Classification

Bayesian decision theory allows for expressing the probability of a particular class given an observed x as

P(Cj│x) = P(x,Cj)/P(x) = (P(x│Cj) P(Cj))/P(x)

where P (x, C_j) is the joint probability of x and C_j, P (x | C_j) denotes the conditional probability of x given C_j. In order to establish a lithoclass x (i.e. gas sand, oil sand, brine sand, shale, etc.), a rock diagnostic is performed where, using the available well log data, a set of crossplots is generated utilizing the inversion output attributes (i.e. AI, SI, V_p/V_s, Poisson’s ratio, Lambda-rho, Mu-rho, etc.) and petrophysical logs (i.e. VCL, PHIT, S_w, etc.). The objective is to determine which elastic attributes best separate out the desired lithoclasses in the crossplot analysis. Last, probability density functions (pdfs) are generated from the well log data and are applied to the selected seismic inversion output cubes to produce class and probability cubes. At each sample, there is an associated probability which represents the likelihood of each class defined based on the cross plot analysis explained above. Several examples as well as a more comprehensive explanation of Bayesian statistics is given in the book Quantitative Seismic Interpretation by Avseth et al. (2005).

The next figure shows a map view of pay sand probability horizon slice. The result is part of a reservoir characterization study in the Burgos Basin, which includes a spatial lithology distribution based on lithoclasses generated using rock physics and Bayesian statistics. The lithology prediction was preceded by prestack simultaneous AVO inversion. The study was carried out to determine the lateral extent of the producing gas-saturated sand which has been penetrated twice successfully (Wells A and B) based solely on seismic amplitudes. Well C was a dry hole, which inspired the operator to find a different methodology to accurately characterize the reservoir sand.

The result may be interpreted as: given the information known from wells A and B, there is an 88.0% probability that hydrocarbons exist at Well A, which is where we know the well produced gas; the same goes for Well B, which shows a 61.5% probability. In Well C, we know there are no hydrocarbons and has a 47.6% probability. One may assign a conservative threshold of ~55% – 60% to determine whether gas saturated sand is present or not.

Map view of pay sand probability horizon slice.

The next case study is an example of using Bayesian statistics to reduce the risk of drilling. The patented workflow is discussed in The Interpretation Journal – November 2013, by Leiceaga et al. In this study, seismic-derived elastic properties from simultaneous inversion were successfully correlated with hydrocarbon production in an unconventional reservoir using Bayes’ theorem. A production classification was carried out where all wells were split into three categories (high, medium, and low) with respect to gas production. The classification was based on a 30-day average, initial production (IP) rate, where a medium producer fell within 3–5 MMCF∕day. Moreover, production rates are normalized to the lateral length of the horizontal wells.

The color scheme used in the resulting hydrocarbon production capacity (HPC) volume follows a traffic-light approach where green corresponds to expected high production, yellow to medium, and red to low; nonclassified samples are indicated in black.

The figure shows a hydrocarbon production capacity section at a vertical well where the upper perforation gave high production (green) while the lower perforation set is low (red). The result shows a good correlation with the observed production in both perforated intervals. The result also shows potential zones of high production capacity downdip.

Classification

Map view of pay sand probability horizon slice.

Hydrocarbon production capacity section. The upper perforation gave high production (> 5 MMCF/day) while the lower perforation was low (< 3 MMCF/day).