The scattered seismic waves of fractured porous rock are strongly affected by the wave-induced fluid pressure diffusion effects between the compliant fractures and the stiffer embedding background. To include these poroelastic effects in seismic modeling, we develop a numerical scheme for discretely distributed large-scale fractures embedded in fluid-saturated porous rock. Using Coates and Schoenberg's local-effective-medium theory and Barbosa's dynamic linear slip model characterized by complex-valued and frequency-dependent fracture compliances, we derive the effective viscoelastic compliances in each spatial discretized cell by superimposing the compliances of the background and the fractures. The effective governing equations for fractured porous rocks are viscoelastic anisotropic and numerically solved by the mixed-grid-stencil frequency-domain finite-difference method. The main advantage of our proposed modeling scheme over poroelastic modeling schemes is that the fractured domain can be modeled using a viscoelastic solid, while the rest of the domain can be modeled using an elastic solid. We have tested the modeling scheme in a single fracture model, a fractured model, and a modified Marmousi model. The good consistency between the scattered waves off a single horizontal fracture calculated using our proposed scheme and the poroelastic modeling validates that our modeling scheme can properly capture the fluid pressure diffusion (FPD) effects. In the case of a set of aligned fractures, the scattered waves from the top and bottom of the fractured reservoir are strongly influenced by the FPD effects, and the reflected waves from the underlying formation can retain the relevant attenuation and dispersion information. The proposed numerical modeling scheme can also be used to improve migration quality and the estimation of fracture mechanical characteristics in inversion.

Fluid-saturated porous rocks in a reservoir, which are characterized by a heterogeneous internal structure consisting of a solid skeleton and interconnected fluid-filled voids, are often permeated by much more compliant and permeable fractures. Although the fractures typically occupy only a small volume, they tend to dominate the overall mechanical and hydraulic properties of the reservoir (Liu et al., 2000; Gale et al., 2014). Thus, fracture detection, characterization, and imaging are of great importance for hydrocarbon exploration and production. Seismic waves are widely used for these purposes because their amplitude, phase, and anisotropy properties can be strongly affected by the fractures (Chapman, 2003; Gurevich, 2003; Brajanovski et al., 2005; Rubino et al., 2014). Therefore, appropriate numerical modeling methods are required for the interpretation, migration, and inversion of seismic data from porous media containing discretely distributed fractures.

Biot's poroelastic theory (Biot, 1956a, b) is the fundamental theory to describe elastic wave propagation in fluid porous media, including the dynamic interactions between rock and pore fluid. However, the original theory, assuming a macroscopically homogeneous porous media saturated by a single fluid phase, fails to explain the measured velocity dispersion and attenuation of seismic waves (Nakagawa and Schoenberg, 2007). In recent decades, many researchers have found that if porous media contains mesoscale heterogeneity, a local fluid pressure gradient will be induced at a scale comparable to the fluid pressure diffusion length at the seismic frequency band, thus causing significant velocity dispersion and attenuation (White et al., 1975; Dutta and Odé, 1979a, b; Johnson, 2001; Müller et al., 2008; Norris, 1993; Gelinsky and Shapiro, 1997; Kudarova et al., 2016). Fractures embedded in homogeneous porous background are special heterogeneities, exhibiting strong mechanical contrasts with background. When seismic waves travel through fluid-saturated fractured porous rocks, local fluid pressure gradients will be induced between the fractures and the background in response to the strong compressibility contrast. To return the equilibrium state, fluid pressure diffusion (FPD) occurs between the fractures and the embedding background, which in turn changes the fluid stiffening effect on the fractures and thus their mechanical compliances depending on frequency (Barbosa et al., 2016a, b).

