Summary
The methodology presented in this paper incorporates seismic data, geological knowledge and well logs to produce models of reservoir parameters and uncertainties associated with them. A three-dimensional (3D) seismic dataset is inverted within a geological and stratigraphic model using the geostatistical inversion technique. Several reservoir-scale acoustic impedance blocks are obtained and quantification of uncertainty is determined by computing statistics on these 3D blocks. Combining these statistics with the kriging of the reservoir parameter well logs allows the transformation of impedances into reservoir parameters. This combination is similar to performing a collocated cokriging of the acoustic impedances.
Introduction
Our geostatistical inversion approach is used to invert seismic traces within a geological and stratigraphic model. At each seismic trace location, a large number of acoustic impedance (AI) traces are generated by conditional simulation, and a local objective function is minimized to find the trace that best fits the actual seismic trace. Several three-dimensional (3D) AI realizations are obtained, all of which are constrained by both the well logs and seismic data. Statistics are then computed in each stratigraphic cell of the 3D results to quantify the nonuniqueness of the solution and to summarize the information provided by individual realizations.
Finally, AI are transformed into other reservoir parameters such as Vshale through a statistical petrophysical relationship. This transformation is used to map Vshale between wells, by combining information derived from Vshale logs with information derived from AI blocks. The final block(s) can then be mapped from the time to the depth domain and used for building the flow simulation models or for defining reservoir characterization maps (e.g., net to gross, hydrocarbon pore volume).
We illustrate the geostatistical inversion method with results from an actual case study. The construction of the a-priori model in time, the inversion, and the final reservoir parameters in depth are described. These results show the benefit of a multidisciplinary approach, and illustrate how the geostatistical inversion method provides clear quantification of uncertainties affecting the modeling of reservoir properties between wells.
Methodology
The Geostatistical Inversion Approach.
This methodology was introduced by Bortoli et al.1 and Haas and Dubrule.2 It is also discussed in Dubrule et al.3 and Rowbotham et al.4 Its application on a synthetic case is described in Dubrule et al.5 A brief review of the method will be presented here, emphasizing how seismic data and well logs are incorporated into the inversion process.
The first step is to build a geological model of the reservoir in seismic time. Surfaces are derived from sets of picks defining the interpreted seismic. These surfaces are important sincethey delineate the main layers of the reservoir and, as we will see below, the statistical model associated with these layers, andthey control the 3D stratigraphic grid construction.
The structure of this grid (onlap, eroded, or proportional) depends on the geological context. The maximum vertical discretization may be higher than that of the seismic, typically from 1 to 4 milliseconds. The horizontal discretization is equal to the number of seismic traces to invert in each direction (one trace per cell in map view).
Raw AI logs at the wells have to be located within this stratigraphic grid since they will be used as conditioning data during the inversion process. It is essential that well logs should be properly calibrated with the seismic. This implies that a representative seismic wavelet has been matched to the wells, by comparing the convolved reflectivity well log response with the seismic response at the same location. This issue is described more fully in Rowbotham et al.4
Geostatistical parameters are determined by using both the wells and seismic data. Lateral variograms are computed from the seismic mapped into the stratigraphic grid. Well logs are used to both give an a priori model (AI mean and standard deviation) per stratum and to compute vertical variograms.
The geostatistical inversion process can then be started. A random path is followed by the simulation procedure, and at each randomly drawn trace location AI trace values can be generated by sequential Gaussian simulation (SGS). A large number of AI traces are generated at the same location and the corresponding reflectivities are calculated. After convolution with the wavelet, the AI trace that leads to the best fit with the actual seismic is kept and merged with the wells and the previously simulated AI traces. The 3D block is therefore filled sequentially, trace after trace (see Fig. 1).
It is possible to ignore the seismic data in the simulation process by generating only one trace at any (X, Y) location and automatically keeping it as "the best one." In this case, realizations are only constrained by the wells and the geostatistical model (a-priori parameters and variograms).
Seismic inversion by Newtonian machine learning