Thierry Coléou, Fabien Allo and Raphaël Bornard, CGG; Jeff Hamman and Don Caldwell, Marathon Oil Summary We present a seismic inversion method driven by a petroelastic model, providing fine-scale geological models, in depth, fully compatible with pre-stack seismic measurements. Introduction As part of a two years collaborative R&D project between CGG and Marathon Oil, we have developed a petrophysical seismic inversion workflow to generate fine-scale geomodels that are fully consistent with observed seismic data; the geomodels are not simply conditioned to seismic attributes by seismic-guided mapping but reproduce prestack seismic measurements (Bornard et al., 2005). A key component of the inversion methodology is a Petro-Elastic Model (PEM) that links the reservoir properties stored in the geomodel (e.g., porosity, rock types and fluid saturations) to the elastic response. The new technique is illustrated using a direct porosity inversion case study involving a large North Sea reservoir. : methodology Unlike traditional seismic inversion techniques that solve for elastic properties in time, the Petrophysical Seismic Inversion operates on rock properties in depth. Figure 1: workflow. The workflow for is illustrated in Figure 1. We start from an initial fine-scale geomodel defined from a 3-D stratigraphic grid in depth (left). The PEM is applied to calculate elastic properties in each cell of the geomodel from stored values of porosity, rock type and saturations (middle). Angle-dependent reflectivity series are calculated from the elastic properties through the Zoeppritz equation at each trace location. The reflection coefficient series are then converted from depth to time using the velocities stored in the geomodel. Angledependent 3-D synthetics are finally generated by wavelet convolution (top-right in Figure 1). Perturbations of the properties of the geomodel are introduced using a simulated annealing algorithm to optimise the degree of match between the synthetic and the real angle stacks. After convergence, the final geomodel honours the observed seismic amplitudes, is consistent with the user-specified PEM and integrates inversion-based velocities that ensure coherence between the depth and time domains. It should be noted that changes in the initial model such as the smallscale distribution of rock type, or the modification of the PEM will lead to different solutions. The final models then represent alternative solutions consistent with the seismic data. Post and pre-stack inversion: the value of information Post-stack inversion provides acoustic impedance information I P, often assuming zero-offset reflectivity. One seismic measurement is de-convolved into one localized attribute (I P ) more suitable for reservoir characterisation. It is then transformed into petrophysical variables of interest through statistical or empirical, implicit or explicit, relationships (Doyen, 1988). Data integration techniques such as co-kriging or neural networks are used to drive the interpolation away from well control. However, when the same seismic attribute, I P, is used to predict porosity, netto-gross ratio, fluid type and even permeability, we can question the ability of a single measurement to accurately predict such a multiplicity of variables. It should also be stressed that using several attributes derived from the same original measurement does not provide additional information and that the multiplicity of attributes increases the risk of spurious statistical relationship (Kalkomey, 1997). Working pre-stack or with partial angle stacks introduces more than one seismic measurement and therefore provides an additional degree of freedom for reservoir property prediction. We can expect to access more than one petrophysical parameter in a meaningful manner. Traditional pre-stack inversion provides either compressional and shear impedances (I P and I S ) or the triplet V P, V S and density ( ). The last three variables are not independent as they are the elastic expression of various petrophysical variables, for example, porosity, sorting, fluid type and saturations. First attempts to invert for more than two variables from pre-stack data incorporated very tight constraints for example a priori V P /V S or the use of Gardner s equation (Gardner et al., 1974) to link V P and. However, these relations do not SEG/Houston 2005 Annual Meeting 1355
properly describe the changes that occur when the reservoir is shaling out or when fluids are substituted. Role of the petro-elastic model during inversion Elastic behaviour of the reservoir rock can be predicted through forward modelling using a petro-elastic model. Seismic inversion methods derive V P, V S and values that reproduce reflectivity observed at various angles through the Zoeppritz equation or one of its approximations and a 1-D convolution. Solutions of the ill-constrained inverse problem are not necessarily compatible with the a priori knowledge given by the elastic response of the rock types within the reservoir. A petro-elastic model is sometimes used for inversion quality control or transformation of the inversion results through lithology classification (Ødegaard and Avseth, 2004). Inversion results at seismic scale can also be downscaled to reservoir properties in a second inversion step through a petro-elastic model; this is the concept of inversion of the inversion (Caldwell and Hamman, 2004). In the petrophysical seismic inversion, we use the petro-elastic model during the inversion process. Porosity, saturation or facies values of the cells within the geomodel are optimised so that their combined elastic response reproduces the observed seismic as shown in Figure 2. This prediction ability from a small number of variables illustrates the relatively low