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On the closed solution of a problem coupling fluid infiltration with a hydration reaction

This paper presents a one-dimensional closed-form analytical solution for a moving free boundary problem coupling water infiltration with a hydration reaction, revealing that the infiltration front's evolution transitions from a square-root-in-time behavior at early stages to square-root, linear, or exponential forms at later times depending on reaction consumption.

Original authors: Diego Guevara, Sabrina Roscani, Piotr Rybka, Vaughan Voller

Published 2026-08-21

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CC BY 4.0

Original authors: Diego Guevara, Sabrina Roscani, Piotr Rybka, Vaughan Voller

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep beneath the Earth's surface, in the hidden pores of rock and soil, a quiet but powerful struggle often takes place between fluid and solid. When water seeps into dry ground, it does not merely fill empty spaces; it can trigger chemical changes that reshape the very material it touches. This process is central to how we might store carbon dioxide underground by turning it into stone, or how natural fractures in rock can suddenly open up to allow fluids to flow more freely. The core of this interaction involves two competing forces: the physical push of water moving through tiny gaps in the rock, and the chemical hunger of a reaction that consumes that water as it arrives. Scientists have long known that these two forces are linked, but predicting exactly how the boundary between wet and dry rock moves over time has been a difficult puzzle, usually requiring complex computer simulations to get an answer.

A team of researchers has now solved this puzzle with a precise mathematical description that reveals how the wet front advances through dry rock when a hydration reaction is happening. In their study, they modeled a one-dimensional slice of porous rock where water enters from one side under a steady pressure. As the water moves forward, it encounters a sharp boundary separating the saturated, wet zone from the dry zone behind it. Simultaneously, a chemical reaction begins to consume the water, effectively shrinking the amount of liquid available to push the front further. The researchers derived a single, exact formula that describes the position of this moving boundary at any moment in time, replacing the need for step-by-step computer approximations with a clear, analytical truth.

The solution reveals a fascinating shift in behavior as time passes. At the very beginning, when the water first starts to enter the rock, the chemical reaction has not had time to consume much fluid. During this early phase, the wet front moves forward at a speed determined solely by the rock's porosity, or how much empty space it contains. The distance the water travels grows in proportion to the square root of time, a familiar pattern seen in simple water infiltration where no chemical reaction occurs. However, as time goes on, the chemical reaction begins to eat away at the water, and the behavior of the front changes. The researchers found that the long-term movement of the front depends entirely on the balance between the rock's porosity and the intensity of the water consumption by the reaction.

If the reaction consumes water at a moderate rate relative to the rock's capacity, the front continues to move forward, but its speed adjusts to a new, slower square-root pattern. If the reaction is extremely aggressive, consuming water faster than the rock's structure can support, the front's behavior shifts dramatically. In one specific scenario, the front stops following a square-root pattern and begins to move in a straight line, advancing at a constant speed. In an even more extreme case, where the reaction is overwhelmingly strong, the front's movement accelerates exponentially, growing faster and faster as time goes on. These different outcomes are not guesses or rough estimates; the authors proved mathematically that these are the only possible behaviors for this system, providing a complete map of how the front evolves.

This discovery offers more than just a theoretical curiosity; it provides a reliable tool for understanding real-world geological processes. By knowing exactly how the wet front moves, scientists can better predict when and where the swelling caused by hydration might create enough internal stress to crack the rock, forming new pathways for fluids. The study confirms that while the initial entry of water follows a standard rule, the long-term fate of the infiltration is dictated by the chemical battle between the fluid and the rock. The researchers demonstrated that this complex interplay, which previously required heavy numerical computation to visualize, can be captured in a single, elegant expression that tells the whole story from the first drop of water to the distant future.

Technical Summary: Closed Solution of a Fluid Infiltration–Hydration Coupling Problem

Problem Formulation
The paper addresses a one-dimensional moving boundary problem describing the infiltration of a reactive fluid into a dry porous medium, coupled with a first-order hydration reaction. This model is motivated by environmental remediation processes, such as CO₂ mineralization, where reactive fluids infiltrate subsurface rock masses, potentially leading to "reaction-driven fractures."

