Fracture closure and proppant settling are two fully coupled processes during both shut-in and production. Proppant distribution greatly affects the residual fracture width and conductivity evolution, whereas fracture closure might limit proppant settling and force the proppant to crush or embed into the rock. Modeling fracture closure with proppant settling and embedment is challenging because of the multiple coupled physical processes involved, large time-scale differences, and extreme nonlinearity in the coupling of the processes. Conventional fracture-closure models either use simplified analytical estimates of the stress-dependent permeability of the reservoir or explicitly calculate the fracture width using empirical relationships, without considering the effect of fluid leakoff and dynamic changes in the proppant distribution in the fracture. In this work, we use a novel fully implicitly coupled fracturing/reservoir simulator to study fracture closure and proppant-settling/embedment processes during shut-in and production. This simulator implicitly couples the reservoir (rock deformation and porous flow), fracture (fracturing-fluid flow, proppant transport), and wellbore (slurry distribution, production) domains. During shut-in, a modified Barton-Bandis (Bandis et al. 1983) formula is used to describe the nonlinear relationship between the contact force and the residual fracture aperture considering the dynamic proppant spatial distribution and rock heterogeneity. During production, fracture conductivity is evaluated according to proppant distribution and further fracture closure caused by proppant crushing and embedment. A Newton-Raphson method is applied to solve the coupled system of equations.

Results from the simulations clearly show that typical periods of shut-in after fracturing lead to the formation of proppant banks at the bottom of the fracture in low-permeability, low-leakoff formations. This can lead to near-wellbore tortuosity and poor connectivity between the wellbore and the hydraulic-fracture network. Stress-dependent permeability, likely induced by induced unpropped fractures, is shown to be essential to obtain reasonable values of leakoff and to history match production trends. Proppant embedment is shown to be an important factor controlling production-decline rates in clay-rich shales.

裂缝闭合和支撑剂沉降是关井和生产过程中的两个完全耦合的过程。支撑剂分布极大地影响了残余裂缝宽度和电导率演化，而裂缝闭合可能会限制支撑剂沉降并迫使支撑剂压碎或嵌入岩石中。由于支撑过程涉及多个耦合的物理过程，较大的时标差异以及过程耦合中的极端非线性，因此用支撑剂沉降和嵌入来模拟裂缝闭合是一项挑战。传统的裂缝闭合模型要么使用简化的储层应力相关渗透率分析估算值，要么使用经验关系明确计算裂缝宽度，无需考虑流体泄漏的影响以及裂缝中支撑剂分布的动态变化。在这项工作中，我们使用一种新型的完全隐式耦合的压裂/储层模拟器来研究关井和生产过程中的裂缝闭合和支撑剂沉降/埋入过程。该模拟器将储层（岩石变形和多孔流），裂缝（压裂液流，支撑剂运输）和井筒（浆液分布，生产）域隐式耦合。在关井过程中，考虑到动态支撑剂的空间分布和岩石的非均质性，采用修正的Barton-Bandis（Bandis等，1983）公式来描述接触力和残余裂缝孔径之间的非线性关系。在生产过程中，根据支撑剂分布评估裂缝的导流能力，并根据支撑剂的压碎和埋入作用进一步闭合裂缝。牛顿-拉夫森法被用来求解方程的耦合系统。

模拟结果清楚地表明，压裂后的典型关井期会导致低渗透率，低渗漏地层中支撑剂在裂缝底部形成。这会导致近井眼曲折以及井眼与水力压裂网络之间的连通性差。事实证明，应力诱导的渗透性很可能是由未支撑的裂缝引起的，对于获得合理的渗漏值和历史拟合生产趋势至关重要。支撑剂嵌入被证明是控制富含粘土的页岩中产量下降速率的重要因素。