Fracture density inversion from a physical geological model using azimuthal AVO with optimal basisfunctionsIsabel Varela∗, University of Edinburgh, Sonja Maultzsch, TOTAL E&P, Mark Chapman and Xiang-Yang Li, BritishGeological Survey

SUMMARY

We invert for fracture properties using a model-based inversionmethod. We calculate orthogonal optimal basis functions forazimuthal AVO by the singular value decomposition of a groupof modelled realizaitons of the reflection coefficient. The mod-elled reflection coefficients are computed for different fracturedensities and a level of uncertainty in the velocities and densi-ties used.We apply the method to surface seismic recorded from alaboratory-scale physical geological model. This approach hasthe advantage of knowing the physical properties and the struc-ture of the subsurface under study while using data acquiredwith techniques for field 3D surface seismic. The inversionworks well within a certain offset range. A more accurate re-sult is obtained when less uncertainty (smaller standard devia-tions) are used in the velocities and densities of the modelledreflection coefficients.

INTRODUCTION

Over the last two decades many studies have been publishedboth on fracture characterization from seismic data (surfaceseismic and VSP) and on theories that describe the effect offractures in wave propagation. However very few studies havebeen published on fracture studies from physical earth models(Wang et al., 2007). The geological model and seismic dataset used here were acquired by the Geophysical Key Lab fromChina University of Petroleum.

Azimuthal AVO (AVOZ) has become an important tool in thestudy of fractures, particularly through linearized approxima-tions (Rüger, 1997) that relate the azimuthal variation in theP-wave AVO gradient to the anisotropic parameters γ and δ .Furthermore, Bakulin et al. (2000) substitute the relationshipbetween the anisotropic parameters and fracture density intothe approximation by Rüger (1997), thus solving for the AVOZresponse as a function of fracture density. A limitation of theseapproximations is that they are valid for angles of incidencesmaller than the critical angle. The inversion method presentedhere is based on modelling of the exact AVOZ response thusovercoming the far offset limitations, with the compromise ofrequiring prior knowledge of the velocities and densities of thebackground rock.

BACKGROUND

The works that have most influenced our approach are thoseof Causse and Hokstad (2005), Causse et al. (2007a), Causseet al. (2007b), Riede et al. (2005) and Saleh and de Bruin(2000). Causse and Hokstad (2005) developed a method to

Vp: 1470 m/s

Vp: 1870 m/s

Vp: 2648 m/s

Vp: 2560 m/s

Thickness: 10 m

Thickness: 430 m

Thickness: 492 m

Thickness: 690 m

Thickness: 602 m

Vs: 980 m/s

Vs: 1180 m/s

Vs: 1180 m/s

density: 1.10 gr/cc

density: 1.18 gr/cc

density: 1.45 gr/cc

density: 1.0 gr/cc

Water

Isotropic

Isotropic

Isotropic

Anisotropic

0

1

2

3

4

Vpx: 2960 m/s, Vs1: 2010m/sVpy:3642 m/s, Vs2: 1490m/s

Figure 1: Diagram of physical geological model (Figure modified from Wang et al. (2007)).The numbering on the right corresponds to the interface number.

improve the accuracy of the estimated AVO intercept and gra-dient by constructing a set of optimal basis functions throughsingular value decomposition. They developed the method in adifferent context, for corrections to non-hyperbolic travel-timeand applied it to AVO for a more precise calculation of inter-cept and gradient. Riede et al. (2005) and Causse et al. (2007a)used the same principles and applied them for classifying porefluids and lithological facies from AVO.

Our method uses optimal basis functions to describe the az-imuthal AVO (AVOZ) response. The principles of the methodwere presented by Varela et al. (2007) in a numerical study.Here we present the first application of this method to non-synthetic data.

METHOD: AVOZ WITH SVD FOR FRACTURE INVER-SION

The method uses the general approach that the reflection co-efficient versus offset and azimuth can be written as the sumof orthogonal basis functions multiplied by coefficients. For afixed angle of incidence i, we then have,

Ri(φ) = c1iF1(φ)+ c2iF2(φ)+ c3iF3(φ)+ . . . (1)

where φ is the azimuthal angle, F1, F2, F3 etc, are the basisfunctions and c1, c2, c3 etc, the multiplying coefficients. Theadvantage of this approach being that we can find a set of ba-sis functions such that the multiplying coefficients are relatedto the fracture properties at the interface and allow for theirinversion.