The physical system assumes:

A homogeneous porous medium with porosity $\phi$ .
A sharp infiltration front separating saturated and dry regions (Green-Ampt assumption).
A first-order hydration reaction consuming the infiltrating water.
Darcy flow within the saturated region ( $0 < x < s(t)$ ).

The governing equations combine Darcy's law, mass conservation, and reaction kinetics. The pressure $p(x,t)$ satisfies a Poisson-type equation where the source term represents water consumption by the reaction. The system is closed by a Stefan-type condition at the moving front $s(t)$ , linking the pressure gradient to the front velocity, and boundary conditions fixing pressure at the origin and zero at the front.

Methodology
The authors derive a dimensionless form of the problem, introducing a length scale $\ell^*$ and time scale $t^*$ based on permeability, viscosity, pressure head, and reaction rate. The dimensionless system involves finding the free boundary position $s(t)$ , the pressure field $p(x,t)$ , and the reaction progress variable $\Gamma(x,t)$ .

To obtain a closed-form solution, the authors employ an analytical approach centered on an auxiliary function $J(t)$ , defined as the integral of the reaction progress variable over the saturated domain. By integrating the governing equations and applying Fubini's theorem, they derive a linear ordinary differential equation (ODE) for $J(t)$ . Solving this ODE allows for the explicit determination of the free boundary position $s(t)$ . Once $s(t)$ is known, the reaction variable $\Gamma$ and pressure field $p$ are reconstructed using the inverse of the strictly monotonic function $s(t)$ and integration of the pressure gradient equation.

Key Contributions and Results
The primary contribution is the derivation of an exact, closed-form analytical solution for the free boundary $s(t)$ and the associated fields, valid for all time $t \geq 0$ . The solution is unique and strictly increasing.

The analysis reveals distinct dynamic regimes for the infiltration front based on the relationship between the porosity $\phi$ and the dimensionless water consumption parameter $Y$ (representing the total volume of water consumed per unit volume of media):

Early-Time Behavior: For $t \to 0$ , the solution universally exhibits a square-root-in-time dependence ( $s(t) \sim \sqrt{t/\phi}$ ), regardless of the hydration reaction strength. This matches the classic infiltration behavior in the absence of reaction, as the front advances faster than the reaction can consume water.
Late-Time Behavior: The asymptotic behavior as $t \to \infty$ $t \to \infty$ depends on the sign of $\phi + Y$ $ϕ + Y$ :
- Case $\phi + Y > 0$ : The front advances with a square-root-in-time behavior ( $s(t) \sim \sqrt{t/(\phi+Y)}$ ), but with a modified pre-factor determined by both porosity and reaction consumption.
- Case $\phi + Y = 0$ : The front advances linearly with time ( $s(t) \sim t$ ).
- Case $\phi + Y < 0$ : The front advance exhibits exponential growth.

The paper explicitly demonstrates a "crossover" from the early-time square-root regime to the late-time regime, analytically recovering behaviors previously observed only numerically in literature [2, 9].

Significance
The paper claims that this closed solution provides a rigorous analytical tool for quantifying transport and reaction systems in the Earth's subsurface. Specifically:

It validates and explains the numerical predictions of early and late-time behaviors found in previous studies.
It offers a precise mathematical basis for analyzing the conditions under which reaction-driven fractures may occur, linking the front dynamics to eigenstrains induced by hydration swelling.
It generalizes the classic Stefan problem (in the limit of vanishing specific heat) by incorporating chemical consumption, thereby enriching the mathematical understanding of free boundary problems with reaction terms.

The authors emphasize that while the physics dictates $\phi > 0$ and $Y > 0$ (water consumption), the mathematical analysis holds for a broader range of parameters, including cases where $Y < 0$ (water release), though these are noted as physically distinct scenarios.

Technical Summary: Closed Solution of a Fluid Infiltration–Hydration Coupling Problem

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