Introduction

Gas hydrates are ice-like crystalline solid substances that are composed mainly of methane and water in low temperature and high-pressure environments (Lee and Collett 2012). It occurs in shallow marine sediments of the outer continental margin of the deep offshore basin. The markers of gas hydrate are anomalous seismic bottom simulating reflectors (BSR), gas chimney, and seismic amplitude attenuation or blanking observed in high resolution multi-channel seismic (MCS) data (Sain and Gupta 2008, 2012; Coffin et al. 2007). Gas hydrate sediments show relatively higher seismic velocity than sediments without gas hydrates. These have been identified in multi-channel seismic (MCS) data in the deep offshore Mahanadi basin (Singha et al. 2019; Sain et al. 2012). As gas hydrates occur mainly in fracture filings, veins, and pore filling of the shallow marine sediments in the study area of the Mahanadi basin, the physical properties of gas hydrate sediments change and produce anisotropy consequently (Kumar et al. 2014; Shankar and Riedel 2014). The assessment of a gas hydrate reservoir is done by computing petro-physical parameters such as saturation, porosity and permeability using geophysical well log data (especially P- and S-wave velocity) which are also affected by the several geological factors, anisotropic property and its degree (Shankar and Ridel 2014; Lee and Collett 2012). The anisotropy affects the amplitude of MCS in the azimuth in the transverse component (Satyavani et al. 2013). Anisotropy caused by fractured gas hydrate in Krishna-Godavari and Mahanadi basin was previously reported by Gosh et al. (2010), Kumar et al. (2006), Lee and Collett (2012), Cook et al. (2010) and Shankar and Pandey (2019). Therefore, for accurate assessment and characterization of gas hydrates from seismic velocity and other rock properties, a study of anisotropy from well log data must be required in the gas hydrate bearing sediments of the Mahanadi basin. The anisotropy of rock means variation of a physical property with direction or rock property is direction dependent (Tatham et al 1991; Sill et al. 2012). In sedimentary rocks, two types of anisotropy are prevalent: one is intrinsic type anisotropy defined as platy nature of thin isotropic layers such as shale/clay and another is associated with stress induced anisotropy due to the formation of elongated voids, shape of the particles, voids, and natural fracture alignments (Jaeger and Cook 1979; Prioul et al. 2007; Bidgoli and Jing 2014). The anisotropy due to layering is modeled as vertical transverse isotropy (VTI) whereas stress-induced anisotropy is generally modeled as horizontal transverse isotropy (HTI) in most cases (Wang 2002; Sill et al. 2012). Here we do not consider VTI because it requires a thick overburden and thin shale layering which is not found in the depth of interest below the seafloor in the study area. The presence of vertical fractures/joints and preferred alignments by grain or pore filling is common and thus, it makes sense to assume an HTI medium for a description of sediments in our analysis (Sayers 1994, 2005; Wang 2002; Sill et al. 2013). Moreover, these sediments might be VTI, but the presence of fractures makes HTI a much better description of the sediments since the fractures, not the bedding, control the anisotropy. The sediment’s thickness varies 180–200 m above BSR which mainly contains highly unconsolidated gas hydrate sediments with clay/silt marine sediments. The gas hydrate is present in clay/silt-based fracture medium in the eastern offshore basin of India (Lee and Collett 2012; Sain et al. 2012; Kumar et al. 2014).

For a vertical well, the parameters of an anisotropy medium are defined by the Thomsen’s parameters for a VTI medium (Thomsen 1986, 1999; Sill et al. 2012; Sill 2013).Thomsen’s parameters and fracture density have been calculated for wells at National Gas Hydrates Program (NGHP)-expedition-01 drilling sites of the offshore Mahanadi basin for understanding their effects on rock property.

The objectives of this study are to compute the (a) stiffness coefficients and Thomsen’s parameters (epsilon, gamma and delta) for HTI medium at NGHP-01-19, (b) anisotropy coefficient and fracture density from P-wave velocity and dipole shear data, (c) natural fracture and orientation from formation micro image log (FMI), (d) synthetic model of S-wave estimation, (e) estimation of elastic parameters in isotropic and anisotropic medium and (f) modeling of P-wave and S-wave using Thomsen’s parameters.

Geological setting of study area

The Mahanadi basin is a significant hydrocarbon basin located in the northern side of eastern passive continental margin (ECMI). The offshore basin is in the Bay of Bengal surrounded by the Bengal basin in north east and Krishna-Godavari basin in south-west covering an area of 14,000 sq. km in the sea (Bastia and Nayak 2006; Bastia et al. 2014) as seen in Fig. 1. The margin evolved during Permo-Trassic geologic time due to rifting and break-up of the Gondwana supercontinent (Sastri et al. 1981; Sastri et al. 1973).

