Transp Porous Med DOI 10.1007/s11242-010-9568-x
Seismic Reflection from an Interface Between an Elastic Solid and a Fractured Porous Medium with Partial Saturation Ashish Arora · S. K. Tomar
Received: 1 July 2009 / Accepted: 12 March 2010 © Springer Science+Business Media B.V. 2010
Abstract The theory of Tuncay and Corapcioglu (Transp Porous Media 23:237–258, 1996a) has been employed to investigate the possibility of plane wave propagation in a fractured porous medium containing two immiscible fluids. Solid phase of the porous medium is assumed to be linearly elastic, isotropic and the fractures are assumed to be distributed isotropically throughout the medium. It has been shown that there can exist four compressional waves and one rotational wave. The phase speeds of these waves are found to be affected by the presence of fractures, in general. Of the four compressional waves, one arises due to the presence of fractures in the medium and the remaining three are those encountered by Tuncay and Corapcioglu (J Appl Mech 64:313–319, 1997). Reflection and transmission phenomena at a plane interface between a uniform elastic half-space and a fractured porous half-space containing two immiscible fluids, are analyzed due to incidence of plane longitudinal/transverse wave from uniform elastic half-space. Variation of modulus of amplitude and energy ratios with the angle of incidence are computed numerically by taking the elastic half-space as granite and the fractured porous half-space as sandstone material containing non-viscous wetting and non-wetting fluid phases. The results obtained in case of porous half-space with fractures, are compared graphically with those in case of porous halfspace without fractures. It is found that the presence of fractures in the porous half-space do affect the reflection/transmission of waves, which is responsible for raising the reflection and lowering the transmission coefficients. Keywords Fractured porous medium · Immiscible fluids · Wave propagation · Reflection · Transmission · Amplitude and energy ratios
A. Arora Department of Mathematics, Kanya Mahavidyalya, Jalandhar, 144 001 Punjab, India e-mail: aroraashish−
[email protected] S. K. Tomar (B) Department of Mathematics, Panjab University, Chandigarh 160 014, India e-mail:
[email protected];
[email protected]
123
A. Arora, S. K. Tomar
1 Introduction The study of wave propagation in fluid saturated porous media is of great importance due to its applications in many fields. Biot (1956a,b) seems to be the first to formulate the theory of porous medium saturated by a single fluid. This theory predicts the existence of two compressional and one rotational waves in an infinite elastic porous medium saturated with a single fluid. The first compressional wave is analogous to the compressional wave of classical elasticity, while the second compressional wave is slower in speed than the first and highly attenuated. This slow compressional wave arises due to the presence of fluid in the pores. Using Biot’s theory of poroelasticity, many dynamical problems have been attempted in the past (e.g., Geertsma 1957; Geertsma and Smit 1961; Biot 1962, Deresiewicz and his coworkers 1960, 1962, 1963, 1964, 1967; Yew and Jogi 1976; Stoll 1981; Dutta and Ode 1983; Wu et al. 1990; de la Cruz et al. 1992; Tajuddin and Hussaini 2005; Lin et al. 2005, etc.). Many of the predictions of this theory, including the observation of the slow compressional wave, have been confirmed by both laboratory and field experiments (Berryman 1980, 1981; Plona 1980; Chin et al. 1985; Winkler 1985). Though this theory has theoretical as well as practical importance, it is always limited by an assumption that the porosity is homogeneous. Although this assumption is often applied to acoustic studies but heterogeneity of porosity exists naturally, in the form of fractures and cracks in the medium. For example, in case of huge dams (reservoirs) having fractured porous bed and walls, these fractures not only weaken the rock elastically, but also introduce a high permeability, e.g., they provide easy passage for fluid to flow out of the reservoir. Conceptual model of a fractured porous medium contains two parts—non-fractured porous blocks and interconnected network of fractures. The pores in the non-fractured porous blocks are considered homogeneous and referred as primary pores, while the fractures are those pores, which have opening greater than the primary pores and referred as secondary pores. Thus, due to the presence of fractures, there are two distinct pressure fields in the porous medium, one in the primary pores and the other in the secondary pores. For such a porous medium, double-porosity models are suitable, which is a three-phase system, i.e., solid phase, fluid phase in the pores, and fluid phase in the fractures, with fluid mass exchange between the pores and fractures. The two types of porosities are normally called storage and transport porosities. Storage porosity is of high volume but low permeability and holds most of the fluid underground, whereas the transport porosity is of low volume and high permeability. The transport porosity is treated as being in the form of fractures in the reservoir or joints in the rock mass. Barenblatt and Zheltov (1960a) and Warren and Root (1963) generalize the theory of poroelasticity by considering double porosity and dual permeability in the system. Their papers take explicit note of the fact that real reservoirs tend to be very heterogeneous in both their porosity and permeability characteristics. The theory of fractured porous media is based on the concept of double porosity and several problems in such media have been studied extensively by many researchers. Notable among these are Barenblatt et al. (1960b), Barenblatt (1963), Kazemi (1969), Shapiro (1987), Nilson and Lie (1990), Cho et al. (1991), Bai et al. (1993a,b), Berryman and Wang (1995), Tuncay and Corapciaglu (1995), Bai (1999), Lewallen and Wang (1998). Aifantis and his co-workers (Wilson and Aifantis 1982; Beskos and Aifantis 1986; Khaled et al. 1984) published interesting and important series of papers on saturated fractured porous media. One can note that the final set of governing equations is a direct generalization of Biot’s consolidation theory. Wilson and Aifantis (1984) studied the propagation of waves in a saturated fractured porous medium and showed the existence of an extra compressional wave (which appears due to the
123
Seismic Reflection from an Interface
presence of fractures in the medium), in addition to those encountered in Biot’s theory. Similar inferences were derived in some papers on fissured rocks (see Beskos 1989; Beskos et al. 1989a,b). Tuncay and Corapcioglu (1996a) extended their theory of porous media saturated by two immiscible Newtonian fluids to include the volume fraction of fractures present in the media. They derived macroscopic constitutive relations and mass momentum balance equations by volume averaging the corresponding microscale balance equations. In the absence of volume fractions of the fractures, the constitutive relations reduce to those obtained by Tuncay and Corapciaglu (1997) in the theory of porous media without fractures. Tuncay and Corapcioglu (1996b) studied the propagation of body waves in a fractured porous medium containing two immiscible fluids and showed that there may exist four compressional waves and one rotational wave. Of the four compressional waves, two are analogous to the fast and slow compressional waves of Biot’s theory. The remaining two compressional waves arise due to the presence of fractures and due to the pressure difference between the fluid phases in the porous block. They found that all the four compressional waves are dispersive and attenuated. At limitedly low frequency, all the compressional waves are found to disappear, except one, which is analogous to the fast compressional wave of Biot’s theory. The disappearance of three compressional waves at low frequency is because of their highly attenuated nature. For a specific model, they have also investigated the effects of frequency, saturation, and volume fraction of fractures on the phase speeds and attenuation coefficient of these body waves. They found that the phase speeds of one compressional wave and of transverse wave are not appreciably altered due to the presence of fractures in the medium. Berryman and Wang (2000) extended Biot’s theory of poroelasticity to incorporate the concept of fractures or cracks in the medium, in addition to the generalization to double porosity modeling done in their previous work (Berryman and Wang 1995). The problem of reflection and refraction of plane elastic waves striking at the plane interface between an elastic solid and a poroelastic solid saturated by a single fluid/two immiscible fluids have been attempted by Hajra and Mukhopadhyay (1982), Sharma and Gogna (1992), Tomar and Arora (2006), and Dai et al. (2006). Hajra and Mukhopadhyay (1982) followed Biot’s theory to obtain the amplitude and energy ratios when a plane elastic wave impinges at the interface between an impervious elastic solid and a porous solid saturated by a single fluid. They found that these ratios corresponding to various reflected and transmitted waves depend significantly on the angle of incidence. Sharma and Gogna (1992) extended their problem by taking viscous fluid in the porous half-space and studied the effect of viscosity of the fluid on various reflected and refracted waves, when a plane longitudinal/transverse wave is made incident at the interface. Tomar and Arora (2006) studied corresponding problem to a porous medium containing two immiscible Newtonian fluids using the theory of Tuncay and Corapciaglu (1997) and obtained the amplitude and energy ratios of various reflected and transmitted waves. Later, this problem was extended by Arora and Tomar (2008) by accounting the inertial coupling between fluid–fluid and solid–fluid already explained in the theory developed by Lo et al. (2005). Dai et al. (2006) have studied the reflection and transmission of elastic waves at a plane interface between an elastic solid and a double porosity medium. They discussed the effect of angle of incidence, frequency, fracture permeability, and porosity on the amplitude ratios of various reflected and transmitted waves when a plane longitudinal wave is made incident. They have also discussed the dispersion and attenuation of elastic waves propagating in a double porosity medium. In the present study, we have employed the theory given by Tuncay and Corapcioglu (1996a) to investigate the possibility of plane wave propagation in an infinite fractured
123
A. Arora, S. K. Tomar
porous medium containing two immiscible fluids. We have assumed that the fractures are distributed isotropically throughout the medium. The reflection and transmission coefficients are obtained when a plane longitudinal/transverse wave becomes incident obliquely at a plane interface between a uniform elastic solid half-space and a fractured porous solid half-space containing two immiscible fluids. These coefficients are then used to find the expressions of energy ratios of various reflected and transmitted waves at the interface. The problem is studied numerically by taking granite as an elastic half-space and sandstone as a fractured porous half-space containing non-viscous wetting and non-wetting fluid phases. The results obtained are compared graphically with those relevant to the corresponding problem in non-fractured porous half-space.
