Effects of Background Porosity on Seismic Anisotropy in Fractured Rocks: An Experimental Study

Abstract

Fractures are widely distributed in the subsurface and are crucial for hydrocarbon, CCS, offshore infrastructure (windfarms), and geothermal seismic surveys. Seismic anisotropy has been widely used to characterize fractures and has been shown to be sensitive to background matrix porosities in theoretical studies. An understanding of the effects of background porosity on seismic anisotropy could improve seismic characterization in different fractured reservoirs. Based on synthetic rocks with controlled fractures, we conducted laboratory experiments to investigate the influence that background porosity has on P-wave anisotropy and shear wave splitting. A set of rocks containing the same fracture density (0.06) with varying porosities of 15.3%, 22.1%, 26.1% and 30.8% were constructed. The P- and S-wave velocities were measured at 0.5 MHz as the rocks were water saturated. The results show that when porosity increased from 15.3% to 22.1%, P-wave anisotropy and shear wave splitting exhibited slight fluctuations. However, when porosity continued to increase to 30.8%, P-wave anisotropy declined sharply, whereas shear wave splitting stayed nearly constant. The measured results were compared with predictions from equivalent medium theories. Qualitative agreements were found between the theoretical predictions and the measured results. In the Eshelby–Cheng model, an increase in porosity reduces fracture-induced perturbation in the normal direction of the fracture, resulting in lower P-wave anisotropy. In the Gurevich model, an increase in porosity can reduce the compressional stiffness in parallel directions to a larger extent than that in perpendicular directions, thus leading to lower P-wave anisotropy.

1. Introduction

Fractures are found throughout crustal rocks, and fracture detection is of critical importance for hydrocarbon exploration, water resource management, CO₂ capture and storage, and geo-thermal exploitation. Aligned fractures can cause seismic anisotropy, making it possible to obtain fracture information from seismic data. Fracture-induced anisotropy is an integrated result of various fracture parameters (fracture density, scale, aspect ratio, etc.), fluid properties (fluid modulus and viscosities), and background properties (modulus, porosity, and permeability). Background matrix porosity has been thought to greatly influence P-wave velocity anisotropy [1,2]. Because fractures are present in low, mid, and high porosity hydrocarbon reservoirs, understanding the effect of background porosity on seismic anisotropy could further improve seismic fracture characterization in reservoirs of different porosities.

To date, theoretical and experimental studies have been conducted to relate the physical properties of rock to seismic velocity and anisotropy, and various equivalent medium theories have been proposed [1,3,4,5,6]. These theories were based on different assumptions and were derived using different methods, resulting in different predictions. Thus, there is a desire to verify and calibrate these theoretical models through laboratory experiments. Laboratory experiments typically use synthetic samples with controlled and known rock physical properties and fracture parameters. A number of experimental studies have been conducted to observe elastic wave velocity and anisotropy in fractured rocks [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. These studies have investigated how parameters, such as fracture density, scale, and aspect ratio, can influence elastic wave velocity and anisotropy, providing significant insights into the physical mechanisms of elastic wave propagation in fractured rocks. However, natural rocks have a complex depositional and diagenetic/tectonic history, and fracture parameters (fracture geometry and density) cannot be quantitatively controlled, making them too variable for experimental investigations concerning the effect of a single variable on seismic wave propagation. Thus, synthetic samples must be used in laboratory experiments to observe seismic wave propagation in cracked rocks. Despite variations in the background porosity of fractured reservoirs, few experimental studies, both in laboratory and field observations, have focused on the effect of background porosity on seismic anisotropy in these reservoirs.

Gurevich (2003) discussed P-wave anisotropy variation with background porosity in water-saturated rocks with aligned fractures in a low-frequency range using numerical simulations [1]. When there is no background porosity, P-wave anisotropy (ε) is very small. In the absence of background porosity, fluid is ‘isolated’ in fractures. The ‘isolated’ fluid acts to ‘stiffen’ the fractures and reduce fracture compliance. This results in a relatively low P-wave anisotropy for the overall fractured rock. As the characteristic porosity increases, there is a sharp increase in P-wave anisotropy. When surrounded by pores, the fluid in these fractures can escape during compression; therefore, the fracture is more compliant than it would be in ‘isolated’ conditions. As porosity continues to increase, P-wave anisotropy flattens and decreases gradually. Other theoretical studies have also discussed the influence of porosity on anisotropic parameters [2,23]. However, no experimental studies have been conducted to test and verify this, owing to the difficulties faced in manufacturing artificial fractured rocks with low porosity.

