1 Introduction

In the oil- and gas-producing areas of eastern China, tectonic fractures in reservoirs are closely related to the degree of fault development, and the formation mechanisms of faults and tectonic fractures are relatively consistent (Zhang et al. 2004; Jiu et al. 2013a; Liu et al. 2016). The spatial relationship between higher-order faults and lower-order faults provides insight into predicting fault-related fractures. Challenges related to the exploration and development of fractured oil reservoirs include evaluating the distribution and degree of fracture development in the reservoirs and quantitatively predicting fracture porosity and permeability (Zeng and Li 2009; Hennings et al. 2012). Fractures play a dual role in oil and gas exploration: They increase the permeability of reservoirs and can even provide basic permeability and reservoir porosity within reservoirs (Wang et al. 2016b; Gong and Rossen 2017), but the presence of fractures can complicate oil and gas EOR injection processes and well network deployment (Cook et al. 2008; Laubach et al. 2009; Olson et al. 2009; Wang et al. 2016a; Soleimani 2017).

Similar to faults, fractures exist at various scales (Prioul and Jocker 2009; Strijker et al. 2012; Chen et al. 2016), and the flow capacities of fractures at different scales are different (Fan et al. 2017). To the best of our knowledge, few studies on the quantitative prediction of fractures have focused on the scaling properties of fractures and the evaluation and classification of flow capacity. In regions with multiple groups of fractures and multiple scales of fracture development, the prediction of the predominant direction of fracture-related seepage is a difficult problem (Liu et al. 2015). Many researchers (Mynatt et al. 2009; Brogi 2011; Walker et al. 2011; Jiu et al. 2013b; Fang et al. 2017; Liu et al. 2017a, 2018a; Pan et al. 2017) have focused on the timing, distribution, and formation mechanisms of fractures. In this paper, based on the similarity of faults and fractures, we predict areal fracture densities at different scales and develop a skewed distribution model of fracture apertures from cores. Furthermore, we apply the in situ stress field to analyze the apertures of underground fractures at different scales. By developing a mathematical model to characterize the fracture porosity and permeability, we quantitatively calculate the porosity and permeability of multiperiod and multiscale fractures. Based on the relationship between the aperture and the density of the fractures and the relative magnitudes of the fracture porosity and permeability, the development pattern of fractures in the Jinhu Sag is determined. The results are important for the practical demands of oil field exploration and development.

2 Prediction of the areal density of fractures at different scales

In this paper, we use the reservoirs in the second member of the Funing Formation in the Jinhu Sag as a case study. The Jinhu Sag is located in the western part of the Subei Basin and extends from the Jianhu Uplift in the northwest to the Tianchang Uplift in the south and from the Zhangbaling Uplift in the west to the two low Lingtiangqiao and Liubao Uplifts in the east (Fig. 1a). The Jinhu Sag formed gradually and became deeper in the south and shallower in the north. The southern area is faulted with a northward stratigraphic overlap pattern that developed during the deposition of the Funing Formation (Fig. 1b). The lithologies of the reservoirs in the second member of the Paleogene Funing Formation, which is the primary target formation, are mainly argillaceous siltstone, siltstone and fine sandstone, all of which have low permeability. Many years of exploration has confirmed the existence of large oil and gas resources in these reservoirs. The patterns of the oil and gas development programs in the Shigang, Qiaohekou, Biandong, Minqiao, Yangjiaba and Tongcheng regions are closely related to the spatial distribution and degree of development of fractures (Neng et al. 2009, 2012; Ji et al. 2010; Liu et al. 2015).

Fig. 1
figure 1

a Location of the Jinhu Sag and the in situ stress orientation in the Subei Basin; the red box shows the study area. b Cross-section through the Jinhu Sag, with its location shown in Fig. 1a. The faults in Fig. 1a are normal faults. F1 is the Yangcun fault; F2 is the Shigang fault; F3 is the Tongcheng fault; and F4 is the Cuizhuang fault. JHS Jinhu Sag, GYS Gaoyou Sag, YS Yellow Sea, LSUP Lusu Uplift, SNUP Sunan Uplift, TCS Tianchang Salient, BD Biandong, YJB Yangjiaba and SG Shigang. K2t the Late Cretaceous Taizhou Formation; Ef1 the first member of the Eocene Funing Formation; Ef2–4 the second, third and fourth members of the Eocene Funing Formation; Ed the Eocene Dainan Formation; Es the Eocene Sanduo Formation; Ny the Neogene Yancheng Formation; and Qd the Quaternary Dongtai Formation

