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unit. A brief description of asymptotic calculation is summarized in the next section. Asymptotic calculation In case of Fast P incident wave, the reflection and transmission coefficients from Fast P wave to Fast P wave are denoted as RFF and TFF; the reflection and transmission coefficients from Fast P wave to Slow P wave are denoted as RFS and TFS. They have the following asymptotic forms for normal incident P wave: 1 i H; R FF R0 R1FF 2 1 i H; T FF 1 R0 T1FF 2 1 i H; R FS R1FS 2 1 i H; T FS T1FS 2 where R0 is the zero order classical reflection coefficient, and the first order reflection and transmission coefficients R1FF and T1FF have the form:

R1FF

Z 2 (T1FS R1FS ) ; T1FF Z1 Z 2

Z1 ( R1FS T1FS ) . Z1 Z 2

Here Z1 and Z2 are the acoustic impedance of medium 1 and medium 2 respectively. The first order reflection and transmission coefficients R1FS and T1FS have the form:

2

FS 1

R

A JM2 JE2 ; T1FS D JM2

2

A J M 1 J E1 , D J M1

here, subscript 1 and 2 indicates medium number. Other internal descriptions related to fluid and solid are:

Silin and Goloshubin (2008, 2009) carried out a low frequency asymptotic analysis of Biot’s poroelasticity. They used both fluid flow and scattering mechanisms to derive a frequency dependent reflection. In this case, the reflection and transmission coefficient are expressed as power series of the square root of a dimensionless parameter:

U NZ H i f , K where, Uf is fluid density, N is permeability, K is fluid viscosity, Z is angular frequency, and i is the imaginary

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JE A

D

§ 1 I ·¸ M ¨E fI ; JM ¨ K fg ¸¹ ©

1

(1 I ) K ; K sg

º 2Z Z ª J M1 JM2 1 2 » « ; 2 2 Z «¬ J M 1 J E 1 J M 2 J E 2 »¼ 1 Z 2

Z1 Z 2 J E 1J E 2 ª 1 1 Q b1 2 J M 1 J E1 « J M 1J M 2 «¬ J N J U 2 M 1 1 Q b2 J M 2 2 J E 2 º», J U1 M 2 ¼

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Asymptotic calculations of the Biot’s waves where,

Ef

H2 ; JU H1

4 Ub 1 ; M K P. ; JN 3 Uf Kf K g and Kg . ; K sg K fg 1I K 1 Kg

9 1F

Here Kf is the bulk modulus of fluid, Kg is the bulk modulus of solid grain, K is the dry rock bulk modulus, I is porosity, Uf is the fluid density, and Ub is the bulk density.

Ug g/cc

Pdry Gpa

Kdry Gpa

I \

N darcy

Kf Gpa

Uf g/cc

Kf cp

J E J M J E 2

2

§ J 2 J · E ¨ M JM ¸ . ¨ ¸ JU © ¹

Asymptotic solution also provides the reflection and transmission coefficients in case of Slow P wave as incident wave, where the reflection and transmission coefficients for converted Slow to Slow P wave are denoted as RSS and TSS, and the reflection and transmission for converted Slow to Fast P wave are denoted as RSF and TSF. They have the asymptotic forms below:

F 01S

A summary of all necessary input properties is demonstrated in Table 1. It can be seen that the input parameters are grain bulk modulus Kg, dry rock bulk modulus Kdry, dry rock shear modulus Pdry, fluid bulk modulus Kf, grain density Ug and fluid density Uf. These parameters are routinely used in fluid substitution technique based on Gassmann’s equation. The asymptotic description of the Biot’s model includes two additional parameters (rock permeability N and fluid viscosity Kf). Thus, it allows besides realization of the fluid substitution technique to provide an investigation of the influence of the permeability (fluid mobility) to seismic response. All input parameters can be acquired from log data and laboratory measurements. Hence, it makes the asymptotic description of the Biot’s model more practical for application. Kg Gpa

1

R SS

1

JN

F 01S

1

F 02S

1

F 01S

1

T SS

R SF T SF

JN JN JN

M 2 k02S [ 02S F 02S M 1k01S [ 01S ; M 2 k02S [ 02S F 02S M 1k01S [ 01S M 2 k 02S [ 02S F 01S M 1k 01S [ 01S ; M 2 k 02S [ 02S F 02S M 1k 01S [ 01S

Z 2 (1 R SS T SS ) Z1 Z 2

;

Z1 (1 R SS T SS ) Z1 Z 2

,

where,

k0S

1 vf

J E J M 2 ; F 0S

JM

JE ; [ 0S JM

1

JM

.

Table 1. Input parameters and units for asymptotic solution.