Fig. 1
figure 1

a National gas hydrate program expectation 01 (NGHP-01) drill site map depicting the modified location of the drill sites established during the expectation in the Mahanadi basin (after Kumar et al. 2014). b Modified Bathymetric map depicting the location of the research drill sites established in the Mahanadi basin (after Kumar et al. 2014; Singha et al. 2019)

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The major faults are lying with a dominant orientation ENE-WSW: NNE–SSW and NNW–SSE are parallel/sub-parallel to the present-day coastline of the basin (Das et al. 2010; Fuloria et al. 1992) (Fig. 2). The rift structure of Jurassic age along the eastern continental margin is cutting across older NW–SE-trending in Permian–Triassic age (Sastri et al. 1981; Ramana et al. 1981). Two rivers mainly Mahanadi and Godavari discharge the sediments upto the continental margin in the offshore basin (Sastri et al. 1981; Biksham and Subrahmanyam 1988) and the sediments become thicker reaching up to about 8 km because of the collision of Indian plate with Eurasia in early Miocene age. The geologic age of the sediment in the deep offshore is from upper Cretaceous to recent age (Jagannathan et al. 1983; Fuloria et al. 1992; Fuloria 1993). Many geological features such as the regional fans, cut-and-fill channels and abundant growth faults are observed in offshore due to the deposition of sediments (Bastia et al. 2014; Bastia and Nayak 2006). The deep-water deposits are identified by interpreted high resolution seismic data which indicates the presence of potential hydrocarbon reservoirs in the offshore Mahanadi basin (Bastia and Nayak 2006; Kumar et al. 2014). High sediment rate (20–40 cm/kyr), high total organic matter and geothermal gradients of 35–45 °C/km may show indication of gas hydrate reservoir in the deep offshore where low temperature and high pressure satisfy the favorable condition in the basin (Singha et al. 2019, Sain et al. 2011; Shankar et al. 2013; Kumar et al. 2014 and Collette et al. 2008). The gas hydrate is deposited with clay/silt sediments and fracture zones of Pleistocene age in the deep water in the basin. The evidence of gas hydrates are identified by logging while drilling (LWD) and temperature of core sample by IR camera at NGHP-01 sites in the offshore Mahanadi basin (Kumar et al. 2014; Rai et al. 2020; Sain and Gupta 2012).

Fig. 2
figure 2

Geological cross section in the Mahanadi offshore basin, India near the study area at NGHP-01 sites (after Fuloria et al. 1992)

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Material and method

With the presence of fractures and faults in marine gas hydrate-bearing sediments of the offshore basin (Singha et al. 2019; Collett et al. 2008), we have assumed azimuthal anisotropy for characterization of the hydrate reservoir. The most common approach for modeling stress-induced or azimuthal anisotropy is to consider angular of penny shape fractures for a vertical well (Gupta 1973, Sill et al. 2010, 2012; Sill 2012). The azimuthal anisotropy here is represented by transverse isotropy with a horizontal axis of symmetry (HTI) as shown in Fig. 3a. In the figure, the symmetry axis of HTI is along the direction of the x-axis and the plane (x, z) containing the symmetry axis is referred to as the symmetry axis plane. For a vertical well, S- wave is polarized in two components; one is propagating along symmetric plane while the other propagates along fracture isotropic (Sill 2012; Satyavani et al. 2013). Here, we have used the upper shear mode (VS1) and lower shear mode (VS2) wave of S-dipole data measured by the dipolar sonic tool (DSI) at NGHP-01-19 as shown in Fig. 3b. VS1 mode is considered to propagate along isotropic plane whereas VS2 mode is traveling along the symmetry axis. Dipole shear data were not recorded in the remaining wells at NGHP-01-09 and NGHP-01-08, but all conventional data such as P-wave velocity, density, gamma, resistivity, neutron porosity and formation image log (FMI) are available for these wells. The wells (NGHP-01-19, NGHP-01-09 and NGHP-01-08) are located at a water depth of 1433 m, 1935 m, and 1701 m, and the total drilled depth area 280mbsf, 330mbsf, and 350mbsf for above mentioned wells respectively. Based on log data analysis, the BSR of the three wells is observed at depths of 208 m, 290 m, and 257 m respectively below seafloor (mbsf) at this site of the Mahanadi offshore. Quality of VS1 and VS2 is poor up to a depth of 90 mbsf but the coherence of waveform of upper and lower are of good quality above and below the BSR (~ 1641 m) shown in upper and lower BSR (UBSR and LBSR) in Fig. 3b.

Fig. 3
figure 3

a Graphical presentation of symmetric axis and isotropic plane for an horizontal transverse medium (HTI) where upper dipole shear mode wave (VS1) and lower dipole shear mode wave (VS2) which are along with axis of symmetry (isotropic plane) and perpendicular to the axis of symmetry respectively. b P-wave velocity, VS1, VS2 and density varying with depth in gas hydrates bearing sediments and depth of BSR is 1641 m at NGHP-01-19 site

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The effect of gas hydrates sediments in anisotropy medium (HTI) is investigated by Thomson’s coefficients. Thomsen (1986) first introduced three dimensionless anisotropic parameters. The coefficient, epsilon is related to P-wave anisotropy and gamma relates to S-wave anisotropy while delta is a critical factor that depends on the shape of P- and S-wave surfaces. P-wave anisotropy is the fractional differences between vertical P-wave i.e. VP (90°) velocity and horizontal P-wave velocity i.e. VP (0°) which is earlier defined as epsilon (ε) (Wang 2002) and similarly S-wave velocity is the fractional difference between SV and SH velocity which is earlier defined as gamma (\(\gamma ).\) These parameters are computed from stiffness coefficients which are used for a fractured medium with given fracture density in HTI medium (Hudson 1980).

