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It is well known that amplitude variations of seismic reflected waves and energy loss are usually associated with fluid-saturated porous rock. However, to quantify such effects and predict the seismic signature generated by fluid saturation is a challenge. This paper presents some study on evaluating seismic response from fluid-saturated porous permeable rock due to the conversion from Fast P wave to Slow P wave using asymptotic Biot’s solution. Since fluidsaturated layer also shows high dispersion, frequency dependency is also taken into account. The difference between 0 Hz calculation and 100 Hz calculation represents the difference between the Gassmann’s theory and Biot’s theory. It can be seen that for homogeneous reservoir, there is no big difference between them, while for heterogeneous reservoir the difference can be as high as 10% of the average amplitude. The heterogeneity may be with respect to porosity, permeability and fluid phase between reservoir layers.

reservoir using Biot’s model (1956). Silin and Goloshubin (2008, 2010) derived a low frequency asymptotic solution of Biot’s poroelasticity. For comparison, the asymptotic solution reduces the complexity of the calculations and provides a similar result at seismic frequencies relative to exact Biot’s solution. Figure 1 shows the attenuation coefficients for Fast P wave and Slow P wave as computed both by asymptotic formulas and Dutta and Ode’s computation in solving exact Biot’s model. It can be seen that in seismic frequency range the results are very similar. -3

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Introduction There are many publications for numerical calculations of seismic energy absorption based on Biot’s theory. In particular, Carcione et al. (2003) clearly demonstrated remarkable influence of heterogeneity of porous fluidsaturated rocks to attenuation. Liu et al. (2009) utilized the asymptotic solution on Biot’s theory based on propagator matrix method to calculate reflectivity from multi-layered medium including both Fast P wave and Slow P wave. It was demonstrated that for layer thickness higher than one meter, Slow P wave is hard to detect. However, the energy converted to Slow P wave still has a significant effect on attenuation. And we could assume that all the energy converted to Slow P wave diminishes within the rock layer, therefore, by re-arranging the propagator matrix to only incorporate Fast P wave we could calculate the seismic response for relatively thick poroelastic rocks. Since the reflection coefficients are calculated using the asymptotic Biot’s solution, energy transferred to Slow P wave has already been taken into account. Therefore, the equation of R+T=1 will no longer be satisfied, here R is the reflection coefficient and T is the transmission coefficient. Indeed, R+T1 meter) for Slow P wave to propagate and communicate with Fast P wave, therefore conversion to Slow P wave is counted as seismic absorptions. The reflection and transmission coefficients are all calculated using asymptotic solution of Biot’s theory, thus the portion of energy converts to Slow P wave has already been taken away through each layer.

The reflection and transmission coefficients in asymptotic solution are expressed as power series of the square root of a dimensionless parameter (Silin & Goloshubin, 2008, 2010):

ρ f κω , η where, ρf is fluid density, κ is permeability, η is fluid viscosity, ω is angular frequency, and i is the imaginary ε =i

unit. 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 incidence: 1+ i R FF = R0 + R1FF ε; 2 1+ i ε; T FF = 1 + R0 + T1FF 2 1+ i R FS = R1FS ε; 2 1+ i T FS = T1FS ε; 2 where R0 is the zero-order classical reflection coefficient, and R1FF and T1FF are the first-order reflection and transmission coefficients. Readers are referred to either Liu et al. (2009) or Silin and Goloshubin (2008, 2010) for a detailed description of asymptotic formulas. Propagator matrix method A simplification of the propagator matrix method used in Liu et al. (2009) has been done in this paper. According to the boundary condition in Figure 2, we can obtain the relationships between waveforms at interface j with the corresponding reflection and transmission coefficients. rff is used to represent the reflection coefficient of Fast P wave to Fast P wave while the incident wave is downgoing, and rffup is for 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

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dj(t) uj(t)

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

Figure 2: Schematic plot of wave propagation through layer j at normal incidence. The horizontal displacement corresponds to time delay.

