考虑二次梯度项低渗透油藏水平井压力动态分析

针对低渗油藏中渗流必须克服启动压力梯度的特点，建立了考虑启动压力梯度的盒状油藏水平井不稳定渗流数学模型，并在微分方程中保留了非线性二次梯度项的影响。首先通过对数变换将非线性微分方程线性化，再利用伽辽金有限元方法导出了模型的控制方程组，用直接迭代法进行求解，代入反变换，最后得到井底压力数值解，并与经典的Odeh模型解析解对比验证了该数值算法的正确性。低渗油藏水平井井底压力的计算结果表明：启动压力梯度存在时，水平井压力及压力导数曲线都呈现出上翘趋势，且启动压力梯度越大，曲线上翘越明显，即井底所需压差越大；当充分考虑流体压缩性时，水平井井底压差比忽略二次梯度项的结果偏低，且压缩系数变大，压力解之间的误差将呈非线性增长。

Dynamic pressure analysis of horizontal wells in low permeability reservoirs with the effect of quadratic gradient term

In view of the threshold pressure gradient(TPG) phenomenon that the seepage flow in low permeability reservoirs must overcome,an unsteady flow mathematical model for the horizontal well in box-shaped reservoirs was established with the nonlinear quadratic gradient term remaining in differential equations.Firstly,nonlinear differential equations were linearized through logarithmic transformation,then model controlling equations were derived using the Galerkin finite element method(FEM) and solved by the direct iteration method.After the substitution of the inverse transformation,the numerical solution of bottom pressure was finally obtained and verified by comparison with the classic Odeh solution.Calculation results of bottom hole pressure for the horizontal wells demonstrated that both pressure and pressure derivative curves showed an upwarping tendency as the TPG was considered,the bigger the TPG was,the more distinct the curves upwarped,i.e.the greater the downhole pressure difference was required.With the presence of fluid compressibility,the downhole pressure difference was less than that derived from the model that ignored the nonlinear quadratic gradient term,and the error between them increased nonlinearly with the increase of the fluid compressibility factor.