Statistical integro-differential fracture model (Sid-FM) for isolated fractures with variable apertures and lengths

Abstract

Flow in fractured porous media is associated with high uncertainty, particularly regarding fracture properties and their overall configuration within the domain. This is especially pronounced for disconnected fractures of smaller yet comparable size to the domain. Consequently, ensemble averages are often used to capture this statistical variability and predict the expected behavior. This leads to enormous computational costs, as flow simulations of single realizations with millions of fractures are extremely expensive; and much more so full Monte Carlo studies involving hundreds of realizations. Alternatively, a recently introduced model aims to directly estimate expected flow rates and pressure fields. The model involves few degrees of freedom, leading to low-cost computations. This is achieved by using integro-differential equations involving non-local kernel functions that encompass the statistical information of fractures. So far this statistical integro-differential fracture model (Sid-FM) considers only ensembles with identical fractures having constant aperture and lengths. In this paper Sid-FM is extended to account for arbitrary fracture aperture profiles and reservoirs with fractures following specified length distributions, which is a crucial step towards applications with realistic fractured reservoirs. In a series of numerical experiments, it is demonstrated that the Sid-FM’s predictions are in excellent agreement with Monte Carlo reference data, which are based on many fracture-resolving simulations. The applicability is demonstrated through statistically one-dimensional cases, laying crucial groundwork for 2D and 3D extensions. Future work will focus on further generalizations and extensions such as transport processes and 2D/3D applications.