When the fractures with spacing and length much smaller than the wavelengths are uniformly and regularly distributed, the properties of the fractured rocks are homogeneous at macroscopic scale and can be described by a representative elementary volume (REV). Various effective medium theories are available for estimating the fracture-induced anisotropy, attenuation, and dispersion in a poroelastic context (Hudson, 1981; Thomsen, 1995; Chapman, 2003; Brajanovski et al., 2005; Krzikalla and Müller, 2011; Galvin and Gurevich, 2015; Guo et al., 2017a, b). However, large-scale fractures with much larger spacing and length typically have a more complex discrete distribution rather than a regular one; therefore, the properties of rocks containing such fractures cannot be modeled by the effective medium theory. In contrast, the linear slip model (LSM) (Schoenberg, 1980), which represents individual fractures as nonwelded interfaces with discontinuous displacement tensors, is not limited by the assumption of regular distribution and can be used to model the discretely distributed fractures. Due to the discrete distribution, the effects of large-scale fractures are not uniform and vary spatially, which means that their effects cannot be represented by a single REV. In the framework of LSM, two numerical schemes are available to assess the seismic response of discretely distributed large-scale fractures: the local-effective-medium schemes (Coates and Schoenberg, 1995; Oelke et al., 2013) and the explicit-interface scheme (Zhang, 2005; Cui et al., 2018; Khokhlov et al., 2021). The local-effective-medium scheme uses a very coarse mesh to discretize background media and incorporates the additional effects of fractures within each discretized cell based on LSM; that is, it regards each discretized cell as a REV. The advantage is that it requires no special treatment of the displacement discontinuity conditions on the fractures, which means no additional memory and computation costs. The explicit-interface scheme uses a very fine mesh to discretize fractures and directly treats the displacement discontinuity across each fracture without any equivalent treatment, resulting expensive memory and computation costs.

The common aspect of the aforementioned numerical modeling schemes is that they are all implemented in a purely elastic LSM with real-valued compliance boundaries and represent both the embedding background and fractures as elastic solids; thus, the impact of FPD effects on seismic scattering cannot be accounted for. A dynamic linear slip model incorporating FPD effects should be considered when implementing numerical modeling of seismic waves propagating in fluid-saturated porous rocks containing discretely distributed large-scale fractures. Nakagawa and Schoenberg (2007) developed an extended poroelastic LSM (PLSM) for a single fracture. The proposed model representing both the background and the fracture as poroelastic media can appropriately incorporate the frequency-related effects, but it will also result in a higher computational consumption and more memory requirements. Rubino et al. (2015) proposed a frequency-dependent complex-valued normal compliance for a set of aligned fractures with a separation much smaller than the prevailing seismic wavelength. Despite the ability of including the FPD across the fractures, the model is not suitable for modeling discretely distributed fractures. In the context of viscoelasticity, Barbosa et al. (2016a) developed a viscoelastic linear slip model (VLSM) for an individual fracture with explicit complex-valued and frequency-dependent fracture compliances to account for the impact of FPD on the fracture stiffness. That provides a viscoelasticity-based modeling algorithm for discretely distributed large-scale fractures with smaller computational costs and memory requirements than the poroelasticity-based modeling.

In this paper, we develop a viscoelastic numerical modeling scheme to simulate seismic wave propagation in fluid-saturated porous media containing discretely distributed large-scale fractures. To capture the FPD effects between the fractures and background, we use the local-effective-medium theory based on Barbosa's VLSM to derive the effective anisotropic viscoelastic compliances in each numerical cell by superimposing the compliances of the background and the fractures. The effective anisotropic viscoelastic governing equations of the fractured porous rock are then numerically solved using the mixed-grid-stencil frequency-domain finite-difference method (FDFD) (Hustedt et al., 2004; Operto et al., 2009; Liu et al., 2018). Compared to poroelastic modeling scheme, the main advantage of our modeling scheme is that it uses VLSM-based viscoelastic modeling to account for FPD effects in the domain permeated by fractures, while in the rest fracture-free domain, it uses elastic modeling. To validate that the proposed viscoelastic modeling scheme can capture the impact of FPD effects on seismic wave scattering, we compare the scattered waves of a single horizontal fracture obtained using our proposed modeling scheme with the poroelastic modeling scheme and elastic modeling scheme. Numerical examples of a fractured reservoir are presented to demonstrate that the proposed modeling scheme can properly simulate the wave attenuation and dispersion due to the FPD effects between the fracture system and background. A set of rock physics models were generated by the Marmousi model to test the proposed modeling scheme and code. The scheme can be used not only to study the impact of mechanical and hydraulic of fracture properties on seismic scattering but can also to improve migration quality and the estimation of fracture mechanical characteristics in inversion.

The LSM was originally proposed by Schoenberg (1980) to represent a solid- or fluid-infilled fracture permeated in a pure solid background, and then it was extended by other researchers (e.g., Nakagawa, Barbosa) to represent a poroelastic fracture to include the FPD effects. We briefly review the original LSM and its poroelastic and viscoelastic extensions.

