Rigorous Modelling of Fractures in a Porous Medium

Abstract While fractured formations are possibly the most important contributors to the oil production world-wide, modelling fractured formations with rigorous treatments has eluded reservoir engineers in the past. To-date, one of the most commonly used fractured reservoir model remains the one that was suggested by Warren and Root more than three decades ago. In this paper, a new model for fractures embedded in a porous medium is proposed. The model considers the Navier Stokes equation in the fracture (channel flow) while using Brinkman equation for the porous medium. Unlike the previous approach, the proposed model does not require the assumption of orthogonality of the fractures (sugar cube assumption) nor does it impose incorrect boundary conditions for the interface between the fracture and the porous medium. The proposed model is derived through a series of finite element modelling runs for various cases using Navier Stokes equation in the channel while maintaining Brinkman equation in the porous medium. Various cases studied include different fracture orientations, fracture frequencies, fracture width, and the permeability of the porous medium. Finally, a series of numerical runs also provided validity of the proposed model for the cases for which thermal and solutal effects are important. Such a study of double diffusive phenomena in the context of fractured formations has not been reported before. Introduction The number of oil and gas fields that are affected by fracture flow is increasing1. It is now known that fractures can play a significant role even when the reservoir is not considered to be fractured. Whenever fractures are present, the contrast between fracture and rock matrix creates highly heterogeneous permeability fields, which result in complex saturation distributions. Efficient production of these reservoirs requires careful management of production rates and placement of injection wells. The biggest scientific challenge appears in the areas of scaling up so that fracture distribution becomes useful for reservoir engineers. Fractures constitute some of the most difficult topics of research from both hydrology and petroleum engineering perspectives. They are often associated with unpredictability and variability2. Decades of studies, ranging from single narrow fractures dominating flow3, to wells or drawdowns in a fractured formation(4–5) revealed many fundamental features of structural geology, and hydrogeology. Emphasis has been given to understanding of fracture geometry1, new techniques for conductive fracture detection(6–7), and new tools for computer simulation8. Similar to the studies on hydrodynamic aspects of fractures, solute transport through a fractured medium has received considerable attention for last several decades(9–11). The latest development in this regard has been the introduction of flow channels12. Early findings showed that the amount of coupling can be significantly reduced due to the fracture channeling if the time scales of interest are short when compared to diffusive time scales in the rock matrix system. Dykhuizen13 showed that for longer time scales this reduction in the coupling is greatly reduced. However, experimental work has yet to be conducted to verify mathematical findings. More recently, heat transport through fractures has been addressed in a coupled form in the context of thermal stress14.

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