This study experimentally examined the influence of background porosity on P-wave anisotropy in fractured rocks under water saturation conditions. A set of artificial fractured sandstone samples with the same fracture density (0.06), but different porosities (ranging from 15.3% to 30.8%), were built. Laboratory ultrasonic measurements were conducted to obtain the P- and S-wave velocities and anisotropy. We analysed the variations in P-wave anisotropy and shear wave splitting (SWS) with background porosity. The results were compared with theoretical models and the causes of the variations in P-wave anisotropy and shear wave splitting with porosity were discussed.

2. Equivalent Medium Theories

2.1. High Frequency (Unrelaxed) Moduli

High frequency theories have modelled fractures as “isolated” inclusions. Thus, there was no fluid exchange between inclusions and the background medium. They assumed that if the frequency was higher than the squirt characteristic frequency, $f\gg f_c$, the fluid pressure does not have time to equilibrate between stiff and compliant pores during the half-wave cycle, or the so-called “unrelaxed” state. Cheng (1978, 1993) derived the elastic stiffness tensor of a fractured medium under high frequency (unrelaxed) conditions based on the Eshelby inclusion theory. Mathematically, the Eshelby–Cheng model is given by the following [5,6]:

$$C_{ij}^{eff}=C_{ij}^{\left(0\right)}-{\phi }_f{C}_{ij}^{\left(1\right)},$$

(1)

where $C_{ij}^{\left(0\right)}$ is the background matrix stiffness tensor and ${\phi }_f$ is the fracture porosity, and

$$C_{ij}^{\left(1\right)}=C_{ik}^{\left(0\right)}{A}_{kj},$$

(2)

where $A_{ij}$ relates the applied strain $e_i^A$ to the “stress-free” strain $e_i^T$ of the inclusion:

$$e_i^T={\mathrm{A}}_{ij}{e}_j^A$$

(3)

2.2. Low-Frequency (Relaxed) Moduli

In contrast to high-frequency theories, low-frequency theories consider the fluid exchange between fractures and the surrounding pores and micro-cracks. They assume that if the frequency is lower than the squirt characteristic frequency, the fluid pressure can equilibrate locally between the cracks and adjacent pores. By combining linear slip theory and the anisotropic Gassmann equation [24], Gurevich (2003) derived expressions for a fluid-saturated cracked porous medium under low frequency conditions in the following form:

$$C_{ij}^{sat}=C_{ij}^0+{\alpha }_i{\alpha }_{j}Mi,j=1,\dots 6$$

(4)

where $C_{ij}^0$ is the dry modulus of the fractured medium and ${\alpha }_m$ is Biot’s coefficient, which is as follows:

$${\alpha }_m=1-\frac{\underset{n=1}{\overset{3}{{\displaystyle \Sigma }}}{C}_{mn}^0}{3K_g}$$

(5)

For i = 1, 2, and 3, ${\alpha }_4={\alpha }_5={\alpha }_6=0$. Scalar $M$ is the direct analog of Gassmann’s pore space modulus, which is as follows:

$$M=\frac{K_g}{(1-\frac{K^{\ast }}{K_g})+\phi (\frac{K_g}{K_f}-1)}$$

(6)

where $K_g$ and $K_f$ is the bulk modulus of mineral grains and pore fluid, respectively. φ is the overall porosity of the fractured rock. $K^{\ast }$ denotes the so-called generalized drained bulk modulus, which is defined as:

$$K^{\ast }=\frac{1}{9}\sum _{i=1}^3\sum _{j=1}^3{C}_{ij}^0$$

(7)

3. Sample Preparation

The method of Ding was adapted to build artificial porous sandstones with similar mineral components [17,18], pore structures, and cementation to natural rocks. In this method, powders of silica sand, feldspar, and kaolinite, were mixed in a ball mill for 24 h to ensure homogeneity. Then, these mineral powders were mixed with sodium silicate solution. The mixture was then paved into a mould layer by layer. In order to make the “meso-scale” fractures, touch paper discs were spread on the surface of the mineral powder mixtures when layering the mixture in the mould. Touch paper is a special kind of paper which has been soaked in saltpetre. At high temperatures, touch paper decomposes into gas and leaves nearly no remains.

