Matthew Otwinowski

The examples (Scenarios 1-4 in Section 5.1) illustrate the practical power of the scaling relations. Results for various combinations of physical and chemical parameters characterizing Waste Rock and pile design, can be obtained without repeating the numerical calculations for every new set of entry data.

Analogous estimates can be made also in the presence of bacteria. In the presence of bacteria pyrite oxidation rates are much higher and the critical value of pile size, L^* will be lower. The activation energy of pyrite oxidation by ferric iron is greater than the activation energy of oxidation by oxygen. For this reason the model involving the bacterial oxidation will use a different value of the nonlinear thermokinetic exponent q and additional nonlinearity will be introduced by the temperature dependence of bacterial oxidation [SyT], [Se].

While the sample calculations are performed for pyrite, the same model with only slight modifications can be used for other acid generating minerals. Useful scaling relations can also be derived in the presence of water transport and air convection.

Unlike the existing waste rock models, the kinetic equations used in this study do not contain adjustable parameters. All the parameters entering eqs. (4.1) and (4.2) can be measured in a series of independent experiments. At the same time we do not pretend that the equations (4.1), (4.2) and their steady state solutions provide a complete description of Waste Rock Piles. The kinetic model used for our scaling analysis assumes that:

• the convective transport of oxygen is slow;

• the water flow is slow;

• the pile is large and seasonal temperature changes affect a small external portion of the pile;

• the waste material has a uniform pyrite concentration throughout the pile; and

• water vapour concentration is uniform inside the pile so that the diffusion coefficient D remains constant.

The first two assumptions should be adopted as a practical rule for waste rock management and therefore should be considered as realistic features of waste rock piles. (Different principles have to be adopted when the preemptive leaching is used - see comment on p.24). The first four assumptions are not satisfied in the case of most field-study reports available through MEND. In particular, piles 17, 18a and 18b in the Heath Steele study were small (only 4-5 m high), had high porosity and were not placed on impermeable lining (thus allowing high flux of oxygen from the base). In such cases the oxidation rates are limited not by diffusive oxygen transport but by the convective transport of thermal energy and convective oxygen transport. This is an undesirable situation leading to temperatures above 40°C despite a small pile size. In the case of Mine Doyon only the third assumption is satisfied and the estimated thermal energy stored in the pile is equivalent to about 2.5 years of the thermal activity of the waste material. This results in relatively small seasonal temperature variations at distances greater than 5 meters from the pile surface (see Fig. 4 on p. 33 of [LGI]). Because of the large thermal inertia the seasonal changes in Mine Doyon's dump remain small despite high convection rates. 

We concentrate our attention on diffusive transport because it seems that the convective transport should be eliminated in properly managed waste rock piles. (A model which will properly describe convective effects should be analyzed, however, in order to understand ARD for a broad range of conditions). Even when convective transport is eliminated, the nonlinear effects which were disregarded in existing numerical models, are responsible for thermal catastrophes leading to an abrupt increase in the Acid Generation Rates when the critical value δ^* of the physico-chemical scaling parameter is reached. For large values of the scaling indicator δ, the pyrite oxidation rates with diffusive oxygen transport become as high as in the presence of convection. In order to understand this feature one has to realize that rates of oxygen supply while increasing due to convection, are accompanied by increased rates of energy transport to the surroundings. The scaling analysis can be easily performed also when convective mass and energy transport becomes dominant. One of the main differences is the proportionality of acid generation rates to α instead of α^1/2 - this results in a more dramatic increase of the acid generation rates with respect to the pyrite content of waste rock.

A sample of numerical solutions obtained by the finite elements method is presented in Appendix B. The numerical results confirm the analytical results of the scaling analysis.

Our main purpose was to demonstrate that the chemical parameters measured in small- scale and meso-scale laboratory experiments can be combined with large-scale physical parameters (like pile size, for example) and that a physico-chemical model can produce a realistic description of the large-scale thermo-chemical behaviour of waste rock piles. Having demonstrated that the simple model derived from fundamental physical and chemical principles gives reasonable results, we feel confident that after a further refinement it is possible to construct a realistic and useful waste rock model.