2 Field Equations and Wave Propagation In the theory of Tuncay and Corapcioglu (1996a), the presence of fractures in the porous medium can be thought of as that they contain wetting fluid, whereas the pores in the porous blocks are assumed to contain wetting and non-wetting fluids. Under the assumption of small deformations, the mass momentum balance equations in a fractured porous elastic medium containing two immiscible fluids, are given by ⎡ 1 ρs u¨ s ⎢ a11 + 3 G fr ⎢ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ ρ1 u¨ 1 ⎥ a21 ⎢ ⎢ ⎥ ⎢ ⎥ = ∇⎢ ⎢ ⎢ ⎥ ⎢ ⎢ ρ2 u¨ 2 ⎥ a31 ⎢ ⎢ ⎥ ⎢ ⎣ ⎦ ⎣ ρ f u¨ f a41 ⎡
⎤
⎡
1
1
1
a12 a13 a14 a22 a23 a24 a32 a33 a34 a42 a43 a44
1
⎤⎡
a14 αf a24 αf a34 αf a44 αf
∇ · (G fr ∇us )
⎥⎢ ⎢ ⎥⎢ ⎢ ⎢ ⎢ 0 −1 0 0 ⎥ ⎥ ⎢ c1 (v1 − vs ) ⎢ ⎥⎢ +⎢ ⎥⎢ ⎢ ⎢ 0 0 −1 0 ⎥ ⎢ c2 (v2 − vs ) ⎥⎢ ⎢ ⎦⎣ ⎣ 0 0 0 −1 c3 (v f − vs )
− − − −
⎤ ⎡ ∇ · us ⎤ a13 ⎢ ⎥ ⎢ ⎥ α2 ⎥ ⎥ ⎢ ∇ · u1 ⎥ ⎥ a23 ⎥ ⎢ ⎥ ⎥⎢ ⎢ ⎥ α2 ⎥ ⎥ ⎢ ∇ · u2 ⎥ ⎢ ⎥ a33 ⎥ ⎥ ⎥⎢ ⎢ ⎥ ⎥ α2 ⎥ ⎢ ⎥ ∇ · u f ⎥ ⎢ ⎦ a43 ⎣ ⎦ α2 M
⎤ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦
(1)
Here, the quantities having subscript s, 1, 2, and f refer to the solid phase, nonwetting phase, wetting phase in the pores in porous block, and the wetting fluid phase in the fractures, respectively. Superposed dotes denote the double temporal derivative and various other quantities used in the above matrix equation are given in AppendixA.I. It is assumed that the solid phase of the porous medium is linearly elastic and isotropic, while the fractures are isotropically distributed over the entire porous medium. Note that the expressions of various coefficients in (1) given in Tuncay and Corapcioglu (1996a) are misprint. We have re-visited them and correct forms are presented in the Appendix-A.I. The volume averaged stress in the porous solid skeleton, ts and pressures in the fluids pm , are given by
123
Seismic Reflection from an Interface
⎡
⎤
⎡
ts ⎢ a11 ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ ⎢ p1 ⎥ ⎢ a21 ⎥ ⎢ ⎢ ⎥=⎢ ⎢ ⎥ ⎢ ⎢ ⎢ p2 ⎥ ⎢ a31 ⎥ ⎢ ⎢ ⎦ ⎢ ⎣ ⎣ p f a41
a12 a13 a14 a22 a23 a24 a32 a33 a34 a42 a43 a44
a14 αf a24 αf a34 αf a44 αf
− − − −
⎤ ⎡ ∇ · us ⎤ a13 ⎡ ⎤ ⎢ ⎥ G fr × ⎢ ⎥ ⎥ α2 ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎥ a23 ⎥ ⎢ ∇ · u1 ⎥ ⎢ 0 ⎥ ⎥ ⎥⎢ ⎢ ⎢ ⎥ ⎥ ⎥ α2 ⎥ ⎢ ⎢ ⎥, ⎥ ⎢ ∇ · u2 ⎥ I + ⎢ ⎥ a33 ⎥ ⎢ 0 ⎥ ⎥ ⎥⎢ ⎢ ⎢ ⎥ ⎥ α2 ⎥ ⎥⎢∇ · uf ⎥ ⎣ ⎦ ⎢ ⎥ ⎦ a43 ⎣ ⎦ 0 α2 M
(2)
where I is a unit tensor matrix, = ∇us + (∇us )T − 23 ∇ · us I, ts = αs ts and pm = αm pm (m = 1, 2, f ). Introducing the scalar potentials φi and vector potentials i as u i = ∇φi + ∇ × i , ∇ · i = 0, (i = s, 1, 2, f )
(3)
and considering a plane wave propagating along a unit vector n with the phase speed V (= ω/k, ω and k being the circular frequency and the wavenumber, respectively), as {φi , i } = {Bi , Bi } exp{ık(n · r − V t)},
(4)
into Eq. 1, we obtain the following frequency equations Z1 X 4 + Z2 X 3 + Z3 X 2 + Z4 X + Z5 = 0
and
X 3 (Z 6 X + Z 7 ) = 0,
(5)
where X = V 2 and the various coefficients in these equations are given in Appendix-A.II. In (4), the vector r = x ˆi + y ˆj + z kˆ is a position vector, Bi is a scalar constant and Bi is a vector constant. The non-zero roots of equations contained in (5) will provide us the phase speeds of possible plane waves in the fractured porous medium. It can be seen from equations in (5) that there exist four compressional waves with phase speeds, say V1 , V2 , V3 , and V4 obtained from the four roots of Eq. 51 and one transverse wave propagating with phase speed V5 , obtained from the root of Eq. 52 . It has been shown by Tuncay and Corapcioglu (1996b) that the two of these compressional waves are analogous to the fast and slow compressional waves of Biot’s theory. Out of the remaining two compressional waves, one arises due to the presence of fractures and vanishes in the absence of fractures, whereas the other is associated with the pressure difference between the fluid phases in the porous blocks. Tuncay and Corapcioglu (1996b) have also shown that all the four compressional (P-waves) traveling in the fractured porous medium are dispersive and attenuated. In particular, the wave appearing due to the presence of fractures and the wave appearing due to the variation in capillary pressure at the fluid–fluid interface are highly attenuated and do not exist at low frequencies.
