRC, as shown in Figure6.Figure 6
Schematic diagram for calculating fracture roughness.


## 2.1. Theoretical Basis of Permeability Coefficient of Fractured Rock Mass

Poiseuille derived a theoretical formula to describe the motion of a viscous incompressible fluid in the gap between smooth parallel plates under homogeneous and constant velocity conditions; it is expressed as [18](1)u=gb212νJ,where u is the flow rate (m/s), g is the gravitational acceleration (m/s2), b is the width between smooth parallel plates (m), ν is the kinematic viscosity coefficient of water (m2/s), and J is the hydraulic gradient.Snow conducted a parallel plate fissure seepage test and suggested the popular cubic law for fractures [19, 20]:(2)q=gb312vJ,where q is the flow per unit length (m2/s).The width between two parallel plates can be treated as the crack width of the ideal fractured rock layer. When the liquid is in laminar flow, the average flow rate is given by(3)u=B212⋅γμJ,where B is the aperture of the crack (m), γ is the volumetric weight (N/m3), and μ is the dynamic viscosity of water (Pa·s).If the crack system is treated as a lot of straight cracks and all the apertures are considered equal, then the flow velocity along the line of intersection of the fracture group is given by(4)v=NB212⋅γμJ,where v is the total flow velocity of all the cracks (m/s) and N is the number of the cracks.Therefore, the permeability coefficient can be calculated by(5)K=NB212γμ,where K is the permeability coefficient (m/s).Figure2 shows the schematic of the fracture distribution in the borehole. As shown in the figure, we can always obtain the aperture and angle between the crack and the horizontal direction regardless of the shape of the fracture revealed by the borehole.Figure 2
Schematic map of fracture distribution during the borehole hydraulic pressure test.In the smooth parallel-plate model, the fracture system is treated as multiple equal-width and flat-fracture groups. In real engineering practices, however, the cracks in the fracture system revealed by drilling usually exhibit different inclination angles, a variety of aperture, and rough surfaces. Some of the cracks are even packed with filling materials. Therefore, several additional factors neglected by the smooth parallel plate model need to be considered when analyzing the permeability coefficient in engineering practices. These factors include the inclination angle, roughness, and aperture of the fracture.


## 2.2. Parameter Selection at the Structural Plane

By analyzing the results of the hydraulic pressure test performed in the borehole and the fracture development characteristics of the corresponding section, several factors affecting the permeability of the rock mass can be inferred.The results of the borehole TV (Figure3) can be used to extract basic fracture information, as shown in Figure 4.Figure 3
Picture of the borehole televiewer.Figure 4
Schematic map of the borehole information extraction and calculation method.According to the picture of the borehole televiewer, the altitude difference of the crack can be calculated. Given the known diameter of the drill hole, the inclination angle can be obtained. On the other hand, the aperture could be observed from the picture.


### 2.2.1. Aperture of the Crack

The crack aperture revealed by the borehole is not equivalent to the hydraulic aperture of the seepage flow through the crack. However, these two parameters can be connected to each other based on Barton’s empirical equation [21]:(6)ah=am2JRC2.5,where ah is the equivalent hydraulic aperture (m), am is the mechanic aperture (m), and JRC is the roughness coefficient.Though this equation yields an approximation, the aperture of the crack can still be used to reflect the aperture of the seepage channel to a certain extent, and thus it reflects the seepage capacity of the channel.


### 2.2.2. Inclination Angle Index at the Structural Plane

The distribution of the inclination angle at the structural plane can reflect the average direction of the inclination angle distributed in the study area. This direction can reflect the effective seepage path in the target area in the field of seepage research. There should exist a certain relationship between the distribution index of the inclination angle at the structural surface and the permeability coefficient itself.According to the established model, a change in the inclination angle will affect the length of the seepage path. Therefore, the permeability coefficient associated with this path will be reduced by cosθ owing to the variation in the crack inclination angle. Thus, cos θ is proposed as the inclination angle index at the structural plane.


### 2.2.3. Joint Roughness Coefficient (JRC)

Barton established the straight-edge method of JRC measurement (Figure5) on the basis of the in situ rock mechanics test, by which JRC is determined by measuring the surface fluctuation amplitude of the rock mass structural plane [22].Figure 5
Barton’s straight edge method.The crack development diagram shown in boreholes should theoretically adhere to the triangular function curve, as illustrated in the following function:(7)y=Asinωx+t,where A is the amplitude, ω is the angular velocity, and t is the initial phase.After measuring the amplitude from the drill televiewer, the basic form of the crack curve can be determined using the period of the curve function which can be calculated. Then, the curve could be used as the datum line to measure the fluctuation amplitude. The amplitude will be put on Barton’s straight edge figure to obtain the final JRC, as shown in Figure6.Figure 6
Schematic diagram for calculating fracture roughness.