The boundary condition can be written as:

⎧⎪d j+1 (t ) = tff j ⋅ A ⋅ d j (t − 1) + rffup j ⋅ u j +1 (t ) ⎨ −1 ⎪⎩ A ⋅ u j (t + 1) = rff ⋅ A ⋅ d j (t − 1) + tffup j ⋅ u j +1 (t ) where,

A=e

−α j h j

is the attenuation through layer j,

is the attenuation coefficient (m-1) and

h j is

αj

the layer

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

D j ( z ) = ∑ d j (t ) ⋅ z t

t =0 , n is the total number of samples in the time series of dj(t). And similarly define Uj(z) as the z-transform of uj(t).

Then, we can obtain for any interface j:

⎡ D j +1(s) ⎤ z −1 ⋅ [ M j ] 2×2 ⎢U ( s )⎥ = ⎣ j +1 ⎦ tffup j

⎡ D j ( s )⎤ , ⋅⎢ ⎥ ⎣U j ( s )⎦

where, [Mj] is the 2x2 propagator matrix that communicates the waveforms between layer j and j+1. Each matrix

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Seismic reflection study using asymptotic Biot’s theory element of [Mj] is also in z-transform, i.e., polynomials of z. Thus, for wave propagation in multi-layered media (Figure 3), we can obtain:

⎡ Dk +1(s)⎤ ⎢U (s)⎥ = ⎣ k +1 ⎦

z

−k

⎡ D ( s)⎤ ⋅∏ M j ⋅ ⎢ 0 ⎥ = C ⋅ Nk j =0 ⎣U 0 ( s) ⎦ k

k

∏ tffup

j

⎡ D ( s)⎤ ⋅ ⎢ 0 ⎥, ⎣U 0 ( s )⎦

j =0

where,

C=

z −k

k

and N k = ∏ M j

k

∏ tffup

j =0

j

j =0

and the multiplication between any two matrix elements in [Mj] is a convolution of their polynomial coefficients.

Source D0=1

U0

Layer 0

D1

U1

Layer 1

D2

U2

Layer 2

…… Layer k

Dk

Uk

Dk+1

Uk+1 Layer k+1

Figure 3: Schematic plot of wave propagation through multilayered media at normal incidence. D0 is downgoing Fast P wave in layer 0; U0 is upgoing Fast P wave in layer 0. We do not consider Slow P wave in this study and assume it completely attenuates before reaching the next layer.

Here we present our results of the matrix elements of Mj. For any layer other than the first layer (layer 0), the two by two matrix Mj can be expressed as:

M j (1,1) = (tff j ⋅ tffup j − rff j ⋅ rffup j ) ⋅ A ⋅ z 2 , M j (1,2) = rffup j ⋅ A−1 , M j ( 2,1) = − rff j ⋅ A ⋅ z 2 , M j ( 2,2) = 1 ⋅ A−1 .

M 0 (2,2) = 1 . Finally, by setting D0(z) = 1 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 taken away as seismic absorptions. Examples

Homogeneous vs. heterogeneous reservoirs In this example, we calculate the response from two fluidsaturated layered reservoir models. The original model (Model 1) contents porous permeable sandstones with total thickness being 22 meter and each layer thickness equal to 1 meter. Sandstones are saturated by gas and water. And there is variation in such rock properties as Kdry, udry, porosity and permeability. The parameters are derived from log data and laboratory measurements. Model 1 corresponds to a heterogeneous reservoir. Model 2 is a result of averaging Model 1 and it represents a homogeneous model of the reservoir. The reflectivity series from these two models are plotted for 100 Hz and 0 Hz as well as the difference between these two frequencies as a function of the total distance wave passed in Figure 4. Therefore, the reflection at 44 m (two way propagation) in the figure corresponds to the reflection from the reservoir bottom. It can be seen that for Model 1 the difference between 100 Hz and 0 Hz is substantial. The deviation can be as high as 10% of the average amplitude. However, such difference in Model 2 is much smaller. The deviation is only about 0.1% of the average amplitude, thus is negligible. This experiment tells us that for homogeneous reservoir Gassmann’s equation can satisfy the need in predicting the relationship between fluid and reflection response, however for fluid-saturated heterogeneous reservoir, Gassmann’s equation can lead to about 10% error in computing fluid effect on seismic response. And this error should be taken into account and try to reduce through Biot’s theory. Asymptotic solution of Biot’s theory is a good candidate in doing such a computation since it has simple form and all parameters can be estimated from log data and laboratory analysis. Quantification of the quality factor Q