Determination of stiffness coefficients

In this regard, applying General Hooke’s law for the non-vanishing elastic stiffness coefficients Cij (Ostadhassan et al. 2012) for an HTI medium with respect to the symmetry axis, the stiffness tensor has five independent components which are related with each other. The symmetric six by six matrices with elastic stiffness coefficients for HTI are represented as below (Musgrave 1970).

$${C}_{{i}_{j}}=\left[\begin{array}{cccccc}{C}_{11}& {C}_{13}& {C}_{13}& 0& 0& 0\\ {C}_{{1}{\text{3}}}& {C}_{33}& \left({C}_{33}-2{C}_{44}\right)& 0& 0& 0\\ {C}_{{1}{\text{3}}}& \left({C}_{33}-2{C}_{44}\right)& {C}_{33}& 0& 0& 0\\ 0& 0& 0& {C}_{44}& 0& 0\\ 0& 0& 0& 0& {C}_{55}& 0\\ 0& 0& 0& 0& 0& {C}_{55}\end{array}\right]$$
(1)

In HTI medium, some of stiffness coefficients are equal to each other as follows;

$${\text{C}}_{22} = {\text{C}}_{33} ,{\text{ C}}_{55} = {\text{C}}_{66} ,{\text{ C}}_{32} = {\text{C}}_{23} = {\text{C}}_{33} - 2{\text{C}}_{44} ,{\text{ C}}_{13} = {\text{C}}_{12} ,$$

The polarized S-wave velocities in an HTI medium can be written as (Wang 2002):

$$\rho {{V}^{2}}_{S1}={C}_{44},$$
(2)

and

$$\rho {{V}^{2}}_{S2}={C}_{55},$$
(3)

As vertical the well at NGHP-01-19 site was drilled perpendicular to the sedimentary bedding plane, only three of these five independent moduli using the stiffness coefficients (C33, C44, and C66) can be measured by dipole shear data.

$${\text{C}}_{33} = \rho {\text{V}}^{2}_{{\text{P}}} = \, \lambda \, + 2\mu ,$$
(4)

where, λ is Lame’s constant and μ is shear modulus or rigidity.

Also, in HTI medium (Sil 2012)

$${C}_{55}={C}_{66}=\rho {{V}^{2}}_{S2},$$
(5)

As no core data is available, other stiffness coefficients (C11 and C13) can be estimated from Annie model proposed by (Schoenberg and Sayers 1995). The constants C13 and C11 are given below,

$${\text{C}}_{11} {-}{\text{ C}}_{33} = \, 2\left( {{\text{C}}_{66} - {\text{ C}}_{44} } \right)$$
(6)

and

$${\text{C}}_{13} + 2{\text{C}}_{44} {-}{\text{ C}}_{33} = \, 0$$
(7)

Using Eqs. (6) and (7), we can estimate all five independent stiffness coefficients. Further, Thomsen’s parameters such as gamma, epsilon and delta for an HTI medium can be estimated using these coefficients. The shear modulus is related to coefficient given below.

$${\text{C}}_{44} = \rho V_{S1}^{2} = \mu ,$$
(8)

Estimated stiffness coefficients such as C11, C13, C33, C44 and C55 are the five independent stiffness moduli in HTI medium show lower values in the gas hydrate sediments and the values increase below BSR where free gas is present (Fig. 4). The minimum, maximum and average values of stiffness coefficients are listed in (Table 1) for upper BSR (UBSR) of the depth interval 1550 to 1641 m and lower BSR (LBSR) of depth interval 1641 to 1690 in the well NGHP-01-19.

Fig. 4
figure 4

Stiffness coefficients such as C11, C33, C44, C66 and C13 respectively using P-wave, VS1 and VS2 from DSI tool for an horizontal transverse medium (HTI) in the well NGHP-01-19

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Table 1 Stiffness coefficient such as C11, C13, C33, C44 and C66 using upper and lower dipole shear velocities and P-wave data in above and below BSR
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Estimation of Thomsen’s parameter

Using the following, we have estimated Thomsen’s parameters ε, \(\gamma\) and \(\delta\) given below (Rüger 1998a, b):

$$\upvarepsilon =\frac{{C}_{11}-{C}_{33}}{{2C}_{33}},$$
(9)
$$\gamma =\frac{\left({C}_{55}-{C}_{44}\right)}{{2C}_{44}},$$
(10)

and

$$\delta =\frac{{\left({C}_{13}+{C}_{55}\right)}^{2}-{\left({C}_{33}-{C}_{55}\right)}^{2}}{{2C}_{33}\left({C}_{33}-{C}_{55}\right)},$$
(11)

We notice a relation among these Thomsen’s parameters—first when ε remains constant, δ increases with decrease of γ; when γ is constant, δ will increase with ε; lastly when ε is approximately equal to γ, and then δ generally increases with degree of anisotropy (Ostadhassan et al. 2012; Mavko et al. 1995).

