1* 0D-S Holding A/S mm A Company in the 0degaard & Danneskiold-Samsoe Group

Report 97.487 A/jEX- dK~ " <9x5'(?(5x

3D Seismic Inversion

Energistyrelsen Amaliegade 44 DK-1256 Kpbenhavn K Denmark

March 1997

iBsrammoN or tub document ts unlimith) Prepared by

Klaus Bolding Rasmussen 0D-S Holding A/S, Copenhagen, Denmark Jacob M0rch Pedersen 0D-S Holding A/S, Copenhagen, Denmark

Serge Gluck Compagnie Generale de Geophysique, Massy, France Emmanuelle Juve Compagnie Generale de Geophysique, Massy, France DISCLAIMER

Portions of this document may be illegible in electronic image products. Images are produced from the best available original document Contents Page Dansk sammendrag...... 3 1. 3-D seismic inversion...... 4 1.1 New products onthe world market...... 4 1.2 Seismic inversion...... 4 1.3 Applications of inversion...... 5 2. The Measurements...... 7 2.1 Seismic data...... 7 2.2 The well data...... 7 2.2.1 Deviated wells...... 8 3. ...... 8 3.1 Wavelet estimation...... 9 3.1.1 Wavelet methods...... 9 3.1.2 Deviated wells...... 12 4. The inversion...... 13 4.1 Method...... 13 4.2 Parameterisation...... 14 4.3 Objective function...... 14 4.4 Global search...... 16 4.5 The spectral content of the seismic data related to the inverted seismic traces 16 4.5.1 Initial model / a priori model...... 16 4.6 The results...... 19 4.7 Uncertainties...... 19 5. Reservoir description...... 21 6. The future...... 24 Acknowledgements...... 25 References...... 26

2 Dansk sammendrag

Denne rapport praesenterer resultateme af EF-93 forskningsprojektet "3-D inversion af seismiske data". Arbejdet er udfprt af 0D-S Holding A/S i samarbejde med det franske firma Compagnie Generale de Gdophysique (CGG). Projektet bar f&et stptte af EU under Joule- programmet, under hvilket det indgik i et stprre projekt udfprt af en raekke europaeiske universiteter og forskningscentre: Imperial College, London, UK (projektleder); Birkbeck College, London, UK; Hamburg Universitat, Hamborg, Tyskland; Institut Francais du Petrole, Paris, Frankrig; Observatorio Geoflsico Sperimentale di Trieste, Trieste, Italien samt olieselskabeme Norsk Hydro, Norge; BP, UK og Amoco, UK.

I projektet er der udviklet en ny 3D inversionsmetode, der benytter en global optimeringsteknik til at s0ge efter den bedste undergrundsmodel. Metodens succes skyldes denne avancerede spgestrategi kombineret med en unik parameterisering af inversionsproblemet. Parameteriseringen g0r, at det er muligt at anvende global optimering, uden at regnetiden for inversionsmetoden bliver urimelig lang.

Metoden er under projektforlpbet blevet kommercialiseret, og den bruges allerede som konsulentydelse. Denne konsulentydelse giver arbejde til et betydeligt antal geologer og geofysikere bos 0D-S Holding A/S i forbindelse med 3D inversion for olieselskaber i hele verden. Desuden er der foreg&et et software udviklingsarbejde under EFP-94. 0D-S forventer sig meget af 3D inversionsmarkedet fremover, bade hvad angSr konsulentarbejde og software. Af den grand Abnes der kontor i Houston, Texas ca. 1. maj 1997.

Det EFP-93 projekt, som er beskrevet i denne rapport og i rapporten "ISIS Global Inversion of 3-D Land Seismic Data", 0DS rapport 95.391 af Lars S. Hansen, bar sAledes vaeret bAde en videnskabelig og kommerciel succes, som vil danne basis for det videre geofysiske udviklingsarbejde i 0D-S Holding mange ar frem.

Jacob M0rch Pedersen, Projektleder 1. 3-D seismic inversion

1.1 New products on the world market

The research results from the 3D inversion have been made into products already available on the market. The new products are used both within the EU and in the rest of the world. The two partners in the 3D inversion project have launched two different products onto the market; 0D- S Holding A/S (0DS) has introduced ISIS and Compagnie Generale de Geophysique (CGG) has introduced TDROV. Both products use global optimisation in the 3D seismic inversion.

The two products have been very well received by oil companies, and global inversion is increasingly used for seismic inversion on the European market. There is also a significant worldwide export market. These two products are the only commercial products utilising global optimisation for seismic inversion, giving EU a significant advance internationally. 0DS is a small enterprise (SME) and the research results have already lead to several new jobs and the market is rapidly growing.