And the matrix elements for M0 can be expressed as:

M 0 (1,1) = tff 0 ⋅ tffup0 − rff 0 ⋅ rffup0 , M 0 (1,2) = rffup0 , M 0 (2,1) = −rff 0 ,

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It can be estimated based on reflection amplitude from the reservoir zone bottom (AB at 44 meter in Figure 4) at different frequencies. We have estimated the Q factor for models 1 and 2 using the following equation:

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Seismic reflection study using asymptotic Biot’s theory

ln AB100 − ln AB0 1 = π ⋅ Δt ⋅ f Q

Gassmann’s theory can lead to about 10% error in computing fluid effect on seismic response. This error can be reduced by using asymptotic solution of Biot’s theory. It provides simpler form of solutions while still has reasonable accuracy compared with exact Biot’s theory at seismic frequencies. Therefore it may be suitable for replacing Gassmann’s calculation without increasing computational effort.

,

where, Δt is the total travel time and f is frequency that equals to 100 Hz. So indeed the Q factor calculated here is the Q factor at 100 Hz for models 1 and 2. The results are summarized in Table 2. AB (100Hz) 0.154701 0.133654

Model 1 Model 2

AB (0 Hz) 0.140708 0.133679

Q factor 33 16797

Seismic absorption by fluid oscillation has been evaluated and quantified by estimating their Q factor. For heterogeneous reservoir, the Q factor is realistic (Q=33) and it is much smaller than the Q factor for the homogeneous reservoir. It can be concluded that seismic absorption due to fluid flow is not negligible unless everything can be treated as homogeneous.

Table 2. Reflection amplitudes and computed Q factors.

The Q factor for model 1 is realistic (Q=33) and it is much smaller than the Q factor for the homogeneous model 2.

Acknowledgement

Conclusions

The work has been performed at the University of Houston and Lawrence Berkeley National Laboratory. It has been partially supported by CAGE consortium at the University of Houston and DOE Grant No. DE-FC26-04NT15503.

We have analyzed the reservoir response from two models using both Gassmann’s theory (0 Hz) and Biot’s theory (100 Hz). In case of heterogeneous reservoir the 0.3

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Figure 4: Reflectivity response of model 1 (heterogeneous reservoir) and model 2 (homogeneous reservoir) in terms of permeability and fluid saturation at 100 Hz and 0 Hz. The figures on the most right sides show the difference between 100 Hz response and 0 Hz response. It can be seen that for heterogeneous reservoir, the difference can be as high as 10% of the average amplitude, however for homogeneous reservoir, it is very small and can be considered as negligible.

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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 2010 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. Lowfrequency range: The Journal of the Acoustic al Society of America, 28, no. 2, 168–178, doi:10.1121/1.1908239. Carcione, J. M., H. B. Helle, and N. H. Pham, 2003, White’s model for wave propagation in partially saturated rocks: Comparison with poroe lastic numerical experiments: Geophysics, 68, 1389–1398, doi:10.1190/1.1598132. Dutta, N. C., and H. Ode, 1983, Seismic reflections from a gas-water contact: Geophysics, 48, 148–162, doi:10.1190/1.1441454. 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. Silin, D., and Gennady Goloshubin, 2010, An asymptotic model of seismic reflection from a permeable layer: Transport in Porous Media, on-line Sprigerlink.com

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