改进叠加原理求解低渗储层关井阶段压力响应

Pressure responses in shut-in period by modified pressure superposition principle for low permeability reservoirs

流体在低渗储层中的流动受流固界面作用影响,是低速非达西渗流。拟启动压力梯度常用于描述非达西渗流,但其在关井阶段的作用机制尚不明确。直接应用传统压降叠加原理的关井阶段压力响应计算结果已被证明和数值计算所得规律相反。针对上述矛盾,基于拟启动压力梯度,建立了低渗储层关井阶段试井数学模型。该模型是非齐次的二阶线性微分方程。根据其解的可叠加性,证明和提出了改进的压降叠加原理,即低渗储层关井阶段井底压力变化等于一个考虑拟启动压力梯度时变化值与一个不考虑拟启动压力梯度时变化值的叠加。计算结果显示,开井过程拟启动压力梯度使压力-压力导数曲线后期上翘,作用机制类似于"封闭"边界;关井过程拟启动压力梯度使压力-压力导数曲线后期下掉,作用机制类似于"恒压"边界。计算得到的曲线变化趋势与文献中数值计算结果吻合,验证了改进压降叠加原理的正确性。改进的压降叠加原理解决了传统压降叠加原理在非达西渗流压力恢复测试解释的错误,使得通过压力恢复测试准确识别储层平均拟启动压力梯度成为可能。

The fluids flow in the low permeability reservoir as the low velocity non-Darcy flow, which has a nonlinear section in the small pressure gradient range. The corresponding flow equations are nonhomogeneous or non-linear, leading to the traditional pressure superposition principle is improper anymore. Specifically, its change law of threshold pressure gradient（TPG） on build-up well test typical curves is contrast to the results from numerical method. By introducing pseudo TPG, the established well test mathematical model is a linear system. According to its additivity, we propose a new pressure superposition principle and prove it mathematically. The new principle shows that there are two pressure drop resulting in the total change of bottom hole pressure（BHP） in shut-in period, which are pressure drop in a real well with pseudo TPG and that in a virtual well without pseudo TPG, respectively. Based on the characteristic curves, we also observe that TPG makes the curves value to increase in production or injection process, and to decrease in buildup process, which is same with change law from numerical calculation in literature. In production or injection process, it is resulted from a closed moving-boundary. In buildup process, it is resulted from a supply moving-boundary. The new pressure superposition principle improve the accuracy of pressure interpretation in buildup process, giving exact TPG values to evaluate the flow state.