Identification of sedimentary facies with well logs: an indirect approach with multinomial logistic regression and artificial neural network

Abstract

Taking K-successions of the H-Zone of the Pearl River Mouth Basin as a testing example, we used two kinds of approaches to implement the microfacies identification. One is a direct identification, the other is an indirect approach in which we conducted the lithofacies classification first and then identified the microfacies based on previously estimated lithofacies. Both approaches were trained and checked by interpretations of experienced geologists from real subsurface core data. Multinomial logistic regression (MLR) and artificial neural network (ANN) were used in these two approaches as classification algorithms. Cross-validations were implemented. The source data set was randomly divided into training subset and testing subset. Four models, namely, MLR_direct, ANN_direct, MLR_indirect, and ANN_indirect, were trained with the training subset. The result of the testing set shows that the direct approaches (MLR_direct and ANN_direct) perform relatively poor with a total accuracy around 75%. While the indirect approaches (MLR_indirect and ANN_indirect) perform much better with a total accuracy of around 89 and 82%, respectively. This indirect method is simple and reproducible, and it could lead to a robust way of analyzing sedimentary microfacies of horizontal wells with little core data or even are almost never cored while core data are available for nearby vertical wells.

Appendix

1) Brief introduction to the MLR and the parameter information of the ANN used above

MLR is a generalization of the binary logistic regression. The dependent variable in a binary logistic regression has only two classes, i.e., “TRUE” or “FALSE,” while the dependent variable Y in the MLR has K classes (K > 2), which is denoted as {1, 2, ...K}.

The independent variables can be denoted as the following form:

$$ {\mathbf{X}}_i={\left({x}_{i_0},\cdot {x}_{i_1},\cdot {{x_i}_2}_{,\cdots }{x}_{i_{\mathrm{n}}}\right)}^{\mathrm{T}} $$

In above form, n denotes the amount of the independent variables. Note that ${x}_{i_0}$ is the intercept coefficient, ${x}_{i_0}=1$.

In the MLR model, taking the Kth class as the reference class, there should be

$$ \begin{array}{c}\hfill \ln \left(\frac{\mathrm{P}\left( Y=1\right)}{\mathrm{P}\left( Y= K\right)}\right)={\mathbf{B}}^{\left[1/ K\right]}{\mathbf{X}}_i\hfill \\ {}\hfill \ln \left(\frac{\mathrm{P}\left( Y=2\right)}{\mathrm{P}\left( Y= K\right)}\right)={\mathbf{B}}^{\left[2/ K\right]}{\mathbf{X}}_i\hfill \\ {}\hfill \ln \left(\frac{\mathrm{P}\left( Y= K-1\right)}{\mathrm{P}\left( Y= K\right)}\right)={\mathbf{B}}^{\left[\left( K\hbox{-} 1\right)/\mathrm{K}\right]}{\mathbf{X}}_i\hfill \end{array} $$

In the above form,${\mathbf{B}}^{\left[\mathrm{j}/\mathrm{K}\right]}\cdot ={\upbeta}_0^{\left[\mathrm{j}/\mathrm{K}\right]}\cdot =\left({\upbeta}_0^{\left[\mathrm{j}/\mathrm{K}\right]},\cdot {\upbeta}_1^{\left[\mathrm{j}/\mathrm{K}\right]},\dots {\upbeta}_{\mathrm{n}}^{\left[\mathrm{j}/\mathrm{K}\right]}\right)$ are partial regression coefficients related to X _i , where j ∈ {1, 2 … K − 1}. Note: the [j/K] means with jth class related parameter with the Kth class being the reference. After transformation, there are

$$ \begin{array}{l} P\left( Y=1\right)=\frac{e^{{\mathbf{B}}^{\left[1/ K\right]}{\mathbf{X}}_i}}{1+\sum_{k=1}^{K-1}{e}^{{\mathbf{B}}^{\mathrm{k}/ K\Big]}{\mathbf{X}}_{\mathrm{i}}}}\\ {} P\left( Y=2\right)=\frac{e^{{\mathbf{B}}^{\left[2/ K\right]}{\mathbf{X}}_i}}{1+\sum_{k=1}^{K-1}{e}^{{\mathbf{B}}^{\left[\mathrm{k}/ K\right]}{\mathbf{X}}_{\mathrm{i}}}}\\ {} P\left( Y= K\right)=\frac{1}{1+\sum_{k=1}^{K-1}{e}^{{\mathbf{B}}^{\left[\mathrm{k}/ K\right]}{\mathbf{X}}_{\mathrm{i}}}}\end{array} $$