1.2 Seismic inversion

Seismic data is a very important contribution to oil and gas exploration and optimisation of oil and gas exploitation. Drillings give direct and detailed information about reservoir properties and content, but they are expensive and cover only a small area. Seismic data on the other hand is relatively inexpensive and covers a large area. The drawbacks are that seismic is a remote sensing with limited resolution and it is the changes in acoustic impedance that give rise to the signal, with acoustic impedance being only an indirect reservoir parameter. The present research project is focused on utilising the information in the seismic data and combing it with other sources of information such as the well log measurements from the bore holes.

When oil companies want to perform detailed stratigraphic interpretation of fully processed post stack seismic data, it becomes more and more natural to include seismic inversion and thereby transform the seismic data to acoustic impedance. The acoustic impedance field produced by these methods is meant to help stratigraphic interpretation and reservoir characterisation since: a. it is less oscillatory than the initial seismic data (impedance vs. reflectivity), b. it is more directly correlatable to well log data for lateral prediction of lithology, and fluid content, and c. it produces a high resolution layer framework whose strata are deformed in order to become conformable to seismic reflectors.

Since the transformation of seismic data to impedance is non-unique, some a priori information has to be introduced in order to sample only the acoustic impedance solutions which are geologically realistic. The seismic response of the impedance model is compared with the real

4 EH

seismic data in a volumetric fashion by measuring the error along the mobile impedance model interfaces. The resulting misfit is minimised in order to maximise a probability function of the impedance model parameters. By doing so, the impedance model interfaces tend to orient themselves conformably to the seismic reflectors, thereby averaging out the random seismic noise, and reinforcing laterally weak, but coherent events. This feature contributes to thin layer detection/resolution. More generally, 3D seismic amplitudes may be distorted as a result of poor 3D spatial sampling/binning and a possible low signal-to-noise ratio. These data problems are addressed through 3D inverse modelling coupled to sparse model parameterisation and relevant a priori information.

1.3 Applications of inversion

Inversion can be applied from the first seismic data are collected in the oil/gas exploration to exploitation with the seismic being a help describing the flow properties for better utilisation in a producing reservoir.

The advantages of utilising inversion is:

• High resolution of thin layers.

• Accurate determination of the physical properties and the variations in the subsurface.

A good example of the importance of seismic inversion is detection of sand channels in the Danish central graben. Above the chalk in the Tertiary sand channels is a new play in Denmark. A recent discovery of this type was the largest oil discovery made in Danish waters for 18 years. This type of reservoir is well-known in both the UK and Norwegian sectors. These channels can be difficult to resolve when the seismic data is interpreted, but show clearly as high impedance in the low impedance surroundings. See the figure.

5 Acoustic impedance at flattened horizon -40 ms.

300 400 900 600 Inline

6.600 7333 8.067 8.800 9533 10.270 11.000 Acoustic impedance, xIO6 [kg nr2 s_1]

Figure 1:

Acoustic impedance 40 ms above top chalk in the North Sea Danish sector. The yellow to red indicates high acoustic impedance. The structure seen is a sand channel. It goes from the Ringkjpbing-Fyn high towards the central graben. This inversion result shows how the inversion can help finding possible reservoirs. The inversion is performed with ISIS. ip ^km

2. The Measurements

The primary measurement for seismic inversion is the seismic data together with the data from the .

2.1 Seismic data

The seismic data is a wide spread observation covering a large area. However, as it is a remote observation of the acoustic and elastic properties, it has resolution limits. This resolution limit is specially in contrast to what is needed to describe reservoir properties for flow properties. The flow properties are needed to find the best production strategy for the reservoir.

The work performed in this project has only been dealing with post stack data. This means that the data has been processed to create an image of the subsurface. Multiples have (hopefully) been removed, redundancy in the measurement has been used to attenuate noise. The data has been migrated and shaped to create the best possible image. What has been lost in the processing is the elastic information. The processing has attempted to generate a seismogram of normal incidence (zero offset data) and the appropriate modelling of this data is the convolution model.

The assumption behind the 3D inversion is that the subsurface can be described with a sparse primary reflectivity model, the inversion algorithm requires 3D zero offset migrated seismic data from which multiples have been removed. In spite of the fact that long and oscillatory can be handled, we try to come up with short and near zero phase wavelets, in order to pick the initial model geometry more easily and keep the inversion process time at a minimum. In ISIS, the requirement for a certain phase of the seismic data has been removed so any phase wavelet can be estimated and used.

