1 INTRODUCTION

The Logging-While-Dilling(LWD)acoustic technology was developed in recent years as a replacement of wireline logging(Tang and Cheng, 2004). Many studies have been done on LWD multipole acoustic wave propagation along a borehole surrounded by an isotropic formation(Tang et al., 2002; Cui, 2004; Su et al., 2006; Wang et al., 2009). The simulations of anisotropic formation in the LWD environment have got great progress(Wang et al., 2012; Li et al., 2013). The LWD acoustic technology now has been applied widely in exploration and production of unconventional oil-gas resources(Hao et al., 2012).

The universal presence of cracks and pores in formation can significantly affect the propagation of elastic waves. Tang(2011) and Tang et al.(2012)have recently developed an elastic wave theory for cracked porous rocks and used it to interpret acoustic log data from tight s and and shale gas formations. The theory has also been applied to model multipole acoustic wave propagation along a borehole surrounded by a cracked porous formation(Chen et al., 2012; Chen et al., 2013). In the modeling, Chen et al. simulate the velocity dispersion, attenuation, sensitivity, spectral excitation and waveforms of multipole wave modes for different crack parameters. They also analyzed the difference between the impermeable and permeable borehole wall conditions of wireline logging in a cracked porous formation.

This paper extends the study above to the LWD environment, modeling the acoustic wave propagation in a cracked porous formation in the presence of an LWD acoustic tool that occupies a large portion of the borehole. The multipole elastic wavefield for a cracked porous formation follows from Chen et al.’s formulation; the modeling of an LWD tool in borehole has been described in Tang and Cheng’s book as well as in Su et al.(2006). To the authors’ knowledge, suhc LWD modeling work has not been reported. In this paper, we describe the main characteristics of LWD multipole acoustic logging for tight rocks with low permeability and porosity but with abundant cracks. In the following, we present the calculated waveform characteristics for different pore fluids in tight formation rocks. The modeling results can provide theoretical support for interpreting LWD acoustic data logged in cracked porous formations.

2 LWD ELASTIC WAVEFIELD IN A CRACKED POROUS MEDIUM

In the acoustic LWD situation, the drill collar occupies a large portion of the borehole. The source called a multipole acoustic ring source are implemented as a distribution of the point sources placed on the rim of the collar. We use a pore-crack model in Fig. 1 to analyze the LWD multipole wave characteristics. In this model, an LWD drill collar of radius r_a, r_b, R is centered at the borehole of radius which is surrounded by a cracked porous formation.

2.1 Elastic Wavefield in Inner Fluid

There are only incoming waves traveling from the outer to the inner boundary in the inner fluid. Using cylindrical coordinates(r, θ, z)the inner fluid wavefield(Φ')for the nth multipole is yielded:

where the coefficient A'_n is a function of k and ω, and n is the azimuthal order number which specifies the source type. From the displacement potential in Eq.(1), the radial displacement u'_f and stress σ'_rrf of the inner fluid are given by where f is the radial wavenumber; k is the axial wavenumber, and I_n is the modified Bessel function of the first kind and order n.

2.2 Elastic Wavefield in Drill Collar

Drill collar is a special elastic medium in which exit compressional waves, SH-type shear-waves and SV-type shear-waves. Each of the potentials is the solution of a scalar wave equation

where Φ^tool, χ_tool and Γ_tool are respectively compressional wave potential, SH-type shear-wave potential, and SV-type shear-wave potential, k_p = ω/v_p^tool and k_s = ω/v_s^tool are the compressional and shear wavenumber, and v_p^tool and v_s^tool are the compressional and shear velocity, respectively.

Considering both outgoing waves and incoming waves, the solution of Eq.(3)is

where p_tool = $\sqrt{{{k}^{2}}-{{\left( w/v_{\text{p}}^{\text{tool}} \right)}^{2}}}$ and s_tool = $\sqrt{{{k}^{2}}-{{\left( w/v_{\text{s}}^{\text{tool}} \right)}^{2}}}$ are the compressional and shear radial wavenumbers of the drill collar, I_n and K_n are the modified Bessel function of the first kind and second kind, A_n^tool(B_n^tool), C_n^tool(D_n^tool) and E_n^tool(F_n^tool)are the incoming waves(outgoing waves)amplitude coefficients of compressional wave, SH-type shear-wave, and SV-type shear-wave, respectively.