Furthermore, velocity of Fast P wave and Slow P wave, VF and VS; attenuation coefficients of Fast P wave and Slow P wave, aF and aS (in unit of m-1) can be calculated from:

J ; 1 2

V

F

VS aF

a

vb

vf

M

JE

2H

J E J M

Z

JE

vb

J E J M

S

2

Z vf

9 1F H; 29 0F

J E J M , 2H 2

where,

J

2

Qb

M

Ub

; Qf

M

Uf

; 9

F 0

M

Following the description of Robinson (1967), we demonstrate a normal incident Fast P wave and Slow P wave propagation through porous layered media using the Thomson (1950) and Haskell (1953) propagator matrix method. According to the boundary condition in Figure 1, we can obtain the relationships between all waveforms at interface j with the corresponding reflection and transmission coefficients. We use rff to represent the reflection coefficient of Fast P wave to Fast P wave while the incident wave is downgoing, and rffup to represent the reflection coefficient of Fast P wave to Fast P wave while the incident wave is upgoing. Thus, rff = uj(t+1)/dj(t-1) and rffup = dj+1(t)/uj+1(t). Similar denotations are used for other reflection and transmission coefficients.

;

2

Propagator matrix method

JE

J EJ U

;

The time delay for Fast P wave travel through any layer j is taken to be 1 unit of time. The time delay for Slow P wave travel through any layer j is taken to be W unit of time.

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Asymptotic calculations of the Biot’s waves Thus, W depends on the relative velocity of Fast P wave and Slow P wave, i.e., vfast j Wj . vslow j

It can be seen that for homogeneous fluid saturated media, there is almost no visible Slow P wave effect, while for inhomogeneous fluid saturated media, a significant Slow P wave effect exists. And this Slow P wave effect varies for different frequencies. A longer and stronger Slow P wave

Also, define Dj(z) as the z-transform of dj(t), i.e.,

dj(t), dj’(t)

n

¦d

D j ( z)

j

(t ) z t

Layer j

t 0

n is the total number of samples in the time series of dj(t). And similarly define Dj’(z), Uj(z), Uj’(z) as the z-transform of dj’(t), uj(t), uj’(t), respectively.

j

W j

z Mj tffup j tssup j tsfup j tfsup j

> @

ª D j ( z )º « ' » « D j ( z )» « », «U j ( z ) » «U ' ( z ) » ¬ j ¼

where, the four by four matrix [Mj] is the propagator matrix that communicates the waveforms between layer j and j+1. Each matrix element of [Mj] is also in z-transform, i.e., polynomials of z. Thus, for wave propagation in multilayered media (Figure 2), we can obtain: ª Dk 1 ( z )º « ' » « Dk 1 ( z )» «U k 1 ( z ) » « ' » ¬«U k 1 ( z ) ¼»

k

z

¦W j j 1

k

(tffup j 0

j

tssup j tsfup j tfsup j )

k

j 0

dj(t-1), dj’(t-Ĳ)

uj+1(t), uj+1’(t) Layer j+1

Then, we can obtain for any interface j: ª D j 1 ( z )º « ' » « D j 1 ( z )» «U ( z ) » « j 1 » «U ' ( z ) » ¬ j 1 ¼

uj(t), uj’(t)

dj(t) dj’(t) uj(t) uj’(t)

uj(t+1), uj’(t+Ĳ) dj+1(t), dj+1’(t)

downgoing Fast P wave in layer j downgoing Slow P wave in layer j upgoing Fast P wave in layer j upgoing Slow P wave in layer j

Figure 1: Schematic plot of wave propagation through layer j at normal incidence. The horizontal displacement corresponds to time delay.

ª D0 ( z ) º « ' » D0 ( z ) » , M j « «U 0 ( z ) » « ' » ¬«U 0 ( z ) ¼»

> @

and the multiplication between any two matrix elements in [Mj] is a convolution of their polynomial coefficients. By setting D0(z) = 1, D0’(z) = 0, Uk+1(z) = 0, and Uk+1’(z) = 0, we can obtain U0(z) as the reflectivity series of an impulse Fast P wave traveling through the multi-layered media, with mod conversion to Slow P wave and multiples taken into account. Example 1: Homogeneous vs. inhomogeneous fluid saturation

Figure 2: Schematic plot of wave propagation through multilayered media at normal incidence. D0 and D0’ are downgoing Fast and Slow P wave in layer 0, U0 and U0’ are upgoing Fast and Slow P wave in layer 0, respectively.

