Rémi Moyen* and Philippe M. Doyen (CGGVeritas) Summary Static reservoir connectivity analysis is sometimes based on 3D facies or geobody models defined by combining well data and inverted seismic impedances. However, this is mostly performed from deterministic inversion results that provide limited information on uncertainty and may yield biased estimates of reservoir volume. Here, we present a workflow exploiting stochastic impedance realisations for facies characterisation and connectivity analysis with uncertainty, and illustrate the workflow using seismic and well data from a turbidite sand reservoir. After pre-stack stochastic inversion, we perform seismic facies classification based on cross-plots of inverted elastic attributes, calibrated to log data. Applying the classification to all impedance realisations, we generate multiple seismic sand facies models that can be averaged to produce a sand probability cube, reflecting the level of uncertainty in the inversion process. We then identify all connected sand geobodies in each realisation. We rank the facies realisations based on computed geobody statistics. Finally, we combine the geobodies from all realisations to calculate the probability of connection to given wells and the uncertainty in connected sand volume. Introduction In order to overcome the band-limited nature of deterministic inversion methods, stochastic approaches have been proposed as a way of generating multiple highfrequency realisations of elastic properties from seismic data (Haas & Dubrule, 1994). By construction, the realisations are constrained to match observed seismic data within some noise-dependent tolerance limits, honour conditioning well data and approximately reproduce a spatial variogram model, incorporated as a priori geological information. The multiplicity of models reflects the inherent non-uniqueness of the inverse problem, but allows getting more realistic reservoir heterogeneity models than traditional deterministic inversion (Sancevero et al., 2005), especially when incorporating a spatial continuity model. The availability of multiple realisations of elastic parameters becomes a significant advantage when using inversion results for reservoir modelling and uncertainty analysis. A mapping can often be done between elastic attributes derived from inversion and geological facies in the reservoir. With stochastic inversion, it becomes possible to not only build facies models but also estimate their uncertainty. With multiple facies models, stochastic reservoir connectivity analysis can also be applied to assess uncertainty in drainable reservoir volume for given well trajectories. Stochastic Inversion Methodology Following the work of Buland and Omre (2003), we use a Bayesian framework and a linearised AVA model to calculate a joint posterior distribution for P-wave (I P ) and S-wave (I S ) impedances, constrained by seismic amplitudes measured on a number of input angle stacks. Under certain assumptions described in Escobar et al. (2006), the joint posterior distribution of the impedances can be expressed as a multidimensional Gaussian distribution. We then use a sequential procedure to decompose the joint posterior into a series of local Gaussian posteriors for each seismic trace. The local posteriors are sampled sequentially and repeatedly to generate multiple, high-frequency 3D realisations of elastic properties. The stochastic inversion is performed in a fine-scale stratigraphic grid defined in the time domain, with horizontal sampling fixed by the seismic bin size and vertical columns of cells with variable thickness, typically much smaller than the seismic resolution. The inversion layered framework facilitates the incorporation of geological constraints. In particular, stratigraphic conformance rules and layer-dependent vertical and horizontal variograms can be introduced in the prior model. Non-stationary noise statistics can also be handled by inputting trace-dependent S/N information. This method was tested using data from a deep-water field. The reservoir is a faulted anticline comprising turbidite channels and lobe complexes. One of the challenges in field development has been the assessment of lateral and vertical connectivity between channels, which is a hard task given the relatively low resolution of the seismic data in the area. As we are expressly focusing on fine-scale reservoir characterisation, a stochastic inversion approach is particularly suitable. The data include three angle stacks (16 o, 30 o and 40 o ), dipole sonic and density logs at 4 wells and 3 seismic horizons interpreted over the 4 km 5 km project area. A stratigraphic grid framework was constructed in time by proportional interpolation from the three interpreted horizons to define micro-layers in each macro-interval with maximum time-thickness of 2 ms. The inverted volume corresponds to 300 400 seismic traces at 12.5 m bin spacing and spans a 400 ms time window. This results in a layered model containing about 25 10 6 cells. Five hundred realisations of both I P and I S were generated, as illustrated in Figure 1.