The drill collar displacement and stress elements are calculated by the following formulae:

2.3 Elastic Wavefield in Outer Fluid

There are incoming waves and outgoing waves for the outer fluid. The outer fluid wavefield(Φ)for the nth multipole is yielded

The radial displacement u_f and stress σ_rrf of the outer fluid are given by

2.4 Elastic Wavefield in Cracked Porous Formation

The constitutive equation for the average stress for a cracked porous formation is given by(Tang et al., 2012)

where σ_ij is the average stress; λ = K_d-2μ/3 is Lame’s constant; μ and K_d are the shear and dry bulk modulus of the solid; p = -(α$\nabla $·u+$\nabla $·w)/[β+S(ω)] is the pore fluid pressure; u and w are the solid displacement and relative displacement between fluid and solid, respectively; e = $\nabla $·u; e_ij is strain tensor; ω is angular frequency; and α = 1 - K_d/K_s and β =(α - $\phi $)/K_s + $\phi $/K_f are two parameter combinations. Using S(w)to characterize the squirt flow effect yields where is parameters combination; $\xi $ = [3$\phi $/(4πε)]^1/3 is poreto-crack size ratio; η is pore fluid viscosity, ε is crack density and γ is crack aspect radio; and μ₀ and υ₀ are the shear modulus and Poisson’s ration in the absence of squirt flow, respectively. μ and K_d can be calculated by the Biot-consistent theory(Thomsen, 1985).

Three wave modes(fast-, slow-compressional and shear waves)can propagate in a cracked porous formation. The displacements of these three waves can be expressed as

where Φ_fast and Φ_slow denote the fast- and slow-compressional waves, χ and Γ represent SH-type and SV-type shear-waves, respectively.

The stresses and displacements of the cracked porous formation can be calculated by Eqs.(8)to(10). The potentials are expressed as follows(Chen et al, 2013):

where q_fast, q_slow and q_s are radial wavenumbers of fast-, slow-compressional and shear waves, respectively. where k_fast, k_slow and k_shear are wavenumbers of fast-, slow-compressional and shear waves, respectively. The derivation of these wavenumbers can be found in Chen et al.(2013).

The displacement components, pore stress and stress components are obtained by

2.5 Boundary Conditions

In the theory above, we provide detail theoretical elastic wavefield formulations of different portions in Fig. 1. By employing the boundary conditions, we relate the wave motion of these portions. We use a multipole acoustic ring source on the rim of the collar to model source excitation.

2.5.1 Inner fluid and drill collar boundary

2.5.2 Drill collar and outer fluid boundary

where, σ^s is the acoustic ring source,

The formulation above is a stress source(Cui, 2004). It is worth mentioning that the expression of the source can also be described as a displacement source(Tang and Cheng, 2004).

2.5.3 Outer fluid and formation boundary

Considering the boundary condition as an open-pore case, which means that the borehole pressure equals the pore fluid pressure and there is a free hydraulic exchange between the borehole and formation(Tang and Cheng, 2004).

Substituting the stress and displacement components from Eqs.(1)-(13)into the boundary conditions from Eqs.(14)-(17), we give rise to a matrix equation

The details of components of H are given in Appendix A.

3 DISPERSION AND ATTENUATION CHARACTERISTICS OF LWD MULTIPOLE ACOUSTIC LOGGING IN CRACKED POROUS FORMATION

According to the theoretical formulation above, the main characteristics of LWD multipole acoustic logging for different crack density values and different pore fluids in cracked porous formation are simulated. The model parameters of drill collar and borehole fluid are listed in Table 1. Table 2 shows two sets of formation parameters of different pore fluids.