Furthermore, given an optimal set of basis functions we canget a good approximation of the reflection coefficient with just

Fracture inversion with optimal AVOZ

a few functions. When using the first three basis functions fora fixed angle of incidence i, expanding on the azimuthal angleand using a matrix form we then have,2666664

ri(1)ri(2)ri(3)

...ri(n)

3777775≈

2666664

F1(1) + F2(1) + F3(1)F1(2) + F2(2) + F3(2)F1(3) + F2(3) + F3(3)

......

...F1(n) + F2(n) + F3(n)

3777775

24

c1ic2ic3i

35

(2)or in a more compact form,

R≈ FC

Given the measured reflection coefficients for a wide range ofazimuths, a certain offset i, and the basis functions F we canthen calculate the coefficients c1, c2 and c3 with,

C ≈ (FT F)−1FT R (3)

Realizations of reflection coefficientsIn order to find the basis functions, we create a set of realiza-tions of the reflection coefficient under study. We give a nor-mal distribution to the the rock properties of the layers aboveand below the interface, draw random realizations from thesedistributions and calculate the reflection coefficient for differ-ent azimuths and angles of incidence. We repeat this step forseveral realizations and different values of fracture density. Wethen accommodate the resulting reflection coefficients in a ma-trix R̄ such that every row has a different azimuthal angle andevery column a different realization/angle of incidence.

Optimal basis functionsWe can then calculate the singular value decomposition on ma-trix R̄ (containing the modelled reflection coefficients) suchthat,

R̄ = FDV T (4)

Furthermore if we let C̄ = DV T then we have the desired de-composition such that,

R̄ = FC̄ (5)

where the basis functions F are then the normalized eigenvec-tors or R̄R̄T , and the coefficients from matrix C̄ are related tothe modelled fracture density.

We can now use the basis functions Fi to approximate the mea-sured reflection coefficients R by finding c1, c2 and c3 withequation 3. We can invert for fracture density from the rela-tionship between c1, c2 and c3 and C̄ -as C̄ holds a relationshipwith the modelled fracture density of the realizations in R̄.

THE LABORATORY-SCALE EARTH MODEL

The physical earth model consists of several horizontal isotropiclayers made of an Epoxylite material with a fractured layer inbetween them. The model was acquired at a scale of 1:10000∗,with the frequencies also scaled up by 10000:1. From here on-wards we will refer only to the scaled up dimensions of themodel.

∗1mm of the physical model is 10m

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Geophone locationsSource locationsLive geophones, azi>0Live sources, azi>0Live geophones, azi<0Live sources, azi<0

(b) Sorting by common offset

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Figure 2: Acquisition geometry of source and receiver locations. a)The black dots show theCMP’s of the supergather used, the blue lines connect the source and receivers for one CMP ofthe supergather. b) sorting by common-offset for an offset bin of 1760m. c) Azimuth versusoffset of the supergather. The red circles correspond to the CMP bin selected in a). d)Raypaths for the reflections at the top of the fractured layer for the supergather.

Figure 1 shows the layout and geometry of the model (on theXZ plane) with the numbers on the right referring to the inter-face or seismic reflection. The whole model is submerged in awater tank, under a 10 meter thick water layer where the sourceis shot. There are four layers, three of which are isotropic. TheP- and S-wave velocities, density and thickness of each layerare listed in Figure 1. We analyse the reflection from inter-face 2 between the isotropic layer 2 overlying the fracturedanisotropic layer 3.

The material of the anisotropic layer is a set of expoxy bondedfibre sheets that simulate vertical fractures with high azimuthalP-wave and S-wave anisotropy of 20%, and a fracture densityof 0.2. The elastic constants of this layer, measured by Wanget al. (2007), exhibit a weak orthorhombic symmetry and aregiven by,

CORTi j =

0BBBBB@

12.7040 7.8650 8.1990 0 0 07.8650 19.2330 9.3200 0 0 08.1990 9.3200 22.1620 0 0 0

0 0 0 5.8580 0 00 0 0 0 3.2990 00 0 0 0 0 3.2190

1CCCCCA

(6)

Data extraction and sortingWe work with a portion of the entire 3D seismic data set, asupergather with common midpoints (CMP’s) at the center ofthe model to maximize the offset and azimuth† spread of thedata used. Figure 2a shows all the CMP’s used. Figure 2cshows the offset azimuth spread for the supergather.

†The azimuths for Figure 2c and throughout this paper are measured as arctan( gy−sygx−sx ),

where (gx,gy), and (sx,sy) are the coordinates of the geophones and source locations respec-tively.

Fracture inversion with optimal AVOZ

We sort the data into common offset gathers grouping them ev-ery 50m from 160m to 1960m. The range of offsets for the su-pergather vary from 161m to 2008m. The source and receiversthat are live for a common offset bin of 1760m are shown inFigure 2b. For a more detailed description of the acquisitionprocess and parameters of this data set see Wang et al. (2007).