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2.1 Development features of fractures in the Jinhu Sag

Based on observations and statistical analyses of fractures in cores from more than 50 wells in the Biandong, Shigang, Qiaohekou, Minqiao, Yangjiaba and Tongcheng regions of the Jinhu Sag, we determined that tectonic fractures are well developed in the reservoirs of the second member of the Funing Formation (Ji et al. 2010; Liu et al. 2015). Most of the fractures are nearly vertical (dips of 75°–90°), with some high-angle fractures and few low-angle fractures (Fig. 2). The timing of fracture development is closely related to the timing of fault development (Ji et al. 2010; Liu et al. 2015). Core observations show that tensional fractures, shear fractures and tensional-shear fractures are well developed and that the proportion of each fracture type is different (Fig. 3). The Wubao movement, which occurred during the late stages of the deposition of the Funing Formation, was a period of strong fault activity and the main period of fracture development. The faults and fractures have consistent formation mechanisms, and the formation and distribution of fractures are closely related to the stress field that was present during fault formation, development and activity (Gudmundsson et al. 2010; Savage and Brodsky 2011).

Fig. 2
figure 2

Observations of core fractures from the Jinhu Sag. a Vertical fracture filled with calcite in the B12-3 well, 1604.00 m. b High-angle tensile-shear fracture in the TX33-3 well, 2144.92 m. c High-angle tensile fractures in the Y20 well, 1560.75 m. d Unfilled tensile-shear fracture in the S7 well, 2276.60 m. e Low-angle shear fracture filled with calcite in the M33 well, 1678.05 m. f High-angle parallel shear fractures in the S10 well, 2292.39 m. g Tensile fractures that are well developed around a high-angle shear fracture in the S11 well, 2283.55 m. h High-angle tensile fracture showing bifurcation at the end of the fracture in the TX33-3 well, 2071.11 m

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Fig. 3
figure 3

Statistical results of tensile and shear fractures in different regions of the Jinhu Sag. a Biandong area. b Minqiao area. c Yangjiaba area. d Tongcheng area. e Shigang area. f Qiaohekou area. See Fig. 1a for locations

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2.2 Fault-fracture similarity

2.2.1 Principle of the self-similarity calculation

Rocks rupture in a self-similar manner (Barton and Zoback 1992; Aguilar-Hernández and Ramírez-Santiago 2010). Based on the orientations of fractures in cores, the strike distributions of the faults and fractures are similar in different regions of the Jinhu Sag (Fig. 4). The nearly north–south-oriented tensile principal stress that was present during the late stages of the deposition of the Funing Formation controlled the fault activity. The tectonic fractures, faults and other fractures exhibit statistically significant self-similarity in terms of geometric morphology, tectonic evolution and genetic dynamics. The self-similarity of fractures is comprehensively quantified and characterized by the information dimension D. Closely related to the index of fault development, the fault D value can be used as a comprehensive index of the fault number, scale, pattern, horizontal length and distribution heterogeneity and can be calculated as follows (Matsumoto et al. 1992; Davarpanah and Babaie 2013):

$$D = - \mathop {\lim }\limits_{\varepsilon \to 0} \frac{I(\varepsilon )}{\ln (\varepsilon )}$$
(1)

where \(I(\varepsilon ) = \sum\nolimits_{i = 1}^{N(\varepsilon )} {P_{i} \ln (1/P_{i} )}\), Pi is the probability that each point falls in the ith grid, and ε is the length of the side of the grid (m). In this calculation, by changing unit length (r) of the fractal statistics, we can obtain the information dimension if there is a linear relationship between I(r) and lnr, where \(I(r) = \sum\nolimits_{i = 1}^{N(r)} {P_{i} (r)\ln \left[ {1/P_{i} (r)} \right]}\).