The determinant D of the matrix in Eq.(18)leads to dispersion equation,

By solving the Eq.(19), the phase velocity and attenuation of the wave mode are calculated:

The sensitivity of velocity or attenuation to a model parameter p, which is defined as the normalized partial derivative of the wave’s phase velocity(attenuation)to q(Tang and Cheng, 2004)is

3.1 LWD Monopole Stoneley Waves

Figure 2 shows the Stoneley wave velocity(Fig. 2a) and attenuation(Fig. 2b)in a water-saturated formation for various crack density values. The model parameters are listed in Table 1 and Table 2, respectively. The Stoneley velocity decreases and attenuation increases with crack density increasing from 0.00 to 0.25. The sensitivity of velocity to crack density analysis(Tang and Cheng, 2004)increases with crack density increasing(Fig. 2c). The modeling results show that the LWD Stoneley waves are significantly affected by the presence of cracks in tight rocks.

The sensitive to fluid saturation showing in the cracked porous theory can be utilized for hydrocarbon detection. Therefore, we present the calculated characteristics for different pore fluids in cracked porous formation. The fluid is a mixture of water and gas. The density, modulus and viscosity of the mixed fluid are given by(Brie et al, 1995)

where S_g is the gas saturation. Fig. 3 shows the Stoneley wave velocity(Fig. 3a) and attenuation(Fig. 3b)in a cracked formation(ε = 0.2)for various gas saturations. With gas saturation increasing, the Stoneley velocity decreases and attenuation increases, especially in the low-frequency range. It is noted that the velocity of the LWD Stoneley waves is more sensitive to crack density while the attenuation is more sensitive to gas saturation(Fig. 3a, Fig. 2a), which also can be seen through the comparison between Fig. 2c and Fig. 3c. However, the attenuation is more sensitive to gas saturation(Fig. 3b, Fig. 2b). These characteristics can be used to detect the presence of gas in tight formations.

3.2 LWD Dipole Wave Mode

The wave with azimuthal order number n = 1 in Eq.(18)is the dipole wave. Fig. 4 shows the flexural wave velocity(Fig. 4a) and attenuation(Fig. 4b)in a water-saturated formation for various crack density values. The velocity decreases and attenuation increases with crack density increasing from 0.00 to 0.25. It is also interesting to note that there is a minimum of the group velocity in the low-frequency range(3 kHz), which can produce a phenomenon in the waveform called the “Airy phase”. Meanwhile, a maximum of the attenuation exists among the same frequency range. Compared with the wireline logging(WL), the LWD dipole wave characteristics of the dispersion and attenuation vary very much at the high-frequency range. This is because the drill collar occupies a large portion of the borehole substantially influencing the acoustic wave propagation characteristics(Tang and Cheng, 2004), especially in the high-frequency range. It is also noted that the sensitivity of dipole wave velocity to crack density is severely decreased. It can be increased with frequency increasing because of the drill collar(Fig. 4c). The sensitivity of velocity to crack density is the minimum area coincidently in conventional logging-frequency. From this point of view we also demonstrate the drawback of acoustic dipole in LWD, which shows strong dispersion and low sensitivity to crack density in conventional logging-frequency. Fig. 5 shows the flexural wave velocity(Fig. 5a) and attenuation(Fig. 5b)in a various gas-saturated formation. The modeling results show that with gas saturation increasing, the flexural velocity decreases, but not clearly.

3.3 LWD Quadrupole Wave Mode

The wave with azimuthal order number n = 2 in Eq.(18)is the quadrupole wave. Because the LWD quadrupole tool has become a st and ard technology for shear measurement(Tang and Cheng, 2004), it is particularly important to study the LWD quadrupole wave characteristics. Fig. 6 shows the wave’s velocity(Fig. 6a), attenuation(Fig. 6b) and the sensitivity analysis in a water-saturated formation for various crack density values. It is noted that increasing crack density decreases the velocity and increases the attenuation. The maximum of attenuation curves correspond to the Airy phase in the dispersion curves. The dispersion and attenuation characteristics of LWD quadrupole waves are similar to that of WL dipole waves, which shows some properties common of the S-type guided waves(depending strongly on the S-wave velocity of the formation). Fig. 7 shows the screw wave velocity(Fig. 7a) and attenuation(Fig. 7b)in a cracked formation(ε = 0.2)for various degrees of gas saturation. It is noted that increasing crack density decreases the velocity and increases the attenuation. However, increasing gas saturation has little effect on this S-type guided wave but significantly increases the attenuation.