For m observed cases (i ∈ {1, 2 … m}, the class of each case is known), the probability of each case should be

$$ {l}_i={P}_i^{\left( Y={K}_i\right)},{K}_i\in \left\{1,2,\dots K\right\}, i=1,2,\dots m $$

The likelihood function

$$ \mathrm{L}=\prod_{i=1}^m{l}_i $$

To get the maximum likelihood, there should be

$$ \left(\begin{array}{cccc}\hfill \frac{\partial \mathrm{L}}{\partial {\upbeta}_1^{\left[1/ K\right]}}=0\hfill & \hfill \frac{\partial \mathrm{L}}{\partial {\upbeta}_2^{\left[1/ K\right]}}=0\hfill & \hfill \dots \hfill & \hfill \frac{\partial \mathrm{L}}{\partial {\upbeta}_{\mathrm{n}}^{\left[1/ K\right]}}=0\hfill \\ {}\hfill \frac{\partial \mathrm{L}}{\partial {\upbeta}_1^{\left[2/ K\right]}}=0\hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill \\ {}\hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill \\ {}\hfill \frac{\partial \mathrm{L}}{\partial {\upbeta}_1^{\left[\left( K\hbox{-} 1\right)/ K\right]}}=0\hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill \frac{\partial \mathrm{L}}{\partial {\upbeta}_1^{\left[\left( K\hbox{-} 1\right)/ K\right]}}=0\hfill \end{array}\right) $$

Solving the equation set above with Newton-Raphson iteration, we can get the estimated coefficients ${\mathbf{B}}^{\left[ j/ K\right]}\cdot ={\upbeta}_0^{\left[ j/ K\right]},\cdot {\upbeta}_0^{\left[ j/ K\right]}\cdot =\left({\upbeta}_1^{\left[ j/ K\right]},\dots {\upbeta}_{\mathrm{n}}^{\left[ j/ K\right]}\right),\mathrm{where} j\in \left\{1,\cdot 2\dots K-1\right\}$.

The MLR is a classification method with no bias, which means that whichever class we choose as the reference class, the result will always be the same. We have searched many publications for the viewpoint, while none of the publications shows the detailed proving process of the unbiasness. So we decided to prove it and give the details out.

Proof on the unbiasness of arbitrary reference class:

$$ \mathrm{For}\forall p,\cdot q,\cdot \mathrm{and}\cdot r\in \left\{1,\cdot 2,\cdot \dots K\right\} $$

$$ \ln \left(\frac{\mathrm{P}\left( Y= p\right)}{\mathrm{P}\left( Y= q\right)}\right)={\mathbf{B}}^{\left[ p/ q\right]}{\mathbf{X}}_i $$

(1)

$$ \begin{array}{l} \ln \left(\frac{\mathrm{P}\left( Y= q\right)}{\mathrm{P}\left( Y= p\right)}\right)={\mathbf{B}}^{\left[ q/ p\right]}{\mathbf{X}}_i\\ {}(1)+(2)\Rightarrow \end{array} $$

(2)

$$ {\mathrm{B}}^{\left[ p/ q\right]}{X}_i=-{\mathbf{B}}^{\left[ q/ p\right]}{\mathbf{X}}_i.\mathrm{Also}, $$

(3)

$$ \ln \left(\frac{\mathrm{P}\left( Y= q\right)}{\mathrm{P}\left( Y= r\right)}\right)={\mathbf{B}}^{\left[ q/ r\right]}{\mathbf{X}}_i $$

(4)

$$ \begin{array}{l} \ln \left(\frac{\mathrm{P}\left( Y= r\right)}{\mathrm{P}\left( Y= p\right)}\right)={\mathbf{B}}^{\left[ r/ p\right]}{\mathbf{X}}_i\\ {}(1)+(4)+(5)\Rightarrow \end{array} $$

(5)

$$ {\mathbf{B}}^{\left[ p/ q\right]}{\mathbf{X}}_i=\left({\mathbf{B}}^{\left[ r/ q\right]}\hbox{-} {\mathbf{B}}^{\left[ r/ p\right]}\right){\mathbf{X}}_i $$

(6)