3 Reflection and Transmission Here, we shall discuss the reflection and transmission phenomena of plane elastic waves propagating through an elastic medium and striking at a plane interface separating an impervious elastic solid half-space and a fractured porous solid half-space containing two immiscible fluids. We consider that a train of incident plane wave (longitudinal/transverse) originate at infinity in the uniform elastic half-space and impinging at the interface z = 0 making an angle θ0 with the z-axis. The incident wave will give rise to two reflected waves (P- and SV-wave) in the uniform elastic half-space and five transmitted waves (four P-waves and one
123
A. Arora, S. K. Tomar
(P or SV, θ0) (P, θ1) (SV, θ2)
Elastic Half-Space
X Fractured Porous Half-Space
(SV, γ 3)
(P4, γf) (P2, γ1) Z
(P3, γ2)
(P1, γs )
Fig. 1 Rough sketch of the problem
SV-wave) in the porous elastic half-space. A schematic diagram of the problem in (x − z) plane is shown in Fig. 1. The boundary conditions at the interface z = 0 are given by (see Deresiewicz and Skalak 1963) (te )zz = ts zz + p1 + p2 + p f , (te )zx = ts zx , (u˙ e )x = (u˙ s )x , (u˙ e )z = (u˙ s )z , (u˙ s )z = (u˙ 1 )z , (u˙ s )z = (u˙ 2 )z , (u˙ s )z = (u˙ f )z ,
(6)
where the subscripts e and s, respectively, correspond to elastic half-space and porous halfspace, the quantities (u s )x , (u s )z , (u 1 )x , (u 1 )z , (u 2 )x , (u 2 )z , and (u f )z denote, respectively, the displacement components in porous solid phase, fluid phase in primary pores, fluid phase in secondary pores. The suffixes x and z denote their directions and over dot represents the temporal derivative. From relations in (3), the displacement components along x- and z-directions in the solid and fluid phases, are given by ∂φs ∂φs ∂ψs ∂ψs − , (u s )z = + , ∂x ∂z ∂z ∂x ∂ψi ∂ψm ∂φi ∂φm − , (u m )z = + . (u m )x = ∂x ∂z ∂z ∂x (u s )x =
(7) (8)
where ψs and ψm (m = 1, 2, f ) are the y-components of the vectors s and m , respectively. The vectors 1 , 2 , and f are related to the vector s through the following relations: 1,2 = 1,2 s ,
f = f s ,
(9)
ıc1,2 ıc3 and f = . ıc1,2 + ρ1,2 ω ıc3 + ρ f ω Relevant to the uniform elastic half-space, we borrow the stress–strain relation, the displacement components (u e )x , (u e )z and the wave equations given by Eqs. 25–27 of Arora and Tomar (2008). where 1,2 =
123
Seismic Reflection from an Interface
Now, we consider the following form of potentials for various waves in the two half-spaces: For an incident P/SV-wave propagating in the uniform elastic solid half-space,
φe ae0 e−ık0 αt (10) = exp {ık0 (x sin θ0 + z cos θo )}, ψe be0 e−ık0 βt
where k0 =
ke0 , ke1 ,
when P-wave is incident when SV-wave is incident,
where ae0 denotes the amplitude of an incident longitudinal wave (P), while be0 denotes the amplitude of an incident transverse wave (SV). For the reflected waves in the uniform elastic solid half-space, φe = ae1 exp{ıke0 (x sin θ1 − z cos θ1 − αt)},
(11)
ψe = be1 exp{ıke1 (x sin θ2 − z cos θ2 − βt)},
(12)
where ae1 and be1 denote the amplitudes of the reflected P and SV waves, respectively, ke0 (= ω/α) is the wavenumber corresponding to the incident P-wave and ke1 (= ω/β) is the wavenumber corresponding to the incident SV wave. The quantities α and β are the phase speeds of classical P and SV waves, respectively. For transmitted waves in the porous elastic half-space, φi = ai exp{ı[ki (x sin γi + z cos γi ) − ωt]},
(13)
ψs = a3 exp{ı[k3 (x sin γ3 + z cos γ3 ) − ωt]},
(14)
where as , a1 , a2 , a f , a3 ; ki (ks = ω/V1 , k1 = ω/V2 , k2 = ω/V3 , k f = ω/V4 , k3 = ω/V5 ) are, respectively, the amplitudes and wavenumbers of the four transmitted P-waves and one SV-wave. 3.1 Incidence of a Longitudinal Wave For a plane longitudinal wave incident at the interface, we insert the appropriate potentials from (10)–(14) into the boundary conditions (6), owing to (7)–(9), and the Snell’s law given by, sin γ f sin θ0 sin γ1 sin γ2 sin γ3 sin θ1 sin θ2 sin γs = = = = , = = = α α β V1 V2 V3 V4 V5 we obtain the following set of seven equations into seven unknowns bi j R j = qi , (i, j = 1, 2, ...7),
(15)
ae1 be1 and R2 = , respectively, represent the amplitude ratios corresponding to ae0 ae0 af as a1 a2 a3 the reflected P and SV waves, R3 = , R4 = , R5 = , R6 = , and R7 = ae0 ae0 ae0 ae0 ae0 represent that of corresponding to the transmitted four P and one SV waves, respectively. Expressions for the coefficients bi j and qi are given in Appendix-A.III. Following Achenbach (1973), the rate of transmission of energy per unit area denoted by Pe∗ at the interface z = 0 in the uniform elastic solid, is given by where R1 =
Pe∗ = (te )x z (u˙ e )x + (te )zz (u˙ e )z .
(16)
123
A. Arora, S. K. Tomar
In the fractured porous solid containing two fluids, the rate of transmission of energy per unit area denoted by Pp∗ at the interface z = 0 is given by Pp∗ = ts x z (u˙ s )x + ts zz (u˙ s )z + p1 (u˙ 1 )z + p2 (u˙ 2 )z + p f (u˙ f )z .
(17)
Plugging the appropriate potentials from (10)–(14) into the expressions given by (16) and (17), one can obtain the average energy transmission per unit surface area carrying along various waves. The energy ratios denoted by E i , (i = 1, 2, ...7) give the time rate of average energy transmission corresponding to the respective wave to that of the incident wave. The expressions of these energy ratios E i are given in Appendix-A.IV. 3.2 Incidence of a Transverse Wave In a similar way as in case of incident longitudinal wave, when a transverse wave is made incident at the plane interface, we can obtain a set of seven equations in seven unknowns. These equations can be written as bi j R j = qi . (18) ae1 be1 and R2 = denote the amplitude ratios corresponding to the reflected be0 be0 af as a1 a2 a3 SV and P waves, respectively, R3 = , R4 = , R5 = , R6 = , and R7 = be0 be0 be0 be0 be0 represent that of corresponding to the transmitted four P- and one SV- waves, respectively. The non-vanishing coefficients bi j and the expressions of quantities qi are given in Appendix-A.III. While the expressions of energy ratios E i , (i = 1, 2, . . . , 7) corresponding to the reflected P, reflected SV waves, four transmitted P-waves and one transmitted SV-wave are given, respectively, in Appendix-IV. To check the problem at a glance, we neglect the presence of fractures in the porous medium by taking α f = M = E 2 = 0 and K frm = K fr , one gets a41 = a42 = a43 = a44 = 0. In this case, the boundary condition corresponding to (u˙ s )z = (u˙ f )z is no more required and the remaining equations given by (15) and (18) reduce to those obtained by Tomar and Arora (2006) in the corresponding problem. Moreover, neglecting the presence of non-wetting fluid by taking S1 = A2 = 0, so that a12 = a22 = a32 = 0, one can see that the boundary condition corresponding to (u˙ s )z = (u˙ 1 )z at the interface z = 0, is no more required and the problem reduces to the corresponding problem in double porosity medium, earlier studied by Dai et al. (2006). Here R1 =
4 Numerical Example To study the effect of presence of fractures numerically on the amplitude and energy ratios, the material parameters in the two half-spaces are given in Table 1 and 2 . Here, we have considered the fluids in the porous medium to be non-viscous. Using the values of parameters given in Table 2, we have solved Eq. 5 to find the phase speeds of various waves propagating in the fractured porous medium. The phase speeds (in m/s) are found to be V1 = 41.072, V2 = 8.194, V3 = 2.116, V4 = 0.647, and V5 = 19.619. These phase speeds are used to find the amplitude ratios of various reflected and transmitted waves from Eqs. 15 and 18. Using matrix inversion method through a FORTRAN code, the variations of the modulus of various amplitude ratios with the angle of incidence are computed and shown graphically. These amplitude ratios have also been compared with those obtained by neglecting the fractures in the sandstone medium by taking α f = 0 and
123
Seismic Reflection from an Interface Table 1 Material parameters for elastic half-space (granite)
Parameter
Symbol
Value
Lamé parameter
λ
2.238 × 109 N/m2
Lamé parameter
µ
2.992 × 109 N/m2
Density
ρe
2650 Kg/m3
Velocity of longitudinal wave
α
5.57 × 103 m/s
Velocity of transverse wave
β
3.36 × 103 m/s
Table 2 Material parameters for fractured porous medium (sandstone) Parameter
Symbol
Value
Bulk modulus of fractured medium
K fr
1.44 × 109 N/m2
Bulk modulus of non-fractured blocks
K frm
2.1 × 109 N/m2
Bulk modulus of solid grains
Ks
35 × 109 N/m2
Bulk modulus of non-wetting fluid
K1
0.145 × 106 N/m2
Bulk modulus of wetting fluid
K2
2.25 × 109 N/m2
Shear modulus of solid matrix
G fr
1.02 × 109 N/m2
Density of solid grains
ρs
2650 Kg/m3
Density of non-wetting fluid
ρ1
1.1 Kg/m3
Density of wetting fluid in primary pores
ρ2
997 Kg/m3
Density of wetting fluid in secondary pores
ρf
997 Kg/m3
