Analysis of porosity distribution of large-scale porous media and their reconstruction by Langevin equation

Several methods have been developed in the past for analyzing the porosity and other types of well logs for large-scale porous media, such as oil reservoirs, as well as their permeability distributions. We developed a method for analyzing the porosity logs $\varphi (h)$ (where $h$ is the depth) and similar data that are often nonstationary stochastic series. In this method one first generates a new stationary series based on the original data, and then analyzes the resulting series. It is shown that the series based on the successive increments of the log $y(h)=\varphi (h+\delta h)-\varphi (h)$ is a stationary and Markov process, characterized by a Markov length scale $h_M$. The coefficients of the Kramers-Moyal expansion for the conditional probability density function (PDF) $P(y,h|y_0,h_0)$ are then computed. The resulting PDFs satisfy a Fokker-Planck (FP) equation, which is equivalent to a Langevin equation for $y(h)$ that provides probabilistic predictions for the porosity logs. We also show that the Hurst exponent $H$ of the self-affine distributions, which have been used in the past to describe the porosity logs, is directly linked to the drift and diffusion coefficients that we compute for the FP equation. Also computed are the level-crossing probabilities that provide insight into identifying the high or low values of the porosity beyond the depth interval in which the data have been measured.

Figure 1

The three porosity logs analyzed.

Figure 2

Relative likelihood function for $h_M$ and for well 3.

Figure 3

The drift coefficient $D^{(1)}$ for the three wells.

Figure 4

The diffusion coefficient $D^{(2)}$ for the three wells.

Figure 5

Comparison of the directly evaluated PDFs using the actual data and the PDFs obtained from Eq. (16). Values for $y_1$, from left to right, are $-0.1$, $0.0$, and $0.1$ [measured in units of $y_{\max }(t)$]. For better presentation, the PDFs have been shifted on the horizontal axis.

Figure 6

Comparison of the original data with the reconstructed ones for the three wells.

Figure 7

Frequency of crossing ${\nu }^{+}$ (top) and the average depth interval $l$ for observing $y(h)=\alpha$ again.

Figure 8

The scaling exponents $\xi (q)$ for the porosity logs of the three wells.