Numerical modelling and analysis of solute transport in a single fracture

Abstract

Studies of non-reactive tracer movement in porous formations exhibiting heterogeneity due to variation in hydraulic conductivity have been the subject of considerable research over the past several years. These studies focus on understanding the effective values of solute velocity and macro-dispersion, since these properties strongly influence pollutant migration in aquifers, affecting both dilution of concentrations and first-arrival times of solutes at a given location.

Transport through fractured rocks involves a special kind of heterogeneity, where fracture permeability differs markedly from the surrounding rock matrix. The movement and mixing of solutes in fractured media is of particular environmental interest because contaminants can move rapidly and extensively through fractures, cracks, or fissures in otherwise low-permeability rock.

Scope of the Present Study This study examines the transport behavior of solutes in a single fracture with matrix diffusion. Analyses include both non-reactive and reactive solutes, considering linearly and non-linearly sorbing solutes under the local equilibrium assumption.

A finite difference model was developed to simulate solute transport behavior.

Solute mobility and spread characteristics were analyzed using the first two spatial moments of concentration distribution.

The study was extended to decaying solutes, analyzed using temporal moment analysis.

A one-dimensional approach was adopted along the fracture and perpendicular to it, representing a rock matrix embedded with parallel fractures.

Numerical simulations used a second-order central difference finite difference model ensuring flux continuity at the fracture-matrix interface, with forward Euler for time marching.

Experimental and theoretical results suggest that contaminant transport through fractures is influenced by fracture and matrix properties such as aperture, spacing, water velocity, local dispersivity, matrix porosity, and diffusion coefficient. This highlights the need to relate these parameters when describing solute mobility and spread characteristics.

Objectives Provide expressions for solute velocity at pre-asymptotic and asymptotic stages, analyzing the role of matrix and fracture porosity for non-reactive and linearly sorbing solutes.

Study the effect of fracture-matrix coupling on local-scale dispersion coefficients at both stages, with expressions for non-reactive and linearly sorbing solutes.

Extend analysis to solutes with non-linear equilibrium sorption.

Analyze velocity and macro-dispersion behavior of solute fronts in fractures for first-order decaying solutes.

Chapter Outline Chapter 1: General introduction and scope of the study.

Chapter 2: Literature review on transport of non-reactive and reactive solutes in fractures with matrix diffusion.

Chapter 3: Spatial moment analysis of non-reactive solute transport in a single fracture using a dual-porosity framework. Expressions for solute velocity, dispersivity, and macro-dispersion coefficient in asymptotic regimes are presented.

Chapter 4: Asymptotic behavior of solute velocity, dispersivity, and macro-dispersion coefficient for linearly sorbing solutes. Linear sorption isotherms are considered, with emphasis on matrix and fracture porosity.

Chapter 5: Pre-asymptotic transport behavior of non-reactive and reactive solutes. Time-dependent velocity and dispersivity expressions are derived. Comparisons are made with multiple fractures of varying aperture widths.

Chapter 6: Asymptotic behavior of solute transport for non-linearly sorbing solutes using a Freundlich sorption isotherm. Analysis includes decaying solutes using temporal moment analysis.

Chapter 7: Conclusions drawn from the study.