Determination of fracture density

We can write γ in terms of tangential fracture compliance, ZT (Liu et al. 2000) as,

$$\gamma =\frac{-1}{2}{\mu Z}_{\text{T}},$$
(12)

For the tangential ZT, it is given by (Sayers and Kachanov 1995),

$$\frac{{Z}_{N}}{{Z}_{T}}=1-\frac{\sigma }{2},$$
(13)

where, ZN is the normal fracture compliance and σ is the Poisson’s ratio of the medium. The Poisson’s (\(\sigma\)) ratio can be derived using the VP/VS1 ratio (Mohammed and Zillur 2001; Potter and Foltinek 1997);

$$\sigma =\frac{1}{2}\frac{\left(({{V}_{p}/{V}_{s1})}^{2}-2\right)}{\left(({{V}_{p}/{V}_{s1})}^{2}-1\right)},$$
(14)

The significance of the ratio of ZN/ZT is that high values greater than 0.8 indicate the presence of gas-filled fractures and values less than 0.5 represent water-filled fractures (Sill 2012). In our study area, the free gas is assumed to be present below the BSR in the un-compacted fracture clay/silt with Poisson’s ratio varying from 0.38 to 0.40. Therefore, our computed ZN/ZT using Eq. (13) is approximately 0.8 that indicates gas-filled fractures. Also, we can easily compute the normal fracture weakness parameter; δN (Schoenberg and Sayers 1995) from the values of ZN/ZT, and background P-wave, S-wave and density.

$${\delta }_{\text{N}}=\frac{{Z}_{N}{M}_{b}}{1+{Z}_{N}{M}_{b}},$$
(15)

where, \({M}_{b}=\uplambda +2\upmu ={C}_{33}\)

Further, the fracture density can be calculated using the following equation (Sill 2012; Shaw and Sen 2006):

$$e=\frac{3{\delta }_{N}\left(3-2g\right)}{16},$$
(16)

where, e is the fracture density; \(g\) is the ratio of square of vertical VP and VS wave velocity. A negative value of fracture density shows weak anisotropy zone and positive shows strong anisotropy zone.

By following all the steps and equations described above, we have computed the anisotropic parameters such as ε, γ and δ reaching maximum value below the depth of 1550 m indicating the presence of strong anisotropy in the well NGHP-01-19 (Fig. 5). A high value of fracture density at several depth intervals marked by red circles seen in Fig. 5 is clear indication of presence of anisotropy in the uncomapacted gas hydrate bearing sediments. We consider this as stress induced anisotropy because we have found many fractures and breakouts in the unconsolidated shallow marine sediments in the image log data at NGHP-01 site and the gas hydrates mainly deposited in fracture pore filled in clay/silt dominated fine grain sediments, those will be discussed in the Sect. 4. The stress induced anisotropy is caused by uneven stress in the sedimentary rock that is responsible for azimuthal variation of stress concentration around the borehole (Fang et al. 2013; Sayer 2002; Ruger 1998a, b). As a consequence the share wave velocity varies azimuthally around the borehole (Ruger 1998a; b). The minimum, maximum and average values of ε, γ and δ for the well NGHP-01-19 are listed in Table 2.

Fig. 5
figure 5

Computed Thomsen's parameters for the analysis of anisotropy such as gamma ray log value in API, fracture density, epsilon, gamma and, delta for horizontal transverse medium (HTI) from upper and lower BSR. Higher fracture densities values are marked by red circle are for corresponding to the higher values of Thomson’s parameters such as epsilon, gamma and delta

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Table 2 Thomsen’s parameters such as epsilon (ε), gamma (γ) and delta (δ) computed from stiffness coefficient for an HTI medium for the well NGHP-01-19 in above and below BSR
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Estimation of anisotropy coefficient from S-wave

Anisotropy coefficient from S-wave data is another important parameter for measuring the degree of anisotropy, i.e., whether it is strong or weak. It can be estimated using the velocity relationship of shear-wave velocities (VS1 and VS2) from the well log data. The anisotropy coefficient (Ab) is determined by the following equations (Yan 2002)

(17)

where,

Our data shows that the anisotropy coefficients value increases and the maximum value of anisotropy are about 0.47 in the depth range of 1551 m to 1575 m, thus this is a strong anisotropy zone (Fig. 6). As the values of VS1 are not always higher than the values of VS2 throughout the depth in the well, the anisotropy coefficient from S-wave velocity we are getting sometimes negative values. The intermediate anisotropic zone may have the values in range -0.2 to 0.3. It is also noticed that the values of anisotropy coefficients are high for higher values of fracture density and Thomson’s parameters in free gas zone is higher than that in the gas hydrate zone in the well NGHP-01-19.

Fig. 6
figure 6

Computed anisotropy coefficient using VS1, VS2 and P-wave and maximum value is ~ 0.47 in the strong anisotropy zone of depth interval 1600 to 1645 m as marked by green circle

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Natural fractures observed in FMI log

Natural fractures are joints or discontinuity in the surface, which are expressed by cracks, fissures, or faults in the rock (Twiss and Moores 1992; Zoback et al. 1985). Solid gas hydrates fill most of the natural fracture, converting an open fracture to a sealed fracture. The natural fractures are subdivided into the open fracture, partially open fracture and resistive fracture (Rajabi et al. 2010; Esmersoy et al. 1995). The presence of the natural fractures influences the stress-induced anisotropy in the gas hydrate sediments.