In order to pave the mixture in the mould in layers, a special tool was made (Figure 1). The tool consisted of a frame, a glass plate, and a threaded rod. By rotating the threaded rod, the glass plate can be moved vertically. The procedure of paving the mixture is shown in Figure 2. Afterwards, a predetermined number of discs were carefully placed on top of each mixture layer. Then, the next mixture layer was paved and the same number of discs were placed on it (except for the top layer).

After this, a uniaxial compressing pressure was applied to the mould for 10 min in order to ensure that the mineral grains were compact. The compressing direction was perpendicular to the fractures and layers. During this process, the pressure was adjusted to control the porosity of the samples. During the next 10 h, the sodium silicate in the mixture consolidated. In this way, a block was formed and the consolidated sodium silicate gave the block initial mechanical strength. Next, the block was demoulded and sintered in a muffle oven at 900 °C. During sintering, the touch paper discs decomposed, leaving penny-shaped voids as fractures. The final step was to polish the sample surface for the convenience of obtaining ultrasonic velocity measurements. The size of all the cuboid-shaped samples was 69.9 ± 0.1 mm in the two directions parallel to the fractures and 49.9 ± 0.1 mm in the direction perpendicular to the fractures.

In order to produce artificial rocks with the same fracture densities but different porosities, we made the following modifications to the method of Ding (2014) [17,18]: (1) Different amounts of mineral material mixture were layered in the mould (70 mm × 70 mm) in 37 layers when building the different porosity samples. (2) The mixture of different samples was compressed into the same thickness (50 ± 0.3 mm) by adjusting the applied pressure to form blocks of the same volume. Therefore, the blocks that were made using more mixture would experience higher loading pressure and would have lower porosity. (3) A total of 126 touch paper discs were spread on the surface of each layer (except the top layer) in all the blocks. Fracture density is unitless and is calculated using the following formula:

$$e=\frac{N{a}^3}{V}$$

(8)

where N is the number of fractures, a is the radius of the fractures, and V is the total volume of the sample. Thus, all the blocks had the same volume (70 mm × 70 mm × 50 mm) and contained the same number of fractures (36 layers × 126 fractures/layer), making the fracture density practically identical for the different porosities.

For each fractured sample, a blank sample (containing no penny-shaped fractures) was made as a reference sample to provide background properties. The blank samples were created using the same method as the fractured samples, excluding the introduction of fractures.

Four groups of artificial rocks with different porosities were constructed (Figure 3a). Each group consisted of two samples (one fractured sample containing penny shaped fractures with a fracture density of 0.06 and a corresponding unfractured sample). The fracture diameter was set at 3 mm and the fracture thickness was 0.06 mm.

Table 1. Physical parameters of the three sample groups (water-saturated).

	Group 1		Group 2		Group 3		Group 4
	1-B	1-F	2-B	2-F	3-B	3-F	4-B	4-F
Fracture density	0	0.06	0	0.06	0	0.06	0	0.06
Fracture aspect ratio	/	0.018	/	0.018	/	0.018	/	0.018
Porosity (%)	14.1	15.3	20.6	22.1	25.3	26.1	30.4	30.8
Fracture porosity (%)	/	0.75	/	0.75	/	0.75	/	0.75
Bulk density (g/cm3)	2.22	2.21	2.16	2.15	2.07	2.05	2.03	2.03
Compression pressure (MPa)	186	186	53	53	21	21	15	15

lists the main parameters of all the samples. The porosity was measured using helium porosimetry. In the fractured samples, the fracture porosity was calculated to be 0.75%, but the porosity difference between the fractured and blank (unfractured) samples within each group varied from 0.4% to 1.5% larger or smaller than the fracture porosity, indicating a small difference in background porosity between the fractured and blank samples. This was caused by tiny errors in the manufacturing process. A section obtained by cutting a fractured sample is shown in Figure 3b, where parallelly distributed fractures can be observed. A scanning electron microscope (SEM) image of the section is shown in Figure 3c,d, where pore and fracture structures with controlled geometry are clearly shown.