## 2.2.1. Aperture of the Crack

The crack aperture revealed by the borehole is not equivalent to the hydraulic aperture of the seepage flow through the crack. However, these two parameters can be connected to each other based on Barton’s empirical equation [21]:(6)ah=am2JRC2.5,where ah is the equivalent hydraulic aperture (m), am is the mechanic aperture (m), and JRC is the roughness coefficient.Though this equation yields an approximation, the aperture of the crack can still be used to reflect the aperture of the seepage channel to a certain extent, and thus it reflects the seepage capacity of the channel.


## 2.2.2. Inclination Angle Index at the Structural Plane

The distribution of the inclination angle at the structural plane can reflect the average direction of the inclination angle distributed in the study area. This direction can reflect the effective seepage path in the target area in the field of seepage research. There should exist a certain relationship between the distribution index of the inclination angle at the structural surface and the permeability coefficient itself.According to the established model, a change in the inclination angle will affect the length of the seepage path. Therefore, the permeability coefficient associated with this path will be reduced by cosθ owing to the variation in the crack inclination angle. Thus, cos θ is proposed as the inclination angle index at the structural plane.


## 2.2.3. Joint Roughness Coefficient (JRC)

Barton established the straight-edge method of JRC measurement (Figure5) on the basis of the in situ rock mechanics test, by which JRC is determined by measuring the surface fluctuation amplitude of the rock mass structural plane [22].Figure 5
Barton’s straight edge method.The crack development diagram shown in boreholes should theoretically adhere to the triangular function curve, as illustrated in the following function:(7)y=Asinωx+t,where A is the amplitude, ω is the angular velocity, and t is the initial phase.After measuring the amplitude from the drill televiewer, the basic form of the crack curve can be determined using the period of the curve function which can be calculated. Then, the curve could be used as the datum line to measure the fluctuation amplitude. The amplitude will be put on Barton’s straight edge figure to obtain the final JRC, as shown in Figure6.Figure 6
Schematic diagram for calculating fracture roughness.


## 3. Model Development

Because the size of the water-sealed cavern is much larger than that of the crack in the borehole, the following formula valid for each crack is introduced to simplify the calculation:(8)K=112⋅γμB2cosθ,where γ is the volumetric weight (N/m3), μ is the coefficient of viscosity (Pa·s), θ is the inclination angle of the crack (°).Therefore, by combining this with the empirical equation proposed by Barton, equation (8) can be converted into(9)K=112⋅γμam4JRC5cosθ.The mechanical aperture and the inclination angle index used in the equation can be obtained from the drilling video. These parameters allow us to calculate the permeability coefficient of an arbitrary crack and thus determine the seepage characteristic of any section in the borehole.Based on the calculation, the permeability coefficient of the section in the drilling hole is the sum of the permeability coefficients of cracks in all directions:(10)Km=∑i=1n112⋅ami4JRCi5⋅γμcosθi,where Km is the permeability coefficient of the section (m/s), n is the number of cracks, ami is the mechanic aperture (m), JRCi is the roughness coefficient, γ is the volumetric weight (N/m3), μ is the viscosity (Pa·s), and θi is the inclination angle of the crack (°).