2.2 The well data

The well logs give very detailed measurements of relevant reservoir parameters. In the bore holes, a lot of different parameters are measured on cores, side cores, both with direct or indirect measurements. The measurements are generally more detailed and accurate than the seismic measurements, but are only covering a very small area.

Well log data is used to estimate wavelets and to tie the main time-horizons to the seismic data and for TDROV to estimate the average strata thickness within each macro interval, and to define the impedance extreme of each macro interval. In ISIS, the well logs are also used in a geostatical calculation determining how much of the low-frequency information that can reliably be introduced to the inversion itself.

If many wells are available, a priori model parameter could be made and used as such to sample the model space. For instance, within a given macro geological interval, histograms of the layer thickness and the layer impedance can be constructed and used later on

7 in the inversion to perturb more often the current model with model parameter values which are the more probable.

2.2.1 Deviated wells

The fact that wells are not vertical poses a number of other problems. The main problem is to position the wells correctly in the seismic data where the depth axis is given neither in metres nor in feet but in two-way travel-times. In practice, it is necessary to treat the deviated well carefully and correctly. Fordeviated wells, the well bore trajectory is integrated to the seismic data by using its coordinates (X,Y,Z,T) and tying it to the in situ seismic data. The geological dips may be taken into account beyond certain values for non-horizontal layering.

well log data ----- exponential tit-----

30 layer thickness

Figure 2:

Observation on layer thickness measured in ms compared with an exponential fit.

3. Wavelet

The wavelet is an important part of the seismic inversion. The wavelet is the bridge between the seismic data and the well logs. The wavelet is the mathematical filter that combines the seismic data with the physical measurements in the well. The wavelet is not the original source wavelet and it is not attempted to relate the used wavelet with the source wavelet as the changes taking place through the processing of the seismic data are difficult to describe. The primary method to obtain the wavelet is to correlate the seismic data with the well logs. Methods for estimating the wavelet directly from the seismic data are also available.

8 3.1 Wavelet estimation

The so-called convolutional model upon which the inversion is based requires a wavelet estimate. The wavelet is derived from an acoustic impedance log, as the filter which shapes the well log reflectivity series to the seismic traces at the well location. To do that, the logs must show a good correlation with the seismic data close to the well location. In the first place, some stretching/squeezing operation may be carried out in order to compensate for the offset and small deviations from the vertical. Also, a constant phase shift may be applied in order to compensate for the bulk of the wavelet phase shift without resorting to well logs which may be unreliable. In due course, a residual wavelet estimate may be obtained by shaping the reflectivity series to the seismic trace(s) at the well location. The well known shaping filter derivation is obtained from the seismic traces around the well location which correlate with a first estimate of synthetic seismogram. The derivation is carried out in the time domain where the correlation functions which intervene in the linear system tobe solved come from weighted time windows of seismic and well data.

For deviated wells, the same procedure applies, but the log data is projected to vertical and the seismic data are gathered along the well bore trajectory in order to obtain a corridor stack seismic trace.

3.1.1 Wavelet methods

Three different wavelet estimation methods are available in ISIS: In the least squares methods. the sum of the squared differences (the misfit) between the seismic trace and a synthetic trace, obtained by convolution of the well log reflectivity series and a wavelet, is minimised. This method is able to estimate the amplitude and the phase spectra of the wavelet. The method is based on the assumption that a reasonable tie between the well logs and the seismic data exists and therefore is sensitive to alignment errors between the reflectivity log and the seismic data, especially for the higher frequencies. The spectral wavelet estimation method sets the amplitude spectrum of the wavelet equal to the estimated amplitude spectrum of the seismic data. The phase spectrum of the wavelet is either assumed known, e.g. zero phase or minimum phase, or is estimated from the seismic data (Rasmussen, K. B. 1994). Given the amplitude and the phase spectrum, the wavelet itself can be computed using the inverse Fourier transform. The remaining gain (including the sign) and delay of the wavelet are computed by convolving the wavelet with either the noisy reflectivity log or the guessed reflection coefficient of a significant reflector in the seismic section, and finally comparing this synthetic trace with the seismic data. Finally, the two methods above can be combined: In the hybrid wavelet estimation method, the amplitude spectrum of the wavelet is set to equal the estimated amplitude spectrum of the seismic data. The phase spectrum of the wavelet is assumed constant. The constant phase, the gain (including the sign) and the delay of the wavelet are estimated by computing the values which minimise the misfit between the synthetic data and the seismic data in a least square sense. For all the three methods, a whole suite of wavelets is estimated. The optimal wavelet is found for a range of different wavelet lengths. With increasing wavelet length, the misfit energy between the calculated synthetic seismogram and the seismic section decreases. To define an optimal wavelet length, Akaike's Final Prediction Error (FPE) criterion (Ljung, 1987) is used, resulting in the best trade-off between noise and signal in the estimated wavelet. Akaike's FPE criterion consists of the product of the misfit, which decreases monotonously as a function of