These fractures appear in the formation micro image (FMI) log as a sinusoidal or half sinusoidal as the projection of a planar intersection with a cylindrical borehole. The FMI is a resistivity imaging tool consisting of two perpendicular pairs of caliper arms, with the end of each arm hosting a pad and attached flap. The pads and flaps contain typically 24 resistivity sensors on each pad or flap. The resistivity data from this log can be processed to build up a picture of the wellbore wall based on resistivity contrasts (Ekstrom et al. 1987; Rajabi et al. 2010; Chatterjee and Singha 2018). The dip and azimuth of the fractures for a vertical well can be calculated from a sinusoidal appearance by the following formula:

$$\theta ={\mathrm{tan}}^{-1}h/d,$$
(18)

where, h = length from crest to trough, d = diameter of borehole and θ = dip angle of fracture.

$${\text{Azimuth}}\;{\text{of}}\;{\text{fracture}}\; = \;{\text{dip}}\;{\text{direction}} + 90^\circ \left( {{\text{dip}}\;{\text{azimuth}}\;{\text{is}}\;{\text{calculated}}\;{\text{from}}\;{\text{the}}\;{\text{position}}\;{\text{of}}\;{\text{the}}\;{\text{Trough}}} \right)$$
(19)

The well NGHP-01-19 contains 25 fractures observed from the FMI log in several depth intervals. The dip and orientation of the fractures vary from 11.3° to 36.91° and N166.15° to N360° respectively as demonstrated in Fig. 7a, b. Examples of resistive fracture and dip for NGHP-01-19, NGHP-01-09 and NGHP-01-08 are shown in Fig. 7. The orientations of strong and weak anisotropy are N320° and N351.5° for well NGHP-01-19 respectively as shown in Fig. 8a, b. The well NGHP-01-09 located towards the south holds total 38 fracture sets with dip amount ranging from 10.45° to 38.15° as shown in Fig. 7c, d and orientation N198.6° in strong anisotropy and N136.6° in weak zone as shown in Fig. 8c, d. The least number of fractures are present in well NGHP-01-08 located eastward with dip amount 21.0° shown in Fig. 7e, f and orientation N344.6° in strong anisotropy and N345.7° in weak zone shown in Fig. 8e, f. The number of fractures with dip amount and orientation for three wells are listed in Table 3.

Fig. 7
figure 7

Natural fractures marked by red circle and corresponding the rose diagram for the orientation of the total natural fractures, a and b for well NGHP-01-19, c and d for NGHP-01-09 and e and f for well NGHP-01-08 respectively

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Fig. 8
figure 8

Orientation of strong and weak anisotropy based on the orientation of natural fractures, a and b for well NGHP-01-19, c and d for NGHP-01-09 and e and f for well NGHP-01-08 respectively

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Table 3 Natural fractures for three wells with dip amount and dip orientation at NGHP-01 sites
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Estimation of S-wave velocity

Only the NGHP-01-19 contains S-wave data and synthetic S-wave data are estimated for the remaining wells NGHP-01-09 and NGHP-01-08 respectively. To do so, we first use Castagna’s equation between P-wave velocity and S-wave velocity for gas hydrate sediments of the study area with coefficients that match the predicted shear velocity. The estimated S-wave velocity is matched separately with VS1 and VS2 with changing the coefficients (‘a’ and ‘b’). As, the shear wave velocity was affected by the unconsolidated sediments below the seafloor, based on this statistical approach, we performed the better correlation between the predicted shear wave velocity and the dipole sonic velocity. We obtained for coefficients values 1.452 and 1.031 between estimated S-wave and VS1 with goodness of fit (R2) ~ 0.70 (Fig. 9a). The poorer match was observed between estimated S-wave and VS2 with R2 ~ 0.51 in Fig. 9b. Therefore, VS1 was chosen to establish the equation with P-wave velocity and we have used VS1 for computation of elastic parameters in other wells where no availability of S-wave velocity. Thus we obtained the following modified empirical relationship for gas hydrate-bearing sediments in the Mahanadi offshore;

$${\text{V}}_{{\text{P}}} \, = \, 1.452{\text{V}}_{{\text{S}}} \, + 1.031$$
(20)

where, VP and VS refer to P-wave and S-wave velocities in km/s respectively.

Fig. 9
figure 9

a A synthetic model between P-wave and S-wave (VS1) in km/s with goodness of fit R2 is 0.612. b A statistical linear relationship established between S-wave (VS1) and estimated S-wave from modified Castagna’s empirical relation in with goodness of fit R2 is 0.701

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Using this modified empirical relation, we estimate the S-wave velocity at other well sites where no recorded S-wave data is available.

