Optimal design of an underground mine decline with an associated vent raise Peter Grossman Dept of Mechanical Engineering The University of Melbourne MTNS, Melbourne, July 20121

The research team At The University of Melbourne: Prof Hyam Rubinstein, Dept of Mathematics and Statistics Prof Doreen Thomas, Dept of Mechanical Engineering A/Prof Marcus Brazil, Dept of Electrical and Electronic Engineering Prof David Lee, Dept of Mathematics and Statistics (also University of SA) Peter Grossman, Dept of Mechanical Engineering Prof Nick Wormald, University of Waterloo, Canada The financial support of Rand Mining / Tribune Resources and the Australian Research Council through an ARC Linkage Grant is gratefully acknowledged. MTNS, Melbourne, July 20122

Outline 1.Background to the problem: current industry practice 2.The optimal vent raise location problem 3.The solution and its implementation 4.Future research MTNS, Melbourne, July 20123

Underground mine access In many underground mines, access between the surface and the working areas is provided by a network of declines and drives. This network provides access for equipment and personnel. It is used by the trucks that haul the ore to the surface. MTNS, Melbourne, July 20124

Our team has developed an algorithm for finding near-optimal designs for decline networks. Objective: Minimize total construction and haulage costs. Constraints: – Given topology – Passes through given locations (the access points for the mining levels and the surface portal or breakout from existing infrastructure) – Navigability (maximum gradient, minimum turning circle, etc.) – Avoids certain areas (a buffer zone around the ore body, geologically unstable regions, etc.) bounded by barriers specified by the user MTNS, Melbourne, July Optimally designing decline networks

The software tool DOT We have implemented the algorithm in the software tool DOT. A mine design engineer can use DOT to rapidly generate a number of alternative strategic designs that can form the basis of the final design. MTNS, Melbourne, July 20126

Ventilation – provides fresh air and removes dust and contaminants; – is essential but costly (15-22% of a mine’s operating cost and up to 40% of total energy costs*). The primary ventilation system delivers air from the surface to the vicinity of the working areas. Fans on the surface extract the air from the decline network via vertical vent raises. (Underground booster fans may also be present.) The secondary ventilation system uses auxiliary fans and ducts to deliver air to the working faces. * Popov, 2004; Hardcastle & Kocsis, 2007; quoted by Acuña et al, MTNS, Melbourne, July Mine ventilation

A mine design engineer designs the access network. A mine ventilation specialist adds the vent raises and other ventilation infrastructure. The objective is to find the design with the lowest energy consumption that meets the airflow requirements. Shortcomings of this process: – Solution quality depends on the ventilation specialist’s expertise. – The solution may be far from optimal. – No time to explore designs for alternative scenarios. MTNS, Melbourne, July Current industry practice

The vent raises could simply be added to the access design generated by DOT. Could a better design be achieved by optimally locating the vent raises concurrently with optimising the access design? We investigated this problem for a single gradient- constrained decline and an associated vent raise. The objective was to minimise total construction and haulage costs (but not ventilation operating costs). MTNS, Melbourne, July Optimal vent raise location problem

A decline and a vent raise 10MTNS, Melbourne, July 2012

A decline and a vent raise Disregard the short connectors. Disregard the curvature constraint. Assumptions: – The vent raise is vertical. – The decline gradient is constant with a given upper bound, m 0, on its value. – The access drive points are uniformly spaced vertically with spacing h. – The vent raise connection points are uniformly spaced vertically, also with spacing h. 11MTNS, Melbourne, July 2012

Finding the optimal location Project the problem onto a horizontal plane. Vent raise point: V Access points: A 1,..., A n (in height order) Because of the regular vertical spacing of the access points and the vent raise connection points, the sum of the lengths of the paths along the projected decline from A i to V and from V to A j equals a constant, k, for all i and j, where k ≥ h/m 0. MTNS, Melbourne, July

Hence, k ≥ |A i V| + |A j V| for all i and j, where |.| denotes Euclidean distance in the plane. Define T(V) = max{|A i V| + |A j V|} (max over all i and j). Then k ≥ max{h/m 0, T(V)}. Conversely, given k that satisfies this inequality, a decline can be found that satisfies all of the constraints. To minimize the length of the decline, and hence its construction cost, we seek to minimize k. We can achieve this by choosing V so as to minimize T(V) and setting k = max{h/m 0, T(V)}. MTNS, Melbourne, July Finding the optimal location

Let V 0 be the value of V that minimizes T(V). Three cases can arise: – T(V 0 ) < h/m 0. Then k = h/m 0 and the upper bound on the gradient is achieved. There are values of V that do not minimize T(V) but that yield the same value of k. – T(V 0 ) = h/m 0. Then k = h/m 0 = T(V 0 ) and the upper bound on the gradient is achieved. – T(V 0 ) > h/m 0. Then k = T(V 0 ) and the upper bound on the gradient is not achieved. MTNS, Melbourne, July Finding the optimal location

Minimum bounding circle (MBC) for a set of points: Smallest circle containing all of the points on or inside it. Properties of the MBC: – The MBC of a given set of points is unique. – At least 2 of the points are on the MBC, and the angles of the arcs between any pair of adjacent such points is ≤ π. – The centre of the MBC minimizes the maximum distance to each point. MTNS, Melbourne, July

We have established the following result: Let A 1,..., A n be n points in the plane (n > 1). Let X be the centre of the MBC of A 1,..., A n. Let T(V) = max{|A i V| + |A j V|} where 1 ≤ i ≤ n − 1 and 2 ≤ j ≤ n. Then X minimizes T. Note: The converse is not true in general, i.e., there may also be other points that minimize T(V). There are established algorithms for finding the MBC of a set of points. One such algorithm is Welzl’s algorithm, which runs in time O(n) on average, where n is the number of points. 16 Finding the optimal location MTNS, Melbourne, July 2012

1.Use Welzl’s algorithm to find the centre of the MBC of the horizontal projections of the access points. 2.Calculate the vertical coordinates of the connection points. 3.Use a numerical optimization method such as simulated annealing to modify the horizontal coordinates of each connection point independently to meet other constraints, e.g., curvature, subject to a constraint on how far the vent raise may depart from vertical. 4.Design the decline through the access points and the connection points using a currently available algorithm, e.g., DOT. 5.Design the vent raise by joining the connection points with straight lines. MTNS, Melbourne, July Procedure for designing a decline and a vent raise

Implementing the algorithm We have implemented this algorithm in UMOID (Underground Mine Optimal Infrastructure Designer), a software application based on DOT. Left: UMOID design with no curvature constraint. Right: UMOID design with a constraint on the curvature. Note that the vent raise is connected to an additional point on the surface whose location is specified by the user. MTNS, Melbourne, July

Mathematical theory: – Find necessary and sufficient conditions for a point to minimize T(V). Mining application: – Make the design more realistic by adding connecting drives between the decline and the vent raise. – Develop methods for optimally designing more general ventilation networks. 19 Future research MTNS, Melbourne, July 2012