嵌入不透水球体复合多孔介质渗流场分析

Analysis of Seepage Field for Composite Porous Medium Embedded with Impervious Sphere

嵌入不透水球体的复合多孔介质广泛存在于地下水渗流领域,由于不透水球体的影响,多孔介质中的水在流经不透水球体时发生绕流,且距离球体越近绕流现象越明显。为求解该复合介质的渗流场和等效渗透系数,在球坐标系下采用非线性Brinkman-extended Darcy方程,以无穷远处单向来流均匀和球面不滑移条件作为边界条件,求解出球体周围渗流流速与压强的解析解。分析该解析解表达式可知,此复合介质渗流场流速和压强分布与多孔介质基质的渗透率和地下水动力黏滞系数有关。在此基础上对渗流场流速进行积分,求出了Darcy流平均流速,并将平均流速表达式与Darcy公式表达式进行对比,得出等效渗透系数表达式,并理论证明球体与多孔介质的相对尺寸是影响复合介质等效渗透系数的最主要因素。最后以一定级配的砂作为多孔介质基质,用直径分别为4、6、8 cm的3种有机玻璃球体作为不透水球体,自制试验装置,开展嵌入不透水球体复合多孔介质的常水头渗流试验。试验通过控制多孔介质基质的压实度来保证其渗透系数基本不变,改变填充在多孔介质基质中的不透水球体进行渗流试验,对每一个不透水球体分别进行3组试验,取平均值作为试验结果。试验结果与理论计算的等效渗透系数值误差较小,验证了所推求嵌入不透水球体复合多孔介质渗流场的正确性。

Composite porous media embedded with impermeable spheres are widely used in the field of groundwater seepage. Water flows around impermeable spheres in porous media due to the influence of impermeable spheres,and the closer to the sphere,the more obvious effect is. In order to obtain the seepage field and equivalent permeability coefficient,the nonlinear Darcy Brinkman-extended equation was applied in spherical coordinates. With no-slip conditionon sphere and the unidirectional uniformed flow at infinity as the boundary conditions,the analytical solution of seepage velocity and pressure around the sphere was obtained. By analyzing the expression of the analytical solution,the results showed that the velocity and pressure of the seepage field were related to the permeability of the porous media and the dynamic viscosity coefficient of the groundwater. On this basis,the Darcy-averaged flow was obtained via integral of velocity to calculate the equation of equivalent permeability coefficient,and the theory proved that the relative size of the sphere and the porous medium was the most important factor to affect the permeability coefficient of the composite. Finally,the seepage test was carried out by self-made test equipment by using a certain graded sand as a porous medium with three kinds of organic impervious glass spheres of diameters of 4 cm,6 cm and 8 cm,respectively. In the experiment,the degree of compaction of the porous media was controlled to keep its permeability coefficient almost unchanged,and the seepage experiment was carried out by changing the impervious sphere filled in the porous medium. Three groups of experiments were performed on each impervious sphere,and the mean value was taken as the test result. The equivalent permeability coefficient deviations between test results and theoretical calculation were small,and the correctness of the calculation of porous medium seepage field embedded with impermeable sphere was verified.