The Attractive Distribution Of Effective Permeability During Renormalization For 3D Case

Permeability is one of the important parameters,which is required in many fields of engineering like as reservoir engineering,environmental engineering,geoscience engineering etc.Average permeability values of grid blocks is used to perform the numerical simulation of reservoir.Determining a suitable effective value from these samples is necessary and this effective value is seemed as a single value for an equivalent homogeneous block.The renormalization process is an effective way to calculate the effective permeability.In this study,the renormalization process for the permeability and the attractive distribution of the renormalized permeability are investigated for 3D uncorrelated porous media.The attractive distribution is the invariant one after enough times of renormalization.It is well known that the attractive or fixed distribution is log-normal for 2D case but it isn't for 3D case.It is easy to see that in 3D case,the attractive distribution is p-normal but not the log-normal according the Central Limit theorem.It is discovered that if the permeability samples obey the attractive distribution,the renormalized permeability from infinite system will be independent of the distribution variance and the averaging exponent will also be independent of the mean or variance of the distribution.Making use of this important property,numerical simulation is then performed to approximate the averaging exponent function during the renormalization process.The simulation area includes n cubic elements,and then the system effective permeability is calculated numerically by solving the quasi-Laplace equation.In order to get the accurate results,each element cell needs to be refined enough.The finite analytical method(FAM)is applied here,because this numerical method can provide rather accurate results even in coarse calculation grids.After calculating the effective permeability K numerically,the value of the averaging exponent ?l+1 also has been determined.When performing numerical simulation,the number of the elements n can only take a limited value.In our simulation,we choose n?642 for 2D case and n=323 for 3D case,respectively.Based on the numerical results,the value of ?*seems to be 1/3 and the attractive distribution of the renormalized permeability in 3D case is found to be the truncated 1/3-normal one approximately,which is different from the Lognormal one in 2D case.At the attractive ?*-normal distribution,the averaging exponent used to calculate the effective permeability is found to be just ?*?independent of the distribution variation.This paper also proposes a conjecture about d-dimensional space:for d-dimensional spatially uncorrelated porous media,the renormalized permeability corresponds to the distribution of attractive as p-normal distribution,where p =(d-2)/d.This conjecture is correct for d=0,1,2 and d??;for d?3,according to the numerical results of this paper,it is basically correct.