A theoretical depiction of percolation showing pore spaces (green) in a solid medium (white). Even in a nearly homogeneous medium, in which the pores all have similar sizes and are arranged almost perfectly geometrically, small pore size differences can result in a complex pattern of potential flow paths such as that shown. Credit: Alexis Monnerot-Dumaine (CC BY-SA 3.0)

In the geosciences the flow and transport properties of porous media are typically associated with water resources, hydrogeology, groundwater contaminants and remediation. However, porous media and their processes actually extend from soils and plants at Earth’s surface down to the deep crust. In an article recently published in Reviews in Geophysics, Hunt and Sahimi [2017] suggested an alternative approach and theoretical solutions to difficult problems regarding porous media. The editor asked the authors to explain the significance of porous media and describe developments in this field of research.

What is a “porous medium?”

Porous means having pores or small spaces; these allow liquids or gases to be held there or to pass through. Essentially everything we perceive as solid is actually porous at some scale. Plant matter and soils have spaces between the solid portions that are as large as 0.1 millimeters. Rocks contain pore space between the solid portions, which may be either crystalline or amorphous. Even crystalline matter is typically porous to gases with small enough molecules or atoms, such as helium, which diffuse through the crystal lattice structure.

What kind of substances flow, or are transported, through porous media?

The way that water moves through plant tissue and soil is an example of flow through porous media. Credit: Allen Hunt

Examples include blood flow in bone marrow, nutrients through cell membranes, oil through subsurface rock and sediment, water through soil to plant tissues, fluids through crystallizing melts, trace gases through Earth’s crust, and carbon dioxide through compacted snow.

The scale at which porous media are studied varies from cell membranes at micron scales and below to groundwater contamination at scales of up to tens of kilometers.

Which disciplines within the geosciences have an interest in porous media?

It actually spans most fields within geosciences. In our review, specific examples were drawn from hydrology, soil science, shallow earth geochemistry, geomorphology, and oil and gas prospecting and extraction. Other applications are to plant physiology, ecology, land-atmosphere interactions, compaction of sediments and dewatering phenomena, melting of sea ice, hyporheic exchange in rivers, nutrient transport to trunk streams.

Some other recent Editors’ Vox are good examples of ways in which flow, transport and reaction in porous media are studied in different areas of the geosciences, including the pathways of water at the interface between soil, vegetation and atmosphere, factors that influence the process of mechanical weathering which causes solid rock to break up, and the seismic and elastic properties of the Earth’s continental crust.

What are the main theoretical approaches for studying properties and processes in porous media?

At the fundamental level, all the flow, transport and reaction problems in porous media are governed by the conservation laws – mass, momentum, species, and energy. The question has always been how to solve the governing equations, given the complexity of even the simplest porous media.

In hydrological applications, scientists typically use numerical solutions of standard partial differential equations, such as the Advection-Dispersion Equation (ADE) or Richards’ equation, which are derived considering that the medium is like a smear, or continuum, but using numerical parameters that are calculated from the assumption that the medium is a collection of parallel tubes.

We suggest that many difficult problems are solved more effectively by treating the medium as a network with all its potential branching and intersecting paths, but then using theoretical techniques which were designed to address this complexity, such as percolation and effective-medium.

What have been some recent advances in percolation theory and its applications?

Percolation theory, when applied to network models, accurately predicts two- and three-phase flows in porous media, brine evaporation, and salt precipitation.

In the Atacama Desert, where rainfall levels are as little as 2 millimeters per year, the rate at which soil is produced is less than 1 meter per million years. Credit: Alessandro Caproni (CC BY 2.0)

Percolation theory also provides a foundation for understanding why Archie’s law for the electrical conductivity, often used in oil and gas exploration, varies from place to place, and what that implies about the medium.

Another important percolation theoretical advance is showing that chemical weathering and soil formation rates are proportional to how fast water flows through soil. This explains a factor 3,300 increase in soil formation rates from the Atacama to the New Zealand Alps.

Percolation theory also predicts how vegetation growth rates slow over time, and shows that these rates are proportional to how fast water flows through a plant (transpiration). This discovery solves a half-century old problem: How does the rate of conversion of atmospheric carbon to plant matter depend on how fast plants transpire water?

How can theoretical and applied research be combined to address a particular scientific problem?

An enduring topic of scientific debate is what causes the mass extinctions seen in the geologic record and percolation theory could offer a new insight. Several extinctions are closely correlated with large igneous province (LIP) eruptions, particularly the catastrophic one at the end of the Permian period when the Siberian trap LIP eruptions injected massive amounts of carbon dioxide into the atmosphere, producing deathly high surface temperatures and ocean acidity. But episodic LIP eruptions extend over large time intervals, giving time for increased temperatures, that could increase rates of silicate mineral weathering (when kinetic-limited) to draw down atmospheric carbon dioxide and cool the globe. However, percolation theory reinforces inferences from field studies that the silicate weathering reaction cannot proceed faster than water removes its products. Although increased global atmospheric temperatures often lead to increased precipitation, the aridity of the supercontinent Pangea during the Permian period would have restricted the impact of this negative feedback mechanism, helping to explain the Permian extinction’s severity. Very simply put: no water, no weathering, no carbon dioxide drawdown, no cooling, thus extreme extinction event.

What are some future directions for research about percolation theory and its applications?

One future area for research is to better understand coupled systems such as the mechanisms of transpiration (left) and evaporation (right) for returning water from the soil to the atmosphere. Credit: pxhere (CC0)

Percolation methods really need to be applied to problems with a higher degree of complexity and systems with more components like those characteristic of coupled processes, such as energy and water transport, and coupled investigations, such as between the soil, plants and the atmosphere, or between a riverbed and the subsurface.

Such topics across disciplines and scales have been at the forefront of hydrology at least since the report on Opportunities in the Hydrologic Sciences [National Research Council, 1991] was developed in conjunction with the establishment of the Hydrologic Sciences Program at the National Science Foundation in the early 1990s.

—Allen G. Hunt, Department of Physics and Department of Earth and Environmental Sciences, Wright State University; email: allen.hunt@wright.edu; and Muhammad Sahimi, Mork Family Department of Chemical Engineering and Materials Science, University of Southern California Los Angeles


Hunt, A. G.,Sahimi, M. (2017), The spaces in between, Eos, 98, https://doi.org/10.1029/2018EO085315. Published on 05 December 2017.

Text © 2017. The authors. CC BY-NC-ND 3.0
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