Taking Y = q as the reference class, then for the probability of Y = p,

$$ \begin{array}{l}{\mathrm{P}}^{\left[ p/ q\right]}=\frac{e^{{\mathbf{B}}^{\left[ p/ q\right]}{\mathbf{X}}_i}}{1+\sum_{w=1,\cdot 2\dots q-1,\cdot q+1\dots K}{e}^{{\mathbf{B}}^{\left[ w/ q\right]}{\mathbf{X}}_i}}\\ {}=\frac{1}{e^{\hbox{-} {\mathbf{B}}^{\left[ p/ q\right]}{\mathbf{X}}_i}+\sum_{w=1,\cdot 2\dots q-1,\cdot q+1\dots K}{e}^{\left({\mathbf{B}}^{\left[ w/ q\right]}-{\mathbf{B}}^{\left[ p/ q\right]}\right){\mathbf{X}}_i}}\\ {}=\frac{1}{e^{\hbox{-} {\mathbf{B}}^{\left[ p/ q\right]}{\mathbf{X}}_i}+\sum_{w=1}^K{e}^{\left({\mathbf{B}}^{\left[ w/ q\right]}-{\mathbf{B}}^{\left[ p/ q\right]}\right){\mathbf{X}}_i}-{e}^{\hbox{-} {\mathbf{B}}^{\left[ p/ q\right]}{\mathbf{X}}_i}}\end{array} $$

Considering Eqs. (3) and (6), then

$$ {\mathrm{P}}^{\left[ p/ q\right]}=\frac{1}{\sum_{w=1}^K{e}^{{\mathbf{B}}^{\left[ w/ p\right]}{\mathbf{X}}_i}} $$

(7)

Hence, from Eq. (7), the probability of Y = p has nothing to do with which is the reference class.

Some parameters about the ANN used above are as follows:

Type: multi-layer perceptron; number of hidden layer: 1; hidden layer activation function: tanh; output layer activation function: Softmax.

2) Combination Rules from Lithofacies to Microfacies

In order to illustrate the procedure clearly, we use a programming style language to express the combination rules. The rules are from the common geological consensus and are also modified according to the core observers’ experience:

• 1. Filter thin layers thinner than 0.2 m, for i is from 2 to (n − 1). #This step is to minimize the effect of some individual fluctuation.

If ((Thickness [i] ≤ 0.2) AND (Lf [i + 1] = Lf [i − 1]) AND ((Thickness [i + 1] > 0.2) OR (Thickness [i − 1] > 0.2)), then {Lf [i] = Lf [i + 1]}.

2. Exchange certain codes, for i is from 1 to n. #This step is to combine several similar lithofacies into a collective code.

If ((Lf [i] = Sp) OR (Lf [i] = Fsp) OR (Lf [i] = Fp)), then {Lf2 [i] = B3}.

If ((Lf [i] = Fsf) OR (Lf [i] = Fsm)), then {Lf2 [i] = M2}.

3. Combination step 1, for i is from 1 to n (the reference class of Mf is OS). #This step is to initially classify some distinct microfacies.

If (Lf2 [i] = B3), then {Mf [i] = OB}.

If ((Lf [i]) = Fsm) AND (Thickness [i] > 2)), then {Mf [i] = OB}.

If (Lf [i] = Fm), then {Mf [i] = OM}.

4. Combination step 2, for i is from 2 to (n − 1). #This step is to further identify some microfacies.

If ((Lf2 [i] = M2) AND ((Lf2 [i − 1] = B3) OR (Lf2 [i + 1] = B3)), then {Mf [i] = OB}.

5. Combination step 3, for i is from 3 to n. This step is to classify the prevailed coarsening-upwards sequences.

If ((Lf [i] = Fsf) AND (Lf [i − 1] = Fsm) AND (Lf2 [i − 2] = B3)), then {Mf [i] = OB}.

6. Combination the rest, for i is from 1 to n. #This step is to classify the rest category.

If ((Mf [i] ≠ OB) AND (Mf [i] ≠ OM)), then {Mf [i] = OS}.

Notes: i denotes the ith (from above down) lithofacies unit to be estimated; Thickness denotes the thickness of a certain unit; n denotes the total account of the lithofacies unit; Lf denotes the lithofacies type of a certain unit belongs to; Lf2 denotes the copy of Lf; and Mf denotes the microfacies type of a certain unit belongs to.