We introduce a new geostatistical method to incorporate seismic attribute maps into a three-dimensional (3D) reservoir model. The method explicitly honors the difference in vertical resolution between seismic and well-log data. The method, called sequential Gaussian simulation with block Kriging (SGSBK), treats the seismic map as a soft estimate of the average reservoir property. With this method, the average of the cell values in anyone vertical column of grid cells is constrained by the value of the seismic map over that column. The result is a model that contains vertical variability driven by well logs and the vertical-variogram model and spatial variability driven by the seismic map and the areal-variogram model.
Reservoir models for flow simulation and volumetrics are often built from well logs by use of geostatistical methods. The well logs provide good vertical resolution required for accurate flow simulation, but they represent only a small portion of the reservoir. Seismic data is very complementary because it is areally dense, but vertically sparse relative to well-log data. The goal of the method presented is to integrate seismic data that more closely represents interval-average rock properties, and well-log data that more closely represents point-rock properties. This "volume support" difference is acknowledged and treated in the SGSBK method presented in this paper.
Geostatistical methods that aim to build 3D reservoir models must honor the difference in volume support between well and seismic data, whereas methods for areal two-dimensional (2D) models do not. Averaged or integrated log properties no longer represent point properties but rather interval properties with lower vertical resolution similar to seismic data. The differences in volume support between the log average and seismic data are acceptable, because both represent large volumes of rock. No special treatment for volume support is thus used in areal 2D simulations that use both log and seismic-map data. This special circumstance is not true for 3D models, so any 3D method to incorporate log data and seismic data should address the volume-support problem.
There are several problems associated with integrating seismic and well data for 3D reservoir characterization: the seismic data must be converted from time to depth domain; seismic data is band-limited, whereas well data has both high- and low-frequency components; seismic data must be calibrated to well data; and a well measurement is of quasipoint support, whereas a seismic datum informs a much larger volume of reservoir rock. (The term quasipoint properties is used to represent the properties in a single cell rather than a core plug or smaller.)
Several authors have worked on the calibration issue. Fournier1 and Fournier and Derain2 performed multivariate statistical analysis on a calibration dataset consisting of well logs and nearby seismic traces to establish a nonparametric regression between petrophysical properties and some seismic attributes. This regression is then applied on the seismic data to obtain seismic-derived reservoir properties that are, in turn, incorporated with well information using cokriging (and variants thereof). In their studies, Fournier and Fournier and Derain considered average properties (e.g., average porosity,1 cumulative lithofacies thickness2), because it was not possible to assess vertical distributions of reservoir properties from their limited-time-resolution seismic traces.
Zhu and Journel3 proposed a different use of the well-seismic calibration dataset. In lieu of a regression, the well (hard) and seismic (soft) data are encoded as local prior probability distributions which are then "updated" into posterior distributions during the sequential indicator simulation process.4 Values of the property of interest are drawn randomly from these local posterio~ distributions. This method was found to be superior by Araktingi et al.,5 who applied it to a synthetic seismic dataset.
Similarly, Doyen and Psaila6 used a "seismic likelihood function" constructed from a seismic-lithotype crossplot to modify the local probability distributions generated by the sequential indicator simulation algorithm; the result is lithologic models that are constrained by seismic data.
Xu et al.7 proposed the sequential Gaussian simulation (SGS) with collocated cokriging algorithm as a more efficient, albeit less rigorous, alternative to SGS with full cokriging.8,9 This algorithm requires the correlation coefficient between the well and seismic data, and their cross-covariance model is derived from the covariance model of the well data. Xu et al. showed a 2D study where the algorithm was applied to incorporate well data and seismic two-way travel times to create realizations of the structure top of a salt dome. Yang et al.10 used SGS with collocated cokriging to construct 3D porosity models conditional to both well and seismic data. In one approach, the seismic amplitude was used as soft data; the required correlation coefficient was obtained by crossplotting averaged porosity and absolute seismic amplitude. In another approach, the inverted seismic impedance was used as soft data. Although not explicitly stated by the authors, either an interpolation or simulation procedure was used to populate the simulation grid with soft seismic data, because the vertical resolution of the seismic data is much less than that of the simulation grid.
Gorell1 proposed a method to account for the difference in vertical resolution of seismic and well data. First, the wells are subdivided vertically into correlatable layers. Each layer is then populated with porosity values using 2D geostatistical operations. Finally, linear rescaling is performed on each vertical column of the simulation grid to ensure that a seismic-derived average porosity map is honored. The resulting 3D porosity model honors both the vertical variations at the well locations and average porosity map. This technique can be applied to several different vertical zones of the reservoir with different average porosity maps, and the rescaled results are stacked together at the end. As pointed out by the author, this technique requires that the wells be vertical or the well data may not be honored. In addition, vertical correlation of porosity between layers is only honored indirectly through the interwell-correlation process and the probability density function (pdf) of the point data is distorted, as will be shown later. Burns et al.12 used a similar resealing procedure to improve the description of a 100-ft-thick reservoir.