A New Mathematical Model to Calculate the Multi-Layer Fracture Conductivity
Authors: Hou Tengfei , Shang Litao , Wang Yehan , Xiao Cong , Chen Weihua , Zhang Yang , Zhang Shiling , Liu Fei , Ren Guangcong 
Published: May 2024

Abstract
Unlike conventional oil and gas reservoirs, shale reservoirs develop a variety of weak structural planes, including structural fractures, interlayer bedding fractures, bedding slip fractures and abnormal pressure fractures. These weak structural planes affect the morphology of artificial fractures during volume fracturing, especially the propagation of fracture height. However, for the coexistence of horizontal and vertical fracture case, the existing models fall short in accurately predicting proppant embedment and fracture conductivity.
In this study, new mathematical models are conducted to evaluate the proppant embedment and the complex fractures conductivity coupled with horizontal and vertical fractures. Taking the stimulated volume covered by horizontal and vertical fractures as a whole, we utilize the Carman-Kozeny equation to estimate the permeability of the primary fracture, and the monolayer proppant laying model was employed to forecast the permeability of each branch fracture. Then, the combined permeability is calculated by using Darcy's Law under the condition of multiple fractures flowing in parallel pattern. Furthermore, we apply particle deformation and creeping deformation models to assess the temporal evolution of complex fracture conductivity. Compared with previous mathematical models, the biggest difference of this model is that it needs to consider the number of vertical fractures and horizontal fractures and the proportion of flow rate. Factors influencing the development of proppant embedment and complex conductivity, including number of branch fractures, closure pressure, elastic-plastic properties, proportion of flow rate are investigated. This study also examines the steady state of conductivity.
The findings show that our proposed analytical model aligns well with experimental observations, thereby validating its accuracy and applicability. The fracture conductivity is directly proportional to number of branch fractures, elastic-plastic properties, proportion of flow rate and inversely proportional to closure pressure. The rock's viscosity exerts a minimal effect on fracture conductivity. The relationship of residual aperture and hydration swelling affect the proppant embedment, and lead to the Significant decrease of complex fracture conductivity. Moreover, As the number of horizontal fractures and the proportion of horizontal fracture flow rate increase, it would extend the duration for the complex conductivity to achieve the steady state.
This paper introduces technical innovations through the development of (a) novel analytical models that incorporate the number of branch fractures and the proportion of flow rate to forecast variations in proppant embedment and fracture conductivity, and (b) the conductivity and proppant embedment of complex fractures in shale reservoirs are analyzed at different scenarios. These advanced mathematical models lay the foundation for assessing the particle embedment and conductivity of complex fractures.