Fig. 4
figure 4

Rose diagrams of strike orientations of fractures and faults in different regions of the Jinhu Sag. a Fracture strikes in the Tongcheng area. b Fracture strikes in the Shigang area. c Fracture strikes in the Qiaohekou area. d Fracture strikes in the Yangjiaba area. e Fracture strikes in the Minqiao area. f Fault strikes in the Tongcheng area. g Fault strikes in the Shigang area. h Fault strikes in the Qiaohekou area. i Fault strikes in the Yangjiaba area. j Fault strikes in the Minqiao area. See Fig. 1a for locations

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$${I(}r )= - D\ln (r) + C$$
(2)

In Eq. (2), r is the unit length of the fractal statistics (m); D is the information dimension of the fracture (or fault); and C is the fitting coefficient of fault similarity.

2.2.2 Self-similarity evaluation of fractures in the Jinhu Sag

Fault systems show self-organization (Shaw 2004; Feng et al. 2016). Early faults affect the subsequent development of new faults, and the pattern of later-forming faults reflects the distribution of earlier faults. Faults at different scales that develop during different time periods have a statistically significant similarity (Hirata 1989; Sarp 2014; Wang et al. 2018). In the Jinhu Sag, we statistically analyzed 826 faults at different scales, including the sag-controlling large fault and associated secondary faults. The degree of similarity of fractures can be expressed by the fitting coefficient R2 of Eq. (2) (Sarp 2014; Mirzaie et al. 2015). Using this principle, we developed a computer program to calculate the D values of the lengths of faults (including the scale information of the faults) with different orientations and scales in the 10,838 statistical units (r = 8 km) in the research area to derive the distribution of the D and R2 values for the faults in the Jinhu Sag (Fig. 5a). A high D value usually indicates that the classification of fractures is complicated because the degree of development of fractures is high and a fracture network has formed (Hou 1994). In zones with high D values, the fault blocks are complicated and fragmented, resulting in small and fragmented reservoirs. In the region of fault development, the D values are in the range of 1.4–1.9, whereas at the edge of the sag, the D values are less than 1.0. Generally, if R2 > 0.97, the faults in a unit are considered statistically self-similar (Sarp 2014; Mirzaie et al. 2015). Of the 10,838 units, the R2 values for the D values are > 0.97 for 9169 (84.6%) statistical units, > 0.99 for 7641 (70.5%) statistical units, and < 0.90 for only 596 (5.5%) statistical units (Fig. 5b). These results indicate that the faults in the Jinhu Sag, which have different orientations and scales and formed during different time periods, are statistically self-similar.

Fig. 5
figure 5

a Distribution of fault information dimension (D) in the Jinhu Sag and b cumulative frequency fitting coefficient (R2) of units in the Jinhu Sag

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Studies have shown that due to the influence of higher-order faults, lower-order faults always develop with orientations that exhibit a high degree of similarity to those of the higher-order faults, which is the theoretical basis for predicting fractures of different scales. As shown in Fig. 6, we selected 390 higher-order faults as the framework model and simulated the development of 436 lower-order faults. The results indicate that the lower-order faults generally develop with an orientation that exhibits a high degree of similarity to that of the higher-order faults (R2 is high) and that the formation, development and distribution of secondary faults are closely related to the distribution of the higher-order faults.

Fig. 6
figure 6

Fault similarity evaluation of different directions and scales in the Jinhu Sag

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2.3 Calculation of fractures with different scales

In a statistical unit, fractures nearly always develop at an orientation with a high similarity to that of the fault, and the R2 values between fractures at different scales and the faults in the unit are high. Based on this principle, as shown in Fig. 7, we use Eqs. (1)–(2) to fit and derive the D value (1.6619) and R2 value (0.9991) of the faults in the statistical range of the fractal analysis in a certain unit. The ideal distribution of fractures at the scale of the cores should cause the R2 value to further increase in the unit; namely, the ideal distribution approaches points A and B in Fig. 7. By changing the length of the grid side, ε, we can determine the values of I(ε) that correspond to fractures at different scales and obtain the probability that a fracture will fall in each small grid. Based on the statistics, we calculate the total length of the fractures with different scales in the unit and propose a model to calculate the areal densities of fractures with different scales:

$$\rho_{\text{s}} = d \cdot \text{e}^{ - D\ln (d) + C}$$
(3)

where d is the scale of the fracture (m).