## 4. Model Validation

### 4.1. Calculation of Permeability Coefficient and Comparison with the Measurement Value

According to the actual drilling condition, the measured permeability coefficient is affected by both its own spatial characteristics and the stress from surrounding rocks. Neglecting other influential conditions, the permeability coefficient derived from the crack seepage model based on the borehole data needs to be constrained using the confining pressure which yields a certain level of attenuation. Therefore, based on the comparison of the permeability coefficients obtained from the crack model calculation and those measured from the water pressure test in the borehole (as shown in Table1), we can estimate the extent to which the permeability coefficient is attenuated by the confining pressure. After integrating all data, the comparison charts are obtained, as shown in Figures 7 and 8.Table 1
The results of water pressure tests in the borehole.
Section no.DepthAltitudeLength of test section (m)Permeability rate (Lu)Coefficient of permeability (m/d)Start (m)End (m)Start (m)End (m)15.1315.50124.10113.7310.3710.080.10401215.1325.50114.10103.7310.378.800.09490325.1335.50104.1093.7310.378.870.10484435.0248.4094.2180.8313.380.750.02719548.3261.7080.9167.5313.380.160.00302661.3274.7067.9154.5313.380.400.00739774.4287.8054.8141.4313.380.150.00336886.82100.2042.4129.0313.380.380.00671996.82110.2032.4119.0313.380.190.0053710106.92120.3022.318.9313.380.280.0030211117.32130.7011.91−1.4713.380.500.0107412127.32140.701.91−11.4713.380.240.0070513137.32150.70−8.09−21.4713.380.270.0090614147.12160.50−17.89−31.2713.381.050.0282015157.12170.50−27.89−41.2713.380.300.00403As shown in Figure7, the permeability coefficients calculated from the seepage model based on the borehole data are generally greater than the measured permeability coefficient. This is because the impact from the confining pressure on the crack aperture becomes greater with increasing confining pressure, which results in a decrease in the permeability coefficient. As each crack exhibits different levels of compressive deformation, the degree of change in the permeability coefficient is also different. However, the measured permeability coefficient is apparently larger at the borehole depth of 35.5 m and 160.5 m. The fracture data for these two depth intervals are provided in Table 2, which indicates that the fracture aperture is essentially small and hence the measured permeability coefficient should be relatively low. Therefore, the apparently larger permeability coefficient measure may arise from two aspects: (1) the hydraulic pressure test is not completely accurate and (2) some engineering operation in the borehole, such as flushing borehole, damages the original crack shape and results in inaccurate crack aperture and number.Figure 7
Comparison of the permeability coefficient by measurement and calculation.Table 2
The fracture data for sections 3 and 14 of the borehole.
Section no.DepthLength of test section (m)Inclination angle (°)Inclination (°)Aperture (cm)Start (m)End (m)325.19325.2030.0124.61240.371.04425.37725.4950.11857.53128.490.21725.73725.890.15363.96359.450.39125.84526.060.21570.76358.890.39125.89526.0860.19168.51359.450.26126.08326.2660.18367.77358.890.17426.45326.5640.11156.14275.820.17427.33127.5440.21370.57359.450.21728.76128.9540.19368.77358.890.26129.20329.4880.28575.25270.830.34833.17633.240.06440.6494.150.43533.65733.720.06340.06137.350.21733.67533.7390.06440.45175.570.21733.71633.8040.08849.65189.420.21733.85933.9020.04329.61151.20.21734.47934.5470.06842.3296.370.26114148.082148.0840.00276.92202.090.217148.976148.9780.00245.34163.440.173149.266149.2740.00863.58193.250.781149.38149.3820.00253.54149.630.173150150.0020.00257.36198.770.217150.396150.40.00450.53127.550.347150.875150.8790.00452.7126.440.39150.915150.9170.00249.01152.390.173150.958150.9630.00543.48125.340.52151.82151.8230.00355.11195.460.26153.532153.5340.00260.04128.650.173159.497159.5010.00456.68200.430.347159.589159.5940.00561.69184.420.52160.219160.2250.00627.7693.310.607Because the calculated and measured data are from different lengths of the study section, it is a little inconvenient to compare the permeability coefficient of the whole section of the borehole. A new parameterKn is defined to describe the permeability coefficient per unit depth:(11)Kn=KmH,where Kn is the permeability coefficient per unit depth (/s), Km is the permeability coefficient of the section (m/s), and H is the depth of the section (m).In Figure8, the comparison of the permeability coefficient per unit depth by calculation and measurement clearly highlights the relevance of the two results.Figure 8
Permeability coefficient per unit depth with buried depth.