4. Ultrasonic Measurement

The P- and S-wave velocity measurements were conducted on an ultrasonic bench-top pulse transmission system at room temperature and an atmosphere pressure suitable for water-saturated conditions. The ultrasonic transducers, which were heavily damped with a central frequency of 500 kHz, were excited by a voltage spike, giving rise to broadband ultrasonic pulses. The transmitted P- and S-wave signals were recorded with a digital oscilloscope and a desktop computer. The time sampling interval for all the experiments was 0.04 μs for both P- and S-wave signals. The measurement error was about 0.5% for P-wave velocity and 0.8% for S-wave velocity. Water saturation was obtained by immersing the samples in a water-filled container. The container was placed in a sealed bin to extract the air. To ensure full water saturation, the saturation rate was calculated using the following formula:

$$Sw(water)=\frac{m_{sat}-m_{dry}}{V\phi {\rho }_f}\times 100\%$$

(9)

where $m_{sat}$ and $m_{dry}$ is the wet and dry mass of the sample, $V$ and $\phi$ represent the volume and helium porosimetry measured porosity of the sample, respectively, and ${\rho }_f$ is the bulk density of water. The P-wave velocities were measured both parallel (90°) and perpendicular (0°) to the fractures. The S-wave velocities were measured parallel to the fractures, and polarization was performed in two directions (parallel (S1 wave) and perpendicular (S2 wave) to the fractures) by rotating the transducers. Then, the Thomsen parameter (ε), which represents the difference in P-wave velocity between the parallel and perpendicular directions, was calculated using the following formula:

$$\epsilon (\%)=100\times (\frac{V_P\parallel -V_P\perp }{V_P\perp })$$

(10)

where $V_P\parallel$ and $V_P\perp$ represent the P-wave velocities in parallel and perpendicular directions, respectively. Shear wave splitting (SWS) in the parallel direction can be calculated as follows:

$$SWS(\%)=100\times (\frac{V_{s1}-V_{s2}}{V_{s2}})$$

(11)

where $V_{s1}$ and $V_{s2}$ are fast and slow shear wave velocities. The numerical modelling experiment of wave propagation in anisotropic media by Dellinger and Vernik (1994) showed that if the wave front was propagating parallel or perpendicular to the layering (or in this case, to the fractures) [25], a true phase velocity was measured in laboratory ultrasonic experiments. Therefore, in this study, P- and S-wave phase velocities were measured in both parallel and perpendicular directions. The transmitted signals of the blank samples were analysed using Fourier transform to obtain the dominant frequency and to calculate the wavelength (Figure 4). The ratio of wavelength to fracture diameter (λ/d ratio) was between 5.2 and 10.8 for the P wave and between 3.9 and 12 for the S wave. Based on experimental and numerical studies [21], for this λ/d ratio, both the P and S wave may exhibit Rayleigh scattering. This is undesirable because the equivalent medium criteria are violated. This problem has been frequently encountered in previous experimental studies [7,11]. However, similar to previous studies, we also found some interesting correlations between the experimental results and theoretical predictions.

5. Results and Discussions

5.1. Laboratory Results

Table 2. P- and S-wave velocities in the blank samples (water-saturated).

	Porosity	P-Wave Velocity (m/s)		S-Wave Velocity (m/s)
	(%)	Parallel	Perpendicular	S1	S2
1-B	14.1	3930	3896	2170	2120
2-B	20.6	3442	3420	1857	1820
3-B	25.3	3128	3005	1583	1495
4-B	30.4	2817	2811	1397	1390

shows the P- and S-wave velocities of the blank samples. The P-wave velocity in the parallel direction was slightly higher than that in the perpendicular direction, and the S1 velocity (polarized parallel to the fractures) was slightly higher than the S2 velocity (polarized perpendicular to the fractures). This was due to layering and uniaxial loading during the manufacturing process.

Figure 5 shows the measured P- and S-wave velocities in the water-saturated fractured samples. Significant velocity anisotropy was evident in all the fractured samples, with the faster P-wave being measured parallel and the slower one perpendicular. The measured velocity anisotropy is discussed further in the following paragraphs. When the porosity increased from 15.3% to 22.1%, the P-wave velocities (both parallel and perpendicular) dropped sharply. However, as the porosity continued to increase, the decreasing trend in the perpendicular direction was slower than that in the parallel direction, narrowing the difference between the parallel and perpendicular directions and indicating a decrease in anisotropy. However, when porosity increased from 15.1% to 30.8%, the decrease observed in both S1 and S2 wave velocities was generally similar.