9 the wavelet length, and a factor increasing monotonously as a function of the wavelet length. Akaike's FPE criterion is an estimate of the expected misfit between the seismic and the synthetic data using the estimated wavelet on a statistically independent realisation of the reflection series and noise realisation. Akaike's FPE criterion is therefore the object function to be minimised when determining the wavelet length.

10 f Chosen wavelet

Figure 3:

The wavelet estimation QC plots from ISIS. To the left is shown the synthetic data from the wavelet estimation inserted in the seismic data from the 3D cube. The acoustic impedance well log measurement is overlayed in red. To the right in the top graph is shown the amplitude spectrum of the synthetic data from the estimated wavelet in green and the amplitude spectrum of the seismic data in red. Below that is shown the amplitude (in red) and phase (in blue) spectrum of the estimated wavelet. The last graph in the right column shows the variation with wavelet length of the misfit (in red), the cross-correlation (in blue) and the Akaike estimate of how well the wavelet will do in another location (in green) The black curve is the relative number of parameters. All misfits are normalised to one. The panel in the bottom shows the wavelet suite. Each trace is the optimal wavelet with a given length. The chosen wavelet is marked. It is here 12 samples long.

11 3.1.2 Deviated wells

Deviated wells have to be treated carefully in wavelet estimation. The method applied is based on deviated sections also called supertraces. The well trajectory is very precisely positioned in the real physical world for modem wells, but to utilise the information within seismic inversion, the depth unit of the well has to be changed to two-way travel-time (TWT). The time to depth relation can be difficult. In the figure below is shown how important this point is. The figure shows the seismic data with the synthetic data from the well inserted. The synthetic data is obtained by convolving the reflectivity log from the well with an estimated wavelet. On the panel to the left is shown the result when the sonic log and check shots have been utilised. In the panel to the right is shown how well the synthetic data can match when the time to depth correlation for the deviated well has been treated accurately. To inspect the fit, it is important to find a super trace in the seismic cube at the well trajectory in the time domain. If the log is stretched or squeezed, it will change the time to depth relation and therefore a new supertrace has tobe found before the result is evaluated.

initial final

MFA-11 MFA-11

Figure 4:

A correct and careful treatment of a deviated well can improve the correlation between the log and the seismic . On the panel to the left is shown the correlation when the sonic and check shots have been used, to the right after stretching and squeezing, the log is compared with a super trace. After each stretch and squeeze, the supertrace is updated.

12 4. The inversion

4.1 Method

The seismic inversion method ISIS is based on a revision of the simulated annealing algorithm (Mosegaard and Vestergaard, 1991; Vestergaard and Mosegaard, 1991). This approach is a global optimisation method, which does not use a starting model. The inversion result is thus unbiased by the interpreter.

The seismic inversion approach utilises a sparse parameterisation of interfaces, representing major contrasts in acoustic impedance. The algorithm first defines the presence of interfaces, and when the globally optimised result is found, the acoustic impedance level is defined. This makes it possible to add low-frequency information not present in the seismic data to the seismic inversion result (Cooke and Schneider, 1982).

The seismic inversion is set-up with 5 parameters: (1) The misfit found during the wavelet estimation between the synthetic seismogram and the seismic section is used to determine the extent to which the seismic energy is to be modelled as a first estimate. (2) The extent to which the seismic inversion is constrained by the acoustic impedance log at the well location. (3) An estimate for horizontal continuity of acoustic impedance within layers is derived from the amplitude variations of the seismic section. (4) Vertical threshold for variation in acoustic impedance being interpreted as an interface. All interfaces have a reflection coefficient above this value, but not all interfaces with a reflection coefficient above this value will necessarily be detected. The magnitude of the reflection coefficients is known from the log reflectivity series. (5) Horizontal threshold for variation in acoustic impedance being interpreted as an interface. Analogue to the vertical threshold in point 4.

The cost function to be minimised by the seismic inversion is non-linear as it consists of the error energy between the synthetic and seismic data found by forward convolutional modelling and penalties for both vertical and horizontal interfaces and lateral continuity of acoustic impedance.

