Jafari G Reza, Sahimi Muhammad, Rasaei M Reza, Tabar M Reza Rahimi
Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 19839, Iran.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Feb;83(2 Pt 2):026309. doi: 10.1103/PhysRevE.83.026309. Epub 2011 Feb 22.
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 ϕ(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)=ϕ(h+δh)-ϕ(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.
过去已经开发了几种方法来分析大型多孔介质（如油藏）的孔隙率和其他类型的测井曲线及其渗透率分布。我们开发了一种方法来分析孔隙率测井曲线ϕ(h)（其中h是深度）以及类似的通常为非平稳随机序列的数据。在这种方法中，首先基于原始数据生成一个新的平稳序列，然后分析所得序列。结果表明，基于对数y(h)=ϕ(h + δh)-ϕ(h)的连续增量的序列是一个平稳的马尔可夫过程，其特征在于马尔可夫长度尺度h(M)。然后计算条件概率密度函数（PDF）P(y,h|y(0),h(0))的Kramers-Moyal展开系数。所得的PDF满足福克-普朗克（FP）方程，该方程等同于y(h)的朗之万方程，为孔隙率测井提供概率预测。我们还表明，过去用于描述孔隙率测井曲线的自仿射分布的赫斯特指数H与我们为FP方程计算的漂移和扩散系数直接相关。还计算了穿越概率，这有助于识别在测量数据的深度区间之外孔隙率的高值或低值。
