# A Critical Review of the Modelling Tools for the Reactive Transport of Organic Contaminants
## 3 Modelling
### Stochastic Models  
Stochastic hydrogeology operates on the concept of probability, which is inherently subjective and reflects the level of understanding or confidence in the actual state of affairs within a system that exhibits randomness [411; 412; 413; 414]. Hence, stochastic hydrogeological models must address the probable distribution of input parameters and their associated uncertainties: theoretical uncertainties arising from limited knowledge about the processes impacting model outcomes; measurement uncertainties stemming from instrument accuracy; and uncertainties attributed to spatial and temporal non-uniformity or missing data [415; 416]. The use of stochastic processes to visualise hydrological and subsurface processes is not a new concept. The first publications and reports on this topic appeared in the late 1960s and early 1970s [417; 418; 419; 420; 421; 422; 423; 424]. In the early days of stochastic subsurface research, many efforts were made to realise mathematical equations for effective parameters such as effective conductivity and macrodispersivity in an elegant way, under the premise that these parameters could be used in large-scale flow and transport models. Matheron [419], for instance, is renowned for developing the theory of regionalized variables, which elucidates the statistical relationships among sample points by considering not only their values, but also their spatial arrangement. Consequently, observed values are outcomes governed by specific probability density functions. The framework of linear geostatistics has served as a foundational framework for many geostatisticians and modelers [425; 426]. Beran [420] and Todorovic [422] accurately forecasted and delineated the mathematical modelling of solute transport at the molecular level. Chow and Prasad [423] asserted that natural hydrological systems, such as watersheds, and hydrological processes inherently exhibit stochastic behaviour, implying that their dynamics vary over time in accordance with probabilistic occurrences. The modelling of stochastic processes at the watershed scale has been the focus of numerous investigations [427; 428; 429]. In recent decades, stochastic studies have addressed various topics due to environmental concerns and interest in subsurface contamination: the modelling of structures, unsaturated soil properties, the spatial propagation of fall heights and velocities, and the transport of reactive solutes [430]. It should be emphasized that, thanks to stochastic subsurface hydrology, many processes have been better understood, the most important mechanisms have been identified, and a new paradigm--heterogeneity--has been introduced in subsurface transport studies. The theoretical progress and a deepening of the understanding of subsurface processes as stochastic has been greatly aided by the dynamic development of computers and programming, access to high-resolution data, and the performance of large-scale experiments [431; 432; 154; 433; 155; 434; 156; 157; 158; 159; 160; 161; 162; 163; 164; 165; 166; 167; 168; 169; 170; 171; 172; 173; 174; 175; 176; 1  
guages and machine learning have facilitated the training and education of experts in stochastic modelling methods. The shortcomings that may hinder the broader application of stochastic models in reactive transport were extensively outlined in the work of Cirpka and Valocchi [436]. Among these, inadequate consideration of the processes and properties governing system behaviour stands out as a significant challenge. For instance, stochastic modelling of the plume tends to underestimate the mass in the tail by focusing solely on the arrival of contaminant peaks. This approach may seem unrealistic to many remediation practitioners and experts who are aware that pollutants are released over extended periods, resulting in the formation of elongated plumes. Moreover, models focusing on local pollution should also account for mixing and its dependence on small-scale heterogeneity [437].