# The Propagation of Wetting Fronts Through Porous Media

Lead Research Organisation: University of Birmingham
Department Name: School of Mathematics

### Abstract

This research is concerned with the flow of liquid through solids which are permeated by a network of holes; such materials are known as porous media. Understanding how a liquid will flow through a porous structure is also the key element to a range of industrial phenomena such as:- The recovery of oil from a reservoir. This is currently a highly inefficient process with at most half of the possible oil recovered from each well.- The storage of carbon dioxide deep underground in a porous medium in an attempt to reduce global warming. Here it is vital to ensure that harmful gases will not escape over a period of hundreds of years.- The seeping of a liquid ink-jet droplet into paper where, to ensure quality of the image, it is desirable to know how quickly the paper will absorb the ink.Often a full scale experiment of such a flow is unpractical, expensive and/or dangerous. Consequently theoretical modelling becomes a tool which can be used to probe the dynamics of such flows to ensure safety, insight and optimisation of the appropriate process. There has been intensive academic research in this field from the mid-nineteenth century when Henry Darcy, motivated by the need to provide clean water for the citizens of Dijon, proposed an equation to describe the flow of water through sand. Amazingly this equation is still used for most porous media flows.Our research concerns the motion of a liquid into an initially dry porous medium. This often occurs spontaneously, for example, you can observe a liquid creeping slowly up a biscuit which is placed in a cup of tea. The reason that the liquid climbs up through the porous medium, against the downward pull of gravity, is that the liquid's surface has a stronger affinity for the solid than it does for the rest of the liquid, creating what is known as a capillary force, dragging the liquid further into the solid.Capillary effects occur in a wide range of phenomena, they are responsible for the spherical shape of bubbles, the ability of small animals to walk on water and the tears of wine on a glass. Perhaps the easiest way to observe their power is to place a small cylindrical tube vertically into a bath of water and observe that the water inside the tube is above that of the bath. In fact the simplest model for flow into an initially dry porous medium is obtained by approximating the medium as a bundle of capillary tubes. It is assumed, rather crudely, that the capillary force propelling the front of the liquid into the porous medium is equal to its equilibrium value. This approach was originally proposed by Washburn in 1921. Although describing the incredibly complex structure of a porous medium in this simplified approach leads to surprisingly accurate results, there is a large body of experimental evidence suggesting that it is often inaccurate.The project proposes a transfer of knowledge from the neighbouring field of dynamic wetting, which is concerned with the flow of liquids on solids, to the class of problems we are interested in, namely the flow of liquids into solids. Recent experimental results in the dynamic wetting community have shown how many industrial processes can be optimised in a way which was not previously realised. This was originally discovered in the photographic industry where it was demonstrated how to coat a solid with a liquid layer as fast as possible without ruining the film's quality by entraining air bubbles into it.The theory which predicts this phenomenon has only been thoroughly explored in the field of dynamic wetting in the past few years and has never been applied to flows through porous media. By applying the advanced mathematical model behind this theory to the propagation of a liquid through a network of pores we hope to improve on the current model for describing flow into a dry porous medium and bridge the gap between the two communities.

### ORCID iD

James Sprittles (Principal Investigator)  http://orcid.org/0000-0002-4169-6468

### Publications

10 25 50

Sprittles J (2011) Viscous flow in domains with corners: Numerical artifacts, their origin and removal in Computer Methods in Applied Mechanics and Engineering

Sprittles J (2011) Finite element framework for describing dynamic wetting phenomena in International Journal for Numerical Methods in Fluids

Sprittles J (2012) Coalescence of liquid drops: Different models versus experiment in Physics of Fluids

Shikhmurzaev YD (2012) Anomalous dynamics of capillary rise in porous media. in Physical review. E, Statistical, nonlinear, and soft matter physics

Description A new model to describe how liquids propagate through porous media was developed that accounts, for the first time, for the different modes which a liquid-gas meniscus makes when it travels through such a medium.

The model describes experiments which had eluded understanding for over ten years.
Exploitation Route The findings suggest new ways to develop the modelling of flow through porous media by accounting for multiple modes of propagation, in particular those in which one phase becomes trapped inside the other.
Sectors Chemicals

Electronics

Energy

Environment

Pharmaceuticals and Medical Biotechnology

Description The fundamental contributions which this project has made in the modelling and simulation of capillary flow phenomena will have impact across a range of fields and applications such as in technological flows, like 3D printing or enhanced oil recovery, and numerous environmental phenomena.
First Year Of Impact 2011
Sector Aerospace, Defence and Marine,Chemicals,Energy,Environment,Healthcare,Manufacturing, including Industrial Biotechology
Impact Types Economic

Description Feasibility Study into the Generation of Compound Microdrops for 3D Printing
Amount £48,496 (GBP)
Organisation Engineering and Physical Sciences Research Council (EPSRC)
Department Centre for Innovative Manufacturing in Additive Manufacturing
Country United Kingdom
Start 03/2013
End 09/2013

Description Funding for a PDRA from the John Fell Fund
Amount £44,266 (GBP)
Organisation University of Oxford
Department John Fell Fund
Country United Kingdom
Start 03/2013
End 04/2014

Description Funding to run a workshop on the Micromechanics of Wetting and Coalescence
Amount £12,000 (GBP)
Organisation University of Oxford
Department Oxford Centre for Collaborative Applied Mathematics (OCCAM)
Country United Kingdom
Start 12/2012
End 12/2012

Description Global Research Fellowship
Amount £200,000 (GBP)
Organisation University of Warwick
Country United Kingdom
Start 09/2014
End 10/2019

Description Collaboration with Professor T.D. Blake (Mons)
Organisation University of Mons
Department Laboratory of Surface and Interfacial Physics
Country Belgium
PI Contribution As a result of the work undertaken during the fellowship, I have begun a collaboration with Professor T.D. Blake, an acknowledged world leader in the field of dynamic wetting, with whom I have started to investigate the phenomenon of dewetting in liquid-solid-gas systems.
Start Year 2012

Description Collaboration with the EPSRC Centre for Innovative Manufacturing in Additive Manufacturing (Nottingham & Loughborough)
Organisation Engineering and Physical Sciences Research Council (EPSRC)
Department Centre for Innovative Manufacturing in Additive Manufacturing
Country United Kingdom
PI Contribution The developments of numerical software to simulate industrially-relevant free surface flows, including those in which drops interact with porous substrates, has attracted interest from the Additive Manufacturing Centre who wish to describe the dynamics of liquid drops ejected from inkjet printers in order to build 3D structures. Collaboration has involved a number of discussions and plans for future ties including, in particular, two grant applications.
Start Year 2012

Description Collaboration with the laboratory of Professor Sigurdur Thoroddsen (KAUST)
Organisation King Abdullah University of Science and Technology (KAUST)
Country Saudi Arabia
PI Contribution As a result of the project's developments, a link with Professor Thoroddsen's laboratory in KAUST (Saudi Arabia) has been established. Collaboration involves the design of theory-driven experiments for two phenomena: (a) the impact and spreading of liquid drops on hydrophobic powder beds, and the associated formation of liquid marbles and (b) the early-stages of dynamic wetting on cone-shaped surfaces.
Start Year 2011

Description Impact of Drops on Solids: Modelling and Simulation
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional