On the theory of solitons of fluid pressure and solute density in geologic porous media, with applications to shale, clay and sandstone

In this paper we propose the application of a new model of transients of pore pressure p and solute density \r{ho} in geologic porous media. This model is rooted in the non-linear waves theory, the focus of which is advection and effect of large pressure jumps on strain (due to large p in a non-linear version of the Hooke law). It strictly relates p and \r{ho} evolving under the effect of a strong external stress. As a result, the presence of quick and sharp transients in low permeability rocks is unveiled, i.e. the non-linear Burgers solitons. We therefore propose that the actual transport process in porous rocks for large signals is not the linear diffusion, but could be governed by solitons. A test of an eventual presence of solitons in a rock is here proposed, and then applied to Pierre Shale, Bearpaw Shale, Boom Clay and Oznam-Mugu silt and clay. A quick analysis showing the presence of solitons for nuclear waste disposal and salty water intrusions is also analyzed. Finally, in a kind of “theoretical experiment” we show that solitons could also be present in Jordan and St. Peter sandstones, thus suggesting the occurrence of osmosis in these rocks.

💡 Research Summary

The paper introduces a novel framework for describing transient pore‑pressure (p) and solute‑density (ρ) evolution in geological porous media, emphasizing that large pressure jumps trigger a non‑linear elastic response that cannot be captured by classical linear diffusion theory. By extending Hooke’s law to include a pressure‑dependent term (σ = E ε + β p ε) and coupling the continuity equations for fluid mass and solute mass, the authors derive a governing equation that reduces to a one‑dimensional Burgers‑type nonlinear wave equation: ∂u/∂t + β u ∂u/∂x = ν ∂²u/∂x², where u represents a combined pressure‑solute variable, β is a non‑linear coefficient reflecting the strength of the pressure‑induced strain, and ν is an effective viscosity/diffusivity term. When β is large and ν is small, the non‑linear advection term dominates, producing sharp, self‑steepening fronts—Burgers solitons—that travel much faster than diffusion fronts.

Parameter estimation is performed for several representative rocks: Pierre Shale, Bearpaw Shale, Boom Clay, and Oznan‑Mugu silt/clay. Laboratory measurements and literature values provide permeability (k), porosity (φ), elastic modulus (E), and bulk compressibility, from which β and ν are calculated. Numerical simulations (high‑resolution finite‑difference, 1‑D domain of 100 m) are carried out with an initial pressure‑solute pulse of order 1 MPa and 10⁻³ kg m⁻³. The results show soliton propagation speeds of 10⁻²–10⁻¹ m s⁻¹, maintaining a distinct, asymmetric front over distances of tens of meters, in stark contrast to the 10⁻⁴ m s⁻¹ diffusion speeds predicted by linear theory. Peak pressures can reach 1.5–2 times the initial amplitude, illustrating the strong strain amplification caused by the non‑linear elasticity.

To enable field verification, the authors propose a “soliton detection protocol” that combines high‑frequency pore‑pressure transducers with concurrent electrical conductivity (or direct solute concentration) sensors. A measured front that travels faster than ν/β and exhibits a non‑Gaussian, steep leading edge would be indicative of soliton behavior.

The practical implications are explored through two case studies. First, a nuclear‑waste repository scenario: heat‑induced pressure gradients of up to 5 MPa m⁻¹ can generate solitons in surrounding shale, potentially transmitting stress and fluid over hundreds of meters within decades, thereby challenging conventional safety assessments that assume diffusive transport. Second, saline‑water intrusion in coastal aquifers: the rapid, soliton‑driven migration of brine could outpace predictions based on Darcy diffusion, leading to faster contamination of freshwater resources and oil‑gas storage zones.

A “theoretical experiment” extends the analysis to higher‑permeability sandstones (Jordan and St. Peter). Although sandstones typically have larger ν, the authors demonstrate that under certain stress conditions—e.g., elevated confining pressure or dense micro‑fracture networks—β can increase sufficiently to lower the ν/β ratio, allowing soliton formation. Simulations suggest that when β exceeds ~0.8 Pa⁻¹ m⁻¹, soliton speeds approach 1 m s⁻¹ even in these more permeable rocks, implying that osmosis‑like rapid transport may occur where it was previously thought impossible.

The paper’s strengths lie in (i) a rigorous derivation linking non‑linear elasticity to soliton dynamics, (ii) quantitative parameterization for a range of lithologies, and (iii) a practical field‑testing methodology. Limitations include reliance on 1‑D modeling, uncertainty in β and ν due to limited laboratory data, and the omission of multi‑phase flow, chemical reactions, and anisotropic effects that are common in real reservoirs.

Future work recommended by the authors encompasses (a) inverse‑modeling of field pressure and solute data to refine β and ν, (b) extension to 2‑D/3‑D non‑linear wave simulations that incorporate heterogeneity and anisotropy, and (c) coupling with reactive transport models to assess how chemical interactions modify soliton behavior. Such advancements would broaden the applicability of the soliton framework to groundwater management, carbon‑capture‑and‑storage, and geotechnical risk assessment, offering a more realistic description of fast, large‑amplitude transients in geological media.