# A Model for Simulating the Upward Flow of a Viscous Fluid in a Fracture Network
## 1 Introduction
[18] used a hybrid model to study the leakage of natural fractures, focusing on their theoretical leakage characteristics. [PERSON] et al. [19] established a grid model for hydraulic fracturing simulation, in which there are two groups of mutually perpendicular plane fractures in the ellipse of the simulation region. [PERSON] et al. [20] proposed a new fracture model to simulate complex fracture network expansion in naturally fractured formations. The interaction, coupling, and deformation of hydraulic fractures and natural fractures are considered in the model. [PERSON] et al. [21] established a discrete fracture network based on the normal distribution of  
Figure 1: **(a)** A conceptual cartoon proposed by [PERSON] et al. showing a fold-related crestal fracture corridor with a network of fluid flow pathways [8]. **(b)** The conceptual model proposed by [PERSON] et al., for fluid escape path growth by hydraulic fracturing in a sedimentary basin [11]. -fracture length, and further developed a gas production prediction model for fractured shale reservoirs. When evaluating the reservoir performance of shale oil reservoirs, [PERSON] et al. [22] found that the fractal discrete fracture network model well characterized the fracture characteristics. Such studies have shed light on the effect of parameters such as rock permeability, fluid viscosity, and fluid pressure on the migration process. However, the interactions between the migration process and the shape of the fracture network are rarely straightforward to interpret. The goal of this paper is to develop a comprehensive model to simulate fluid migration in a structural fracture network with heterogeneous properties by fully considering the role of a rock matrix on the flow dynamics in the fracture, as well as the feedback of the fluid flow to the shape of the fractures. This model simplifies the pressure-driven upflow in each fracture to a fully developed laminar flow between two infinite plates, and hence employs 1D Navier-Stokes equations to describe the mass and momentum conservation of the fluid flow in the vertical direction. In the lateral direction, we assume the surrounding rock matrix to be a set of spring matrices and achieve a balance between the pressure of the fluid and the elastic force of the rock matrix using the linear deformation law proposed by [PERSON] et al. [23]. This balance is dynamic, and hence the larger the pressure is, the larger is the corresponding fracture aperture. In this work, we validate our model by reproducing a thickness profile of a fluid-driven crack, which is measured in the experiment [24], and then apply the model to describe the effects of various physical parameters of rock matrix and liquid as well, as structural fracture heterogeneity, on the steady-state migration that is reached with an equilibrium between liquid loss through the rock matrix and continuous replenishment from the source rock [25]. We first study the role of the thermal and dynamic parameters of the fluid and the rock matrix in the upflow in a single fracture using the model. Imposing suitable boundary conditions, we apply either an \"open\" or \"closed\" status to the outlet of the fracture in order to simulate the upflow in both the sealed fracture and the fracture connecting the reservoirs or the \"sweet spots\" [9]. At the same time, the study of the transition from open state to closed state is helpful in providing theoretical guidance for the new technology of oil exploitation--temporary-plugging-and-diverting fracturing (TPD fracturing). TPD fracturing technology is widely used in the re-fracturing of low production wells and completed wells. It can increase production by artificially sealing existing fractures and creating new fracture pathways in the formation [26; 27; 28]. We then apply our model to a bifurcated fracture that includes three connected channels to evaluate the effect of heterogeneous properties and the boundary conditions of the channels on the upflow within the whole fracture.