Computation of elastic parameters

Elastic properties in an isotropic elastic medium are affected by stress causing anisotropy (Ruger 2002). Stress-induced anisotropy does influence the elastic properties under the state of hydrostatic pressure. In other way, stress-induced anisotropy can be applied for the spatially varying anisotropic elastic constants that are required for the forward modeling of wave propagation in a borehole under a given pressure condition, i.e. hydrostatic pressure (Fang et al. 2013). In the study area, the observed stress is normal followed by the hydrostatic pressure (Singha et al. 2019) whereas the velocity and density are not enough high to influence the elastic properties as unconsolidated sediments contains mostly muds (Winkler 1996). We examine various elastic parameters such as young’s modulus, bulk modulus, shear modulus, and Poisson’s ratio in both media i.e. isotropic and anisotropic. The rock elastic parameters are seen to have sharp changes in Thomsen’s parameters as well as fracture density in the strong and weak anisotropy zones.

Isotropic medium

The elastic moduli have been estimated from VP, VS and density (ρ) log data using the equations provided by (Potter and Foltinek 1997).

$${\text{Shear}}\; {\text{modulus}},\;\mu ={V}_{s}^{2}\rho$$
(21)
$${\text{Young's}}\; {\text{modulus}}\; Y=2*\mu \left(1+\sigma \right)$$
(22)
$${\text{Bulk}}\; {\text{modulus}},\; K=\rho *\left({V}_{P}^{2}-\frac{4}{3}{V}_{S}^{2}\right)$$
(23)

Anisotropic medium

The elastic moduli have been computed from P- wave and S-wave relates stiffness coefficient for HTI medium such as C33, C44, and C55 by Wang 2002.

$${\text{Shear modulus}},\;\mu = {\text{C}}_{44}$$
(24)
$${\text{Young's}} \; {\text{modulus}}, \;{\text{ Y }} = \, 2*{\text{C}}_{44} *\left( {1 + \sigma } \right),$$
(25)
$${\text{Bulk}}\;{\text{modulus}},\;{\text{ K}} = {\text{C}}_{33} - {\text{C}}_{55} *\left( {4/3} \right)$$
(26)

The elastic parameters were calculated from log data for NGHP-01-19 in both the mediums while from synthetic shear data for NGHP-01-09 and NGHP-01-08 in only isotropic medium because of unavailability of S-wave velocity (Figs. 10, 11 and 12). To fully describe TI, we need both vertical and horizontal section of well which will improve anisotropy estimation. As we are using only vertical velocity, therefore, elastic properties obtained from these logs are isotropic in nature. No such significant differences are observed in the values of the elastic parameters for the isotropic and anisotropic medium at NGHP-01 wells shown in Fig. 10. But, the elastic parameters are seen to be increasing from UBSR to LBSR while the Poisson's ratio decreases. Elastic properties of gas hydrate-bearing sediments are similar to those of the unconsolidated sediments. From the above analysis, it is clear that if the normal fracture weakness increases then fracture density also increases. Therefore, the rock elastic coefficient values decrease at that depth zone because rock elastic moduli values are directly proportional to the stiffness coefficient values. The values of the stiffness coefficients are inversely proportional to the fracture density values. The low range of elastic modulus values gives a high range of fracture density values showing a strong range/zone of anisotropy and vice versa as clearly noticed in Fig. 10 marked by the red circles.

Fig. 10
figure 10

Changes of elastic parameter such as shear modulus, young’s modulus and bulk modulus with fracture density at the depth of anisotropy in the well NGHP-01-19. The lower elastic values (denoted by L) are found across the higher fracture density values indicating strong anisotropy

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Fig. 11
figure 11

The computed elastic parameters such as shear modulus, young are modulus and bulk modulus for NGHP-01-09 using synthetic shear wave data with BSR depth 2225 m. The lower values of elastic parameters are denoted by L at the depth intervals where fractures are observed from FMI log data to indicate presence of stress induced anisotropy

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Fig. 12
figure 12

The computed elastic parameters such as shear modulus, young’ s modulus and bulk modulus for NGHP-01-08 using synthetic shear wave data with BSR depth 1957 m. The lower values of elastic parameters are denoted by L at the depth intervals where fractures are observed from FMI log data to indicate presence of stress induced anisotropy in the well