Figure 1: 3D view of a single realisation of inverted P- and S-impedances. High S-impedances values, leading to low Vp/Vs ratio and thus low Poisson s ratio, indicate sand facies (see Figure 2) and show one of the main channel systems in the lower reservoir section, and a smaller one on the top-left part of the pictures. Facies characterisation with uncertainty Figure 2 depicts a cross-plot of P-impedance versus Poisson's ratio constructed from the well log data and colour-coded according to the shale volume fraction, V sh. Sands (i.e., points with low values of V sh ) can clearly be discriminated from shales based on their low Poisson s ratio values. Inversion results are consistent with the wellderived cross-plot and also show that a good sand / shale discrimination is achieved from the Poisson s ratio. simple sand polygon, the facies prediction workflow could be improved by using a Bayesian classification procedure, as explained for example in Coulon et al., 2006. Following the method described by Coulon et al. (2006), we define a sand cross-plot polygon, as shown in Figure 2, and use it to classify each realisation of elastic properties; i.e., each point of each realisation is classified as sand or shale depending on whether or not it falls inside the sand polygon. Using this simple procedure, we can generate as many facies realisations as we have impedance ones, as illustrated in Figure 3. Although not directly constrained by the wells, the realisations match closely the log-derived facies. Once generated, the facies realisations can be ranked using different criteria, for example according the calculated netto-gross ratio. This enables the selection of individual realisations, representing for example P 10, P 50 and P 90 scenarios, for more detailed reservoir modelling. Moreover, by counting in each cell the number of realisations classified into each facies, we can compute a facies probability cube, as shown in Figure 4 (top) and evaluate the uncertainty in seismic lithology prediction. The thresholded probability cube highlights the most likely sandy areas in the turbidite complex. Instead of using a Figure 2: Cross-plot of P-impedance (m/s g/cm 3 ) versus Poisson's ratio, colour-coded according to V sh. Points inside the polygon are assumed to belong to the sand facies, and points outside to the shale facies. Stochastic connectivity analysis To better assess the volume of sand drainable by individual wells, we perform a stochastic sand connectivity analysis.
Connectivity of reservoir bodies is also a better proxy for dynamic flow properties than purely volumetric measurements (McLennan and Deutsch, 2005; Hird and Dubrule, 1998). Computing geobodies, i.e. sets of connected pay cells, is a well-known problem (e.g., Deutsch, 1998) with many variations. However, geobody extraction methods are usually applied to compute connectivity on a single property model and, for this reason, can afford to be computationally expensive. These methods cannot easily be applied to stochastic models where a large number of facies realisations are considered. We have therefore implemented a new stochastic connectivity algorithm suitable for multiple facies realisations handling. extracting geobodies from multiple facies realisations, we can also compute a sand well-connection probability cube. For example, given an existing or planned well trajectory, we compute in each cell the number of realisations for which the cell is connected to the well. Figure 4 (bottom) shows the probability of sand connection to at least one of the three wells. It clearly shows that the channel sands on the left of the top picture are divided into small nonconnected bodies that cannot be drained from the existing wells. In addition to well connection probabilities, we can also compute the volume of sand connected to a specific well from each facies realisation and output a connected sand volume histogram for uncertainty analysis. Figure 3: Vertical section intersecting three wells and depicting two individual facies realisations generated using the P- impedance versus Poisson's ratio cross-plot and the sand polygon defined in Figure 2. In our method, we extract automatically geobodies from each individual realisation. Once computed, the geobodies can be used to define flow-aware ranking criteria, for example by sorting realisations according to the combined volume of their 10 biggest geobodies, or to the mean tortuosity (defined as the ratio between the surface and the volume of a geobody) of their 10 biggest geobodies. By Figure 4: Top: sand probability cube computed from 500 realisations (probabilities under 30% are not displayed). Bottom: probability of sand connection to at least one well. The sand bodies visible on the left of the top picture are not connected to any well and thus are not drainable from the current well configuration. Comparison with deterministic inversion Compared to deterministic inversion, the stochastic approach has several advantages. First it can be readily used for subsurface uncertainty analysis. Second it removes the statistical bias which may occur when computing net sand volume or other statistics from deterministic inversion
results. As an illustration of this flaw of averages (Mukerji and Mavko, 2005), we have computed an estimate of sand volume from the mean P- and S-impedances, obtained by averaging all corresponding realisations. In terms of spatial frequency content, the posterior means are an analogue for deterministic inversion in that the mean impedance results are band-limited and do not exhibit the high frequency details visible in individual realisations. Figure 5 (bottom left corner) shows the facies model obtained by applying the sand cross-plot polygon to the inverted I P and I S means. The corresponding sand volume is depicted just above. This deterministic sand volume estimate is much lower than predicted from the range of histogram values corresponding to the multiple realisations (top right corner). This apparent bias is a direct result of the smooth, band-limited nature of deterministic inversion compared to stochastic inversion. Conclusions We have illustrated how multiple realisations of I P and I S produced by stochastic seismic inversion can be used for uncertainty analysis and cascaded stochastic simulation of facies. In particular, we compute facies probability cubes for uncertainty analysis and design realisations ranking criteria which can be used for P 10 P 50 P 90 geological scenarios analysis. We have also developed a stochastic connectivity analysis workflow. From multiple impedance realisations, we automatically extract the geo-bodies connected to specific wells and compute connected sand probabilities and connected sand volume information. The stochastic connected sand volume analysis is particularly useful to assess uncertainty in drainable reservoir volume for existing or planned new wells. 40 P50 Number of realisations Sand volume from inversion mean P10 P90 0 Sand volume Figure 5: Comparison of the facies cubes derived from individual impedance realisations and from the mean of all impedance realisations (considered as an analogue to a deterministic inversion). Top: histogram of the total sand volume constructed from 500 facies realisations. Bottom: facies cube derived from the mean of all impedance realisations (left) and from one impedance realisation (right). Sand volume from deterministic inversion is significantly smaller than from any realisation and the geometry of the sand bodies is very different.
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