4 LWD MULTIPOLE ACOUSTIC WAVEFORM CHARACTERISTICS

The goal of LWD acoustic measurement is to determine formation compressional and shear velocities(Tang and Cheng, 2004). For this reason, we show the LWD multipole acoustic waveform characteristics in connection with formation cracks. The model parameters are listed in Table 1 and Table 2, respectively.

4.1 LWD Monopole Waveform

Figure 8 shows the LWD monopole waveforms in a fluid-filled borehole surrounded by a cracked porous formation with crack density values ranging from 0.00 to 0.25. We assign a 9 kHz center-frequency source to the rim of the tool.(The source implementation follows from Cui’s approach(Cui Z W, 2004)). The source-toreceiver distance is 5 m. Because the waveforms are dominated by the S-wave and Stoneley waves with strong amplitude, we multiply the early portion of the data that contains the collar and formation P-waves by a factor of 100 for visualization. The P- and collar-waves overlap for this fast formation is tight. We then increase the collar-wave attenuation(Q^-1 = 0.2)in the simulation to simulate the effects of an acoustic isolator between the source and receiver. The waveforms are displayed from crack density values ranging from 0 to 0.25 for comparison. We also simulate the effects of water saturation(S_g = 0, black) and gas saturation(S_g = 1.0, red)on the waveforms. It is noted that the velocities of P-, S-, and Stoneley waves decrease with increasing crack density. Meanwhile, the Stoneley-wave attenuation increases the amplitude decrease as crack density increases. It is also noted that the P-wave arrival time for gas saturation, as denoted by a reline, lags behind that for water saturation(denoted by a black line). This well demonstrates that gas saturation lowers the P-wave velocity for a tight formation containing cracks. The higher the crack density value, the more pronounced this phenomenon becomes. In comparison, the S-wave velocity shows little changes, showing S waves are insensitive to formation fluid. A remarkable phenomenon is that Stoneley-wave amplitude is dramatically different between the two saturation conditions, the amplitude being strongly attenuation by gas. In addition, P wave amplitude is also significantly attenuated by gas saturation. A useful conclusion from the modeling is that one can use compressional-to-shear velocity ratio(V_P/V_S) and P- and Stoneley-wave amplitude from the LWD measurement for hydrocarbon detection in tight formations.

4.2 LWD Quadrupole Waveforms

Figure 9 shows the LWD quadrupole waveforms excited by a 3 kHz source in a fluid-filled borehole surrounded by a cracked porous formation with crack density values ranging from 0 to 0.25. The source-to-receiver distance is 3 m. The S-wave velocity decreases with increasing crack density. Both water(black) and gas(red)saturation conditions are calculated. The wave onset for both situations is the same, showing that S-wave velocity is insensitive to fluid saturation, consistent with the monopole modeling result of Fig. 8. It is noted that the later dispersive portion of waveform corresponding to the “Airy phase” is severely attenuated by gas saturation, in agreement with the attenuation modeling of Fig. 7b and Fig. 7d. This phenomenon can be used as a characteristic for hydrocarbon detection from LWD quadrupole wave data. Fig. 8 Simulation of LWD monopole waveforms for different crack density values and saturation conditions for water(black) and gas(red)Fig. 9 Simulation of LWD quadrupole waveforms for different crack density values and saturation conditions for water(black) and gas(red)

5 FIELD DATA EXAMPLE

The crossplot method of the compressional-to-shear velocity ratio(V_P/V_S)versus compressional-wave slowness is usually used in the fluid identification of the acoustic data interpretation. Tang et al.(2011)has used this method to interpret acoustic log data from tight s and and shale gas formations.

We use a field data of acoustic LWD(the tool only has the monopole acquisition mode)to compare with the theoretical modeling results above. Fig. 10 shows the LWD acoustic data of a horizontal well with a fast formation. We get a high quality data because of the successful construction. Track 1 shows the monopole acoustic waveform data(VD images). Track 2 shows the slowness-time coherence processed of the waveform. The data is respectively processed to give the compressional(DTP) and shear(DTS)slowness curves in track 3.