BUILDING ROCK MODELS A & B

We use the Chapman (2003) poroelastic theory to calculate theelastic constants of the fractured layer with different fracturedensities.

We create two groups of realizations of the reflection coeffi-cient, referred to as models A and B. For each group we varythe standard deviation of three parameters in each layer: P-wave velocity, S-wave velocity and density. We assign a nor-mal distribution to each one of these rock properties with themeans and standard deviations listed in Table 1. The mean foreach of these properties will remain the same in both models,and the standard deviations are smaller for Model B.

V p± std (m/s) V s± std (m/s) Density ±std (gr/cc)

Model A Top layer 2737.3±50 1002.8±50 1.15±0.05Fractured layer 4139.4±50 2058±50 1.447±0.05

Model B Top layer 2737.3±10 1002.8±10 1.15±0.01Fractured layer 4139.4±10 2058±10 1.447±0.01

Table 1: Mean and standard deviations for distribution of rock properties used to create real-izations of models A and B. The properties listed here are those of the unfractured rock.

Modelled reflection coefficientsThe two groups of reflection coefficients are calculated as fol-lows,

1) Let the modelled fracture densities have a uniform distri-bution starting at 0 until 0.3 in steps of 0.01, for a total of 30entries. 2) Let the angles of incidence have a uniform distribu-tion starting at 1o until 55o in steps of 1o, and the azimuths gofrom −180o every 1o until 180o.

3) For each fracture density draw a realization from the P-wave, S-wave and density distributions of the top and the bot-tom layers and calculate the resulting reflection coefficient.Repeat this step 20 times (drawing a different realization eachtime) for the same value of fracture density, azimuth and angleof incidence.

4) Accommodate the resulting reflection coefficients in a ma-trix R̄ such that every row has a different azimuthal angle andevery column a different realization/angle of incidence. Ma-trix R̄ has then 361 rows (equal to the number of azimuths) and33000 columns (number of angles of incidence (55)× numberof fracture densities (30) × number of realizations (20)).

The resulting realizations for three different angles of inci-dence are shown in Figure 3, where as the angles of inci-dence increase, the anisotropic effect from the fractures be-comes stronger.

OPTIMAL BASIS FUNCTIONS

We calculate the singular value decomposition of matrix R̄ forboth models A and B. The singular values for Model A are

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oo o

Figure 3: Modeled reflection coefficients using random variables from realizations drawnfrom Table 1 for Model A. Blue, cyan and red curves correspond to low, medium and highfracture densities respectively.

shown in Figure 4b. The resulting optimal basis functions formodel A are shown in Figure 4a, where it is obvious that asthe index of the basis function increases so does its frequency.The basis functions for Model B have the same amplitude asthose of Model A, but differ in the sign of F1 and F3.

AVOZ with new basis functionsFigure 4c shows the estimated fit of the measured reflectioncoefficient using two, three and nine basis functions at the lo-cations (azimuths) where there are measurements. Figure 4dshows the estimated reflection coefficients for all azimuths us-ing two (dashed line) and three (solid line) basis functions. Ad-ditionally it shows the expected reflection coefficient (calcu-lated using the elastic constants and physical properties givenby Wang et al., 2007) and the measured reflection coefficients.

In general, for offsets between 1000m (θ = 33o) and 1500m(θ = 45o) the fit using the optimal basis functions is very good.For offsets smaller than 1000m (θ < 33o) there are no consis-tent azimuthal changes in the measured reflection coefficientsand the fit becomes meaningless.

At offsets greater than 1500m (θ > 45o) the modelled reflec-tion coefficient reaches the critical angle predicting a strongincrease in amplitudes. Since in the measured reflection coeffi-cients we don’t observe the strongly marked amplitude changespredicted for angles of incidence greater than the critical angle,the data from this offset range should be taken carefully, andshould perhaps not be given as much weight.

FRACTURE DENSITY INVERSION

In order to invert for fracture density from the inverted coeffi-cients, we first find the coefficient from the modelled realiza-tions with the highest correlation to the fracture density mod-elled. We calculate the correlation coefficient for each of theranges of angles of incidence discussed above, as well as forall angles of incidence as shown in Table 2.

We find that for all the ranges considered C̄1 is the coefficientwith the highest correlation to fracture density (see Table 2).We thus base the inversion from the inverted coefficients tofracture density entirely on the relationship between c1 (fromthe data) and C̄1 (from the models).

Figures 4e and 4f show the maps of C̄1 versus angle of inci-dence and fracture density. By mapping the inverted values ofc1 for different offsets into C̄1 (also shown in figures 4e and4f) we can invert for fracture density.