Figure 6 shows the P-wave anisotropy parameter (ε) and SWS in all the blank samples. Both the ε parameter and SWS decreased with increasing porosity in the blank samples. Because the higher porosity samples suffered lower uniaxial loading (Table 1), they exhibited weaker anisotropy.

We aimed to investigate the effect of background porosity on fracture-induced P-wave anisotropy and SWS. Because the blank samples exhibited transverse isotropy, we inferred that the background matrix in the fractured samples had the same anisotropy as the blank samples. In fractured samples, the measured anisotropy was a superposition of background matrix anisotropy and fracture-induced anisotropy. To model the anisotropy caused by fractures, the anisotropy of the background matrix had to be considered. Therefore, we approximately represented the fracture-induced P-wave anisotropy and SWS by subtracting the measured ε parameter and SWS of the blank samples from that of the fractured ones. Figure 7 depicts the measured results of the fracture-induced ε parameter and SWS, as well as the theoretical predictions. When porosity increased from 15.3% to 22.1%, both the measured ε parameter and SWS exhibited small fluctuations. In this porosity range, the ε parameter and SWS are not so sensitive to porosity changes. However, when porosity increased to 26.1%, P-wave anisotropy dramatically declined from 4.5% to 3.5%, whereas SWS remained fairly constant. Afterwards, as porosity increased to 30.8%, P-wave anisotropy continued to decline from 3.5%to 2.2%, but SWS fluctuated very slightly.

5.2. Modelling Insights and Discussions

The inputs for equivalent medium theories are given in Table 1 and

Table 3. Main inputs for equivalent medium theories.

Sample Number	Background Matrix Modulus (GPa)				Grain Modulus (GPa)		Fluid Modulus (GPa)
	${\mathit{K}}_{\mathit{d}\mathit{r}\mathit{y}}$	${\mathit{\mu }}_{\mathit{d}\mathit{r}\mathit{y}}$	${\mathit{K}}_{\mathit{s}\mathit{a}\mathit{t}}$	${\mathit{\mu }}_{\mathit{s}\mathit{a}\mathit{t}}$	${\mathit{K}}_{\mathit{s}}$	${\mathit{\mu }}_{\mathit{s}}$
1-F	7.05	7.41	20.36	10.45	30.6	24.5	2.15
2-F	5.52	5.82	15.66	7.45	30.6	24.5	2.15
3-F	4.69	4.14	13.33	5.19	30.6	24.5	2.15
4-F	3.12	3.11	10.83	3.96	30.6	24.5	2.15

. The Eshelby–Cheng theory (high frequency) predicted P-wave anisotropy to be close to the measured results [3,4]. Although the expressions of the Eshelby–Cheng theory do not contain background porosity, its predicted P-wave anisotropy showed a decreasing trend with increasing background porosity. The Eshelby–Cheng theory is expressed by Equations (1) and (2). Note that in Equation (1), ${\phi }_f$ refers to the fracture-occupied porosity, not the background porosity, and in all the fractured samples, the fracture porosities are practically identical. The predicted P-wave anisotropy fluctuations were mainly dominated by parameter $A_{ij}$ in Equation (2), which relates the applied strain to the “stress-free” strain of the inclusion. $A_{ij}$ is expressed as follows:

$$A_{ij}=B_{ij}^{-1}{L}_{kj}^{\prime }$$

(12)

where:

$$B_{ij}=G{S}_j+2H{S}_{ij}+{\lambda }_b+2{\mu }_b{\delta }_{ij}$$

(13)

$$L_{ij}=-G-2H{\delta }_{ij}$$

(14)

$$H={\mu }^{\prime }-{\mu }_b$$

(15)

$$G={\lambda }^{\prime }-{\lambda }_b$$

(16)

$$S_j={\displaystyle \sum _{i=1}^3{S}_{ij}}$$

(17)

where ${\lambda }_b$ and ${\mu }_b$ are the lame constants of the background medium, and ${\lambda }^{\prime }$ and ${\mu }^{\prime }$ are the lame constants of the inclusion material (water). $S_{ij}$ is the Eshelby S tensor, which relates the strain field (due to an ellipsoidal inclusion) to the “stress-free” strain in the inclusion. For very thin fractures (with a small aspect ratio), $S_{ij}$ can be simplified as follows:

$$S_{11}=S_{22}=\pi \alpha (9-5R)/16$$

(18)

$$S_{33}=1-\pi \alpha R/2$$

(19)

$$S_{12}=S_{21}=-\pi \alpha (3-7R)/16$$

(20)

$$S_{13}=S_{23}=-\pi \alpha R/4$$

(21)

$$S_{31}=S_{32}=1-2R-\pi \alpha (3-5R)/4$$

(22)

$$S_{55}=S_{44}=2-\pi \alpha (3-2R)/2$$

(23)

$$S_{66}=\pi \alpha (3+R)$$

(24)

where $\alpha$ is the fracture aspect ratio, and

$$R=\frac{{\mu }_b}{{\lambda }_b+2{\mu }_b}$$

(25)