Fig. 7
figure 7

Computational model of multiscale areal fracture density

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We use the principle for calculating the fracture D value and compile the corresponding algorithm. After obtaining the data on the faults, we simulate the distribution of the areal fracture density. As shown in Fig. 8, the region with a dense distribution of secondary faults along the main fault is an area of high fracture density. We use the areal density of fractures at the thin-section scale (fracture scale d > 10−5 m) to calculate the porosity; this parameter somewhat reflects the reservoir and corrosion capability of fractures. We use the linear density of fractures at the core scale (fracture scale d > 0.5 m) to calculate the magnitude and direction of the permeability and use this parameter as a reference for the subsequent deployment of a well network. The areal fracture density at the thin-section scale is 4–12 m/m2, and the areal fracture density at the core scale is 0.25–0.70 m/m2.

Fig. 8
figure 8

Distribution of multiscale areal fracture density. a Thin-section scale (d > 10−5 m) and b core scale (d > 0.5 m). ρrmin is the areal fracture density at the thin-section scale (m/m2), and ρrmax is the areal fracture density at the core scale (m/m2)

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3 Prediction of the current apertures of fractures

3.1 Model for the distribution of fracture apertures

The apertures of fractures statistically follow a skewed distribution (Nelson 2001), and the median, mode and mean of the apertures are commonly different. In a unit, a few fractures with large apertures can dominate the fracture permeability direction. The length of a fracture is positively correlated with the aperture of the fracture (Nelson 2001). We assign different apertures to the areal densities of fractures at different scales that were calculated in the previous section to develop a skewed distribution model of fracture apertures at different scales and orientations in the unit.

The statistical analysis of the scales and apertures of more than 800 faults in the Jinhu Sag (Fig. 9) indicate that both exhibit strongly skewed distributions. By establishing a skewed distribution model, we assign different apertures to fractures at different scales. We then use the program to perform probability coupling between the apertures of the fractures in the cores and thin sections. The detailed scheme is as follows: Using the statistical analysis of the scales of the faults in the unit, we apply Fig. 9b to determine the probability distribution of the scales of the fault and use the aperture model in Fig. 9a to obtain the average aperture of the fractures in each statistical unit, baver. We then use Eq. (3) to determine the areal density of fractures with various scales in the Jinhu Sag (the total length). The length distributions of all the statistical units are normalized according to Fig. 9b. Through cyclic iteration, we divide the total length of the fractures in each unit into m segments. The length distribution is consistent with the skewed distribution model of the distribution. We assign the probability function of the apertures of fractures in the cores and thin sections into m segments of fractures with different scales. The large-scale fractures have large apertures, and the small-scale fractures have small apertures, which simultaneously satisfy the following equation:

$$b_{{\text{aver}}} = \frac{{\sum\nolimits_{i = 1}^{m} {b_{i} l_{i} } }}{{\sum\nolimits_{i = 1}^{m} {l_{i} } }}$$
(4)

where li is the length of the ith segment of the fracture (m); bi is the aperture of the ith segment of the fracture (m); and m is the number of segments of the fracture. If m is sufficiently large, it is statistically significant. In this paper, the value of m is 8260.

Fig. 9
figure 9

a Skewed distribution model of fracture apertures and b skewed distribution model of fault lengths

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Because of the probability coupling between fracture length and aperture, the fracture apertures in each unit are consistent with a skewed distribution, and the distribution of fracture apertures in the entire study area is also consistent with the frequency distribution shown in Fig. 9a.

3.2 Aperture properties of fractures analyzed by simulations of the stress field

3.2.1 Numerical simulation of the in situ stress

The fracture density is related to the paleostress field and fault activity, and the size and direction of the present horizontal principal stress determine the opening and permeability of unfilled fractures. Underground natural fractures have very small apertures, and they are very difficult to measure. The in situ stress field is the key to predicting the apertures of underground fractures, and the finite element method (FEM) has produced good results in the simulation of the in situ stress field (Jiu et al. 2013b; Liu et al. 2017b, 2018b; Zhao and Hou 2017). The FEM requires an established geological model, a mechanical model and a mathematical model (Jiu et al. 2013b; Liu et al. 2017b, 2018b). We developed a geological model based on a three-dimensional structure map of the second member of the Funing Formation, and triaxial fracturing experiments performed on rocks from the Biandong, Minqiao, Tongcheng and Shigang regions are used to determine the mechanical parameters of the rock formations. These include the Poisson’s ratio, elastic modulus, uniaxial compressive-shear strength and internal frictional angle. The results are combined to develop a mechanical model of the Jinhu Sag. Because of the mechanical, physical and mathematical principles used in the FEM simulation, the geological model was meshed with triangular elements and subdivided into a series of nodes and elements. The fault zone and target stratum were meshed with a fine grid, and the surrounding rocks were meshed with a coarse grid. The model was subdivided into 304,125 elements and contained 66,566 nodes, and each element and node was endowed with different mechanical attributes.

