Page 90 - Machine Learning for Subsurface Characterization

Shallow neural networks and classification methods Chapter 3 75

0
where i is an index that identifies the Gaussian distribution, T 2 ¼ log (T 2 ); g i is
the probability distribution function of a Gaussian distribution with mean μ i and
standard deviation σ i ; α i represents the proportion of pore volumes representing
the constituent Gaussian distribution with respect to total pore volume, such that
α 1 + α 2 + α 3 ¼ 1; and A is the amplitude parameter. In our study, the shale sys-
tem exhibits NMR T 2 distributions having either one or two peaks. We fit the T 2
distributions using a modified version of Eq. (3.2) expressed as
2
X

fT 0 ¼ ðÞg i μ , σ i , T 0 (3.3)
2 α i i 2
i¼1
Compared with Eq. (3.2),Eq. (3.3) does not implement the amplitude
parameter A and α 1 + α 2 6¼ 1. When using Eq. (3.3), six parameters are required
to fit the T 2 distribution response at each depth. The six parameters are μ i , σ i , and
α i , for i ¼ 1 and 2. The reliability of the fitting is expressed in terms of the coef-
2
ficient of determination R formulated as
2 RSS
R ¼ 1 = TSS (3.4)
where
n
2
X
0
RSS ¼ f i, fit T f i T 0 (3.5)
2 2
i¼1
and
n
i
X h 2

0
TSS ¼ f i T fT 0 (3.6)
2 2
i¼1
where n ¼ 64 is the number of bins into which the original T 2 distribution (cor-
responding to a depth) is discretized, f i (T 2 ) represents the ith discretized T 2 dis-
0
0
tribution measurement, f i, fit (T 2 ) represents the fit to the ith discretized T 2

distribution computed using the Eq. (3.3), and fT 2 0 is the mean of the 64 dis-
cretizations of the original T 2 distribution for the given depth. RSS is the sum of
squares of the residuals, and TSS is the total sum of squares proportional to the
variance of the data. T 2 distributions acquired at 416 depth points in the shale
system were fitted with Eq. (3.3) to estimate the six characteristic fitting param-
eters for each depth point. In doing so, the 64 bins of NMR T 2 are transformed to
six logs, which were used for training and testing the second ANN-based pre-
dictive model. For T 2 distribution with single peaks, α 2 ¼ μ 2 ¼ σ 2 ¼ 0. Figs. 3.3
and 3.4 show the results of fitting for randomly sampled depth points. T 2 dis-
2
tributions were fitted at median R of 0.983 (Fig. 3.3). Only 12% of the depths
2
were fitted with R lower than 0.95.
Normalized root mean square error (NRMSE) in synthesizing a specific T 2
2
bin across all the depths is used together with R of synthesizing all the 64 bins
at a specific depth to assess the accuracy of fitting and predicting the NMR T 2
distributions. NRMSE for any specific discretized NMR T 2 bin is expressed as