Schoenberg (1980) presented the original LSM in the context of elasticity, representing both the background and the fracture as elastic solids. The original LSM assumes that across a fracture surface the stresses are continuous while the displacements are discontinuous. The discontinuous displacement vector of a horizontal fracture is linearly related to the stress tensor through the fracture compliance, which can be written as

Nakagawa and Schoenberg (2007) presented a PLSM in the framework of poroelasticity, representing the fracture as a highly compliant and porous thin isotropic, homogeneous layer embedded in a much stiffer and much less porous background (Nakagawa et al., 2007; Barbosa et al., 2016a). Similar to the classic LSM, the PLSM assumes that across a fracture surface the stresses are continuous while the displacements are discontinuous. The discontinuous displacement components for a horizontal fracture are (Nakagawa and Schoenberg, 2007)

Barbosa et al. (2016a) derived a VLSM that accounts for the FPD effects between a fracture and background and the resulting stiffening effect impact on the fracture. The background is assumed to be not impacted by the FPD and can be represented by an elastic solid, whose properties are computed according to Gassmann's equation (Gassmann, 1951). By representing fractures as extremely thin viscoelastic layers, the poroelastic effects were incorporated into the classical LSM through complex-valued and frequency-dependent compliances. These compliances characterize the mechanical properties of the fluid-saturated fracture.

The discontinuous displacement components of the VLSM (Barbosa et al., 2016a) for a horizontal fracture are

In the low-frequency limit, the two frequency-dependent and complex-valued parameters become

In the high-frequency limit, the two frequency-dependent and complex-valued parameters become

According to Barbosa et al. (2016a), there are two distinct frequency regimes due to the FPD effect, and the characteristic frequency for the transition between the two regimes is

In this section, we focus on the implementation of seismic modeling of fluid-saturated porous media containing discretely distributed large-scale fractures in the 2-D case. We develop a viscoelastic modeling scheme based on the VLSM and local-effective-medium theory (Coates and Schoenberg, 1995) to incorporate the FPD effects between fractures and background. To validate that the proposed viscoelastic modeling scheme can capture the impact of FPD effects on seismic wave scattering of fractures, we outline the implementation of the poroelastic modeling scheme using an explicit application of the PLSM.

Complex-valued and frequency-dependent

To incorporate the VLSM into viscoelastic finite-difference modeling algorithms, we adopt Coates and Schoenberg's local-effective-medium theory (1995) to account for the property of each fracture. We first provide the specific derivation of the effective viscoelastic–anisotropic stiffness matrix of the numerical cell by superimposing the compliances of the background and the fractures. The porous background is assumed to be unaffected by the FPD in the presence of fractures because of the small amount of diffusing fluid and large compliance contrast between background and fluid. Thus, the rock background can be represented by an elastic homogeneous solid, and its strain tensor

Snapshots of the wavefield components

If we assume that the interface of the fracture is normal to the

Comparison of 1-D seismograms components

Since the local-effective-medium theory assumes that the real structure of the fractured porous rock is substituted by ideal continua, the balance equations of classical continuum mechanics can be applied without considering the discontinuity at the fracture interfaces, and the constitutive equations can be characterized by the effective viscoelastic stiffness. Combined with the effective complex-valued and frequency-dependent tilted transversely isotropic (TTI) viscoelastic stiffness, the 2-D frequency-domain second-order heterogeneous governing equations with a perfectly matched layer (PML) of fractured porous rock can be expressed as

Snapshots of the wavefields components

Comparison of 1-D seismograms components

In time domain, the governing equations are integral differential equations, which require special processing for the convolution operations, resulting in high computational costs. Although the problem can be relieved by memory functions, it still requires high memory requirements. Instead, the governing equations can be straightforwardly solved using FDFD. To efficiently and accurately model seismic wave propagation in fluid-saturated fractured porous rock, we solve the second-order heterogeneous governing equations with the mixed-grid-stencil FDFD method (Jo et al., 1996; Hustedt et al., 2004). The mixed system of governing equations is formulated by combining the classical Cartesian coordinate system (CS) and the 45° rotated coordinate system (RS):

Schematic diagram of the fractured reservoir model with a set of aligned horizontal fractures:

To improve the modeling accuracy of the mixed-grid stencil, the acceleration term

Seismogram components

In order to assess the FPD effects on the seismic response, a similar procedure can be adopted in the implementation of elastic modeling by replacing the frequency-dependent fracture compliances with its low- or high-frequency limit compliances. The main advantage of our VLSM-based modeling scheme over poroelastic modeling schemes is that the fractured domain can be modeled using a viscoelastic solid, while the rest of the domain can be modeled using an elastic solid.