### 4.2. Trend Analysis of the Permeability Coefficient

The confining pressure is the most direct and influential factor affecting seepage flow through cracks. The relationship between the confining pressure and the permeability coefficient has been studied by many domestic and international researchers. The relationship between the permeability coefficient per unit depth and confining pressure (or buried depth) should be built to contribute to the prediction of the permeability coefficient of different sections of a borehole.The first study on this relationship was performed by Louis [23], who believed that normal stress has a considerable influence on the fracture surface while the influence of shear stress can be neglected. He tested the coupling relationship between normal stress and seepage flow and obtained an exponential relationship between the total stress and the permeability coefficient as a general conclusion. Later, the author revised this relationship and believed that “effective stress” should be used instead of total stress. In other words, there exists an exponential relationship between the effective stress and the permeability coefficient. Since then, many researchers have studied normal stress and seepage, and they have obtained a plethora of results. Currently, researchers believe that there exists a certain correlation between the stress and the permeability coefficient, which can be expressed by exponential, power, polynomial functions, etc. [24–28].According to the previous research results, the data are fitted by power function, as shown in Figure9.Figure 9
Comparison of fitting curves of the permeability coefficient per unit depth with buried depth.The function of the fitting curve could be obtained by the software Origin, as illustrated in the following:(12)y=1.04×10−5x−1.37,(13)y=3.05×10−6x−1.13,where function (12) is the fitting curve of the permeability coefficient per unit depth by calculation and function (13) is the fitting curve of the permeability coefficient per unit depth by measurement.Thus, the permeability coefficient per unit depth evidently varies based on the buried depth in an approximate power function. The calculation data are larger than the measurement data in near-surface areas on account of lower confining pressure, which produces some error when the buried depth is smaller than 60 m. However, it matched precisely when buried depth was greater than 60 m.Therefore, the aforementioned method can be used to estimate the permeability coefficient of any section in the well. Simultaneously, the permeability coefficient of the borehole can be estimated using the fracture characteristics in the borehole. If a large deviation of measured data from the calculated value arises, it is then necessary to conduct further studies to investigate the faults and caves around the borehole.


## 4.1. Calculation of Permeability Coefficient and Comparison with the Measurement Value

According to the actual drilling condition, the measured permeability coefficient is affected by both its own spatial characteristics and the stress from surrounding rocks. Neglecting other influential conditions, the permeability coefficient derived from the crack seepage model based on the borehole data needs to be constrained using the confining pressure which yields a certain level of attenuation. Therefore, based on the comparison of the permeability coefficients obtained from the crack model calculation and those measured from the water pressure test in the borehole (as shown in Table1), we can estimate the extent to which the permeability coefficient is attenuated by the confining pressure. After integrating all data, the comparison charts are obtained, as shown in Figures 7 and 8.Table 1
The results of water pressure tests in the borehole.
Section no.DepthAltitudeLength of test section (m)Permeability rate (Lu)Coefficient of permeability (m/d)Start (m)End (m)Start (m)End (m)15.1315.50124.10113.7310.3710.080.10401215.1325.50114.10103.7310.378.800.09490325.1335.50104.1093.7310.378.870.10484435.0248.4094.2180.8313.380.750.02719548.3261.7080.9167.5313.380.160.00302661.3274.7067.9154.5313.380.400.00739774.4287.8054.8141.4313.380.150.00336886.82100.2042.4129.0313.380.380.00671996.82110.2032.4119.0313.380.190.0053710106.92120.3022.318.9313.380.280.0030211117.32130.7011.91−1.4713.380.500.0107412127.32140.701.91−11.4713.380.240.0070513137.32150.70−8.09−21.4713.380.270.0090614147.12160.50−17.89−31.2713.381.050.0282015157.12170.50−27.89−41.2713.380.300.00403As shown in Figure7, the permeability coefficients calculated from the seepage model based on the borehole data are generally greater than the measured permeability coefficient. This is because the impact from the confining pressure on the crack aperture becomes greater with increasing confining pressure, which results in a decrease in the permeability coefficient. As each crack exhibits different levels of compressive deformation, the degree of change in the permeability coefficient is also different. However, the measured permeability coefficient is apparently larger at the borehole depth of 35.5 m and 160.5 m. The fracture data for these two depth intervals are provided in Table 2, which indicates that the fracture aperture is essentially small and hence the measured permeability coefficient should be relatively low. Therefore, the apparently larger permeability coefficient measure may arise from two aspects: (1) the hydraulic pressure test is not completely accurate and (2) some engineering operation in the borehole, such as flushing borehole, damages the original crack shape and results in inaccurate crack aperture and number.Figure 7
Comparison of the permeability coefficient by measurement and calculation.Table 2
The fracture data for sections 3 and 14 of the borehole.
Section no.DepthLength of test section (m)Inclination angle (°)Inclination (°)Aperture (cm)Start (m)End (m)325.19325.2030.0124.61240.371.04425.37725.4950.11857.53128.490.21725.73725.890.15363.96359.450.39125.84526.060.21570.76358.890.39125.89526.0860.19168.51359.450.26126.08326.2660.18367.77358.890.17426.45326.5640.11156.14275.820.17427.33127.5440.21370.57359.450.21728.76128.9540.19368.77358.890.26129.20329.4880.28575.25270.830.34833.17633.240.06440.6494.150.43533.65733.720.06340.06137.350.21733.67533.7390.06440.45175.570.21733.71633.8040.08849.65189.420.21733.85933.9020.04329.61151.20.21734.47934.5470.06842.3296.370.26114148.082148.0840.00276.92202.090.217148.976148.9780.00245.34163.440.173149.266149.2740.00863.58193.250.781149.38149.3820.00253.54149.630.173150150.0020.00257.36198.770.217150.396150.40.00450.53127.550.347150.875150.8790.00452.7126.440.39150.915150.9170.00249.01152.390.173150.958150.9630.00543.48125.340.52151.82151.8230.00355.11195.460.26153.532153.5340.00260.04128.650.173159.497159.5010.00456.68200.430.347159.589159.5940.00561.69184.420.52160.219160.2250.00627.7693.310.607Because the calculated and measured data are from different lengths of the study section, it is a little inconvenient to compare the permeability coefficient of the whole section of the borehole. A new parameterKn is defined to describe the permeability coefficient per unit depth:(11)Kn=KmH,where Kn is the permeability coefficient per unit depth (/s), Km is the permeability coefficient of the section (m/s), and H is the depth of the section (m).In Figure8, the comparison of the permeability coefficient per unit depth by calculation and measurement clearly highlights the relevance of the two results.Figure 8
Permeability coefficient per unit depth with buried depth.