In this study, based on data from earthquake focal mechanisms, hydraulic fracturing, borehole breakouts and experiments on the sound velocity of oriented cores (Figs. 1a, 10, Table 1), the direction of the maximum horizontal principal compressive stress (σH) was determined. The boundary conditions in the stress simulation included the stress direction, stress magnitude and displacement constraints determined via the FEM. Hydraulic fracturing and the acoustic emission (AE) method are used to estimate the magnitude of the in situ stress (Healy and Zoback 1988; Seto et al. 1997; Jie et al. 2005). When a core is removed from the ground, due to differences in the three-direction principal stresses in the core, the core will develop microcracks that are positively correlated with the stress. The in situ stress can be analyzed based on loading data. In the Jinhu Sag, the values of σH and σh calculated by the AE method for the depth range of 1600–2000 m were 30.1–44.4 MPa and 25.7–30.4 MPa, respectively (Fig. 10e, f). Based on hydraulic fracturing (Table 1) and the AE method (Fig. 10e, f), compressive stresses of 38 and 28 MPa were applied to the ENE and NNW boundaries of the model, respectively. After debugging the above approach, the appropriate displacement was applied to the geological model to prevent it from undergoing rotation and rigid displacement and to facilitate the simulation. The Y-direction constraint was applied to the SSW boundary of the model, the X-direction constraint was applied to the WSW boundary, and the Z-direction constraint was applied to the bottom boundary. The distributions of the maximum, intermediate and minimum principal stresses were determined using a 3D FEM numerical simulation (Fig. 11).

Fig. 10
figure 10

The direction of the present maximum horizontal principal stress within the Jinhu Sag. a Derived from hydraulic fracturing (N = 18). b Derived from sound velocity experiments in oriented cores (N = 42). c Derived from borehole breakouts in the Subei Basin (N = 15). d Derived from focal mechanisms and sliding faults in the Subei Basin (N = 22). Curves of in situ stress measurements based on the initial pressures of acoustic emissions for samples in the Shigang area (1921.11 m): e vertical direction and f horizontal direction. See Fig. 1a for locations

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Table 1 The size and direction of in situ stresses derived from hydraulic fracturing in the Jinhu Sag
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Fig. 11
figure 11

Distributions of in situ stresses in the Jinhu Sag. a Minimum principal stress. b Intermediate principal stress. c Maximum principal stress (Negative values represent compressive stress)

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3.2.2 Apertures of underground fractures

In general, the current stress field no longer generates new fractures, and the linear density and occurrence of fractures essentially remain unchanged. However, under the influence of three-dimensional compressive stress, fractures can partially close, decreasing the seepage capability. Willis-Richards et al. (1996) and Jing et al. (1998) considered the influences of normal stress and shear stress on the aperture of a fracture and derived an equation to calculate the aperture of a fracture within the in situ stress field as follows:

$$b = \frac{{b_{\text{0}} }}{{1 + 9\sigma '_{\text{n}} /\sigma_{{\text{nref}}} }} + b_{{\text{res}}}$$
(5)

where b0 and b are the original and current aperture of the fracture (m), respectively; σ′n is the effective normal stress (MPa); bres is the residual aperture of the fracture (m); and σnref is the corresponding effective normal stress (MPa) when the fracture aperture decreases by 90% (unit: MPa) and is a coefficient related to lithology. Some researchers (Durham and Bonner 1994; Willis-Richards et al. 1996) have provided test values for different conditions. The results from Durham and Bonner (1994) show higher values of σnref that can exceed 200 MPa, whereas the results from Willis-Richards et al. (1996) show that the value of σnref is a constant between 10 and 26 MPa. Qin (2002) tested the permeability of low-permeability sandstone in eastern China and found that, for samples with a uniaxial compression strength of 30–50 MPa, σnref should be considered to be 30 MPa. In addition, σ′n in Eq. (5) is the effective normal stress, which can be determined by stress simulation, and b0 can be determined by the distribution of fracture apertures in Fig. 9a and Eq. (4).

