双重介质部分射开井压力动态模型的统一构造解法

A Unified Constructive Solution for the Pressure-Transient Model of Partially Penetrating Wells in Dual-Porosity Reservoirs

摘要: 裂缝性碳酸盐岩储层天然裂缝发育、厚度大，而生产井射开段一般仅几米，研究其部分射开井压力解具有重要意义。为此，本文建立双重介质油藏部分射开井渗流模型：径向外边界可取定压、封闭或无穷大三种，垂向边界由四种Dirichlet-Neumann组合构成；径向边界决定三类贝塞尔型相似核函数，垂向边界决定对应的Sturm-Liouville特征函数、特征值与归一化因子。本文首次将相似构造法引入该模型，通过Laplace变换与有限Fourier正余弦变换，将渗流偏微分方程化为二阶常微分方程边值问题，进而以两个线性无关解为基础构造相似核函数，并利用边界条件确定其中的待定系数，从而建立适用于各边界组合的统一解析框架，实现全部工况在Laplace空间下解析解的统一表达。

Abstract: Fractured carbonate reservoirs are characterized by well-developed natural fractures and large formation thickness, whereas the perforated intervals of production wells are generally only a few meters long. Therefore, investigating the pressure solutions for partially penetrating wells in such reservoirs is of considerable significance. To this end, a dual-porosity reservoir flow model for partially penetrating wells is established in this paper: the radial outer boundary may be of constant-pressure, no-flow, or infinite type, while the vertical boundaries consist of four Dirichlet-Neumann combinations. The radial boundary determines three types of Bessel-type similarity kernel functions, and the vertical boundary determines the corresponding Sturm-Liouville eigenfunctions, eigenvalues, and normalization factors. For the first time, the similarity construction method is introduced into this model. By applying the Laplace transform and the finite Fourier sine and cosine transforms, the seepage partial differential equation is reduced to a boundary-value problem for a second-order ordinary differential equation; the similarity kernel function is then constructed from two linearly independent solutions, with the undetermined coefficients determined by the boundary conditions. In this way, a unified analytical framework applicable to all boundary combinations is established, yielding a unified expression of the analytical solutions in the Laplace domain for all cases.