Mathematical model and its solution for water-altering-gas (WAG) injection process incorporating the effect of miscibility, gravity, viscous fingering and permeability heterogeneity
水交替气（WAG）注入是一种旨在提高油藏采收率的技术，该技术受到重力、粘性指进和渗透率非均质性等因素的影响。这些参数的综合效应在WAG注入过程中不可忽视。本文提出了一个数学模型，用于模拟WAG注入过程中油的开采，并考虑了混相变化、粘性指进、重力和渗透率非均质性在倾斜层状油藏中的综合效应。首先，解释了控制方程和分数流函数，然后利用这些函数和守恒方程开发了一个数学模型，模拟了注水和溶剂的流动。该模型以准线性一阶偏微分方程的形式呈现，并在二维（2-D）中利用矢量微积分进行了解析求解。通过应用波理论到实际的恒定压力边界条件，模型在不同时间生成了不同的波形，提供了位移前沿位置的压力和饱和度。
通过解析解计算了不同时间的总体积通量和突破时间。所提出的解析解可以快速有效地预测层状多孔介质中的不同参数。最后，将解析解的结果与高分辨率数值模拟的结果进行了验证，显示出在突破时间、饱和度和不同时间不同波的位移位置的压力方面具有极好的一致性。这个解析解将为工程师提供节省时间和成本的指导，帮助他们分析不同时间的饱和度和压力分布，并预测油的采收率。它还将提高对WAG注入过程中多相流物理的理解。
CMG软件应用情况：
在本文中，CMG-GEM（Compositional simulator）作为商业模拟软件被用于进行数值模拟。该软件用于模拟WAG注入过程中的多相流动，以及在不同渗透率非均质性条件下的高分辨率数值模拟，以验证所提出的数学模型和解析解的准确性。
Abstract
The oil recovery from the water-alternating-gas (WAG) injection process is significantly impacted by gravity, viscous fingering, and the permeability heterogeneity of the reservoir. Therefore, the combined effect of these parameters cannot be neglected in the WAG injection process. This article presents the development of a mathematical model of oil recovery and its solution for the WAG injection process that takes into account the combined effects of miscibility change, viscous fingering, gravity, and permeability heterogeneity in an inclined stratified reservoir. First, the governing equations and fractional flow functions were explained in relation to the effects of gravity, permeability heterogeneity, viscous fingering, and miscibility change in an inclined stratified porous medium. Then, a mathematical model was developed using fractional flow functions and conservation equations for both injected water and solvent. The model was generated in the form of a quasi-linear first-order partial differential equation, which was solved analytically in two dimensions (2-D) utilizing vector calculus. Next, this model was solved analytically by applying wave theory to practical constant pressure boundary conditions, which generate distinct waves at different times to provide pressure and saturation at the displacement front location. The total volumetric flux and breakthrough time are calculated from the analytical solution at various times. The presented analytical solutions can be used to predict different parameters for stratified porous media in a fast and efficient way. Finally, the results of analytical solution validated with high-resolution numerical simulation for a wide range of permeability heterogeneity, which shows excellent agreement for breakthrough time, saturation, and pressure versus displacement location of different waves at different times. This analytical solution will save time and money by offering guidance to engineers for analyzing the saturation and pressure distribution at different times and predicting oil recovery. It will also improve the understanding of the physics underlying the multiphase flow WAG injection process in heterogeneous reservoirs.
作者单位：
科威特石油公司（Kuwait Oil Company, KOC）
