Abstract

Most analyses of fluid flow in porous media are conducted under the assumption that the permeability is constant. In some “stress-sensitive” rock formations, however, the variation of permeability with pore fluid pressure is sufficiently large that it needs to be accounted for in the analysis. Accounting for the variation of permeability with pore pressure renders the pressure diffusion equation nonlinear and not amenable to exact analytical solutions. In this paper, the regular perturbation approach is used to develop an approximate solution to the problem of flow to a linear constant-pressure boundary, in a formation whose permeability varies exponentially with pore pressure. The perturbation parameter α_D is defined to be the natural logarithm of the ratio of the initial permeability to the permeability at the outflow boundary. The zeroth-order and first-order perturbation solutions are computed, from which the flux at the outflow boundary is found. An effective permeability is then determined such that, when inserted into the analytical solution for the mathematically linear problem, it yields a flux that is exact to at least first order in α_D. When compared to numerical solutions of the problem, the result has 5% accuracy out to values of α_D of about 2—a much larger range of accuracy than is usually achieved in similar problems. Finally, an explanation is given of why the change of variables proposed by Kikani and Pedrosa, which leads to highly accurate zeroth-order perturbation solutions in radial flow problems, does not yield an accurate result for one-dimensional flow.

Article Highlights

• Approximate solution for flow to a constant-pressure boundary in a porous medium whose permeability varies exponentially with pressure.

• The predicted flowrate is accurate to within 5% for a wide range of permeability variations.

• If permeability at boundary is 30% less than initial permeability, flowrate will be 10% less than predicted by constant-permeability model.

中文翻译：

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应力敏感油藏一维流至恒压边界的摄动解

摘要

多孔介质中流体流动的大多数分析都是在渗透率恒定的前提下进行的。但是，在某些“应力敏感”的岩层中，渗透率随孔隙流体压力的变化足够大，以至于需要在分析中加​​以考虑。考虑到渗透率随孔隙压力的变化，使压力扩散方程呈非线性，不适合精确的解析解。在本文中，使用规则的扰动方法来开发渗透率随孔隙压力成指数变化的地层中流向线性恒压边界的问题的近似解决方案。扰动参数α _d定义为初始渗透率与流出边界处渗透率之比的自然对数。计算零阶和一阶扰动解，从中找到流出边界处的通量。然后，确定的有效磁导率，使得当插入到用于数学上线性问题的解析解，它产生一个磁通即精确到在至少一阶α _d。当相比于问题的数值解，结果有5％的准确度出来的值α _d大约为2-精度范围比通常在类似问题中所达到的范围要大得多。最后，给出了一个解释，说明了为什么Kikani和Pedrosa提出的变量变化会导致径向流问题中的高精度零阶扰动解，而不能产生一维流的精确结果。

文章重点

• 在渗透率随压力呈指数变化的多孔介质中，流至恒压边界的近似解。

• 对于大范围的渗透率变化，预测的流量精确到5％以内。

• 如果边界处的渗透率比初始渗透率低30％，则流量将比恒定渗透率模型预测的流量低10％。