In the normal fracture direction (direction three), since $\alpha$ is so small ($\alpha =0.018$), ${\mathrm{S}}_{33}\approx 1$ and ${\mathrm{S}}_3\approx 1$. Because the shear modulus of the inclusion material (water) ${\mu }^{\prime }$ is zero, $B_{33}\approx {\lambda }^{\prime }$ and $L_{33}\approx {\lambda }_b+2{\mu }_b-{\lambda }^{\prime }$. With increasing porosity, $L_{33}$ decreases, whereas $B_{33}$ stays nearly constant. This leads to a decrease in $A_{ij}$. As a result, the fracture-induced perturbation in the normal fracture direction decreases with increasing porosity. However, as the aspect ratios of the fractures were extremely small in this study, the fracture-induced perturbations were nearly negligible in the parallel direction of the fractured samples. As the fracture-induced perturbations decreased with increasing porosity in the perpendicular direction, the stiffness difference between the parallel and perpendicular directions also decreased. This led to the declining trend observed for the predicted P-wave anisotropy with increasing porosity.

In Figure 7b, we can see that the Eshelby–Cheng theory predicted that SWS would be higher than the measured results. The shear moduli of S₁ ($C_{44}$) and S₂ ($C_{66}$) in the Eshelby–Cheng theory are as follows:

$$C_{44}=\mu [1-\frac{{\phi }_f(\mu -{\mu }^{\prime })}{\mu +2(\mu -{\mu }^{\prime })S_{44}}]$$

(26)

$$C_{66}=\mu [1-\frac{{\phi }_f(\mu -{\mu }^{\prime })}{\mu +2(\mu -{\mu }^{\prime })S_{66}}]$$

(27)

As ${\mu }^{\prime }=0$ in our study, SWS can be approximately expressed as follows:

$$\begin{array}{l}SWS=\frac{C_{66}-C_{44}}{2C_{44}}\approx \frac{4\mathrm{e}}{3-2R}\\ \epsilon =\frac{C_{11}^{sat}-C_{33}^{sat}}{2C_{33}^{sat}}=\frac{1}{2}(\frac{C_{11}^{dry}+{\alpha }_1^{2}M}{C_{33}^{dry}+{\alpha }_3^{2}M}-1)\\ {\alpha }_3\\ {\gamma }^{\mathrm{sat}}\approx \frac{4}{3}e\\ SWS\end{array}$$

(28)

where e is the fracture density and ${\phi }_f=2\pi \alpha \mathrm{e}$. As the value of R in the four rocks was close (between 0.25 and 0.3), the SWS predictions were similar and not sensitive to changes in porosity.

In the theoretical predictions of [1], with increasing porosity, P-wave anisotropy showed a decreasing trend, whereas SWS remained practically constant. This trend was similar to the measured results. However, the theoretical predictions were higher than the measured results. For water-saturated conditions, the theoretically predicted P-wave anisotropy of [1] was as follows:

$$\epsilon =\frac{C_{11}^{sat}-C_{33}^{sat}}{2C_{33}^{sat}}=\frac{1}{2}(\frac{C_{11}^{dry}+{\alpha }_1^{2}M}{C_{33}^{dry}+{\alpha }_3^{2}M}-1)$$

(29)

where direction three is the normal fracture direction. $C_{11}^{sat}$ and $C_{33}^{sat}$ are the P-wave moduli in the parallel and perpendicular directions, respectively, in water-saturated conditions. $C_{11}^{dry}$ and $C_{33}^{dry}$ are the P-wave moduli in the parallel and perpendicular directions, respectively, in dry conditions, and

$$C_{33}^{dry}=\frac{3\mu (\lambda +\mu )}{3\mu (\lambda +\mu )+4(4{\mu }^2+4\lambda \mu )e}{C}_{11}^{dry}$$