Time–frequency distributions of the middle trace for

Seismogram components

The poroelastic modeling means that we numerically solve Biot's equations and adopt an explicit implementation of the PLSM across each fracture instead of using the effective media theory. Hence, the poroelastic modeling can naturally deal with the FPD between fracture and background and account for its impact on wave scattering. To verify the effectiveness of the viscoelastic modeling based on VLSM, we compared the results obtained from the viscoelastic scheme with those obtained from the poroelastic scheme. Although it is difficult to implement an explicit application of PLSM for an arbitrarily orientated fracture, it is relatively straightforward for a horizontal or vertical fracture. In the following, we outline the poroelastic modeling for a single horizontal fracture embedded in an isotropic homogeneous background with an explicit implementation of the PLSM. In frequency the domain, the governing equations for an isotropic poroelastic media in the absence of fractures can be written as (Biot, 1962)

In this section, we apply different numerical modeling schemes on three fractured models to examine the FPD effects on seismic wave scattering. We mainly focus on the amplitudes and phases of the scattered and reflected waves.

Here, we numerically simulate the scattering of seismic waves from a single fracture embedded in a homogeneous background. The model measures 2000 m

Physical properties of the materials employed in the numerical modeling.

Figure 1 shows the complex-valued and frequency-dependent fracture normal compliance

Figure 2 shows the 280 ms snapshots of the displacement fields for the single horizontal fracture model models. The displacement fields are calculated by the PLSM-based poroelastic modeling, the VLSM-based viscoelastic modeling, the LVLSM-based elastic modeling, and the HVLSM-based elastic modeling, respectively. The asterisk represents the source and the blue line represents the fracture. To make the small scattered wave visible, the large amplitude is clipped; thus, the transmitted compressional waves (

Time–frequency distributions of the middle trace for

We consider the poroelastic modeling as a reference scenario because it can naturally incorporate the FPD effects. Figures 2 and 3 suggest very good agreement between the

Schematic diagram of the fractured reservoir model with a set of aligned inclined fractures:

Seismogram components

Time–frequency distributions of the middle trace for

Seismogram components

Time–frequency distributions of the middle trace for

The proposed modeling scheme is also applicable to the inclined fracture. Figure 4 shows the 280 ms snapshots of the displacement fields for the single inclined fracture model. Figure 5 is the comparison of 1-D seismograms at (1200, 0 m). Figures 4 and 5 show that both the scattered P and S waves of a single inclined fracture are strongly affected by the FPD effects.

In addition to a single fracture, we are more interested in the scattering behavior of discretely distributed fracture system. To this end, we designed two fractured reservoir models containing a set of regularly distributed aligned horizontal fractures and a set of randomly distributed aligned horizontal fractures, respectively, as illustrated in Fig. 6. There are 200 horizontal fractures spread over a space of 200 m, each extending 500 m. The material properties of the fracture, background (yellow region), and underlying (green region) formation are given in Table 1. The model size, grid interval, and source location are the same as those in the previous numerical examples. Though a set of aligned horizontal fracture structures is not practical in the actual subsurface, it helps to illustrate the impact of FPD effects on the amplitude and phase of scattered waves of fractures.

Figure 7 presents the seismograms of the fractured reservoir model with a set of regularly distributed aligned horizontal fractures. The scattered compressional wave (

To show the trend of frequency-dependent attenuation and dispersion, time–frequency distribution of the middle trace was computed for three modeling schemes. Figure 8 clearly shows that the frequency content and energy of the scattered and reflected waves calculated by VLSM tend to decrease strongly, while the frequency content and energy calculated by HVLSM and LVLSM remain steady. The impact of FPD effects on the

In addition to regularly distributed fractures, our proposed modeling scheme can also simulate the wave scattering of randomly distributed fractures. Figure 9 presents the seismograms of the fractured reservoir model with a set of randomly distributed aligned horizontal fractures. Figure 10 presents the time–frequency distributions of the middle trace for three modeling scheme cases in Fig. 9. Due to the random distribution of aligned fractures, the fractured reservoir exhibits a stronger heterogeneity, resulting in a more prevalent diffracted wave (coda wave) in Fig. 9 than in Fig. 7. Except for the diffracted wave, the scattered and reflected waves in the random distribution case are similar to those in the regular distribution case due to the FPD effect. The two fractured reservoir models suggest that the scattered waves from the bottom of the fractured reservoir are attenuated and dispersed by the FPD effects, and the reflected waves can retain the relevant attenuation and dispersion information.

