Research Paper Analytical study of fluid flow modeling by diffusivity equation including the quadratic pressure gradient term Mahdi Abbasi a , Mojtaba Izadmehr b , Masoud Karimi b , Mohammad Sharifi a,⇑ , Alireza Kazemi a a Department of Petroleum Engineering, Amirkabir University of Technology (Polytechnic of Tehran), P.O. Box: 15875-4413, Tehran, Iran b Department of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran 11365-9465, Iran article info Article history: Received 5 November 2016 Received in revised form 5 March 2017 Accepted 1 April 2017 Keywords: Diffusivity equation Well testing Fluid flow Porous media Quadratic term Nonlinear abstract Diffusivity equation which can provide us with the pressure distribution, is a Partial Differential Equation (PDE) describing fluid flow in porous media. The quadratic pressure gradient term in the diffusivity equa- tion is nearly neglected in hydrology and petroleum engineering problems such as well test analysis. When a compressible liquid is injected into a well at high pressure gradient or when the reservoir possess a small permeability value, the effect of ignoring this term increases. In such cases, neglecting this param- eter can result in high errors. Previous models basically focused on numerical and semi-analytical meth- ods for semi-infinite domain. To the best of our knowledge, no analytical solution has yet been developed to consider the quadratic terms in diffusivity PDE of one-dimensional unsteady state fluid flow in rectan- gular coordinates and finite length. Due to the resulting errors, the nonlinear quadratic term should also be considered in the governing equations of fluid flow in porous media. In this study, the Fourier transform is used to model the one- dimensional fluid flow through porous media by considering the quadratic terms. Based on this assump- tion, a new analytical solution is presented for the nonlinear diffusivity equation. Moreover, the results of linear and nonlinear diffusivity equations are compared considering the quad- ratic term. Finally, a sensitivity analysis is conducted on the affecting parameters to ensure the validity of the proposed new solution. The results demonstrate that this nonlinear PDE is also applicable for hydrau- lic fractured wells, and well test analysis of fractured reservoirs. Ó 2017 Elsevier Ltd. All rights reserved. 1. Introduction Pore pressure distribution in a single phase porous media is highly important for researchers in the fields of hydrology and pet- roleum reservoir engineering and also applicable for well testing analysis. This pressure distribution which has received particular interest in geophysical science, is obtained based on the transient fluid flow through porous media by diffusivity equation in a nonlinear form [1–3]. Due to the compressibility of fluids, a quadratic pressure gradient term appears in the governing PDE, which leads to high nonlinearity in diffusivity [4]. Many researchers have solved this equation by neglecting the nonlinear part of the equation to linearize it for various boundary conditions. This approach has been widely used in pressure tran- sient analysis of porous media for small pressure gradient condi- tions; however, in many actual field applications, the working pressure of injection fluid into porous media is high enough that linear form of diffusivity equation deviates from the reality. Such application can be observed in many reservoir operations such as hydraulic fracturing, large drawdown flows, slug testing, drill- stem testing, and large pressure pulse testing [5,6]. In these cases, the nonlinear form of pressure distribution equation should be applied through the quadratic pressure gradient term to eliminate the errors rising from the linearization. Even though the impor- tance of this nonlinear pressure term has been noticed in well test applications, modern well test analysis does not consider its impact on their interpretation [7]. Odeh and Babu [8] used the analytical solution of nonlinear dif- fusivity equation in three categories including constant rate inner boundary, infinite outer boundary and no wellbore storage effects. Their findings indicate that for the most of the reservoir engineer- ing processes, linear form of the PDE can be used directly with small errors without quadratic pressure gradient term. However, in the case of high pressure injection, linear form of PDE causes high deviation from the reality and thus the quadratic term should be considered in the equation to solve it more accurately. http://dx.doi.org/10.1016/j.compgeo.2017.04.001 0266-352X/Ó 2017 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail address: m_sharifi@aut.ac.ir (M. Sharifi). Computers and Geotechnics 89 (2017) 1–8 Contents lists available at ScienceDirect Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo