Mathematical and numerical modelling of limestone dissolution

Abstract

The dissolution of calcium carbonate in limestone results in karst morphologies and karst caves over a long length of time. Although the geochemical process is simple, the dissolution can cause serious issues such as sinkholes depending on the geological and/or environmental conditions. Thus, a simulation predicting its dissolution processes is valuable for understanding the geological risk of karst environments. This paper presents a simple model of the dissolution of calcium carbonate in limestone to simulate the geochemical alteration of the karst terrain. The model is formulated by using an advection, chemical reaction and diffusion system and is discretised through the finite-difference method. The numerical simulations provide spatio-temporal dissolution patterns of calcium carbonate. In this work, some numerical simulations for intact rock, rock with internal cracks and rock with a cavity were conducted based on the proposed model. The simulation results indicate that the expansion of cavities depends on the direction of groundwater flow and the distribution of internal cracks and cavities.

Simulation of three-dimensional chemical dissolution of limestone

“…Although the method proposed in this study is based on the approach of Matsubara and Yamada [11], this is not the only technique for analysing the geomorphological characteristics of karstic caves. Moreover, the occurrence of the caves and fractures were not included in the proposed model.…”

“…where 1 and 2 are the reaction rates for the forward and reverse reactions, respectively. The Reaction in Equation 1was also employed for a 2D model by Matsubara and Yamada [11]. Following the study, this equation was adopted as the basic chemical reaction to formulate subsequent mathematical expressions in this study [15].…”

“…Therefore, herein, the reaction in (1) is mathematically formulated using the reaction-diffusion system to demonstrate the complex dissolution patterns of karstic caves. Moreover, advection terms are introduced for the ionic and liquid parameters in this system to account for groundwater flow according to using the following equations [11]:…”

“…where G, D, and R are the arbitrary variable, diffusion term, and reaction term, respectively. The advective transport is then expressed as Equation (11).…”

“…Therefore, in this study, we investigate the temporal and spatial morphological patterns of calcium carbonate dissolution and present 3D mathematical and numerical models based on the approach by Matsubara and Yamada [11]. That is, a 3D technique based on the advection, reaction, and diffusion systems is proposed.…”