dimensionality of the problem. The use of the petro-elastic model establishes the necessary link between V P, V S and. This link is more complex than the implicit linear relationship obtained by co-kriging, more stable than a statistical fit when data are sparse or biased and has physical meaning. Figure 2: Well calibration. Comparison between measured (black) and predicted (green) V P, V S and and, on the right, between synthetic seismic (blue) and real traces surrounding the well (grey) for three angles. Scale of the inversion Working at the geological scale, which is a scale fine enough to represent the significant heterogeneities of the reservoir, we ensure that the geomodel is consistent with the seismic data while preserving the known rock types and their elastic relationships. A simple case with a binary mix of alternating thin sand and shale layers cannot be resolved by the band-limited seismic data alone. A priori geological knowledge expects a bi-valued acoustic impedance distribution (one value for shale and one for sand) while post-stack inversion in the seismic bandwidth provides a continuous distribution corresponding to the different aggregates of rock types. In this simple binary case, the petrophysical inversion will provide the expected bi-valued acoustic impedance distribution. In our stratigraphic framework, the finest scale, V3 in Figure 3, is the target scale and is defined by the geologic layering of the initial model with a typical layer thickness of one meter or less. It is the scale at which the petro-elastic models are considered valid. The intermediate V2 scale is the scale at which acoustic or elastic inversions are commonly performed. A layer at that intermediate scale corresponds to a pile of consecutive fine-scale layers with geological significance, typically within two sequence boundaries or maximum flooding surfaces. The order of magnitude for this seismic scale is about ten meters. The coarsest scale, V1 in Figure 3, is defined by macrohorizons, such as the interpreted seismic events, which are calibrated to well markers and give the structural framework of the geomodel. These three vertical scales provide nested partitions of the subsurface in such a way that a fine-scale layer belongs to a single layer at coarser scale. Why use a layered model? They are several reasons why we use a layered model during the inversion instead of the regular time sampling of the seismic data. First, the stratigraphy is better modelled, both for geological description and flow unit definition, as relatively homogeneous layers with property contrasts between layers. Also, seismic amplitude, despite being regularly sampled in time, is better modelled with accurate positioning of important contrasts beyond the seismic sample. This is observed when we consider seismic event interpretation, interval attribute computation and seismic facies analysis (Coléou et al., 2003). Another reason is that the earth is not graduated in TWT and inversion results need to be converted to depth. In depth, as well as TWT, the stratigraphic layering system is preserved. This is true irrespective of the data acquisition domain, PP or PS time. Furthermore, a layered model enables accounting for small position adjustments, often much smaller than a seismic sample, in the different seismic cubes to compensate for residual NMO or to model time shifts and eventually compaction-induced depth shifts between 4D vintages. SEG/Houston 2005 Annual Meeting 1356
Inversion in TWT or Depth domain? The also addresses the seismic time-to-depth conversion challenge. Traditionally, seismic inversion results are depth-converted prior to integration with the geomodel and inconsistencies often exist between the various manipulated velocities. These velocities are generated at different scales, which does not facilitate their comparison and reconciliation. We have horizon-based velocities used for depthing at large scale (V1), inversion-derived velocities at seismic resolution scale (V2) and velocities coming from the forward petroelastic modelling of the geological information at small scale (V3). By incorporating time-to-depth conversion as part of the inversion, consistency between the time depth relationships and the velocities stored in each of the cells of the geomodel is maintained throughout the entire inversion process. This is achieved using a multi-scale, multi-axis stratigraphic grid system as illustrated in Figure 3. V1 V2 V3 ~100m ~10m ~1m Figure 3: Velocity section at different scales inside the same geomodel. computer architectures, working with a large number of angle stacks for pre-conditioning and inversion is no longer a problem, even for extremely large models. S SS R FS Figure 4: composite sections of 6 seismic gathers from 6 angle stacks: raw seismic (S), Zoeppritz-compliant synthetic seismic (SS), residuals (R) and geostatistical filtered seismic (FS). Application to a North Sea field The has been applied to porosity inversion on a large North-Sea reservoir composed of deep-water deposited sandstones. Six seismic angle stacks were used during the inversion of the fine-scale geomodel derived from an exploration well. Residual analysis Inversion using AVO/AVA information implies the choice of the number of cubes used during the inversion. We can invert the full gathers or limit the number of cubes to as little as two partial stacks, i.e.: nears and fars. There is obviously a trade-off between computational efficiency and our ability to understand and filter the energy within the partial stacks that is not Zoeppritz-compliant. Unlike with post-stack inversion where small residuals are considered a must, residuals in pre-stack inversion are quite often significant. We believe that working with a larger number of seismic