Using Eq. (5), we can calculate the apertures and the means of underground fractures that have different scales. Figure 12 shows the distribution of the average fracture apertures. Due to the influence of the fractures, the in situ stress and the initial aperture (scale) of the fractures, the distribution of the fracture apertures is discontinuous and has large local variations. At the edge of the sag, the fracture apertures are large, whereas in the center of the sag, the fracture apertures are small. The fracture apertures are concentrated in the range of (6–20) × 10−5 m.

Fig. 12
figure 12

Distribution of average fracture apertures in the second member of the Funing Formation

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4 Model for calculating the porosity and permeability of fractures at different scales

In the individual statistical units, fractures at different scales have different flow capacities, and long fractures with large apertures usually play dominant roles in fluid flow. The flow capacities of fractures at different scales can be evaluated using an appropriate model. By developing a model of the micro-unit, we can calculate the porosity and permeability of the fracture. As shown in Fig. 13, we use the fracture as the reference object to establish a static coordinate system (O-ENWS), and the geodetic coordinates of the reference object are used to establish a dynamic coordinate system (O-XY). We define θ as the angle between the OX axis in the horizontal plane and true east, and this term represents the rotation angle of the dynamic coordinate system. By adjusting the value of θ, we can calculate the permeabilities of fractures in various orientations in the dynamic coordinate system. We define θ as negative when the OX axis is oriented toward the northeast and as positive when it is oriented toward the southeast. We use this model to evaluate the flow capacities of fractures at different scales.

Fig. 13
figure 13

Calculation model of fracture porosity and permeability

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The calculation of the fracture porosity in this paper does not consider corrosion of the fracture surface. Using the calculation model for the parameters of multiscale and multiperiod fractures in Fig. 13, we derive the corresponding algorithm and obtain the formula to calculate the porosity of a fracture φ:

$$\varphi = \frac{{\rho_{{\text{rmin}}} \cdot R^{2} }}{{\sum\nolimits_{i = 1}^{m} {l_{i} } }} \cdot \sum\limits_{i = 1}^{m} {(b_{i} \cdot l_{i} )} \times 100\%$$
(6)

In complicated geological conditions, fractures in reservoirs are affected by multiple episodes of tectonic activity, and the occurrence, areal density and aperture values are often different in each group of fractures. Different parts of a single fracture can also have different apertures. When multiple fractures are present in a unit, the equation to calculate the direction of maximum permeability, θmax, is as follows:

$$\theta_{ \hbox{max} } = \arctan \left( {\frac{{\sum\nolimits_{j = 1}^{n} {\sum\nolimits_{i = 1}^{m} {b_{ji}^{3} l_{ji} n_{{{\text{x}}j}} } } }}{{\sum\nolimits_{j = 1}^{n} {\sum\nolimits_{i = 1}^{m} {b_{ji}^{3} l_{ji} n_{{{\text{y}}j}} } } }}} \right)$$
(7)

The maximum permeability of the fracture, Kmax, can be expressed as follows:

$$K_{ \hbox{max} } = \sum\limits_{j = 1}^{n} {\sum\limits_{{i\text{ = 1}}}^{m} {\left[ {\frac{{b_{ji}^{3} l_{ji} }}{12} \cdot \frac{{(n_{{xj}} \sin \theta_{ \hbox{max} } + {n}_{yj} \cos \theta_{ \hbox{max} } )^{2} + {n}_{{zj}}^{2} ({n}_{{xj}} \cos \theta_{ \hbox{max} } - {n}_{yj} \sin \theta_{ \hbox{max} } \text{)}^{2} }}{{\sqrt {n_{{xj}}^{2} + n_{{yj}}^{2} } }}} \right]} } \; \cdot \frac{{\rho_{{\text{rmax}}} \cdot R}}{{\sum\nolimits_{i = 1}^{m} {l_{i} } }}$$
(8)

In Eqs. (6)–(8), R is the length of the side of the unit (m); ρrmin is the areal density of fractures at the thin-section scale (m/m2); ρrmax is the linear density of fractures at the core scale (m/m2); n is the number of fractures; and nxj and nyj are the components of the unit normal vector of the jth group of fracture planes along the X- and Y-axes, respectively.

