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Lecture Series in Porous Media

Concepts, Methods and Applications Focused on Geological Systems Engineering

Lecture Series in Porous Media
This book is 40% completeLast updated on 2026-09-09

First-principles lecture notes on flow in porous media: porosity, permeability, upscaling, and the mass, momentum, and energy balances behind Darcy's law. Currently spans 8 chapters. More chapters to be added in future versions.

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About

About

About the Book

Lecture Series in Porous Media is a concise, self-contained companion to the open YouTube lecture series of the same name. It builds the physics of flow and transport in porous media from first principles: starting from what porosity and permeability actually mean, and ending with the coupled mass, momentum, and energy balances that govern flow in the subsurface. The emphasis throughout is on derivation and understanding, not recipes. Each result is developed step by step from the underlying conservation law, with every assumption made explicit, so you can see exactly where Darcy's law, the cubic law, Carman–Kozeny, the Navier–Stokes equations, and the porous-media heat-transport equation come from, and where they break down.

This book is a work in progress. It currently spans 8 chapters: from porosity and permeability through the mass, momentum, and energy balances. Further chapters (e.g. multiphase flow, numerical methods, etc.) are planned. Everyone who buys now receives all future updates for free.

Author

About the Author

Hadi Hajibeygi

Hadi Hajibeygi is Professor of Geo-Energy Solid and Fluid Mechanics at Delft University of Technology (TU Delft). His teaching and research focus on the modeling and simulation of flow, transport and mechanics in porous media, with applications across geothermal energy, subsurface storage, and the energy transition. He teaches courses including Dynamics of Solids and Fluids, Multiphase Flow in Porous Media, Grand Challenges in Earth Sciences, Subsurface Storage, Advanced Reservoir Simulation, and Fluid Dynamics. Hadi runs an open lecture series on porous-media physics and simulaion on YouTube, of which this book is the written companion. Hadi holds PhD with medal from ETH Zurich and was a postdoctoral scholar at Stanford University before joining TU Delft in 2013 as professor. He was named the Most Innnovative Teaching Talent of TU Delft in 2018 and the Lecturer of the Year of the Faculty of Civil Engineering and Geosciences of TU Delft in 2026.

Contents

Table of Contents

Table of Contents

  • 1. Introduction to Porous Media and Porosity
    • 1.1 Motivation
    • 1.2 Definition of Porosity
    • 1.3 The Representative Elementary Volume (REV)
    • 1.4 Measuring Porosity
    • 1.5 Closing Remarks
  • 2. Darcy's Law and Permeability
    • 2.1 Motivation
    • 2.2 Darcy's Law
    • 2.3 Measuring Permeability: Core Flooding
    • 2.4 The Bundle-of-Tubes Model: A Single Capillary Tube
    • 2.5 Hydraulic Radius and Non-Circular Pore Geometries
    • 2.6 Comparison with Experiments: Carman–Kozeny and Tortuosity
    • 2.7 Validity of Darcy's Law: Reynolds Number and Non-Darcy Flow
    • 2.8 Solving Darcy's Sand Pack: The Governing Equation
    • 2.9 Closing Remarks
  • 3. Upscaling of Permeability in Porous Media
    • 3.1 Motivation: Scale Separation in Subsurface Systems
    • 3.2 Analytical Upscaling of Layered Media
    • 3.3 Upscaling of General Heterogeneous Media
    • 3.4 Closing Remarks
  • 4. Conservation Laws I: Mass Balance
    • 4.1 Motivation
    • 4.2 The Reynolds Transport Theorem
    • 4.3 Mass Balance for Free Flow
    • 4.4 Mass Balance in Porous Media
    • 4.5 Incompressible Flow
    • 4.6 Closing Remarks
  • 5. Conservation Laws II: Momentum Balance
    • 5.1 Motivation
    • 5.2 From the Reynolds Transport Theorem to Momentum
    • 5.3 Forces Acting on the Fluid
    • 5.4 Local (Differential) Form
    • 5.5 Decomposition of the Stress Tensor
    • 5.6 The Navier–Stokes Equations
    • 5.7 Momentum Balance in Porous Media: Darcy's Law
    • 5.8 Closing Remarks
  • 6. Poiseuille Flow: Pressure-Driven Channel Flow
    • 6.1 Motivation
    • 6.2 Tube Flow: Setup and Assumptions
    • 6.3 Mass and Momentum Balance for Tube Flow
    • 6.4 Velocity Profile and Permeability of a Single Tube
    • 6.5 Poiseuille Flow Between Parallel Plates
    • 6.6 Closing Remarks
  • 7. Flow in Porous Media: Practice I
    • 7.1 Practice Problem 1: Porosity of a Bulk Sample
    • 7.2 Practice Problem 2: Permeability via Carman–Kozeny
    • 7.3 Practice Problem 3: Two-Layer 1D Flow
    • 7.4 Closing Remarks
  • 8. Conservation Laws III: Energy Balance
    • 8.1 The General Energy Balance Equation
    • 8.2 Application to Atmospheric Thermodynamics
    • 8.3 Application to Geothermal Systems
    • 8.4 Closing Remarks

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