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Results

For a horizontal transversely isotropic medium the stiffness coefficient C11, C13, C33, C44 and C55 are increasing from depth interval 1400 to 1690.32 m for the well NGHP-01-19. The average values of the stiffness coefficients are 3.738 GPa, 3.534 GPa, 3.754 GPa, 0.0984 GPa and 0.090 GPa respectively in the sediments above the BSR. The average stiffness values are slightly increased below the BSR which are 4.025 GPa, 3.593 GPa, 4.035 GPa, 0.197 GPa and 0.191 GPa respectively. The Thomsen parameters are changing from UBSR to LBSR having an average value of − 0.0027 to − 0.0013, − 0.00026 to − 0.00029 and − 0.004 to − 0.002 respectively. The maximum value of epsilon is 0.022 showing strong zone of anisotropy for the gas-hydrate bearing zone and the minimum value is − 0.053 showing a weak anisotropy zone. The maximum value of epsilon in the gas hydrate stability zone (GHSZ) is 0.07 and the maximum value in the free gas zone is 0.19 showing a considerable amount of anisotropy in the gas hydrate-bearing zone and is strong in the underlying the free gas zone. Similarly, the maximum value of gamma in GHSZ is 0.01 and 0.5 in free gas zone showing anisotropy in free gas zone is stronger than GHSZ. The fracture density calculated from Thomsen’s parameters showing higher values ranging from 0.039 to 0.076 are the strong anisotropy zones (Table 4). The fracture density values are varying from − 0.0731 to 0.0761 having an average value of − 0.0055 in the depth interval of 1493.11 m to1690.32 m. The natural fractures have been observed using FMI log at same depth of higher fracture density having azimuth/orientation ranging from N166.15° to N360° in the depth interval of 1493.11 to 1690.32 m for the well NGHP-01-19.The natural dips for the well NGHP-01-19, NGHP-01-09 and NGHP-01-08 are 16.3°, 21.0° and 17.1° respectively. Further, the orientations of strong anisotropy are N320.4°, N344.6°, and N198.6° and the orientations of weak anisotropy are N351.5°, N345.7°, and N136.6°. The values of the elastic parameters such as shear modulus, young’s modulus and bulk modulus increase from 291.60 to 580.38 MPa, 268.33 to 565.38 MPa and 3634.57 to 3780.55 MPa from the UBSR to LBSR and the value of Poisson’s ratio slightly decreases from 0.486 to 0.474 in the depth range of 1493.11 to 1690.32 m for the well NGHP-01-19. It is observed that the values of elastic parameters decrease while fracture density increases and vice-versa which are shown in Table 4.

Table 4 Correlation of fracture density with natural fracture orientations and elastic parameters (Shear, Young’s and Bulk moduli) for the depth interval of strong anisotropy zone at well NGHP-01-19
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For other two wells, the values of elastic parameters increase from 119.53 to 143.97 MPa, 354.36to 423.97 MPa and 3392.88 to 3680.26 MPa respectively in the depth interval 1935.02 to 2258.87 m from UBSR to LBSR and whereas, the value of Poisson’s ratio changes from 0.483 to 0.480 for well NGHP-01-09 for the corresponding depth of intervals. The values of elastic modulus are varying from 125.69 to 164.05 MPa, 372.82 to 485.43 MPa and 3760.69 to 4021.62 MPa respectively in depth interval of 1700.08 to 2031.86 m with the corresponding value of Poisson’s ratio is varying from 0.484 to 0.480 for well NGHP-01-08.

Discussion

High velocities (~ 1.64 km/s) are achieved with high gas hydrate concentrations (~ 10–12%) just above the BSR depth (~ 208mbsf) and lower velocity (1.44 km/s) is found at depth ~ 1650 m just below it because of the presence of free gas in the well NGHP-01-19. The clay/silt layer which is also present in the un-compacted sediments may also be responsible for intrinsic causes of generating the anisotropy. One could assume a VTI medium for analyses of the anisotropy parameters but because of small thickness of shale layer we did not consider VTI medium in our study. Presence of natural fracture, pore filling gas hydrates and as well as change of mechanical behavior play a major role in generating stress-induced anisotropy and therefore, the analysis of anisotropy was done by assuming an HTI medium in the study area. The anisotropy is dominantly stress-induced than shale based in the gas-hydrate bearing sediments in the wells NGHP-01-19. These natural fractures are filled with solid gas hydrate or free gas, free water or pore filling solid matrix. The value of normal fracture weakness (δN) i.e. inverse of stiffness coefficient is high (high fracture density) showing the degree of anisotropy to be high or strong (Sill et al. 2012) shown in Fig. 10.

Crossplot between elastic parameters and fracture density

We have tried to classify the degree of anisotropy in well NGHP-01-19 based on the cross plot between the fracture density and elastic parameters. From Fig. 13, it is observed that the low value of elastic parameters and high value of fracture density (> 0.03) demarcate for strong anisotropy (SA) zone marked by the red box; intermediate values with fracture density (range − 0.02 to 0.03) represent for medium anisotropy (MA) zone marked by the yellow box and high elastic values and low fracture density (< − 0.02) indicate weak anisotropy (WA) zone marked by the green box. Thus, the fracture density is inversely proportional to the rock elastic parameters. The anisotropy is more sensitive to shear modulus and young’s modulus than bulk modulus because anisotropy changes with size and shapes rather than with its volume. Therefore, we obtain sharp changes in shear and young’s modulus from weak to high anisotropy zone above and below the BSR.

Fig. 13
figure 13

Cross plot between fracture density between elastic parameters a fracture density vs. shear modulus, b fracture density vs. young’s modulus and c fracture density vs. bulk modulus. Three type anisotropy zones classified which strong anisotropy (SA) mark by red colour, followed by medium anisotropy (MA) by yellow colour and weak anisotropy (WA) by green colour with corresponding depth range for well NGHP-01-19

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Effect of gas hydrate and the relation of Thomsen’s parameters

The cross-plots between fracture density with epsilon and gamma giving high correlations suggests that these values are directly proportional to each other while fracture density with epsilon is more correlated (R2 ~ 0.99) than gamma suggesting that P-wave anisotropy is more effective than S-wave anisotropy.In this area the P-wave velocity is more affected by the anisotropy properties as shown in Fig. 14.