To validate the effectiveness of our proposed modeling scheme in a more practical underground fractured reservoir, we replace a set of aligned horizontal fractures in the original model with a set of aligned inclined fractures, as illustrated in Fig. 11. Figure 12 presents the seismograms of the fractured reservoir model with a set of regularly distributed aligned inclined fractures, and Fig. 13 shows the time–frequency distributions of the middle trace for three modeling schemes. Figures 14 and 15 present the seismograms of the fractured reservoir model with a set of randomly distributed aligned inclined fractures and the time–frequency distributions of the middle trace for three modeling schemes, respectively. All results of PLSM-based modeling capture the influence of FPD effects on the amplitude and phase of scattered waves, validating the effectiveness of our proposed modeling scheme. Figures 12 and 14 also show the different scattering characteristics of the randomly and regularly distributed incline fractures: many coda waves are generated by the randomly distributed fractures due to a stronger heterogeneity.

We test the proposed VLSM-based modeling scheme on a more complex modified Marmousi model. To modify the Marmousi model, we generate a porosity model, permeability model, and discrete large-scale fracture system; transform the original P-wave velocity and density into the fluid-saturated bulk and shear modulus of the background by a constant Poisson ratio of 0.5; and finally obtain the grain bulk modulus, the frame bulk, and shear modulus of the background through the Gassmann equation and empirical formula (

The physical properties and elastic modulus models of the modified Marmousi model.

Figure 17 shows the snapshots of displacement fields at 1000 ms. The figure clearly shows the scattered P and S waves by the discretely distributed large-scale fractures. The results with such a complex model clearly verify the numerical implementation and the code. We also calculate the seismograms of the displacement shown in Fig. 18. The seismograms obtained by our proposed modeling scheme present the scattered seismic waves by the discrete fractures.

Snapshots of the wavefields components

Seismogram components

In this work, we have developed a numerical modeling scheme including FPD effects for discretely distributed large-scale fractures embedded in fluid-saturated porous rock. To capture the FPD effects between the fractures and background, the fractures are represented as Barbosa's VLSM with complex-valued and frequency-dependent fracture compliances. Using Coates and Schoenberg's local-effective-medium theory and Barbosa's VLSM, we derive the effective anisotropic viscoelastic compliances in each spatial discretized cell by superimposing the compliances of the background and the fractures. The effective governing equations of each numerical cell are expressed by the derived effective compliances and discretized by the mixed-grid-stencil FDFD method. The proposed modeling scheme can be used to study the impact of mechanical and hydraulic fracture properties on seismic scattering. The main advantage of our proposed modeling scheme over poroelastic modeling schemes is that the fractured domain can be modeled using a viscoelastic solid, while the rest of the domain can be modeled using an elastic solid.

The scattered P wave of a fluid-saturated horizontal fracture calculated by VLSM-based modeling is strongly affected by the FPD effects, while the scattered S wave is less sensitive, which is consistent with the result of PLSM-based modeling. However, the LVLSM-based modeling overestimates the scattered P wave and the HVLSM-based modeling underestimates the scattered P wave. The numerical results show that the proposed VLSM-based modeling can include the FPD effects and thus accurately estimate the scattered wave of the horizontal fracture. The results of the fractured reservoir models show that the amplitudes of the scattered waves from the top of the fractured reservoir are affected by the fluid stiffening effects due to the FPD effects. The scattered waves from the bottom of the fractured reservoir are also attenuated and dispersed by the FPD effects in addition to the fluid stiffening effects, and the reflected waves can retain the relevant attenuation and dispersion information. Randomly distributed fractures can also result in a different scattering characteristic than regularly distributed fractures; i.e., a large number of coda waves are generated due to increased inhomogeneity. The results of the modified Marmousi model clearly show the scattered waves by the discretely distributed large-scale fractures and verify the proposed numerical modeling scheme. The proposed numerical modeling scheme is expected not only to improve the estimations of seismic wave scattering from discretely distributed large-scale fractures but can also to improve migration quality and the estimation of fracture mechanical characteristics in inversion.

We define coefficient vectors

We formulate

The expression of

The nine coefficients of the CS stencil for the submatrix

The numerical simulation data presented in this paper were generated by solving the mixed system of governing Eq. (27). These data can be obtained by contacting the corresponding author.

YQ, XC, and XL planned the campaign; YQ and QZ wrote the numerical simulation programs; YQ, XC, and CF analyzed the numerical cases; YQ and XC wrote the manuscript draft; YQ, XC, QZ, XL, and CF reviewed and edited the manuscript.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The authors gratefully acknowledge comments and suggestions from editor Florian Fusseis, referee Nicolas Barbosa, and an anonymous referee.

This research has been supported by the National Natural Science Foundation of China (grant nos. 42374163 and 41874143) and the Key Program of Natural Science Foundation of Sichuan Province (grant no. 2023NSFSC0019).

This paper was edited by Florian Fusseis and reviewed by Nicolas Barbosa and one anonymous referee.