(30)

where $\lambda$ and $\mu$ are Lame parameters of the dry background matrix. Equation (29) can be rewritten as follows:

$$\epsilon =\frac{1}{2}(\frac{C_{11}^{dry}+{\alpha }_1^{2}M}{\frac{3\mu (\lambda +\mu )}{3\mu (\lambda +\mu )+4(4{\mu }^2+4\lambda \mu )e}{C}_{11}^{dry}+{\alpha }_3^{2}M}-1)$$

(31)

Firstly, with increasing porosity, the value of $\frac{3\mu (\lambda +\mu )}{3\mu (\lambda +\mu )+4(4{\mu }^2+4\lambda \mu )e}$ increases from 0.62 to 0.75. According to Equation (6), when the porosity increases, scalar M decreases greatly. Since ${\alpha }_1$ is larger than ${\alpha }_3$ (Equation (5)), ${\alpha }_1^{2}M$ decreases more than ${\alpha }_3^{2}M$. As a result, with increasing porosity, these two factors cause the $C_{11}^{sat}$ to decrease more than $C_{33}^{sat}$, resulting in a decrease in P-wave anisotropy. The predicted P-wave anisotropy was higher than the measured results. Potential explanations for this phenomenon include: (i) Under an experimental frequency of 0.5 MHz, the fluid did not have enough time to flow between the pores and fractures, thus the system did not reach pressure equilibrium in the pore space. This equilibrium is assumed by the Gurevich theory. This can cause a “stiffening” effect on the fractures, reducing P-wave anisotropy and causing the measured results to be lower than the theoretical predictions. (ii) Interactions between the fractures were not included in the Gurevich theory, which has been shown to reduce the overall anisotropy in a fractured medium in theoretical studies [5,6].

In the Gurevich theory, the expression of SWS is as follows:

$$SWS=\frac{C_{55}^{sat}-C_{44}^{sat}}{2C_{44}^{sat}}$$

(32)

where:

$$C_{55}^{sat}=\mu$$

(33)

$$C_{44}^{sat}=\mu (1-{\Delta }_T)$$

(34)

$${\Delta }_T=\frac{\mu Z_T}{1+\mu Z_T}$$

(35)

$$Z_T=\frac{16(\lambda +2\mu )e}{3\mu (3\lambda +4\mu )}$$

(36)

In combining Equations (32) and (36), the following can be obtained:

$$SWS=\frac{4e}{3}(1-\frac{\lambda }{3\lambda +4\mu })$$

(37)

where $e$ is the fracture density. The term $\frac{\lambda }{3\lambda +4\mu }$ was extremely small in the four fractured samples (less than 0.077); therefore, ${\gamma }^{\mathrm{sat}}\approx \frac{4}{3}e$. Compared with P-wave anisotropy, the $SWS$ prediction was relatively close to the measured results.

Since the fracture aspect ratio is 0.02 in the fractured samples, we assumed that no partial or complete closure of the fractures occurred during the passage of ultrasonic waves. The above analysis is based on this assumption. For cases with smaller aspect ratios, we believe the partial or complete closure of fractures should be considered.

6. Conclusions

In this study, we investigated the effects of background porosity on seismic anisotropy in fractured rocks by conducting laboratory experiments. P- and S-wave velocities were measured on a set of synthetic samples with the same fracture density but different porosities (ranging from 15.1% to 30.8%) under water-saturated conditions. The results exhibited a significant effect of background porosity on P-wave anisotropy for the measured porosity range. The P-wave anisotropy fluctuated slightly when porosity increased from 15.1% to 22.1%, and then declined sharply when porosity continued to increase to 30.8%. However, SWS stayed fairly constant for the entire porosity range. The results were compared with predictions from equivalent medium theories. In both the Eshelby–Cheng model and the Gurevich model, increasing porosity reduces P-wave anisotropy, but has little influence on SWS. The experimental results show good qualitative agreement with the theoretical predictions.

Understanding the influence of background porosity on seismic anisotropy could improve the characterization of different fractured reservoirs. The findings of this study have potential applications for fracture detection in different areas with different reservoir physical properties. However, considering the inevitable differences between synthetic samples and natural rocks, further studies concerning this topic using field observations are required, such as post-survey and post-drilling calibrations of the seismic anisotropy after VSP.