Fractal Analysis of Permeability of Unsaturated Fractured Rocks

Abstract

A physical conceptual model for water retention in fractured rocks is derived while taking into account the effect of pore size distribution and tortuosity of capillaries. The formula of calculating relative hydraulic conductivity of fractured rock is given based on fractal theory. It is an issue to choose an appropriate capillary pressure-saturation curve in the research of unsaturated fractured mass. The geometric pattern of the fracture bulk is described based on the fractal distribution of tortuosity. The resulting water content expression is then used to estimate the unsaturated hydraulic conductivity of the fractured medium based on the well-known model of Burdine. It is found that for large enough ranges of fracture apertures the new constitutive model converges to the empirical Brooks-Corey model.

1. Introduction

Modeling water flow in unsaturated fractured rocks has received considerable attention in the last two decades. One of the main reasons for focussing on the study of liquid flow in this type of media is that spontaneous capillary imbibition is an important fundamental phenomenon existing extensively in a variety of processes such as oil recovery, polymer composite manufacturing, soil science, and hydrology. The other reason is that deep disposal in crystalline rocks is considered to be an effective mean of isolating radioactive wastes from the biosphere. The study of basic transport processes has long been recognized because of which one can heighten the comprehensive understanding of physical phenomena, such as permeability [1, 2], heat transfer [3–5], and sorption [6, 7]. Many parameters such as the porosity, size of pore, and tortuosity of capillaries are very important for fluid flow in hard rocks. These parameters, however, are closely related to the geometric architecture of hard rocks. Cai et al. have analyzed the natural fractured trace maps representing a wide variety of scales, geological settings, and lithologies [8]. Cai and Yu have reported the density of different sized fault segments within the San Andreas fault zone which is fractal [9]. The distributions of fracture aperture and fracture spacing are self-similar over a well-defined range of apertures in the Cajon Pass scientific drill hole which was found by Barton and Zoback [10]. Based on the assumption that the fracture pattern is self-similar, the Sierpinski carpet was often employed to simulate porous media [11]. The conceptual constitutive model proposed by Wu and Yu [12] had used its fractal dimension to the parameters of the Brooks-Corey constitutive model [13] through the Sierpinski carpet. A Sierpinski space was also adopted to characterize the spatial distribution of a drainage network in the Gardon basin, France [14].

From the above review it is shown that a mechanistic model has not yet been established. In this paper, we attempt to develop a mechanistic model for unsaturated flow in fractured hard rocks based on the method using the specific fractal to describe fractured rock. The expressions of the proposed constitutive model are closed form and easy to evaluate. Another important feature, the tortuosity fractal dimension 𝐷_𝑇, which affected other model parameters and should not be neglected while it was often neglected in the past investigation, is considered.

Now fractal theory has a wide variety of applications in sciences and engineering fields such as thermal science [15–19], fluid science [1, 2, 6], and industrial construction engineering [20]. For example, Moussa [2] has systematically investigated the transport of porous media based on fractal theory. The work [2] is open. Xiao et al. [3–5, 15, 16, 18, 19] have done much outstanding work on heat transfer of fluids by using fractal technique. In our work, we derive the analytical expressions for the relative hydraulic conductivity of fractured rock while taking into account the effect of pore size distribution based on the fractal geometry theory.

2. Construction of the Fractal Model

The model is presented in Figure 1. It has been shown that the cumulative size distribution of contact spots on engineering surfaces is similar to islands on earth and pores in porous rock which follows the fractal scaling law [2]. 𝑁𝐿≥𝐷_𝑛=𝐷_𝑛,max𝐷_𝑛^𝑑𝑓, (1) where 𝑑_𝑓 is the fractal dimension for pores, 𝐷 is diameter, 𝐿 is the length scale, and 𝑁 is the total number of pores whose sizes equal to and greater than 𝐷_𝑛. The number of pores whose sizes range from 𝐷_𝑛 to 𝐷+𝑑𝐷_𝑛 is −𝑑𝑁=𝑑_𝑓𝐷^𝑑𝑓_𝑛,max𝐷^−(𝑑𝑓_𝑛⁺¹⁾𝑑𝐷_𝑛 (2) when water flow through the pores of porous rock, the capillaries may be tortuous. These tortuous capillaries could be expressed by fractal equation [21]. Consider 𝐿_𝑎𝐷_𝑛=𝐿^𝐷𝑇₀𝐷^1−𝐷𝑇_𝑛, (3) where 𝐷_𝑇 is the tortuosity fractal dimension and lies in the range 1<𝐷_𝑇<2, which represents the extent of convoluted ness of capillary pathways for fluid flow through a medium. Note that for a straight capillary path 𝐷_𝑇=1, and a higher value of  𝐷_𝑇  corresponds to a highly tortuous capillary. Let the diameter of a capillary in the medium be 𝐷_𝑛 and let its tortuous length along the flow direction be 𝐿_𝑎(𝐷_𝑛). 𝐿₀ is representative length of channels. With a straight capillary, 𝐿_𝑎(𝐷_𝑛) = 𝐿₀. The total volume of pores from 𝐷_𝑛,min to 𝐷_𝑛,max can be obtained from (1) as 𝑣_tol=^𝐷𝑛,max_𝐷𝑛,min𝜋4𝐷²_𝑛𝐿_𝑎()=−𝑑𝑁_{𝐷𝑛,max𝐷𝑛,min}𝜋4𝐷²_𝑛𝐿_𝑎𝑑_𝑓𝐷^𝑑𝑓_𝑛⋅max𝐷^−(𝑑𝑓_𝑛⁺¹⁾𝑑𝐷_𝑛=𝜋43−𝑑_𝑓−𝐷_𝑇𝐿^𝐷𝑇₀𝑑_𝑓𝐷^3−𝐷𝑇_𝑛⋅max×𝐷1−^𝑛,min𝐷_𝑛,max^{(3−𝑑𝑓−𝐷𝑇)}, (4) where 𝑣_tol is the whole volume of the pores, (−𝑑𝑁) is given by (2).