The seismic inversion generates three independent primary results:

(1) A globally optimised acoustic impedance model

(2) A layer interface section

(3) A probability section

(4) Confidence limits on acoustic impedance

and 3 secondary results are also generated: (5) A synthetic seismic section, (6) An error section, and (7) a reflectivity section.

The layer interfaces are defined by a relative change in acoustic impedance, i.e. the reflection coefficient above a certain threshold, and therefore do not need to define geological boundaries.

13 As the seismic inversion is a global method, a probability section can be generated. This section illustrates the probability of a layer interface location from all possible inversion results and provides information about the confidence of the estimated layer interfaces or interpreted boundaries from the acousticimpedance section. The probability section depends on the signal- to-noise level for the seismic data.

4.2 Parameterisation

In TDROV, the notion of continuous layers is introduced. The model parameters M(I,T), where I is impedance and T reflection time (see figure 5), are obtained by sampling the continuous impedance fields of the macro model. Vertical sampling within each macro interval is carried out by interpolating strata in order to represent the layer geometry adequately. In essence, the sampling interval is the average thickness of all the layer thicknesses derived from the well logs for this particular macro interval. Alternatively, this parameter can be experimental and chosen according to criteria based on the seismic bandwidth, the density of reflectors and the character of the well logs. Conformity of the model strata to seismic reflectors may be preserved ornot. The resulting strata are then laterally sampled every A bins in X and Y. The sampling interval A is chosen in such a way as to linearly approximate the continuous impedance field. As a result, the continuous impedance field has been discretized to yield the model parameters I and T at every node of the resulting sparse 3D grid. Typical values for A are 3 to 4 bins.

ISIS does not use continuous layers, but has still both a dense and sparse parameterisation. The sparse parameterisation is again giving the structure of the subsurface while the dense parameterisation is giving an estimate of the physical properties for each seismic data sample.

4.3 Objective function

The inversion follows a model based approach with the model parameters being sparse and constrained. Starting with a simple and approximate macro model, the program is able to produce a finely stratified broad-band impedance model by iteratively updating the current impedance model on a global basis. The volumetric impedance perturbations update the results and the strata interface are deformed, which are accepted/rejected according to the Metropolis algorithm. The Metropolis algorithm is implemented within a simulated annealing schedule.

We assume that a mathematical model, i.e. the convolutional model free of multiples and transmission losses, can establish a relationship between the fully processed post stack seismic data (random variable D) and the unknown model parameters (random variable M(I,T)). The model response F(M) is quasi linear as a function of an impedance model perturbation DI, but absolutely non linear as a function of a model parameter reflection time perturbation AT.

Hence, for a given real seismic sample Dl, the impedance model response can be obtained from the model parameters M by means of a non-linear functional Fl(M) using ID synthetic seismograms. At any sample index L, we look for M such that D[=Fl(M). Inverting the data set D for the model parameters, M, is done by solving for M the set of non-linear equations

14 D=F(M) by pertaining all the data of the seismic block. This is done by minimising a distance function C(M) in a Least Absolute Deviation (LAD) sense or in a Least Square Deviation (LSD) sense. TDROV uses LAD avoiding overweighing of large residuals, as any least squares criteria will do. This is carried out iteratively by a global optimisation method which is best suited to non-linear problems and LAD criteria. The optimal impedance model Mopt is reached when the functional C(M) cannot be further decreased in the model space. ISIS uses LSD which fits with an assumption that the noise in the seismic data are Gaussian distributed.

In order to cope with the very nature of seismic data, being insufficient and noisy, some other function which penaltilizes the deviations of the current model parameters with respect to an a priori model have been added to C(M). The a priori model may be defined as probability distributions if sufficient well information is available. The a priori model is considered in terms of layer impedance and its lateral continuity. The aim of these soft constraints is to achieve a model in which layer impedance may be laterally smooth and within a priori impedance bounds L and H, whereas the functional D=F(M) makes M consistent and conformable with the seismic data. It should be noted that all the a priori information bears on impedance since it is much less sensitive than reflection time and this is the way the low- frequency content of the impedance, not present in the seismic data, is transmitted to the optimal model.

The problem to be solved (in vector notation) is: Find M(I,T) such that C(M) is minimum, with:

C(M)= | | D-F(M) | 11 + Wi | | Mi-Mi pri I 11 + Wc I I GRADL(Mi) - GRADL(Mr pri) I 11 L < Mi < H

The various vectors stand for:

D,F Real and synthetic 3D seismic samples Mi Impedance model parameters Mi pri A priori impedance model parameters GRADL (Mi) Strata lateral impedance gradient Wl.Wc Weight vectors of the a priori model contribution for impedance and impedance lateral continuity L, H Vectors of low and high impedance bounds

In ISIS, the inversion is controlled by five parameters, each connected to a term in the non­ linear cost function:

• The seismic signal. Signal to noise ratio.