Volume fraction of solid phase
αs
0.87
Volume fraction of fractures
αf
0.03
Volume fraction of wetting fluid phase
α1
0.02
Volume fraction of non-wetting fluid phase
α2
0.18
A dimensionless material parameter
F
0.8
K fr = K frm = 1.44 × 109 N/m2 . Figures 2a–e depict the variation in modulus of amplitude ratios corresponding to various reflected and transmitted waves with the angle of incidence, when a longitudinal wave is made incident. Dotted curves in the Figs. 2b–d correspond to the variation in modulus of amplitude ratios corresponding to the transmitted P-waves in the porous medium, when the presence of fractures is neglected. Figure 2a shows the variation in the modulus of amplitude ratios corresponding to the reflected P and SV waves. It can be noticed that when the angle of incidence θ0 is less than 35◦ or greater than 73◦ , the amplitude ratio corresponding to the reflected P-wave dominates over the amplitude ratio corresponding to the SV-wave. Near the angle θ0 = 58◦ , the reflection of P-wave does not take place. While, in the range 35◦ ≤ θ0 ≤ 73◦ , the amplitude ratio corresponding to the reflected SV-wave dominates over the amplitude ratio corresponding to the reflected P-wave and no reflection of SV-wave takes place at normal as well as at grazing incidences. Amplitude ratios corresponding to four transmitted longitudinal waves are shown, respectively, by the curves labeled with P1, P2, P3, and P4 in Figs. 2b–e. It is noticed from these figures that the presence of fractures has significant effect on the amplitude ratios corresponding to the transmitted P1, P2, and P3 waves. No effect of presence of fractures is observed on the amplitude ratios corresponding to all the transmitted longitudinal waves at grazing incidence as these waves
123
A. Arora, S. K. Tomar
(a) 1.00
(d) 0.10 Transmitted P3
Reflected P
0.08
Amplitude ratios
Amplitude ratios
0.80 Reflected SV
0.60
0.40
0.20
0.06 -------- Non-fractured medium Fractured medium
0.04
0.02
0.00
0.00 0
10
20
30
40
50
60
70
80
90
0
10
20
30
40
50
60
70
80
90
80
90
Angle of incidence
Angle of incidence
(b) 0.20
(e)
0.10
Transmitted P1 Transmitted SV
0.08
Amplitude ratios
Amplitude ratios
0.16
0.12 -------- Non-fractured medium _____ Fractured medium
0.08
0.06
0.04
0.04
0.02
0.00
0.00
Transmitted P4
0
10
20
30
40
50
60
70
80
90
Angle of incidence
(c)
0
10
20
30
40
50
60
70
Angle of incidence
0.40
Amplitude ratios
Transmitted P2
0.30
0.20 -------- Non-fractured medium Fractured medium
0.10
0.00 0
10
20
30
40
50
60
70
80
90
Angle of incidence
Fig. 2 (Incident P-wave) Variation of a R1 and R2 , b R3 × 103 , c R4 × 104 , d R5 × 104 , e R6 × 105 and R7 × 103 with angle of incidence
disappear there. The effect of presence of fractures on the amplitude ratios corresponding to the P1, P2, and P3 waves is maximum at normal incidence and it goes on decreasing with the increase of the angle of incidence. In Fig. 2e, the wave corresponding to the curve P4 appears due to the presence of fractures and it disappears when the fractures in the porous medium
123
Seismic Reflection from an Interface
(a) 1.00
(d) 1.60 Transmitted P3 Transmitted P Transmitted SV
0.80
Energy ratios
Energy ratios
1.20 0.60
0.40
0.80 -------- Non-fractured medium Fractured medium
0.40 0.20
0.00
0.00 0
10
20
30
40
50
60
70
80
0
90
10
20
Angle of incidence
40
50
60
70
80
90
80
90
Angle of incidence
(b) 0.05
(e) 2.00
Transmitted P1
Transmitted SV
1.60
Energy ratios
0.04
Energy ratios
30
0.03
0.02
-------- Non-fractured medium ______ Fractured medium
1.20
0.80
0.40
0.01
Transmitted P4
0.00
0.00 0
10
20
30
40
50
60
70
80
90
Angle of incidence
(c) 0.16
0
10
20
30
40
50
60
70
Angle of incidence
Transmitted P2
Energy ratios
0.12
0.08 -------- Non-fractured medium Fractured medium
0.04
0.00 0
10
20
30
40
50
60
70
80
90
Angle of incidence
Fig. 3 (Incident P-wave) Variation of a E 1 and E 2 , b E 3 , c E 4 × 106 , d E 5 × 102 , e E 6 × 103 and E 7 × 102 with angle of incidence
are neglected. Amplitude ratio corresponding to the transmitted SV-wave attains maximum value near θ0 = 50◦ and it vanishes at normal as well as at grazing incidences. At grazing incidence, no transmission of waves is found to occur. The effect of presence of fractures in the porous medium is noticed on the amplitude ratios corresponding to the transmitted lon-
123
A. Arora, S. K. Tomar
(a)
(d) 0.12
2.5
Transmitted P3
Amplitude ratios
Amplitude ratios
2.0
Reflected P
1.5
1.0
0.08
-------- Non-fractured medium Fractured medium
0.04 0.5 Reflected SV
0.00
0.0 0
10
20
30
0
40
10
(b)
20
30
40
Angle of incidence
Angle of incidence
(e)
0.25
0.03 Transmitted SV
Transmitted P1
Amplitude ratios
Amplitude ratios
0.20
-------- Non-fractured medium ______ Fractured medium
0.15
0.10
0.02
Transmitted P4
0.01 0.05
0.00
0.00 0
10
20
30
40
Angle of incidence
(c)
10
20
30
40
Angle of incidence
0.50
Transmitted P2
0.40
Amplitude ratios
0
-------- Non-fractured medium ______ Fractured medium
0.30
0.20
0.10
0.00 0
10
20
30
40
Angle of incidence
Fig. 4 (Incident SV-wave) Variation of a R1 and R2 , b R3 × 103 , c R4 × 104 , d R5 × 104 , e R6 × 104 and R7 × 102 with angle of incidence
gitudinal waves only, while the amplitude ratio corresponding to the transmitted transverse wave is not influenced at all. This is obvious √ because the phase speed of the transverse wave obtained using Eq. 52 is given by V5 = −Z 7 /Z 6 ,√ which in the case of non-viscous fluids (i.e., when c1 = c2 = c3 = 0), reduces to V5 = G fr /ρs , the speed independent of the fracture parameters.
123
Seismic Reflection from an Interface
(a) 1.00
(d) Reflected SV
2.00
Reflectet P Transmitted P3
1.60
Energy ratios
Energy ratios
0.80
0.60
0.40
0.20
1.20
-------- Non-fractured medium Fractured medium
0.80
0.40
0.00
0.00 0
10
20
30
40
0
10
Angle of incidence
(e)
(b) 0.06
Energy ratios
Energy ratios
30
40
0.20
Transmitted P4
0.16
Transmitted P1
0.04
-------- Non-fractured medium Fractured medium
0.02
20
Angle of incidence
0.12
0.08 Transmitted SV
0.04
0.00
0.00 0
10
20
30
40
Angle of incidence
0
10
20
30
40
Angle of incidence
(c) 0.16 Transmitted P2
Energy ratios
0.12
-------- Non-fractured medium Fractured medium
0.08
0.04
0.00 0
10
20
30
40
Angle of incidence
Fig. 5 (Incident SV-wave) Variation of a E 1 and E 2 , b E 3 , c E 4 × 106 , d E 5 × 102 , e E 6 × 102 and E 7 with angle of incidence
Variations of the modulus of energy ratios corresponding to various reflected and transmitted waves against the angle of incidence, for longitudinal incident wave are shown in Figs. 3a–e. It is found that the pattern of variations of various energy ratios is similar to those
123
A. Arora, S. K. Tomar
of the corresponding amplitude ratios. This is obviously acceptable because the energy ratios are proportional to the square of the corresponding amplitude ratios. In the model containing fractured porous medium and model of porous medium containing no fracture, it has been verified that the energy balance relation is satisfied at the interface, i.e., the sum of the energy ratios is found to be unity showing that there is no dissipation of energy at the interface during transmission. In case of an incident transverse wave, Figs. 4a–e show the variation in the modulus of amplitude ratios corresponding to various reflected and transmitted waves. The angle θ0 = 37◦ is found to be the critical angle in this case. The amplitude ratio corresponding to the reflected SV-wave is found to be greater than that of corresponding to the reflected P-wave in the range 0◦ ≤ θ0 15◦ , while in the range 15◦ ≤ θ0 ≤ 37◦ , the amplitude ratio corresponding to the reflected P-wave dominates over the amplitude ratio corresponding to the reflected SV-wave. The presence of fracture is found to affect the amplitude ratios corresponding to the transmitted P1, P2, and P3 waves significantly here again, but contrary to what we had in case of incidence longitudinal wave. Here, the effect is minimum (almost zero) at normal incidence and it increases with increase of the angle of incidence. The curve labeled with transmitted P4 in Fig. 4e shows the variation in amplitude ratio corresponding to the wave appearing due to presence of fractures in the medium. This wave is not encountered in the porous medium without fractures and automatically disappears when the presence of fractures are neglected. It is found that at normal incidence, only SV-wave is transmitted. Figures 5a–e depict the variation in the modulus values of the energy ratios corresponding to various waves with the angle of incidence, when a transverse wave is made incident at the interface. Here again, the pattern of variations of various energy ratios is similar to those of the corresponding amplitude ratios. Presence of fractures has significant effect on the energy ratios corresponding to transmitted P1, P2, and P3 waves and the sum of energy ratios at the interface is found to be unity.