Figure 1 

The model presented.

Similarity to the state above, the volume 𝑣_{(𝐷𝑛,min𝑥<𝐷𝑛)}of pores/particles from 𝐷_𝑛 to 𝐷_𝑛,max can be obtained from (1) as 𝑣_{(𝐷𝑛,min<𝐷𝑛)}=𝜋4𝐿^𝐷𝑇₀𝑑_𝑓𝐷^𝑑𝑓_𝑛⋅max^𝐷𝑛,_{𝐷𝑛,min𝑥}𝐷^{2−𝑑𝑓−𝐷𝑇}_𝑛𝑑𝐷_𝑛=𝜋43−𝑑_𝑓−𝐷_𝑇𝐿^𝐷𝑇₀𝑑_𝑓𝐷^𝑑𝑓_𝑛⋅max𝐷^{3−𝑑𝑓−𝐷𝑇}_𝑛^𝐷𝑛,_𝐷𝑛,min. (5) The effective saturation 𝑆 of volume 𝑣_{(𝐷𝑛,min𝑥<𝐷𝑛)} can be obtained. Consider 𝑣𝑆=^{(𝐷𝑛,min<𝐷𝑛)}𝑣_tol=^{𝐷𝑛,max𝐷𝑛,min3−𝑑𝑓−𝐷𝑇}_𝑛𝐷𝐷_{𝑛,𝐷𝑛,min}𝐷^{3−𝑑𝑓−𝐷𝑇}_𝑛=𝐷^{3−𝑑𝑓−𝐷𝑇}_𝑛−𝐷^{3−𝑑𝑓−𝐷𝑇}_𝑛,min𝐷^{3−𝑑𝑓−𝐷𝑇}_𝑛,max−𝐷^{3−𝑑𝑓−𝐷𝑇}_𝑛,min. (6)

Assuming the immiscible fluid flow in reservoir rocks, three relatively important forces are considered: the capillary pressure can be expressed. Consider 𝛽𝑝=2𝜎cos𝜌𝑔𝐷_𝑛, (7) where 𝜎 is surface tension between the wetting and non-wetting fluids, 𝜌 is the water density, 𝑔 is the gravity acceleration, 𝐷_𝑛 is diameter of a pore, and 𝛽 is contact angle between the extraneous water and solid.

The relative permeabilities are usually expressed in terms of water saturation 𝑆_𝑤. The saturation curve for the proposed model of fractured rock was derived. Consider 𝑆_𝑤=𝑝^{3−𝑑𝑓−𝐷𝑇}_𝑐−𝑝^{3−𝑑𝑓−𝐷𝑇}_𝑐min𝑝^{3−𝑑𝑓−𝐷𝑇}_𝑐,min−𝑝^{3−𝑑𝑓−𝐷𝑇}_𝑐,max, (8) where 𝑝_𝑐max=2𝜎cos𝛽𝜌𝑔𝐷_𝑛,min,𝑝_𝑐min𝑥=2𝜎cos𝛽𝜌𝑔𝐷_𝑛,max. (9)

3. The Permeability of Unsaturated Fractured Rocks

The Burdine and Mualem models are the two most widely used models predicting relative hydraulic conductivity. For the particular state of flow in fractured hard rock, the Burdine model seems to be more consistent and to be often adopted. The expression of the Burdine model is 𝐾_𝐷(𝑠)=𝑠²∫^𝑠₀𝑑𝑠/𝑝²_𝑐∫¹₀𝑑𝑠/𝑝²_𝑐, (10) where 𝐾_𝐷(𝑠) is the relative hydraulic conductivity. Inserting (8) to (10) we obtain the following form for 𝐾_𝐷(𝑠)∶𝐾_𝐷(𝑠)=𝑝^{𝑑𝑓+𝐷𝑇}_𝑐⁻³−𝑝^{𝑑𝑓+𝐷𝑇−3}_𝑐max𝑝^{𝑑𝑓+𝐷𝑇−3}_𝑐,min𝑥−𝑝^{𝑑𝑓+𝐷𝑇}_𝑐⁻³_max²×𝑝^{𝑑𝑓+𝐷𝑇}_𝑐⁻⁵−𝑝^{𝑑𝑓+𝐷𝑇−5}_𝑐max𝑝^{𝑑𝑓+𝐷𝑇−5}_𝑐min𝑥−𝑝^{𝑑𝑓+𝐷𝑇−5}_𝑐max. (11) The expressions of (6) and (10) represent the proposed constitutive model for fractured hard rocks. Note that all model parameters are determined by geometric parameters of 𝐷_𝑛,max,,𝐷_𝑛,min𝑥 and residual water content 𝛽.

