The method of micro-sized Graded Proppant Placement (GPP) has been proposed in the industry to increase the size of the stimulated zone in naturally fractured reservoirs. The models of determining the Optimal Proppant Packing Ratio (OPPR) and the optimal injection schedule have also been established. However, all the previous studies neglected the impact of Proppant Embedment into fracture wall and Proppant Deformation (PEPD), and therefore significant discrepancies between modeling results and experimental results have been observed. In this study, models of calculating PEPD are established and incorporated into the GPP model, and the influences of proppant embedment and deformation on OPPR, optimal injection schedule and predicted post-stimulation productivity increase are investigated. Conductivity Correction Factor (CCF) is introduced to find the OPPR and the results are compared with the results obtained from previous models which are based on Permeability Correction Factor (PCF).

The results show that smaller proppant Elastic Modulus or smaller rock Elastic Modulus leads to bigger OPPR and lower PCF. The CCF -based OPPR matches better with previous experimental results than the PCF -based value. The study also shows that with consideration of PEPD, OPPR is significantly larger; the optimal proppant concentrations at different stages of graded proppant injection are accordingly larger, and the predicted folds of productivity increase after stimulation is much lower.

The results from this study can be used to optimize the planning of GPP. For the stimulations of soft formations like CBM, shale etc. or the stimulations where less-rigid proppants like silica sand or walnut shell are used, the findings of this study are even more important. If the proppants and the formation rock are assumed to be rigid bodies and PEPD are neglected in such situations, 32.5% lower of proppant concentration will be caused, which will consequently lead to under-propping of the fracture and the productivity will not be maximized supposedly. Meanwhile, the predicted folds of productivity increase will be overestimated by 27.5% maximally. The larger the stimulation zone, the stronger the influence of PEPD. PEPD must be considered when planning a large-scale GPP in soft formations.

This study is a continuation of the studies conducted by the authors earlier. The major novel elements are that PEPD are considered, the influences of PEPD are investigated, and fracture conductivity is used instead of permeability to find the optimal proppant packing ratio.

在工业中已经提出了微尺寸的分级支撑剂放置（GPP）的方法，以增加天然裂缝储层中的受激带的大小。还建立了确定最佳支撑剂装填率（OPPR）和最佳喷射时间表的模型。然而，所有先前的研究都忽略了支撑剂嵌入裂缝壁和支撑剂变形（PEPD）的影响，因此，在建模结果和实验结果之间发现了显着差异。在这项研究中，建立了计算PEPD的模型并将其纳入GPP模型，并研究了支撑剂的嵌入和变形对OPPR，最佳注入时间表和预测的增产后生产率提高的影响。

结果表明，较小的支撑剂弹性模量或较小的岩石弹性模量会导致较大的OPPR和较低的PCF。与基于PCF的值相比，基于CCF的OPPR与先前的实验结果更好地匹配。研究还表明，考虑到PEPD，OPPR明显更大；因此，分级支撑剂注入的不同阶段的最佳支撑剂浓度较大，而增产后的预测生产率提高倍数要低得多。

这项研究的结果可用于优化GPP的计划。对于CBM，页岩等软地层的增产，或使用硅砂或核桃壳等较不坚硬的支撑剂的增产，这项研究的结果更为重要。如果将支撑剂和地层岩假定为刚性体，而在这种情况下忽略了PEPD，则将导致支撑剂浓度降低32.5％，从而导致裂缝的支撑不足，并且生产力可能无法最大化。 。同时，生产率的预期增长倍数将被最大高估27.5％。刺激区域越大，PEPD的影响越强。当规划软地层的大规模GPP时，必须考虑PEPD。

这项研究是作者先前进行的研究的延续。主要的新颖因素是考虑了PEPD，研究了PEPD的影响，并使用裂缝导流率代替渗透率来找到最佳的支撑剂堆积比。