We use crossplot analysis with the slowness data in track 3 to interpret the theory(Fig. 11). The green data of Fig. 11 are from 3283 m to 3309 m. The lithologic character of this gas containing section is in common, with abundant cracks, but little changes of the crack density values. The Stoneley-wave of this section is attenuated. The variation of gas saturations lead to the different of V_P/V_S. The orange data of Fig. 11 are from 3321 m to 3324 m with high gas saturations. The severe changes of the compressional-wave slowness shows the crack variations of this section. The processed results are in agreement with the attenuation modeling of Fig. 2 and Fig. 3.

As an example, Fig. 12 shows LWD monopole acoustic waveforms of different gas saturations. We pick up three depths(a: 3308.30 m; b: 3295.35 m; c: 3286.51m)from the same lithologic character section(3283~3309 m). The red data of Fig. 11 shows that the gas saturation of these three depths increases, respectively. The characteristics of waveforms show that the gas saturation lowers the velocity and amplitude of P-wave while the S-wave velocity shows little changes. As we all know, the porosity of shale is usually below 10%. It is difficult to interpret the waveform characteristics above using the previous poroelastic wave theory. The result of Fig. 12 shows that the cracks cause the field-data waveform to be very sensitive to gas saturation which is in agreement with the cracked porous theory. This phenomenon can be used as a characteristic for the exploration and development of shale gas resources.

6 CONCLUSIONS

We have analyzed the characteristics of LWD multipole acoustic waves(Stoneley, flexural and quadrupole waves)for a cracked porous formation as a function of crack density and gas saturation. The conclusions are as follows:

(1)The velocity of LWD Stoneley waves in cracked-porous tight formation is more sensitive to crack density while the attenuation is more sensitive to gas saturation.

(2)The minimum of the group velocity of LWD dipole and quadrupole called the “Airy phase” is corresponding to the maximum of the attenuation. The later dispersive portion of waveform corresponding to the “Airy phase” is severely attenuated by gas saturation, which can be used as a characteristic for hydrocarbon detection from LWD wave data.

(3)The variation of the crack density or gas saturation can significantly change the P- and S-wave characteristics of waveforms. These wave characteristics can be utilized for hydrocarbon detection from LWD acoustic data.

ACKNOWLEDGMENTS

This research was supported by Natural Science Foundation of China(41204070, 41174088), 973 program(2014CB239006) and the Fundamental Research Funds for the Central Universities(14CX06076A).

APPENDIX A

We describe here the detail components of in Eq.(18). To make the formula more simple, we use the following combinations of Bessel functions:

The detail expressions for the elements of H are given as follows:

Where the other unlisted elements are zero; k is the axial wavenumber; ω is the angular frequency; I_n and K_n are the modified Bessel function of the first and second kind and order n; r_a, r_b and R are, respectively, the collar inner radius, collar outer radius and the borehole radius; v_f, v_p^t and v_s^t are, respectively, the borehole fluid velocity, collar P-wave and collar S-wave velocity; k_f = $\frac{w}{{{v}_{\text{f}}}}$, $k_{\text{P}}^{t}=\frac{w}{v_{\text{P}}^{t}}$ and $k_{\text{S}}^{t}=\frac{w}{v_{\text{S}}^{t}}$ are borehole, the collar compressional, and shear acoustic wavenumbers; k_fast, k_slow and k_shear are, respectively, wavenumbers of fast-, slow-compressional and shear waves of cracked porous formation; f = $\sqrt{{{k}^{2}}-k_{\text{f}}^{2}}$, p₁ = $\sqrt{{{k}^{2}}-{{\left( k_{\text{P}}^{t} \right)}^{2}}}$, and s₁ = $\sqrt{{{k}^{2}}-{{\left( k_{\text{S}}^{t} \right)}^{2}}}$ are borehole, the collar compressional, and shear acoustic radial wavenumbers; q_fast = $\sqrt{{{k}^{2}}-k_{\text{fast}}^{2}}$, q_slow = $\sqrt{{{k}^{2}}-k_{\text{slow}}^{2}}$ and q_s = $\sqrt{{{k}^{2}}-k_{\text{shear}}^{2}}$ are, respectively, acoustic radial wavenumbers of fast-, slow-compressional and shear waves of cracked porous formation; μ^t = ρt ·${{\left( v_{\text{S}}^{t} \right)}^{2}}$ is the collar shear modulus, where ρt is the density of the collar; λ, μ are the lame constants of cracked porous formation.