5 Conclusion A problem of wave propagation in the fractured porous medium containing two immiscible fluids has been studied. It has been observed that in an infinite fractured porous medium, there exist four compressional waves and one shear wave, traveling with distinct phase speeds as compared to three compressional waves and one shear wave in a porous medium without fractures. The phenomenon of reflection and transmission of plane waves has been studied when a plane elastic wave (longitudinal/transverse) after propagating through the uniform elastic half-space strikes at the interface between an elastic half-space and a fractured poroelastic half-space containing two immiscible fluids. When a plane elastic wave impinges at the interface, two waves are reflected back in an elastic medium (one longitudinal and one transverse), whereas four compressional waves (P1, P2, P3, P4) and one shear (SV) wave are transmitted in a fractured porous medium containing two fluids. Wave designated by P4 is new and not encountered in the non-fractured porous medium containing two immiscible fluids. It is observed that when a longitudinal or a transverse wave is made incident: (1) Fractures have a significant effect on the amplitude and energy ratios corresponding to the transmitted P1, P2, and P3 waves, though the effect is different at different angles of incidence. (2) The amplitude ratio corresponding to the transmitted transverse wave is not affected at all, by the presence of fractures.
123
Seismic Reflection from an Interface
(3) In the case of a longitudinal incident wave near an angle θ0 = 60◦ , only the transverse wave is found to reflect and no reflection of longitudinal wave takes place. (4) At grazing incidence of longitudinal wave, no transmission phenomena take place. (5) In case of normally incident transverse wave, only transverse wave is found to transmit. (6) It has been verified that the energy is not dissipated at the interface, as is expected in case of non-dissipative medium. Acknowledgment Authors are thankful to the Department of Science and Technology (DST), New Delhi, Government of India for providing financial aid under Grant No. SR/S4/MS:469/07 to complete this work
Appendix-A.I The symbols used in matrix Eq. 1 are given as follows:
c1 = a11 A3 =
a12 A3 =
a13 A3 =
a14 A3 =
a21 A3 = a22 A3 = a23 A3 = a24 A3 = a31 A3 =
α 2p S12 µ1
, c2 =
α 2p (1 − S12 )µ2
, c3 =
α 2f µ2
. K p kr 1 K p kr 2 Kf
T Ks E 3 α 2p αs K 1 K 2 (α f +E 2 K fr )+(−1+α f ) pc α1 α2 K fr +αs2 K 2 (−1+S1 )2 αs + αs K 2 (−1 + α f − E 1 K fr )(−1 + S1 )2 + α2 αs K 1 (−1 + α f + αs − E 1 K fr )S12 + α p (−1 + α f )αs2 K 1 K 2 + (−1 + α f )(α2 K 1 + α1 K 2 )K fr + αs α1 K 2 (α f + E 2 K fr ) pc (−1 + S1 )2 + K 1 (−1 + α f )K 2 (−1 + α f − E 1 K fr ) + α2 (α f + E 2 K fr ) pc S12 , T Ks (1 − α f )α p α1 E 3 K 1 (α2 α f pc S1 − E 1 K 2 K fr − α2 E 1 K fr pc S1 ) αs −α p α1 E 3 K 1 (α p E 2 K fr + α p )K 2 + α2 pc S1 (E 2 K fr + 1) ,
T α2 K 2 K s (α f − 1) E 1 E 3 K fr α p K 1 + α1 pc (1 − S1 ) − α 2p E 3 K 1 αs × (E 2 K fr + α f ) + 1 + α1 E 3 pc (−α p E 2 K fr + α 2f + αs − 1) + S1 α p (E 2 K fr + 1) − 1 , Tαf K s (1 − α f )E 3 K 2 (S1 − 1)(E 1 K fr + α p ) α p K 1 + α1 pc (1 − S1 ) αs −(1 − α f )E 3 K 1 S1 (E 1 K fr + α p )(α p K 2 + α2 pc S1 ) −(E 2 K fr + α f ) α p α1 E 3 pc K 2 (1 − 2S1 ) + α p (α1 K 2 + α2 K 1 )(E 3 pc S12 − αs ) + α 2p E 3 K 1 K 2 − α1 α2 αs pc , T K 1 (α p K 2 + pc S1 α2 )E 3 αs (E 2 K fr + 1)K s − (1 − α f )K fr , T α1 K 1 −αs (E 2 K fr + 1)E 3 K s (α p K 2 + α2 pc ) + E 3 pc K 2 (1 − S1 )2 × −α p (E 2 K fr + α f ) + (1 − α f )(−E 1 K fr − α p ) , T α2 K 2 K 1 α p E 3 K s (E 2 K fr + 1) + S1 (1 − S1 )E 3 pc α p (E 2 K fr + α f ) −(1 − α f )(−E 1 K fr − α p )αs α f , T α f K 1 (α p K 2 + α2 pc S1 ) E 3 K s (E 2 K fr + 1) − (E 2 K fr + α f )αs , T K 2 E 3 α p K 1 + α1 pc (1 − S1 ) −(1 − α f )K fr + αs (E 2 K fr + 1)K s ,
123
A. Arora, S. K. Tomar
a32 A3 = T α1 K 1 K 2 E 3 α p K s (E 2 K fr + 1) + S1 (1 − S1 ) pc α p (E 2 K fr + α f ) −(1 − α f )(−E 1 K fr − α p ) , a33 A3 = T α2 K 2 E 3 K s (E 2 K fr + 1)(α p K 1 + α1 pc ) − S12 pc K 1 α p (E 2 K fr + α f ) −(1 − α f )(−E 1 K fr − α p ) , a34 A3 = T α f K 2 (1 − S1 )α1 E 3 K s pc + α p K 1 (E 3 K s − αs ) (E 2 K fr + 1) −(1 − S1 )α1 αs pc (E 2 K fr + α f ) ,
a41 A3 = T E 3 α 2p K 1 K 2 (−K fr + αs K s ) + K s pc α1 α2 K fr + αs2 K 2 (S1 − 1)2 + αs K 2 (−1 + α f − E 1 K fr )(S1 − 1)2 + α2 αs K 1 (−1 + α f + αs − E 1 K fr )S12 + α p αs2 K 1 K 2 K s + K fr α1 K 2 (K s − pc (S1 − 1)2 ) + α2 K 1 (K s − pc S12 ) + αs K s α1 K 2 pc (S1 − 1)2 + K 1 K 2 (−1 + α f − E 1 K fr ) + α2 pc S12 , a42 A3 = −T α1 K 1 K s (α p K 2 + α2 pc S1 )E 3 E 1 K fr , a43 A3 = −T α2 K 2 K s α p K 1 + α2 pc (1 − S1 ) E 3 E 1 K fr ,
E3 K s (−E 1 K fr − α p ) (1 − S1 ) α p K 1 + α1 pc (1 − S1 ) K 2 a44 A3 = T α f αs − 1 − αs Ks + S1 (α p K 2 + pc α2 S1 )K 1 − (α1 K 2 + α2 K 1 )α p (E 3 pc S12 − αs ) αs + α p E 3 K 2 α1 pc (1 − 2S2 ) + α p K 1 − pc αs α1 α2 , A3 = T (α1 K 2 + α2 K 1 ) −α p E 3 K s (E 2 K fr + 1) + E 3 pc S12 −α f (1 + αs ) + α p E 2 K fr −α f E 1 K fr +(α f − 1) α p K 1 K 2 E 3 (αs − E 1 K fr ) − α1 E 3 pc K 2 (E 1 K fr − αs ) +α2 αs K 1 E 3 pc S12 + (1 − 2S1 ) −α f α1 K 2 E 3 pc (1 + αs ) +α1 K 2 E 3 pc (α p E 2 K fr + 1) + (E 2 K fr + 1) α2 E 3 pc (K 1 S12 − α1 K s ) + S1 (S1 − 2) α1 αs K 2 E 3 pc (α f − 1) + α1 K 2 K fr E 1 E 3 pc +α p K 1 K 2 E 3 (α 2f − 2α f + 2) + α1 E 3 pc S1 K 2 (2α f E 1 K fr + S1 ) +α 2p K 1 K 2 K fr E 2 E 3 , −1 α p = 1 − αs − α f , T = α1 α2 α f αs K 1 K 2 K s K fr ,
1 − αf 1 − αf 1 1 αs αs2 E1 = . − , E = − , E = F − 2 3 Ks K frm K frm K fr Ks K fr K s is the bulk modulus of solid grains; K fr and K frm are the bulk moduli of the fractured and non-fractured porous media, respectively; αi is the volume fraction of phase i; S1 is the saturation of the non-wetting fluid phase; K 1 , K 2 , and K 3 are the bulk moduli of non-wetting fluid, (P (S ) = p ∗ − p ∗ , p ∗ wetting fluid, and wetting fluid in the fractures, respectively; Pcap cap 1 1 2 i is the intrinsic averaged pressures of fluid phase i) is the derivative of the capillary pressuresaturation relation with respect to S1 ; G fr is the shear modulus of the solid matrix; and F is a material property associated with the changes in volume fraction of fractures. The quantity ∂M M is given by = R(P f − P2 ) with P2 and P f as the pressures in the respective fluid ∂t ∂M phases. We note that ρ2 is equal to the mass transfer rate of wetting fluid phase between ∂t the primary pores and secondary pores. K p is the intrinsic permeability of the non-fractured
123
Seismic Reflection from an Interface
porous medium whereas K f is that of the fractures. The quantities kri and µi are the relative permeability and viscosity of the fluid phase i, respectively. The quantities ui , vi , and ρi are the displacement vectors, velocity vectors, and averaged densities of the phase i, respectively. Sn = αn /α p , (n = 1, 2) is the saturation parameters with S1 + S2 = 1. Note that the symbols E 1 , E 2 and E 3 used under this Appendix have nothing to do with energy ratios.