Fig. 14
figure 14

Cross plot correlation between fracture density and Thomson’s parameters a delta vs epsilon, b delta vs gamma, c fracture density vs epsilon and d fracture density vs gamma. These correlations show P-wave anisotropy is slightly higher than S-wave anisotropy while fracture density is collinear with Thomson’s parameters such as epsilon and gamma

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The anisotropy parameters are substantially influenced by the presence of fracture in gas hydrate sediments. The large negative values of epsilon and delta with positive gamma represents either dry solid gas hydrate or free gas filled in the crack or fracture of the sediments, whereas small negative values represent wet filled cracks. The positive correlation between epsilon and gamma may be caused by an organic mixture of silt/clay-bearing sediments along with micro-fracture of the layer.

Velocity modeling using Thomsen’s parameter

P-wave anisotropy is slightly more sensitive to S-wave anisotropy because of P-wave velocity in liquid phase as well as solid phase have greater sensitivity than S-wave velocity (shear modulus in fluid is zero) as reported in Wang 2002. Generally, P-wave anisotropy is slightly higher than S-wave anisotropy (Tsvankin 1996 and Peacher et al. 2003).

The simplified direction-dependent P-wave, SV- and SH-wave velocity for the angle of fracture angle 0°, 16.3° and 90° at well NGHP-01-19 as was given by Thomson’s (1986) as follows;

$${\text{V}}_{{\text{P}}} (\theta ) \approx {\text{V}}_{{{\text{P}}0}} (1 + \delta {\text{sin}}^{2} \theta {\text{cos}}^{2} \theta + \varepsilon {\text{sin}}^{4} \theta )$$
(27)
$${\text{V}}_{{{\text{SV}}}} (\theta ) \, \approx {\text{ V}}_{{{\text{S}}0}} \left[ {1 + \left( {\frac{{{\text{V}}_{{{\text{P}}0}} }}{{{\text{V}}_{{{\text{S}}0}} }}} \right)^{2} (\varepsilon - \delta )\sin^{2} \theta \cos^{2} \theta } \right]$$
(28)
$${\text{V}}_{{{\text{SH}}}} \approx {\text{ V}}_{{{\text{S}}0}} (1 + \gamma {\text{sin}}^{2} \theta )$$
(29)

where,

$${\text{V}}_{{{\text{P}}0}} = \sqrt {\frac{{{\text{C}}_{33} }}{\rho }} ;\;{\text{V}}_{{{\text{S}}0}} = \sqrt {\frac{{{\text{C}}_{44} }}{\rho }}$$

To estimate the velocities at NGHP-01-19 sites, we considered three cases of different angle which are at minimum (0°), maximum (90°) and the third one, the average dip of fracture (16.3°) for strong anisotropy followed by using all the computed stiffness and Thomsen’s parameters. Due to availability of velocity along the direction of 0°, 90°, and 16.3°, we have chosen 0°, 16.3° and 90° for velocity modeling in anisotropy medium. In the Fig. 15a, it is observed that Vp(90°) is more influenced and effected by anisotropic parameters than Vp(16.3°) in the gas hydrate and free gas sediments near BSR. The value of Vp(90°) has increased 1.5% to 2.5% whereas, the value of Vp(16.3°) has enhanced little bit ranging from 0.2 to 0.35% relative to P-wave velocity. Direction dependent shear wave velocities such as VSV (0°), VSV (16.3°), VSV (90°) and VSH (0°), VSH (16.3°), VSH (90°) are less effective in the presence of the anisotropy parameters as observed in the Fig. 15b, c.

Fig. 15
figure 15

Modeled a P-wave, b SH and c SV-wave using estimated stiffness coefficients and Thomsen’s parameters for fracture angle minimum (0°), maximum (90°) and orientation of strong anisotropy (16.3°) relative to symmetry axis at NGHP-01-19

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Conclusions

We estimated the Thomsen’s parameters accomplished with five stiffness coefficients assuming an HTI medium using upper and lower mode shear wave data at NGHP-01 site in the offshore Mahanadi basin. The anisotropic parameters along with fracture density and coefficient of anisotropy reveal a strong anisotropy zone near above and below the BSR because of the presence of gas hydrates and free gas or pore filling fracture. The maximum fracture density reaches 0.07 while the coefficient of anisotropy is 0.32 indicating a strong anisotropy zone. The strong anisotropy zones are also correlated with the natural fractures which are observed from FMI data in the study area. The elastic parameters of un-compacted marine sediments are also affected by stress-induced anisotropy reducing lower values of elastic modulus in the gas hydrate and free gas zones. The cross-plot analysis reveals that the strong, medium and weak anisotropy are classified with elastic parameters. The P-wave velocity has been modified by the inclusion of gamma and delta for accurate assessment of gas hydrates and has increased its value up to 2.8% from log velocity for 90° angle of fracture relative to symmetry axis. The stress induced an isotropy in gas hydrate bearing marine sediments is mostly caused by natural fractures/joints, change of mechanical behavior and pore-filling in clay/silt marine sediments in the study area.