5 Distributions of fracture porosity and permeability

5.1 Distribution of fracture porosity

Based on Eqs. (1)–(8), we use programs written in Visual C++6.0 to develop the Quantitative Prediction of Multiscale Fractures (QPMF) system and to perform a quantitative evaluation of the development model with the fracture parameters from the oil- and gas-bearing areas. The algorithm parameters used in this system are as follows: The unit length of the fractal statistics (r) in the Jinhu Sag was 8000 m; the fracture lengths used in the areal density of fractures at the thin-section scale and core scale were 5 × 10−5 and 0.5 m, respectively; the length of the side of the unit (R) was 10,000 m; the areal fracture density at different scales and fracture opening used for calculation are shown in Figs. 8 and 9a. We adopt the areal density of fractures at the thin-section scale to calculate the distribution of fracture porosity (Fig. 14). The calculated fracture porosity does not consider corrosion/dissolution on the fracture surfaces. Compared with the aperture statistics of Nelson (2001), Anders et al. (2014) and Zeng et al. (2013), the prediction results in Fig. 14 are relatively low. The fracture porosity is closely related to the areal density (Fig. 8a) and the region with high porosities of (35–95) × 10−3%. The fracture porosity is high in the region of secondary faults near the main faults. Along the edge of the sag, the fracture porosity is low in the region of weakly developed secondary faults and areas of low structure.

Fig. 14
figure 14

Distribution of fracture porosity in the second member of the Funing Formation

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5.2 Distribution of fracture permeability

As shown in Figs. 15 and 16, the fracture permeability of the reservoirs in the second member of the Funing Formation is as high as (20–120) × 10−3 μm2. Few secondary fractures are located near the large faults, but they have large apertures and strong flow capacities. Therefore, this type of fault usually plays a role in determining the main permeability directions of the fractures (Fig. 16). The paleostress field that affected the formation and activity of faults (fractures) is important for controlling the direction of primary permeability, and the current maximum horizontal principal stress (ENE) plays a role in changing the main seepage direction. The fractures that trend NE-ESE (45°–120°) are relatively open. Thus, in each unit, the flow capacities of the fractures with these orientations are stronger, so the main direction of fracture permeability generally converges toward the ENE.

Fig. 15
figure 15

Distribution of fracture permeability in the second member of the Funing Formation

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Fig. 16
figure 16

Orientation distribution of maximum fracture permeability in the second member of the Funing Formation

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6 Results and discussion

6.1 Comparative validation of simulation results

A comparison between the linear density and porosity of fractures in the cores and the predicted results shows that the consistency is relatively good in the Tongcheng, Shigang, Qiaohekou, Yangjiaba and Minqiao regions but not in the Biandong region where the prediction has a relatively large error. This error may be related to the reservoirs in the second member of the Funing Formation, which in this area includes mixed deposits of lacustrine limestone and sandstone. Compared to low-permeability sandstone, these rocks are more brittle and prone to fracture. Therefore, the areal fracture density in the observed core is much higher than in the predicted results. The simulation results generally reflect the differences and variations in the fracture parameters in different regions fairly well (Table 2).

Table 2 Correlation between numerical simulations and core measurements in the Jinhu Sag
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6.2 Development patterns of fractures in the Jinhu Sag

Previous studies (Ji et al. 2010; Liu et al. 2015) have used fine-scale geological models for multiparameter simulations of fractures, and the results indicate that the areal density and aperture are different for fractures in different regions. Fractures in the Minqiao and Yangjiaba regions have large apertures and exhibit a high density, whereas fractures in the Shigang area have small apertures and exhibit a relatively high density. Fractures in the Qiaohekou area exhibit a low density and relatively large apertures and are typically low-porosity and high-permeability fractures. Fractures are moderately developed in the Tongcheng area. Therefore, using the relationships between the areal densities and apertures of the fractures in the different statistical units in these regions, we combine the distributions of the fracture porosity and permeability to develop classification criteria for the patterns of fracture development (Table 3). We preliminarily divide the fractures in the Jinhu Sag into five types (Figs. 17 and 18):

Table 3 Classification criteria for different fracture development patterns in the Jinhu Sag
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Fig. 17
figure 17

Distribution of areal fracture density and aperture of different development patterns. F-I is the low-porosity and low-permeability fracture type; F-II is the transitional fracture type; F-III is the high-porosity and low-permeability fracture type; F-IV is the low-porosity and high-permeability fracture type; and F-V is the high-porosity and high-permeability fracture type

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Fig. 18
figure 18

Fracture development patterns in the second member of the Funing Formation in the Jinhu Sag. (A) High-porosity and high-permeability fractures. (B) Low-porosity and high-permeability fractures. (C) High-porosity and low-permeability fractures. (D) Transitional fractures. (E) Low-porosity and low-permeability fractures. (F) No fractures developed in the area

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  1. 1.