Appendix-A.II Re(Z 1 ) = −[c1 c2 {ρ1 + ρ2 + ρs } + c2 c3 ρ1 {ρ2 + ρ f + ρs } + c1 c3 ρ2 {ρ1 + ρ f + ρs } − ρ1 ρ2 ρ f ρs ω2 ]/ω2 , Im(Z 1 ) = [(c1 + c2 + c3 ){ρ1 ρ2 ρ f ω2 } + ω2 {c1 ρ2 ρ f ρs + c2 ρ1 ρ f ρs + c3 ρ1 ρ2 ρs } − c1 c2 c3 {ρ1 + ρ2 + ρ f + ρs }]/ω3 , ∗ Re(Z 2 ) = [c1 c2 [ρ f {a22 + a33 + 2(a12 + a13 + a23 ) + a11 }
∗ + (ρ1 +ρ2 +ρ f )a44 ]+c2 c3 [ρ1 {a33 +a44 +2(a13 +a14 +a34 )+a11 } ∗ + (ρ2 +ρ f +ρs )a22 ]+c3 c1 [ρ2 {a22 +a44 +2(a12 +a14 +a24 )+a11 }
+ (ρ1 + ρ f + ρs )a33 ] − ω2 [ρs (ρ1 ρ2 a44 ∗ + ρ1 ρ f a33 + ρ2 ρ f a22 ) + a11 ρ1 ρ2 ρ f ]]/ω2 ,
∗ Im(Z 2 ) = [c1 c2 c3 {a22 + a33 + a44 + 2(a12 + a13 + a14 + a23 + a24 + a34 ) + a11 }
− ω2 {(c1 + c2 + c3 )(ρ1 ρ2 a44 + ρ1 ρ f a33 + ρ2 ρ f a22 ) + (2c1 ρ f ρ2 a12 + 2c2 ρ1 ρ f a13 + 2c3 ρ1 ρ2 a14 + c1 ρs (ρ2 a44 + ρ f a33 ) + c2 ρs (ρ1 a44 + ρ f a22 ) + c3 ρs (ρ1 a33 + ρ2 a22 )) ∗ + a11 (c1 ρ2 ρ f + c2 ρ1 ρ f + c3 ρ1 ρ2 )}]/ω3 ,
Re(Z 3 ) = [c1 c2 {2a14 a24 + 2a24 a34 + 2a14 a34 − 2a23 a44 − 2a12 a44 − 2a13 a44 2 2 2 ∗ + a14 + a24 + a34 − a22 a44 − a33 a44 − a11 a44 }
+ c2 c3 {2a12 a24 + 2a12 a23 + 2a23 a24 − 2a22 a34 − 2a13 a22 − 2a14 a22 2 2 2 ∗ + a12 + a32 + a42 − a22 a33 − a22 a44 − a22 a11 } + c1 c3 {2a13 a23 2 2 2 + 2a23 a34 + 2a13 a34 − 2a24 a33 − 2a12 a33 − 2a14 a33 + a13 + a23 + a43
∗ 2 2 2 − a22 a33 − a33 a44 − a11 a33 } − ω2 {ρ2 ρ f a12 + ρ1 ρ f a13 + ρ1 ρ2 a14 2 2 2 + ρs (ρ1 a34 +ρ2 a24 +ρ f a32 − ρ1 a33 a44 − ρ2 a22 a44 − ρ f a22 a33 ) ∗ (ρ1 ρ2 a44 + ρ1 ρ f a33 + ρ2 ρ f a22 )}]/ω2 , − a11 2 2 ∗ 2 Im(Z 3 ) = [c1 {ρ1 (a33 a44 − a34 )+ρ2 (a22 a44 − a24 +a44 a11 − 2a14 a24 +2a12 a44 − a14 ) ∗ 2 2 2 + ρ f (a22 a33 +a21 a33 +a33 a11 − 2a13 a23 − a23 − a13 )+ρs (a33 a44 − a34 )}
2 ∗ 2 +a11 a44 − a13 a44 − a34 a41 +a34 a13 − a14 − a43 a14 ) + c2 {ρ1 (a33 a44 − a34
2 ∗ 2 2 + ρ2 (a22 a44 −a24 )+ρ f (a22 a33 +2a22 a13 +a22 a11 −2a12 a23 −a23 −a12 )
2 ∗ 2 2 + ρs (a22 a14 −a24 )}+c3 {ρ1 (a33 a44 +a33 a11 +2a33 a14 −2a34 a13 −a34 −a13 ) ∗ 2 2 + ρ2 (a22 a44 + a22 a11 + 2a22 a14 − 2a12 a24 − a24 − a12 ) 2 2 + ρ f (a22 a33 − a23 ) + ρs (a22 a33 − a23 )}]/ω,
123
A. Arora, S. K. Tomar 2 ∗ ∗ 2 2 Re(Z 4 ) = ρ1 {a34 a11 − a33 a44 a11 + a44 a13 − 2a34 a41 a13 + a33 a14 }
2 ∗ ∗ 2 2 + ρ2 {a24 a11 − a22 a44 a11 + a44 a12 − 2a41 a24 a12 + a22 a14 } 2 2 2 + ρs {a34 a22 + a23 a44 + a24 a33 − a22 a33 a44 − 2a23 a34 a42 }
2 ∗ ∗ 2 2 + ρ f {a23 a11 − a22 a33 a11 + a33 a12 − 2a13 a23 a12 + a22 a13 },
2 2 2 2 ∗ a22 + a23 a44 + a24 a33 − a22 a33 a44 − a23 a34 a42 − a24 a32 a43 + a34 a11 Im(Z 4 ) = [c1 {a34 ∗ 2 − a33 a44 a11 + a34 a21 − a21 a33 a44 − a23 a34 a41 + a31 a23 a44 − a24 a43 a31
2 2 + a33 a41 a24 +a34 a12 −a33 a44 a12 −a34 a41 a13 +a44 a13 −a34 a42 a13 +a32 a44 a13 } 2 2 2 + c2 {a34 a22 + a23 a44 + a24 a33 − a22 a33 a44 − a23 a34 a42 − a24 a32 a43
∗ 2 ∗ 2 − a22 a44 a11 +a24 a11 +a44 a12 +a23 a44 a12 −a41 a24 a12 −a24 a43 a12 +a21 a32 a44 2 2 − a34 a42 a21 −a22 a31 a44 +a34 a41 a22 +a24 a31 +a32 a41 a24 +a24 a13 −a44 a12 a13 } 2 2 2 ∗ + c3 {a34 a22 +a23 a44 +a24 a33 −a22 a33 a44 −a32 a34 a42 −a24 a32 a43 −a22 a33 a11
