A fusion approach to identify reservoir facies based on rock physics modeling

Abstract

Rock physics modeling is the process of finding a relationship between seismic data and well logs. The concluded model can play a crucial role in seismic reservoir characterization studies as well as prediction of reservoir geomechanical parameters. The present study is aimed at proposing a fusion approach based on rock physics modeling to identify reservoir facies. First, we determine suitable rock physics models to predict lithology type of formation under study (the Asmari Formation) in the Mansuri oil field, south of Iran. On the basis of pore fluid type, each of the identified lithologies is divided into two classes as oil saturated facies and water saturated facies. Then we use Support Vector Machine classifier to identify desired facies. The classification is performed in two stages: single well analysis and multi-well analysis. Finally, optimistic ordered weighted averaging method is employed to fuse the results of more than one training wells.

Appendix

Solving convex optimization problem

$$ \mathrm{Minimize}\kern0.5em \frac{1}{2}{\left\Vert w\right\Vert}^2 $$

Subject to (for any i=1, … , n)

$$ {y}_i\left({w}^T.{x}_i + b\right)\ge 1 $$

(1)

The above constrained problem can be expressed as

$$ \mathrm{Minimize}:\ L\left(w,\ b,\ \lambda \right) = \frac{1}{2}{\left\Vert w\right\Vert}^2-{\displaystyle \sum_{i=1}^N}{\lambda}_i\left[{y}_i\left({w}^T.{x}_i + b\right) - 1\right] $$

(2)

Where λ _i ≥0 are the Lagrange multipliers. By setting the derivatives of L with respect to w and b to zero, we have:

$$ \frac{\partial L\left(w,b,\lambda \right)}{\partial w}=w - {\displaystyle \sum_{i=1}^N}{y}_i{\lambda}_i{x}_i=0\to w={\displaystyle \sum_{i=1}^N}{y}_i{\lambda}_i{x}_i $$

(3)

$$ \frac{\partial L\left(w,b,\lambda \right)}{\partial b} = {\displaystyle \sum_{i=1}^N}{y}_i{\lambda}_i=0 $$

(4)

Substituting results from Eqs. (3) and (4) into Eq. (2) gives:

$$ W\left(\lambda \right) = {\displaystyle \sum_{i=1}^N}{\lambda}_i - \frac{1}{2}{\displaystyle \sum_{i=1}^N}{y}_i{y}_j{\lambda}_i{\lambda}_j\left({x}_i,{x}_j\right) $$

(5)

As it is obvious, the optimal λ _i is the one that maximizes the function (4). It can be expressed as:

$$ \mathrm{Maximize}:\ W\left(\lambda \right) = {\displaystyle \sum_{i=1}^N}{\lambda}_i-\frac{1}{2}{\displaystyle \sum_{i=1}^N}{y}_i{y}_j{\lambda}_i{\lambda}_j\left({x}_i,{x}_j\right) $$

(6)

$$ \begin{array}{ll}\mathrm{Subject}\ \mathrm{t}\mathrm{o}:{\displaystyle \sum_{i=1}^N}{\lambda}_i{y}_i=0,\ {\lambda}_i\ge 0\hfill & i=1,\ 2,\ 3, \dots,\ n\hfill \end{array} $$

(7)

This leads to the decision function bellow:

$$ f(x) = \mathrm{sign}\left({\displaystyle \sum_{i,j=1}^N}{\lambda}_i{y}_i\left({x}_i,{x}_j\right) + b\right) $$

(8)