    Low-porosity and low-permeability fractures have low porosity and permeability values and are mainly located in Yangcun, Jinhu, and along the edge of the sag. The sand bodies in these regions are thin and have poor lateral continuity, high clay contents and low degrees of brittleness. These fractures are mainly present in areas where faults are poorly developed. The faults are long but are low in density, and secondary faults are few.

  2. 2.

    High-porosity and high-permeability fractures have high porosity and permeability values and large apertures and are located in regions of high fracture density. The regions with these fractures experienced large differences in principal paleostresses and have a consistent fracture distribution. These fractures are mainly distributed along the western slope and in the Minqiao area, and the current burial depth is moderate to relatively shallow (1500–2600 m). The reservoirs are mainly composed of very brittle siltstone and fine sandstone with low permeability and medium-sized pores that have a consistent distribution.

  3. 3.

    Low-porosity and high-permeability fractures have low porosity and high permeability values, relatively large fracture apertures and a low fracture density. They are located in regions of low paleostress due to fault activity, including near the Shigang fault, the Yangcun fault and some of the main faults. The fractures are mainly large, and their permeability is often dominated by a few fractures that have large apertures.

  4. 4.

    High-porosity and low-permeability fractures have high porosity and low permeability values, relatively small apertures and a high fracture density. They are located in regions of high paleostress release due to fault activity. The fractures are mainly small, densely distributed and located at moderate burial depths (1900–2800 m). They are mainly located in regions of high fracture density at the intersections of small secondary faults (e.g., Lvliang and north of Anle) and south of the Yangcun and Shigang faults.

  5. 5.

    Transitional fractures have moderate porosity and permeability values, and they are located in the transitional regions between the high-porosity and low-permeability fracture type and the low-porosity and high-permeability fracture type and in areas where the low-porosity and low-permeability fracture type transitions to other types of fractures. These fractures are mainly developed in the lower developmental zone of a fault, where secondary faults developed, but the scale is small, and the shapes of the faults are basically the same.

7 Conclusions

  1. 1.

    Fractures always develop in orientations similar to those of nearby faults, and this relationship can be used to predict the areal densities of fractures that have different scales. Furthermore, fracture apertures can be determined from observations of fractures in cores and thin sections. Fractures in reservoirs have scaling properties, and fractures at different scales have different effects on reservoir volume and fluid flow. We use the in situ stress field to calculate the apertures of underground fractures, to develop a predictive model for the porosity and permeability of multiperiod and multiscale fractures and to evaluate the flow capacities of fractures at different scales.

  2. 2.

    The minimum horizontal principal stress in the second member of the Funing Formation in the Jinhu Sag is 28–32 MPa and is oriented ENE to E–W (75°–90°), and the maximum horizontal principal stress is 37–39 MPa and is oriented NNW to N–S (345°–360°). The distribution of fracture apertures is discontinuous and shows significant local variation, with most of the fracture apertures in the range of (6–20) × 10−5 m. The fracture porosity of the reservoirs in the second member of the Funing Formation is (35–95) × 10−3%, and the permeability is (20–120) × 10−3 μm2. The main fractures control the direction of the primary permeability, the paleostress played a role in controlling the direction of fracture-related seepage, and the in situ stress field has changed the main seepage direction of the fractures to the ENE.

  3. 3.

    The fracture density is related to the paleostress field and fault activity, and the opening and permeability of unfilled fractures are determined by the size and direction of the present horizontal principal stress. Controlled by the paleo- and modern stress field and the lithofacies, the porosity and permeability of the fractures vary among different areas in the Jinhu Sag. The fractures in the Jinhu Sag can be divided into the following five patterns: (1) low-porosity and low-permeability fractures with small apertures and a low fracture density; (2) high-porosity and high-permeability fractures with large apertures and a high fracture density; (3) high-porosity and low-permeability fractures, which are mostly small-scale fractures with relatively small apertures and a relatively high density; (4) low-porosity and high-permeability fractures, which are mostly large-scale fractures with relatively large apertures and a relatively low fracture density; and (5) intermediate transitional fractures.