• Horizontal continuity. Standard deviation between acoustic impedance samples.

» Trustworthiness of the prior model. Standard deviation between the acoustic impedance values of the subsurface model and the prior model

• Vertical continuity. Standard deviation between acoustic impedance samples.

15 # Sparseness of the sparse parameterisation. Threshold value for reflection coefficients.

These five terms form a non-linear cost function which is minimised during the iterative search for the best subsurface model.

4.4 Global search

The minimisation of C(M) is implemented by Simulated Annealing (SA), (Rothman 1986), an iterative Monte Carlo process based on the Metropolis algorithm, which is suited to the non­ linear and multi modal nature of C(M). In SA, the minimisation of C(M) is treated as the maximisation of a Gibbs probability distribution over the model space M which is defined as a Markov Random Field (MRF) (Geman & Geman 1984). In our parameterised impedance model, this means that the probability for a given model parameter to have a given value will depend only on the value of the model parameters located in a certain neighbourhood. In other words, we do not need to know the full model M to predict the model value at a given node, butonly its local values in the predefined neighbourhood. The neighbourhood defined here is a rectangular volume, made of 3 squared patches of impedance strata defined by 9 nodes each. Typically, the neighbourhood of a single model parameter comprises several hundreds of seismic samples, thus providing a good statistical estimation of seismic amplitude related to this model parameter. In addition to the inherent robustness of the method due to the sparse parameterisation, SA is relevant to our particular problem since, at the start-up of the SA process, all impedance models are equally possible. Therefore, the picking of the input horizons does not need to be accurate since it is going to be perturbed. The same goes for impedance.

4.5 The spectral content of the seismic data related to the inverted seismic traces

The seismic frequency bandwidth is limited on the low and high frequency sides. To the extent that the seismic data are perfect, the used parameter estimation technique could transform it to absolute impedance by using a single calibration point, in fact, an integration constant. Since the seismic data is insufficient and noisy, the impedance calibration of the seismic data has to be carried out by many calibration points to compensate for the lack of low-frequency seismic data and make up for the non-existent high-frequency seismic data. This is implemented by a variety of techniques. Here, we have chosen to calibrate the inverted seismic traces by looking for an impedance solution within an impedance corridor, since this technique does not constrain the results too much. It is difficult to compensate for the lack of high-frequency information of the seismic data (connected to the thin layers) without making the inversion preparation very heavy. However, a lot of high-frequency information may be obtained from an efficient inversion. This is achieved here by a sparse parameterisation and a global search of the model parameters in the highly non-linear model space.

4.5.1 Initial model / a priori model

In TDROV, it can be assumed that the 3D initial model geometry is inaccurately represented by a priori impedance information. It can be from prospects with reduced well control.

16 The macro model geometry is defined by a few 3D main time horizons. The impedance within each macro layer may vary laterally and vertically, and impedance constraints may be set to keep the optimised impedance model laterally smooth and within given boundaries (figures 5 and 6).

M(I,T)

In TDROV, the seismic data cube is separated into areas with A bins on each subcube. The prior model is defined on the subcubes.

17 TWT (ms)

Figure 6:

In TDROV, the input data for the inversion is generated based on the well log. The Figure shows the input model. The response from the input model is compared with the seismic data. The grey shaded area shows the limits on the acoustic impedance during the inversion. wm

In ISIS, the prior model is generated from a low-frequency filtered version of the logs interpolated following a few seismic horizons. The filtering of the logs is decided from an analysis on the log data. This analysis can determine the amount of log information that can reliably be interpolated between the wells.

4.6 The results

The resulting impedance model may be validated by comparison with well logs, since they are not input explicit to the program. The end product of the program consists of impedance maps and cross-sections together with the geometry of the strata interfaces. Here, we display the initial (figure 6) and optimal (figure 10) model parameters (parameter traces) at four well locations of the inverted square kilometre of seismic data. At every well location, the initial and final model response is compared with the corresponding real seismic trace. The match does not have to be perfect because it is also influenced by the fulfilment of the impedance model constraints. Actually, it is the match between the well impedance log and the inverted seismic traces that should be as good as possible. It should be noted that the impedance logs have not been corrected although some of the wells are deviated or distorted by mud reading and caving. In figure 10, it can be seen that the resolution is in the order of the seismic time sampling interval (2 ms) and that these very thin layers can be picked automatically by consistent model interfaces.

