Lawrence Berkeley National Laboratory Lawrence Berkeley National Laboratory Title Theoretical Studies of Non-Newtonian and Newtonian Fluid Flow through Porous Media Permalink https://escholarship.org/uc/item/6zv599hc Author Wu, Y.S. Publication Date 1990-02-01 eScholarship.org Powered by the California Digital Library University of California Lawrence Berkeley Laboratory e UNIVERSITY OF CALIFORNIA EARTH SCIENCES DlVlSlON Theoretical Studies of Non-Newtonian and Newtonian Fluid Flow through Porous Media Y.-S. Wu (Ph.D. Thesis) February 1990 TWO-WEEK LOAN COPY This is a Library Circulating Copy which may be borrowed for two weeks. r- +. .zn Prepared for the U.S. Department of Energy under Contract Number DE-AC03-76SF00098. :0 DISCLAIMER I I This document was prepared as an account of work sponsored ' : by the United States Government. 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Lawrence Berkeley Laboratory is an equal opportunity employer Theoretical Studies of Non-Newtonian and Newtonian Fluid Flow through Porous Media Yu-Shu Wu (Ph.D. Thesis) Department of Materials Science and Mineral Engineering University of California and Earth Sciences Division Lawrence Berkeley Laboratory University of California Berkeley, California 94720 February 1990 This work was supported by the U.S. Department of Energy under Contract No. DE-AC03-76SF00098. - ii - Theoretical Studies of Non-Newtonian and Newtonian Fluid Flow though Porous Media Yu-Shu Wu Department of Material Sciences and Mineral Engineering University of California, Berkeley Berkeley, California 94720 Abstract A comprehensive theoretical study has been carried out on the flow behavior of both single and multiple phase non-Newtonian fluids in porous media. This work is divided into three parts: (1) development of numerical and analytical solutions; (2) theoretical studies of transient flow of non-Newtonian fluids in porous media; and (3) applications of well test analysis and displacement efficiency evaluation to field prob- lems. A fully implicit, integral finite difference model has been developed for simulation of non-Newtonian and Newtonian fluid flow through porous media. Several commonly-used rheological models of power-law and Bingham plastic non-Newtonian fluids have been incorporated in the simulator. A Buckley-Leverett type analytical solution for one-dimensional, immiscible dis- placement involving non-Newtonian fluids in porous media has been developed. Based on this solution, a graphic approach for evaluating non-Newtonian displacement efficiency has been developed. The Buckley-Leverett-Welge theory is extended to flow problems with non-Newtonian fluids. An integral method is also presented for the study of transient flow of Bingham fluids in porous media. In addition, two well test analysis methods have been developed for analyzing pressure transient tests of power-law and Bingham fluids, respectively. Applications are included to demonstrate this new tech- nology. The physical mechanisms involved in immiscible hsplacement with non- Newtonian fluids in porous media have been studied using the Buckley-Leverett type analytical solution. The results show that this kind of displacement is a complicated pro- cess and is determined by the rheological properties of the non-Newtonian fluids and the flow conditions, in addition to relative permeability data. In another study, an idealized fracture model has been used to obtain some insights into the flow of a power-law fluid in a double-porosity medium. For flow at a constant rate, non-Newtonian flow behavior in a fractured medium is characterized by two-parallel straight lines on a log-log plot of injection pressure versus time. Transient flow of a general pseudoplastic fluid has been studied numerically and it has been found that the long time pressure responses tend to be equivalent to that of a Newtonian system. P. A. Witherspoon Thesis Committee Chairman .iv . Table of Contents ... List of Figures ............................................................................................................ Vlll List of Tables ............................................................................................................. xvii Nomenclature .............................................................................................................xviii Acknowledgements ................................................................................................... xxv Chapter 1. Introduction .............................................................................................. 1 Chapter 2. Literature Review .................................................................................... 7 2.1 Introduction ............................................................................................... 7 2.2 Non-Newtonian Fluids ............................................................................... 9 2.3 Previous Work on Laboratory Studies and Rheological Models .............. 2.4 Previous Work on Analysis of Flow through Porous Media ..................... 2.5 Conclusions from Literature Review ......................................................... Chapter 3. Rheological Model ................................................................................... Chapter 4 . Mathematical Model ................................................................................ 4.1 Introduction ............................................................................................... 4.2 Governing Equations for Non-Newtonian and Newtonian Fluid Flow ................................................................................ 4.3 Constitutive Equations ............................................................................... 4.4 Numerical Model ....................................................................................... 4.5 Treatment of Non-Newtonian Behavior .................................................... 54 4.5.1 Power-Law Fluid .......................................................................... 55 4.5.2 Bingham Fluid .............................................................................. 57 4.5.3 General Pseudoplastic Fluid ......................................................... 59 Chapter 5. Analytical Solution for Immiscible Displacement .................................. 61 5.1 Introduction ............................................................................................... 61 5.2 Mathematical Formulation ........................................................................ 64 5.3 Analytical Solution for Non-Newtonian and Newtonian Displacement ........................................................................... 65 5.4 Graphical Evaluation of Linear Displacement Solution ............................ 71 5.5 Verification of the Numerical Method ...................................................... 78 Chapter 6 . Immiscible Displacement of a Newtonian Fluid by a Power-Law Non-Newtonian Fluid .................................................... 84 6.1 Introduction ............................................................................................... 6.2 Evaluation of Analytical Solution ............................................................. 6.3 Displacement of a Newtonian Fluid by a Power-Law Non-Newtonian Fluid ................................................................................ 6.3.1 Effect of Non-Newtonian Rheological Properties ........................ 6.3.2 Effect of injection Rates ............................................................... 6.3.3 Effect of Gravity ........................................................................... 6.3.4 Conclusions ................................................................................... Chapter 7 . Immiscible Displacement of a Bingham Non-Newtonian Fluid by a Newtonian Fluid ............................................................................... 7.1 Introduction .......................................................................................... 108 7.2 Evaluation of Analytical Solution .......................................................... 109 7.3 Displacement of a Bingham Non-Newtonian Fluid by a Newtonian Fluid ................................................................................ 112 7.3.1 Effect of Non-Newtonian Rheological Properties ........................ 113 7.3.2 Effect of Injection Rates ............................................................... 124 7.3.3 Effect of Gravity .........................................................................