Fracture inversion with optimal AVOZ

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Azimuth (deg)MeasuredApprox. with 2 functionsApprox. with 3 functionsApprox. with 9 functions

Two Fi, rms= 0.91462e-02Three Fi, rms=0.90046e-02MeasuredModelled

Fra

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Figure 4: a) Basis functions, b) singular values, c) estimated reflection coefficients with dif-ferent number of basis functions, d) estimated reflection coefficients with two and three basisfunctions compared to the modelled and measured reflection coefficients, e) and f) maps ofC̄1 versus angle of incidence and fracture density (the inverted C1’s are plotted on top). AllA), b), c), d), e) and f) correspond to Model A.

The inverted fracture density per common offset gather forboth models is shown in Figure 5. In general, the errorbarsbecome smaller as the offset increases. For the range of anglesof incidence of interest the mean of the inverted fracture den-sity becomes stable around 0.2. Nevertheless as the offsets getcloser to 1500m (close to the critical angle 45o) the invertedfracture density starts increasing, away from its true value.

For the range of offsets greater than 1500m the errorbars of theinverted fracture density are smaller for both models (and par-ticularly for Model B) as shown in figures 5a and b. However,the errors of the fit using the optimal basis functions increasesignificantly, as shown in Figure 5c.

As expected, the error of the fit decreases as the number offunctions used increases. This effect can be seen from figures5c and 5d where the errors in the fits using two, three, five andnine functions are shown.

Model A Model BRange θ C̄1 C̄2 C̄3 C̄1 C̄2 C̄3

all 0.435 -0.2897 -0.0718 -0.458 -0.293 0.0757θ < 33 0.299 -0.218 -0.0296 -0.3809 -0.275 -0.0494

33 > θ < 45 0.598 -0.0762 -0.0509 -0.6326 -0.047 0.013945 > θ 0.896 -0.7962 -0.1355 -0.9154 -0.8160 0.1574

Table 2: Correlation coefficient between C̄i’s and the modelled fracture density for models Aand B.

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Two Fi’’sThree Fi’sFive Fi’sNine Fi’s

Two Fi’’sThree Fi’sFive Fi’sNine Fi’s

Figure 5: a) Inverted fracture density for all offsets for Model A. b) same as A but for ModelB. c) RMS errors between measured and estimated reflection coefficients using two, three,five and nine basis functions. d) an expanded detail of b). The data is plotted for all measuredoffsets for reflection 2 (at the top of the fractured layer).

DISCUSSION AND CONCLUSIONS

The method for inverting fracture density from azimuthal AVOusing a set of optimal basis functions is applicable to seismicdata provided prior knowledge of the velocities and densitiesof the medium.

We compare the inversion for two different levels of uncer-tainty (standard deviations) in the velocities and densities ofthe modelled reflection coefficients. The inversion results aremore precise and accurate when less uncertainty is used in therock properties of the modelled reflection coefficients.

The inversion for fracture density of the physical geologicalmodel comes entirely from the coefficient that accompaniesthe function with the cos(2φ) behaviour. In contrast to Varelaet al. (2007) where the inversion is based on all three functions,and mainly on the cos(4φ) behaviour. This shows the versa-tility of the method, as in each application the inversion willbe based on the coefficient or coefficients that have the highestcorrelation with the modelled fracture density.

The data presents a distinct behaviour in three different off-set/angle of incidence ranges. At short offsets (θ < 33o) thereare no azimuthal variations and thus the fit and inversion be-come meaningless. For very large offsets (θ > 45o) the datareaches the critical angle, with the modelled amplitudes in-creasing strongly, unlike the measured ones. Due to this dis-crepancy the inversion from this range is given less weight. Forthe middle range of offsets (33o < θ < 45o) there are clear az-imuthal changes in the measured amplitudes, a good fit of theamplitudes with the orthogonal basis functions, and invertedfracture density values that can be validated from the geolog-ical model. This is the range for which, on this data set, theinversion is optimal. In general, due to the data behaviour inthese three ranges, the final inverted fracture density valuesshould be related to the offset/angle of incidence range theycorrespond to.

ACKNOWLEDGMENTS

We are grateful to the Geoscience Research Center of TOTALE & P for supporting the development of this project. To theEAP sponsors and the BGS University Funding Initiative forsupporting I. Varela’s graduate studies.

Fracture inversion with optimal AVOZ

REFERENCES

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Riede, M., E. Causse, A. Van Wijngaarden, A. Buland, J.Dutzer, and R. Fillon, 2005, Optimized AVO analysis byusing optimal linear approximation: 75th Annual Interna-tional Meeting, SEG, Expanded Abstracts, 222–225.

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