The novel constitutive model has some similarities with the well-known Brooks-Corey model, which is 𝑆₀=(𝑝_𝑐/𝑝_𝑑)^−𝜆 and 𝐾_𝐷(𝑠)=𝑆^3+𝜆/2, where 𝑝_𝑑 is the reciprocal of air entry pressure and 𝜆 is a model parameter related to pore size distribution.

When 𝐷_𝑇=1, 𝜆=2−𝑑_𝑓, and 𝑝_𝑐,max≪𝑝_𝑐,min𝑥, 𝑝_𝑐,max is neglected here, the model derived here is simplified to the Brooks-Corey model. Comparisons of the proposed and the Brooks-Corey models for three different ranges of fracture apertures are depicted in Figure 2. The assumed parameters are 𝑑_𝑓=1.8, 𝐷_𝑛,max=10⁻¹ cm, and 𝐷_𝑛,min𝑥=10⁻³ cm, 10⁻⁵ cm, and 10⁻⁷ cm. According to Figure 2, the Brooks-Corey model seems to be adequate to describe the hydraulic properties of fractured rocks for large ranges of fracture apertures and low values of pressure head.

(a)   Effective saturation

(b) Relative hydraulic conductivity

Figure 2 

Comparison between the Brooks-Corey model and the new relations for different ranges of fracture apertures when 𝐷𝑇=1.

The relationship between the fractal dimension and effective saturation is very important in the study of unsaturated flow in fractured hard rocks. The geometric parameters and physical constants used for the analysis are 𝛽=0,𝜎=72.25dy/pa. In order to analyze the influence of the fracture density we consider different values of 𝐷_𝑇. With (8) and (11), the relationship is obtained (Figure 3). With the physical constants 𝐷_𝑛=0.1cm,  𝐷_𝑛⋅max=1cm, and 𝐷_{𝑛⋅min𝑥}=0.01cm, the relationship between 𝑑_𝑓 and 𝑆₀ can also be determined (Figure 4).

(a) 𝐷𝑛,min=10−3 cm

(b) 𝐷𝑛,min=10−7 cm

Figure 3 

The new relations for 𝐾𝐷(𝑆) and 𝐷𝑇.

Figure 4 

The new relations for 𝑆0 and 𝑑𝑓.

In order to test the proposed invasion depth model, it is crucial to correctly determine the fractal dimensions 𝐷_𝑇 and 𝑑_𝑓.

According to the following formula proved recently by Yu and Li [22, 23], 𝐷𝜙=^𝑛,min𝐷_𝑛,max^{𝐷𝐸−𝑑𝑓}, (12) where 𝐷_𝐸 is the Euclidean dimension and 𝐷_𝐸=2 is used in this work. The maximum pore diameter can be calculated based on the model of square arrangement of particles.

According to the following formula proved recently by Jiangchao Cai, where 𝐾 is permeability, 𝑘 is Kozeny constant, which considers the tortuosity of capillaries and pore non-uniformity. And the parameters 𝐾 and 𝑘 are all proved [23]. Consider 𝐷_𝑇=1+ln𝑡_avln𝐿₀/𝑡_av,𝐷_𝑛,max=21−𝜙𝜙𝐾𝑘𝜙𝜙+1−𝜙𝜋4,1−𝜙−1 (13) where 𝑡_av=1211+21−𝜙+1−𝜙−1²+1−𝜙/41−,𝐿1−𝜙₀𝑡_av=𝑑_𝑓−121−𝜙𝜙𝜋𝑑_𝑓2−𝑑_𝑓𝐷_𝑛,min𝐷_𝑛,max. (14) From (12) and (13), the relationship among 𝜙, 𝐷_𝑇, and 𝑑_𝑓 is determined (Figure 5).

Figure 5 

The relationship among 𝜙,𝐷𝑇, and 𝑑𝑓.

4. Conclusions

In this paper, with the consideration of pore size distribution and tortuosity of capillaries, a new fractal model for relative hydraulic conductivity of fractured rock is developed. The derived constitutive model is an effort to understand and characterize unsaturated flow in fractured rocks. The expressions of water content and relative hydraulic conductivity curves have analytical closed forms. The parameters can be completely determined by the geometry of the fractal model. Every parameter of the proposed formulas of calculating relative hydraulic conductivity of fractured rock has a clear physical meaning. The fractal model can reveal the mechanisms of hydraulic conductivity for fluids flow through unsaturated rocks. The tortuosity fractal dimension 𝐷_T which affected other model parameters and should not be neglected while it was often neglected in the past investigation is considered in this paper.