In the figures in the reservoir description section, is shown several ISIS results on different data sets. The examples with data from Agip, UK and Texaco, UK are blind well tests showing the capability of a globally optimised inversion algorithm of predicting a well. In the blind well test cases, the well has not been used neither for wavelet estimation nor in the inversion itself.

4.7 Uncertainties

There will always be uncertainties of the inversion result and it is important to quantify these uncertainties. The full representation of the uncertainty is the a posteriory distribution, giving the probability of different results. In practice, this type of result is impossible to work with. Therefore, it is important to capture the important part of the uncertainty, quantify it and present it so that it is easy for the geophysicist and geologist at the oil company to use. Otherwise, it will never be used. In ISIS, the uncertainty is quantified as the marginal distribution for the structural outline and the uncertainty of the estimated acoustic impedance for the given structure. In the figures are shown first the seismic data, then the estimated acoustic impedance and last the probability for the structural out-line of the inversion result

19 iwrw Figure 7:

The seismic data. The data is a VSP data set processed for P upgoing only. Data is by courtesy of Norsk Hydro.

Tiacel 0 ■ 1.3e+04

8.0e+03

3.0B4O3

Figure 8:

The estimated acoustic impedance. The well log is actually outsidethe seismic data, but the well log data has here been moved so that the level of the acoustic impedance can be compared. The inversion is performed with ISIS. Data is by courtesy of Norsk Hydro.

i| 0.9125

' 0.825

0.7375 i 0.65

0.5625

0.475

0.3875

Probability for the structure of the subsurface determined by the inversion. The inversion is performed with ISIS. Data is by courtesy of Norsk Hydro.

20 wm

5. Reservoir description

Most often the objective of performing seismic inversion is to obtain a better reservoir description. In this section is shown some of the results obtained with TDROV and ISIS.

The objective of the inversion of the Paris Basin seismic land data is to locate and delineate a stack of thin, laterally discontinuous Triassic fluviatile sandstone used as a gas storage. In figure 10, the various reservoir members are located between the strata interfaces nos. 23 and 26.

The seismic amplitude frequency spectrum for these data extends from 20 to 80 Hz. Amplitude data is affected by distortions coming from acquisition and binning. In figure 6 is shown a sample of the initial information provided for the inversion at 4 well locations distributed over a square kilometre area of the Paris Basin. Basically, an impedance solution is to be found within the impedance corridors that are depicted at the various well locations. We also show the real seismic traces at these locations together with the initial model seismic response. Figure 10 shows the inverted seismic traces at the same locations as in figure 6. The main point about these results is that the new layer interfaces (not present in figure 6), which are located at random at the beginning of the inversion process, are eventually positioned at the same stratigraphic level at the three wells. This allows the direct mapping of the impedance of the 40 strata composing the processed data, (in figure 10, every other interface is displayed).

21 i

Figure 10:

The inversion is performed with TDROV. The log is shown in red. To the le model and its response. To the right of the log is shown the final result and ii compared with the seismic trace. Data is by courtesy of Gaz de France.

I wm

The two inversion methods have been tested on several data sets. The ISIS results are reported in "ISIS Global Inversion of 3-D Land Seismic Data" by Lars S. Hansen, 0DS report 95.391. Two ISIS results are shown in the next figures. They are blind tests as the well log information shown in the displays has not been used, neither in the wavelet estimation nor in the inversion itself.

1.5e+07

1.3e+07

1.0e+07

8.0e+OG

5.6e+06

Figure 11:

Well prediction. The well inserted in the acoustic impedance was not used in any step of the inversion process. The figure thereby shows to what extent a seismic inversion can predict a new drilling in the area. The inversion is performed with ISIS. Data is by courtesy of Agip, UK.

23 Figure 12:

The left panel shows the synthetic traces inserted in the seismic data. The right panel illustrates the acoustic impedance result derived without using the inserted acoustic impedance log in the 3D seismic inversion. The principal aim of the study was to delineate the extent of a Tertiary reservoir. The secondary objective was to identify the nature and extent of Lower Palaeocenesand bodies. The inversion is performed with ISIS. Data is by courtesy of Texaco Limited.