In this example, we calculate the response from two types of fluid saturated multi-layered media. Both media are porous permeable sandstone fully saturated by some fluid. However, one is only saturated by water, which corresponds to a homogeneous fluid saturation, the other one is alternatively saturated by gas and water (Table 2), which leads to inhomogeneous fluid saturation. The reflectivity series from these two types of media are plotted as a function of frequency (Figure 3 and Figure 4).

effect exists for lower frequency, and shorter and weaker Slow P wave effect exists for higher frequency. Thus, it can be expected in seismic profile some energy will appear below an inhomogeneous fluid saturated reservoir for low frequencies, but disappear for higher frequencies. Such phenomenon is similar to the low frequency shadows observed by instantaneous spectral analysis technique, demonstrated by Castagna et al. (2003). Hence, we think

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Asymptotic calculations of the Biot’s waves that Slow P wave may be a major cause for the low frequency shadows. Furthermore, it is well known that low frequency shadows are always associated with gas reservoir. We think the reason for this association is because that free gas in the gas reservoir will induce some degree of inhomogeneity by the form of gas bubbles, and consequently enhances the Slow P wave effect. This effect appears in a seismic profile like shadows beneath the reservoir zone at lower frequencies. At higher frequencies, since Slow P wave attenuates quickly, those shadows would disappear. We must confess that it is not adequate to draw a conclusion at this stage, and more evidence towards this postulate will need to be carried out. Example 2: Permeable vs. impermeable media Both models in Figure 3 are porous permeable media, the permeability were taken to be 2 darcy (Table 2). If the media becomes low permeable, Slow P wave effect is significantly weakened (Figure 4). Conclusions Dynamic modeling on multi-layered media was applied based on asymptotic description of Boit’ model and propagator matrix technique. The reflectivity series as a function of frequency for different types of fluid saturation and permeability are obtained. A strong Slow P wave effect is observed for low frequency, high permeability, and inhomogeneous fluid saturated media. Furthermore, due to the similarity between Slow P wave phenomenon and low frequency shadows observed by instantaneous spectral analysis technique, we think that Slow P wave may be the major cause for these shadows.

Figure 3: Reflectivity series vs. Frequency for an impulse Fast P wave travel through 7 layers of (a) homogeneous only water saturated media (with rock property change alternatively); (b) inhomogeneous gas, water alternatively saturated media (Table 2).

Acknowledgement The work has been performed at the University of Houston and Lawrence Berkeley National Laboratory. It has been partially supported by RQL consortium at the University of Houston and DOE Grant No. DE-FC26-04NT15503. The authors are thankful to Dr. John Castagna and Dr. Chris Liner for remarks and suggestions. Kg [Gpa] 38 38 38 …

Ug [g/cc] 2.65 2.65 2.65

Kdry [Gpa] 1.46 1.46 1.46 …

Pdry [Gpa] 1.56 1.56 1.56

Figure 4: Same as Figure 3 (b), only the permeability is changed to 0.5 darcy.

I 0.3 0.3 0.3 …

N [darcy] 2 2 2 …

Kf [Gpa] 0.025 2.42 0.025 …

Uf [g/cc] 0.15 1 0.15 ...

Kf [cp] 0.01 1 0.01 …

Table 2: Input parameters for the porous permeable inhomogeneous gas, water alternatively saturated media. One-way travel time for Fast P wave in each layer is 0.1 ms. Reflectivity results for 7 layers are plotted in Figure 3 (b).

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EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2008 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES Biot, M. A., 1956a, Theory of propagation of elastic waves in a fluid-saturated porous solid. 1. Low-frequency range: Journal of the Acoustical Society of America, 28, 168–178. ———, 1956b, Theory of propagation of elastic waves in a fluid-saturated porous solid. 2. Higher frequency range: Journal of the Acoustical Society of America, 28, 179–191. Carcione, J. M., H. B. Helle, and N. H. Pham, 2003, White’s model for wave propagation in partially saturated rocks: Comparison with poroelastic numerical experiments: Geophysics, 68, 1389–1398. Castagna, J. P., S. Sun, and R. W. Siegfried, 2003, Instantaneous spectral analysis: Detection of low-frequency shadows associated with hydrocarbons: The Leading Edge, 22, 120–27. Dutta, N. C., and H. Ode, 1979a, Attenuation and dispersion of compressional waves in fluid-filled porous rocks with partial gas saturation (White model)—Part 1: Biot theory: Geophysics, 44, 1777–1788. ———, 1979b, Attenuation and dispersion of compressional waves in fluid-filled porous rocks with partial gas saturation (White model)—Part 2: Results: Geophysics, 44, 1789–1805. ———, 1983, Seismic reflections from a gas-water contact: Geophysics, 48, 148–162. Haskell, N., 1953, The dispersion of surface waves in multilayered media: Bulletin of the Seismological Society of America, 43, 17–34. Robinson, E. A., 1967, Multichannel time series analysis with digital computer programs: Holden-Day. Silin, D., and G. Goloshubin, 2008, Seismic wave reflection from a permeable layer: Low-frequency asymptotic analysis: Proceedings of the International Mechanical Engineering Congress and Exposition. ———, 2009, A low-frequency asymptotic model of seismic reflection from a high-permeability layer: Lawrence Berkeley National Laboratory Report. Thomson, W., 1950, Transmission of elastic waves through a stratified solid medium: Journal of Applied Physics, 21, 89–93.

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