cubes, with more redundancy of the data, provides a better way to filter seismic energy that is considered as noise such as multiples or residual NMO. This energy is somewhat coloured, with spatial, temporal and offset dependencies and therefore not efficiently filtered by the brute stacking process. Pre-stack geostatistical filtering (Hoeber et al., 2003) operating in 4 dimensions (Inline, Crossline, Time and Offset or Angle) proved to have a significant impact on the inversion results (Freudenreich et al., 2004). It is proving quite efficient at reducing the dipping energy in the gathers prior to inversion as seen in the composite sections of seismic gathers in Figure 4. With the use of massively parallel Figure 5: Inverted porosities are displayed on layers and vertical sections, intersecting at a well location, extracted from the finescale geomodel, with higher porosity values in red. The oilsaturated sands are displayed in green and two seismic sections from the near cube are displayed on the edges. SEG/Houston 2005 Annual Meeting 1357
The inversion revealed spatial variations in the porosity hidden in the amplitude volumes by the strong AVO response to the hydrocarbon content. Conclusions A petro-elastic model is the link between rock properties and seismic data and should be at the core of seismic calibration. Through the, the fine-scale geomodel is reconciled with the seismic data, not by simply guiding the interpolation in between wells, but by guaranteeing the reproduction of the seismic amplitude for all angles after forward modelling. This is a particularly interesting feature for further developments towards quantitative time-lapse simultaneous inversion. The model response is directly optimised for reservoir property changes, in depth and at the scale of the flow units, the natural variables and domain to express the production-induced constraints. Such constraints are necessary to control the simultaneous inversion of different seismic vintages. They do not come from wave equation as in the case of simultaneous pre-stack inversion where Zoeppritz equation dictates the behaviour of reflectivity across angles. The constraints across the acquisition times come from the knowledge of the production-induced changes in the reservoir, usually expressed in terms of saturation and pressure changes and translated into elastic property constraints ( V P, V S and ) through a petroelastic model. Aknowledgements The authors would like to thank Marathon Oil Company and Compagnie Générale de Géophysique for the support and permission to publish this work and coworkers of both companies for valuable input. References Bornard R., Allo F., Coléou T., Freudenreich Y., Caldwell D.H. and Hamman J.G.: to determine more accurate and precise reservoir properties, SPE 94144, SPE Europec, Madrid, 13-16 June 2005. Caldwell D.H. and Hamman J.G.: IOI A method for fine-scale, quantitative description of reservoir properties from seismic, paper B027 EAGE 66 th Conference and Exhibition, Paris, 7-10 June 2004. Coléou T., Poupon M. and Azbel K.: Unsupervised seismic facies classification: A review and comparison of techniques and implementation, The Leading Edge, Vol. 22, No. 10, pp. 942-953, October 2003. Doyen P.M.: Porosity from seismic data: A geostatistical approach, Geophysics, Vol. 53, No. 10, pp. 1263-1275, October 1988. Freudenreich, Y., Reiser, C. and Helgesen, J.: Preconditioning workflow for optimised reservoir characterisation by stratigraphic inversion, Petex 2004 Conference, London, 23-25 November 2004. Gardner G., Gardner L. and Gregory A.: Formation velocity and density The diagnostic basis for stratigraphic traps, Geophysics, Vol. 39, No. 6, pp. 2085-2095, December 1974. Hoeber, H., Coléou, T., LeMeur, D., Angerer, E., Lanfranchi P. and Lecerf, D.: On the use of geostatistical filtering techniques in seismic processing, SEG 73 rd Ann. Intern. Mtg., Dallas, 26-31 October 2003. Kalkomey C.T.: Potential risks when using seismic attributes as predictors of reservoir properties, The Leading Edge, Vol. 16, No. 3, pp. 247-251, March 1997. Ødegaard E. and Avseth P.: Well log and seismic data analysis using rock physics templates, First Break, Vol. 23, pp. 37-43, October 2004. SEG/Houston 2005 Annual Meeting 1358
EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2005 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 Bornard, R., F. Allo, T. Coléou, Y. Freudenreich, D. H. Caldwell, and J. G. Hamman, 2005, to determine more accurate and precise reservoir properties: SPE 94144. Caldwell, D. H., and J. G. Hamman, 2004, IOI A method for fine-scale, quantitative description of reservoir properties from seismic: 66th Annual Conference, EAGE, Extended Abstracts, B027. Coléou, T., M. Poupon, and K. Azbel, Unsupervised seismic facies classification: A review and comparison of techniques and implementation: The Leading Edge, 22, 942-953. Doyen, P. M., 1988, Porosity from seismic data: A geostatistical approach: Geophysics, 53, 1263-1275. Freudenreich, Y., C. Reiser, and J. Helgesen, 2004, Preconditioning workflow for optimised reservoir characterisation by stratigraphic inversion: Petex 2004 Conference. Gardner, G., L. Gardner, and A. Gregory, 1974, Formation velocity and density The diagnostic basis for stratigraphic traps: Geophysics, 39, 2085-2095. Hoeber, H., T. Coléou, D. LeMeur, E. Angerer, P. Lanfranchi, and D. Lecerf, 2003, On the use of geostatistical filtering techniques in seismic processing: 73rd Annual International Meeting, SEG, Expanded Abstracts, 2024-2027. Kalkomey, C. T., 1997, Potential risks when using seismic attributes as predictors of reservoir properties: The Leading Edge, 16, 247-251. Ødegaard, E., and P. Avseth, 2004, Well log and seismic data analysis using rock physics templates: First Break, 23, 37-43.