6. The future

The method we present here is flexible since it can be adapted to the quality of the seismic data through a requested degree of lateral impedance continuity and a choice of the impedance sampling parameters, i.e. the number of strata and the lateral sampling interval. The impedance results obtained by inversion are dependent on the amount of a priori information which is input into the inversion, and the complexity of the impedance field to be restored. The method can contribute to reservoir characterisation owing to its good resolution resulting from the sparse parameterisation coupled to the volumetric seismic data handling. New possibilities for automated impedance mapping, of possibly thin layers, have been realised. They work with subhorizontal prospects like in the case with the Gaz de France data or mild tectonics provided that the reflectors are reasonably continuous. In future, it is planned to introduce some a priori knowledge on the impedance model geometry in order to map acoustic impedance along impedance model interfaces related to discontinuous reflectors.

24 One future possibility already started to be explored with ISIS is direct estimation of porosity from the seismic data (Rasmussen and Maver, 1996). From seismic data acoustic impedance can be estimated by inversion. However, porosity is one of the main parameters used to characterise a reservoir. Porosity can be derived as a relationship between acoustic impedance and porosity. The relationship is dependent on lithology, which means that to apply a non­ varying linear transformation to the acoustic impedance results derived from seismic data to estimate porosity is only valid for a single lithology. Thus, in order to derive porosity from acoustic impedance results, a detailed interpretation is needed to define the individual lithologies. A seismic inversion technique can therefore be proposed to derive porosity directly from seismic data. The technique takes into account the lithology dependent relation between porosity and acoustic impedance without requiring the user to make a detailed interpretation of the seismic data and specifying the transformation between acoustic impedance and porosity for each lithology.

The correlation between acoustic impedance and porosity has a negative slope that is approximately constant and not lithological dependent. However, the porosity intercept of the linear relation is lithological dependent. The general negative slope between porosity and acoustic impedance is derived from a statistical analysis performed on the available porosity and acoustic impedance logs at the well locations. The varying intercept with lithology is not specified, as the algorithm will adopt the correct porosity intercept for each lithology by constraining to a low-frequency porosity model which is specified from available well logs. This assumes that the variation in porosity of the different lithologies of the subsurface can be described by a model extrapolated from well logs.

To invert the seismic data directly for porosity a non-linear globally optimised seismic inversion technique is used. This full 3D multi trace technique does not directly include well logs or seismic horizons and therefore unbiased results are generated from the post-stack and migrated seismic data.

The inversion of seismic data directly for porosity is performed by utilising the estimated wavelet, the low-frequency porosity model and by specifying 5 inversion parameters. The wavelet is derived as the operator between the reflectivity log and the seismic data without making any assumptions about phase, polarity, length and time delay. A full suite of wavelets is estimated and from this suite the optimal wavelet is chosen. A constant operator is then estimated between the reflectivity log and the porosity log represented as a difference in porosity from sample to sample. The chosen wavelet is multiplied with this constant operator. The low-frequency porosity model is built by extrapolating all the available low-pass filtered porosity logs along interpreted seismic horizons.

Acknowledgements

The authors acknowledge Gaz De France, Norsk Hydro, Denerco, Agip and Texaco for permission to publish the data.

25 References

Cooke, A. and Schneider, W. A. 1982. Generalised linear inversion of reflection seismic data. Geophysics 48,665-676.

Genian, S. and Geman, D. 1984, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images- IEEE transactions on pattern analysis and machine intelligence. PAM 6

Hansen, L. S. 1995 ISIS Global seismic inversion of 3-D Land Seismic Data. 0DS report 95.391.

Ljung, L. 1987. System identification: Theory for the user. Prentice Hall.

Maver, K. G. and Rasmussen K. B. 1995. Seismic Inversion for Reservoir Delineation and Description SPE 29798.

Maver, K. G. and Rasmussen, K. B. 1995: Seismic inversion for reservoir delineation and description. Society of Engineers (SPE), Bahrain, March 1995,267-276.

Mosegaard, K. and Vestergaard, P. D. 1991. A simulated annealing approach to seismic model optimisation with sparse prior information. Geophysical Prospecting 39, 599-611.

Rasmussen, K. B. 1994. Maximum Likelihood Estimation of the parameters of Nonminimium Phase and Noncausal ARMA Models. IEEE Transactions on signal processing vol. 42 No. 1 January 1994.

Rasmussen, K. B. and Maver, K G. 1996. Direct inversion for porosity of post stack seismic data, SPE 35509.

Rothman, D. H. 1986 Automatic estimation of large residual statics correction Geophysics 51.

Vestergaard, P. D. and Mosegaard, K. 1991. Inversion of post-stack seismic data using simulated annealing. Geophysical Prospecting 39,611-624.

26