## 4.2. Trend Analysis of the Permeability Coefficient

The confining pressure is the most direct and influential factor affecting seepage flow through cracks. The relationship between the confining pressure and the permeability coefficient has been studied by many domestic and international researchers. The relationship between the permeability coefficient per unit depth and confining pressure (or buried depth) should be built to contribute to the prediction of the permeability coefficient of different sections of a borehole.The first study on this relationship was performed by Louis [23], who believed that normal stress has a considerable influence on the fracture surface while the influence of shear stress can be neglected. He tested the coupling relationship between normal stress and seepage flow and obtained an exponential relationship between the total stress and the permeability coefficient as a general conclusion. Later, the author revised this relationship and believed that “effective stress” should be used instead of total stress. In other words, there exists an exponential relationship between the effective stress and the permeability coefficient. Since then, many researchers have studied normal stress and seepage, and they have obtained a plethora of results. Currently, researchers believe that there exists a certain correlation between the stress and the permeability coefficient, which can be expressed by exponential, power, polynomial functions, etc. [24–28].According to the previous research results, the data are fitted by power function, as shown in Figure9.Figure 9
Comparison of fitting curves of the permeability coefficient per unit depth with buried depth.The function of the fitting curve could be obtained by the software Origin, as illustrated in the following:(12)y=1.04×10−5x−1.37,(13)y=3.05×10−6x−1.13,where function (12) is the fitting curve of the permeability coefficient per unit depth by calculation and function (13) is the fitting curve of the permeability coefficient per unit depth by measurement.Thus, the permeability coefficient per unit depth evidently varies based on the buried depth in an approximate power function. The calculation data are larger than the measurement data in near-surface areas on account of lower confining pressure, which produces some error when the buried depth is smaller than 60 m. However, it matched precisely when buried depth was greater than 60 m.Therefore, the aforementioned method can be used to estimate the permeability coefficient of any section in the well. Simultaneously, the permeability coefficient of the borehole can be estimated using the fracture characteristics in the borehole. If a large deviation of measured data from the calculated value arises, it is then necessary to conduct further studies to investigate the faults and caves around the borehole.


## 5. Conclusions

A method for estimating the permeability coefficient was proposed considering the aperture, inclination angle, and joint roughness coefficient. During the prefeasibility stage or under the condition with the limited borehole, the method could help reduce the number of hydraulic pressure tests and save a large amount of cost. Further work should be done on more drill holes to establish a theoretical model of generality.The conclusions of the study are listed as follows:(1)
Based on the crack aperture, inclination angle, and JRC in the borehole data, we can estimate the seepage flow through the cracks in a target section. The estimated value reveals a similar trend to that of the measured permeability coefficient. The fitting curve of the permeability coefficient per unit depth with buried depth approximates a power function.(2)
During the prefeasibility stage, the permeability coefficient of the borehole can be estimated based on the fracture characteristics in the borehole. The permeability coefficients of different areas in boreholes can be estimated by the abovementioned method, which greatly reduces the number of hydraulic pressure tests and consequently reduces a lot of costs.(3)
The method could be used for the judgment of faults or caves. If there exists a large deviation between the measured data and the calculated value, it is necessary to further investigate the faults and caves around the borehole.



---

*Source: 4219847-2020-06-10.xml*