A Non-Linear Flow Model for Porous Media Based on Conformable Derivative Approach

Gang Lei,Nai Cao,Di Liu,Huijie Wang

doi:10.3390/en11112986

Abstract

Prediction of the non-linear flow in porous media is still a major scientific and engineering challenge, despite major technological advances in both theoretical and computational thermodynamics in the past two decades. Specifically, essential controls on non-linear flow in porous media are not yet definitive. The principal aim of this paper is to develop a meaningful and reasonable quantitative model that manifests the most important fundamental controls on low velocity non-linear flow. By coupling a new derivative with fractional order, referred to conformable derivative, Swartzendruber equation and modified Hertzian contact theory as well as fractal geometry theory, a flow velocity model for porous media is proposed to improve the modeling of Non-linear flow in porous media. Predictions using the proposed model agree well with available experimental data. Salient results presented here include (1) the flow velocity decreases as effective stress increases; (2) rock types of “softer” mechanical properties may exhibit lower flow velocity; (3) flow velocity increases with the rougher pore surfaces and rock elastic modulus. In general, the proposed model illustrates mechanisms that affect non-linear flow behavior in porous media.

Highlights

Ever since Henry Darcy (1865) developed his famous linear flow model, based on a series of sand pack experiments, the linear flow through porous media has drawn tremendous attention in various scientific and engineering field [1,2]
Li provided contradictory evidence for the threshold pressure gradient [8]. He suggested that the threshold pressure gradient measured in labs can be probably ascribed to the difficulty in measuring lower flow velocity, and the false phenomenon of the existence of threshold pressure gradient is strengthened by the skin effect
In order to analyze essential controls on non-linear flow in tight porous media, the effects of relevant parameters on average flow velocity are studied in detail

Summary

Introduction

Ever since Henry Darcy (1865) developed his famous linear flow model (the classical Darcy’s law), based on a series of sand pack experiments, the linear flow through porous media has drawn tremendous attention in various scientific and engineering field [1,2]. Reynolds number or pressure gradient, which could be used to well describe low velocity non-linear flow [5,6,7]. They concluded that there is no flow in tight porous media when the pressure gradient is beyond the certain value (i.e., threshold pressure gradient). Li provided contradictory evidence for the threshold pressure gradient [8] He suggested that the threshold pressure gradient measured in labs can be probably ascribed to the difficulty in measuring lower flow velocity, and the false phenomenon of the existence of threshold pressure gradient is strengthened by the skin effect

Objectives

Results

Conclusion