2 ∗ 2 + a23 a11 +a33 a12 +a33 a24 a12 −a23 a31 a22 −a23 a24 a12 −a32 a24 a13 +a34 a22 a13 2 2 + a22 a13 − a32 a21 a13 −a21 a32 a43 +a21 a33 a42 +a22 a31 a43 −a22 a41 a33 +a23 a41
− a23 a31 a42 }]/ω, ∗ 2 2 2 (a22 a33 a44 −a34 a22 +a23 a34 a42 −a23 a44 +a24 a32 a43 −a24 a33 )−a12 (a21 a33 a44 Z 5 = a11 2 − a34 a21 +a23 a34 a41 −a31 a23 a44 +a24 a43 a31 −a33 a41 a24 )
+ a13 (a21 a32 a44 −a34 a42 a21 − a22 a31 a44 + a34 a41 a22 2 + a24 a31 − a32 a41 a24 ) + a14 (a21 a33 a42 2 + a22 a31 a43 + a23 a41 − a21 a32 a43 − a22 a41 a33 − a23 a31 a42 ),
Re(Z 6 ) = [ρ1 ρ2 ρ f ρs ω4 − (ρ1 c2 c3 + ρ2 c1 c3 + ρ f c1 c2 )ρs ω2 + (c1 + c2 + c3 )(c1 c2 c3 − (ρ2 ρ f c1 + ρ1 ρ f c2 + ρ1 ρ2 c3 )ω2 ) + (ρ2 c12 + ρ1 c22 )ρ f ω2 − c1 c2 c3 (c1 + c2 )]/ω4 , Im(Z 6 ) = [((ρ1 ρ2 c3 + ρ1 ρ f c2 + ρ2 ρ f c1 )ρs + ρ1 ρ2 ρ f (c1 + c2 + c3 ))ω + c12 (c2 ρ f + c3 ρ2 ) + c12 (c1 ρ f + c3 ρ1 ) − (c1 + c2 + c3 )(c1 c2 ρ f + c2 c3 ρ1 + c1 c3 ρ2 ) − c1 c2 c3 ρs ]/ω3 , Re(Z 7 ) = [(ρ1 c2 c3 + ρ2 c1 c3 + ρ f c1 c2 )G fr + (ρ1 ρ2 c32 − ρ1 ρ2 ρ f G fr )ω2 − c1 c2 c32 ]/ω2 , Im(Z 7 ) = [(c32 (c1 ρ2 + c2 ρ1 ) − (ρ1 ρ2 c3 + ρ1 ρ f c2 + ρ2 ρ f c1 )G fr )ω2 + c1 c2 c3 G fr ]/ω3 , ∗ = a + 4G . where a11 11 3 fr
Appendix-A.III
α2 λ =− − sin2 θ0 , + 2 cos2 θ0 , b12 = 2 sin θ0 µ β2
∗ a11 + a21 + a31 + a41 α 2 2G fr sin2 θ0 , = − 2 µ µ V1
b11 b13
123
Seismic Reflection from an Interface
b14 = b16 = b21 = b27 = b32 = b43 = b57 = b76 = q1 = = b11 = b13 b14 = b15 =
= b17
b23 =
= b31
= b41
a13 + a23 + a33 + a43 α 2 , µ V32
a14 + a24 + a34 + a44 α 2 2G fr α2 sin θ0 , b17 = − sin2 θ0 , 2 µ µ V4 V52 α2 α2 2G fr 2 sin 2θ0 , b22 = 2 − 2 sin θ0 , b23 = − sin2 θ0 , sin θ0 β µ V12 G fr α 2 2 − − 2 sin θ0 , b31 = b47 = sin θ0 , µ V52 α2 α2 2 − sin θ , b = b = − sin θ , b = − sin2 θ0 , b41 = cos θ0 , 0 33 42 0 37 β2 V52 α2 α2 2 b53 = b63 = b73 = − sin θ0 , b54 = − − sin2 θ0 , 2 V1 V22 α2 (1 − 1 ) sin θ0 , b65 = − − sin2 θ0 , b67 = (1 − 2 ) sin θ0 , V32 α2 − − sin2 θ0 , b77 = (1 − 3 ) sin θ0 , V42
λ 2 + 2 cos θ0 , q2 = sin 2θ0 , q3 = − sin θ0 , q4 = cos θ0 , q5 = q6 = q7 = 0. µ
2 λ β 2 − − 2 sin θ0 , b12 = sin 2θ0 , +2 µ α2 ∗
2 a11 + a21 + a31 + a41 β 2G fr − sin2 θ0 , µ µ V12
a12 + a22 + a32 + a42 β 2 , µ V22
a13 + a23 + a33 + a43 β 2 a14 + a24 + a34 + a44 β 2 , b = , 16 µ µ V32 V42 β2 G fr sin θ0 β 2 2 − sin θ , b = 2 sin θ − sin2 θ0 , b22 = cos 2θ0 , 0 0 21 µ α2 V52 β2 2G fr G fr β 2 2 2 − sin θ0 , b27 = − − 2 sin θ0 , sin θ0 µ µ V52 V12 β2 b47 = sin θ0 , b32 = cos θ0 , b33 = b42 = − sin θ0 , b37 = − sin2 θ0 , V52 β2 β2 2 − sin θ , b = b = b = b = − sin2 θ0 , 0 43 63 73 53 α2 V12 a12 + a22 + a32 + a42 µ
α2 , b15 = V22
123
A. Arora, S. K. Tomar
b54 b67
β2 β2 2 =− − sin θ , b = (1 −
) sin θ , b = − − sin2 θ0 , 0 1 0 57 65 V22 V32 β2 = (1 − 2 ) sin θ0 , b76 = − − sin2 θ0 , b77 = (1 − 3 ) sin θ0 . V42
q1 = sin 2θ0 ,
q2 = − cos 2θ0 ,
q3 = cos θ0 ,
q4 = sin θ0 ,
q5 = q6 = q7 = 0.
