<p>Reservoir permeability is a critical parameter for oil production estimation, exhibiting high spatial variability which suggests treatment as a random function. The present paper focuses on the study of steady state flow of a single phase fluid in a two-dimensional heterogeneous formation with random permeability characterized by a log-normal distribution with coefficient of variation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\gamma _k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> and correlation length <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>. The aim of this study is to determine the relationship between the flux variance <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\sigma _q^2$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>σ</mi> <mi>q</mi> <mn>2</mn> </msubsup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\gamma _k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation>. Previously available analytical solutions to this problem have primarily relied on perturbation-based methods, which are approximate and applicable for relatively small values of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\gamma _k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation>. This study employs the Monte Carlo method to investigate fluid flow in reservoirs with stochastic permeability where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\gamma _k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> ranges from 0.1 to 4.0. The results show that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\sigma _q^2$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>σ</mi> <mi>q</mi> <mn>2</mn> </msubsup> </math></EquationSource> </InlineEquation> exhibits a maximum at a certain critical value of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\gamma _k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation>, denoted as <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\gamma _c$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mi>c</mi> </msub> </math></EquationSource> </InlineEquation>. For highly heterogeneous formations <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\gamma _k > \gamma _c$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>γ</mi> <mi>k</mi> </msub> <mo>></mo> <msub> <mi>γ</mi> <mi>c</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, this study shows that the fluid flux follows the Poisson distribution and flux variance is proportional to its mean value <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\langle q \rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <mi>q</mi> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation>. Such behavior is characteristic of random variables that follow a Poisson distribution. It is well known that the Poisson distribution is a discrete probability distribution and this study shows that for the case <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\gamma _k > \gamma _c$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>γ</mi> <mi>k</mi> </msub> <mo>></mo> <msub> <mi>γ</mi> <mi>c</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, a network of discrete preferential flow channels forms. Such channels are not associated with any extended geological objects like faults or fractures and are caused by a random medium. The possibility of detecting such channels is assessed through the analysis of pressure build-up curves. The process of water-oil displacement is also studied for the case of highly heterogeneous reservoirs. Water breakthrough time shown as a function of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\gamma _k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> illustrates the noticeable influence of the statistical characteristics of permeability on the oil displacement process, that can be used in history matching process.</p>

Particularity of fluid flow in highly heterogeneous stochastic porous media

期刊论文
发表日期：

2026年3月18日

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Posvyanskii D.V.

摘要

Reservoir permeability is a critical parameter for oil production estimation, exhibiting high spatial variability which suggests treatment as a random function. The present paper focuses on the study of steady state flow of a single phase fluid in a two-dimensional heterogeneous formation with random permeability characterized by a log-normal distribution with coefficient of variation $\gamma _k$ $γ_{k}$ and correlation length $\lambda $ $λ$ . The aim of this study is to determine the relationship between the flux variance $\sigma _q^2$ $σ_{q}^{2}$ and $\gamma _k$ $γ_{k}$ . Previously available analytical solutions to this problem have primarily relied on perturbation-based methods, which are approximate and applicable for relatively small values of $\gamma _k$ $γ_{k}$ . This study employs the Monte Carlo method to investigate fluid flow in reservoirs with stochastic permeability where $\gamma _k$ $γ_{k}$ ranges from 0.1 to 4.0. The results show that $\sigma _q^2$ $σ_{q}^{2}$ exhibits a maximum at a certain critical value of $\gamma _k$ $γ_{k}$ , denoted as $\gamma _c$ $γ_{c}$ . For highly heterogeneous formations $\gamma _k > \gamma _c$ $γ_{k} > γ_{c}$ , this study shows that the fluid flux follows the Poisson distribution and flux variance is proportional to its mean value $\langle q \rangle $ $⟨ q ⟩$ . Such behavior is characteristic of random variables that follow a Poisson distribution. It is well known that the Poisson distribution is a discrete probability distribution and this study shows that for the case $\gamma _k > \gamma _c$ $γ_{k} > γ_{c}$ , a network of discrete preferential flow channels forms. Such channels are not associated with any extended geological objects like faults or fractures and are caused by a random medium. The possibility of detecting such channels is assessed through the analysis of pressure build-up curves. The process of water-oil displacement is also studied for the case of highly heterogeneous reservoirs. Water breakthrough time shown as a function of $\gamma _k$ $γ_{k}$ and $\lambda $ $λ$ illustrates the noticeable influence of the statistical characteristics of permeability on the oil displacement process, that can be used in history matching process.