Appendix-A.IV E 1 = −R12 , E2 = E4 = E6 = E 1 = E 3 = E 4 = E 6 =
2 4G fr α2 β2 α a11 2 2 − sin θ0 , E 3 = R3 2 − sin2 θ0 , + 2 β 3µ µ V1 cos θ0 V12 a22 β 2 α2 a33 β 2 α2 2 2 2 R4 − sin θ0 , E 5 = R5 − sin2 θ0 , 2 2 2 µV2 cos θ0 V2 µV3 cos θ0 V32 2 a44 β 2 α2 G β α2 fr 2 2 R6 − sin2 θ0 , E 7 = R7 − sin2 θ0 , 2 2 2 µV4 cos θ0 V4 µV5 cos θ0 V52 1 β2 2 2 R1 − sin2 θ0 , E 2 = R 2 , cos θ0 α 2
β2 β2 4G fr 2 a11 R3 − sin2 θ0 , + µ 3µ V12 cos θ0 V12 β 2 a22 β2 β 2 a33 β2 2 2 2 R4 − sin θ0 , E 5 = R 5 − sin2 θ0 , 2 2 2 µV2 cos θ0 V2 µV3 cos θ0 V32 β2 β 2 a44 β2 β2 2 2 2 G fr R6 − sin θ0 , E 7 = R 7 − sin2 θ0 . 2 2 2 µ V5 cos θ0 V52 µV4 cos θ0 V4
−R22
1 cos θ0
References Achenbach, J.D.: Wave Propagation in Elastic Solids. North-Holland Publishing Co., Amsterdam (1973) Arora, A., Tomar, S.K.: The effect of inertial coupling on seismic reflection amplitudes. Geophys. Prospect. 56(5), 643–654 (2008) Bai, M.: On equivalence of dual-porosity poroelastic parameters. J. Geophys. Res. 104, 10461–10466 (1999) Bai, M., Elsworth, D., Roegiers, J.-C.: Modeling of naturally fractured reservoirs using deformation dependent flow mechanism. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 30, 1185–1191 (1993a) Bai, M., Elsworth, D., Roegiers, J.-C.: Multiporosity/multipermeability approach to the simulation of naturally fractured reservoirs. Water Resour. Res. 29, 1621–1633 (1993b) Barenblatt, G.I.: On certain boundary value problems for the equations of seepage of a liquid in fissured rocks. J. Appl. Math. Mech. 27, 513–518 (1963) Barenblatt, G.I., Zheltov, Yu.P.: Fundamental equations of filtration of homogeneous liquids in fissured rocks, Sov. Phys. Doklady 5, 522–525 (1960a) [English translation of: Doklady Akademii Nauk SSSR. 132, 545–548] Barenblatt, G.I., Zheltow, I.P., Kochina, T.N.: Basic concept in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech. 24, 1286–1303 (1960b) Berryman, J.G.: Confirmation of Biot’s theory. Appl. Phys. Lett. 37, 382–384 (1980)
123
Seismic Reflection from an Interface Berryman, J.G.: Elastic wave propagation in fluid saturated porous media. J. Acoust. Soc. Am. 69, 416–424 (1981) Berryman, J.G., Wang, H.F.: The elastic coefficients of double porosity models for fluid transport in jointed rock. J. Geophys. Res. 100, 24611–24628 (1995) Berryman, J.G., Wang, H.F.: Elastic wave propagation and attenuation in a double-porosity dual-permeability medium. Int. J. Rock Mech. Min. Sci. 37, 63–78 (2000) Beskos, D.E.: Dynamics of saturated rocks-I: equations of motion. J. Eng. Mech. ASCE 115, 983–995 (1989) Beskos, D.E., Aifantis, E.C.: On the theory of consolidation with double porosity-II. Int. J. Eng. Sci. 24, 1697–1716 (1986) Beskos, D.E., Vgenopoulou, I., Providakis, C.P.: Dynamics of saturated rocks-II: body forces. J. Eng. Mech. ASCE 115, 996–1016 (1989a) Beskos, D.E., Providakis, C.P., Woo, H.S.: Dynamics of saturated rocks-III: Rayleigh waves. J. Eng. Mech., ASCE 115, 1017–1034 (1989b) Biot, M.A.: General solutions of the equations of elasticity and consolidation for a porous material. J. Appl. Mech. 23, 91–95 (1956a) Biot, M.A.: Theory of propagation of elastic waves in a fluid saturated porous solid. J. Acoust. Soc. Am. 28, 158–191 (1956b) Biot, M.A.: Mechanics of deformation and acoustic wave propagation in porous media. J. Appl. Phys. 33, 1482–1498 (1962) Chin, R.C.Y., Berryman, J.G., Hedstrom, G.W.: Generalized ray expansion for pulse-propagation and attenuation in fluid-saturated porous media. Wave Motion 7, 43–65 (1985) Cho, T.F., Plesha, M.E., Haimson, B.C.: Continuum modeling of jointed porous rock. Int. J. Numer. Anal. Methods Geomech. 15, 333–353 (1991) Dai, Z.-J., Kuang, Z.-B., Zhao, S-X.: Reflection and transmission of elastic waves at interface between an elastic solid and a double porosity medium. Int. J. Rock Mech. Min. Sci. 43, 961–971 (2006) de la Cruz, V., Hube, J., Spanos, T.J.T.: Reflection and transmission of seismic waves at the boundaries of porous media. Wave Motion 16, 323–338 (1992) Deresiewicz, H.: The effect of boundaries on wave propagation in a liquid-filled porous solid-I. Bull. Seism. Soc. Am. 50, 599–607 (1960) Deresiewicz, H., Levy, A.: The effect of boundaries on wave propagation in a liquid-filled porous solid-II: Transmission through a stratified medium. Bull. Seism. Soc. Am. 57, 381–392 (1967) Deresiewicz, H., Rice, J.T.: The effect of boundaries on wave propagation in a liquid-filled porous solid-III: Reflection of plane waves at a free plane boundary. Bull. Seism. Soc. Am. 52, 595–626 (1962) Deresiewicz, H., Rice, J.T.: The effect of boundaries on wave propagation in a liquid-filled porous solid-V: Transmission across plane interface. Bull. Seism. Soc. Am. 54, 409–416 (1964) Deresiewicz, H., Skalak, R.: On uniqueness in dynamic poroelasticity. Bull. Seism. Soc. Am. 53, 783–789 (1963) Dutta, N.C., Ode, H.: Seismic reflection from a gas-water contact. Geophysics 48, 162–184 (1983) Geertsma, J.: The effect of fluid pressure decline on volumetric changes of porous rocks. Trans. AIME 210, 331–340 (1957) Geertsma, J., Smit, D.C.: Some aspects of elastic wave propagation in a fluid saturated porous solids. Geophysics 26(2), 169–181 (1961) Hajra, S., Mukhopadhyay, A.: Reflection and refraction of seismic waves incident obliquely at the boundary of a liquid-saturated porous solid. Bull. Seism. Soc. Am. 72, 1509–1533 (1982) Kazemi, H.: Pressure transient analysis of naturally fractured reservoirs with uniform fracture distribution. Soc. Petrol. Eng. J. 9, 451–462 (1969) Khaled, M.Y., Beskos, D.E., Aifantis, E.C.: On the theory of consolidation with double porosity-III: a finiteelement formulation. Int. J. Numer. Anal. Methods Geomech. 8, 101–123 (1984) Lewallen, K.T., Wang, H.F.: Consolidation of a double-porosity medium. Int. J. Solids Struct. 35, 4845–4867 (1998) Lin, C.-H., Lee, V.W., Trifunac, M.D.: The reflection of plane waves in a poroelastic half-space saturated with inviscid fluid. Soil Dyn. Earthq. Eng. 25(3), 205–223 (2005) Lo, W.-C., Sposito, G., Majer, E.: Wave propagation through elastic porous media containing two immiscible fluids, Water Resour. Res. 41, W02025 (2005). doi:10.1029/2004WR003162 Nilson, R.H., Lie, K.H.: Double-porosity modelling of oscillatory gas motion and contaminant transport in a fractured porous medium. Int. J. Numer. Anal. Methods Geomech. 14, 565–585 (1990) Plona, T.J.: Observation of a second bulk compressional wave in a porous medium at ultrasonic frequencies. Appl. Phys. Lett. 36, 259–261 (1980) Shapiro, A.M.: Transport equations for fractured porous media. In: Bear, J., Corapcioglu, M.Y. Advances in Transport Phenomena in Porous Media, pp. 406–471. Martinus Nijhoff, Dordrecht (1987)
123
A. Arora, S. K. Tomar Sharma, M.D., Gogna, M.L.: Reflection and refraction of plane harmonic waves at an interface between elastic solid and porous solid saturated by viscous liquid. Pure and Appl. Geophys. 138, 249–266 (1992) Stoll, R.D.: Reflection of acoustic waves at a water-sediment interface. J. Acoust. Soc. Am. 70, 149–156 (1981) Tajuddin, M., Hussaini, S.J.: Reflection of plane waves at boundaries of a liquid filled poroelastic half-space. J. Appl. Geophys. 58(1), 59–86 (2005) Tomar, S.K., Arora, A.: Reflection and transmission of elastic waves at an elastic/porous solid saturated by two immiscible fluids. Int. J. Solids Struct. 43, 1991–2013 (2006) [Erratum, ibid 44, 5796–5800 (2007)] Tuncay, K., Corapciaglu, M.Y.: Effective stress principle for saturated fractured porous media. Water Resour. Res. 31, 3103–3106 (1995) Tuncay, K., Corapcioglu, M.Y.: Wave propagation in fractured porous media. Transp. Porous Media 23, 237–258 (1996a) Tuncay, K., Corapcioglu, M.Y.: Body waves in fractured porous media saturated by two immiscible newtonian fluids. Transp. Porous Media 23, 259–273 (1996b) Tuncay, K., Corapciaglu, M.Y.: Wave propagation in poroelastic media saturated by two fluids. J. Appl. Mech. 64, 313–319 (1997) Warren, J.E., Root, P.J.: The behavior of naturally fractured reservoirs. Soc. Pet. Eng. J. 3, 245–255 (1963) Wilson, R.K., Aifantis, E.C.: On the theory of consolidation with double porosity. Int. J. Eng. Sci. 20, 1009–1035 (1982) Wilson, R.K., Aifantis, E.C.: A double porosity model for acoustic wave propagation in fractured porous rock. Int. J. Eng. Sci. 22, 1209–1217 (1984) Winkler, K.W.: Dispersion analysis of velocity and attenuation in Berea sandstone. J. Geophys. Res. 90, 6793–6800 (1985) Wu, K.Y., Xue, Q., Adler, L.: Reflection and transmission of elastic waves from a fluid-saturated porous solid boundary. J. Acoust. Soc. Am. 87, 2349–2358 (1990) Yew, C.H., Jogi, P.N.: Study of wave motion in fluid saturated porous rocks. J. Acoust. Soc. Am. 60, 2–8 (1976)
123
