Table of Contents
Dewatering of a surface or underground mine can be a significant production cost. Total costs include both the capital costs for wells, pumps, and water-disposal facilities and the operating costs for pumping, treatment and disposal of the water. Such costs are minimized when the present value of the capital and operating costs is minimized. The dewatering can be accomplished with various configurations of wells, rates of pumping, and operating schedules.
Design of an optimal dewatering system can be accomplished by coupling the techniques of operations research to a ground-water flow model. Operations research is a quantitative and systematic approach that mathematically formulates an optimization problem in terms of an objective function and a set of constraints.
Design of an Optimal Dewatering System
Description of Example Problem
A circular open pit mine is to be excavated to a depth of 750 ft below the original ground-water table over a 10-year period with progressive mine stages at 2-year intervals as shown on Figure 1. A dewatering system is to be designed such that the sum of its capital and operating costs is minimized and there is to be no passive ground-water inflow to the pit at any time during the 10-year mining period. The constraint is the condition that the dewatering system produces a “dry” pit.
Objective Functions
The objective function for this problem can be expressed as
C = ∑ Cap + ∑ Op…………………………………………………………………………(1)
where
C = present value of all costs [$],
Cap = present value of a capital cost [$], and
Op = present value of an operating cost [$].
The present value of a capital or operating cost is given by
Cap = cap · PVF(i,t)………………………………………………………………….(2)
and
Op = op · PVF(i,t)……………………………………………………………………(3)
where
cap = a capital cost at time t [$],
op = an operating cost at time t [$], and
PVF(i,t) = present-value factor for cost at discount rate i and time t [$].
The present-value factor is given by (Grant and Ireson, 1964)
PVF(i,t) = 1/(1 + i)t…………………………………………………………..(4)
where
i = discount rate per time unit [ ], and
t = number of time units from present [ ].
The operating cost of a well is principally the cost of the electrical power to run the pump.
where
Q = pumping rate [ft³/s],
γ = unit weight of water [lbs/ft³],
hd = hydraulic head of point at which water is discharged [ft],
h = pumping water-level elevation in well [ft],
Δt = time period [s],
E = electro-mechanical efficiency of pump and motor [ ],
k = conversion factor to units of kilowatt-hours [kwh/(ft·lb)], and
Ck = cost per kilowatt-hour of electrical power [$/kwh].
Simulation of Ground-Water Levels
The pumping water-level elevation in a well is related to its pumping rate and the pumping from all other wells in the dewatering system.
where
h = hydraulic head [ft],
K’ij = effective hydraulic conductivity [ft/s],
kR = relative hydraulic conductivity [ ],
Ss = effective specific storage [ ],
Xi = a coordinate [ft], and
W = all sources or sinks in terms of discharge per unit volume [ft³/s/ft³].
where
Kij = hydraulic conductivity under Darcian flow conditions [ft/s],
b = coefficient of non-Darcian flow conditions [s²/ft²], and
q = ground-water flux in the direction of flow [ft/s].
where
Ss — specific storage due to elasticity of matrix and compressibility of water [ft-¹],
Sy = specific yield at water table [ ],
V = specified volume of media [ft³], and
A = area of water table within specified volume [ft²].
where
Q = pumping rate of well [ft³/s],
j = index for pumping well [ ],
m = number of pumping wells [ ],
δ = Dirac delta function [ft-³],
xj = x coordinate of pumping node in finite-element grid [ft],
yj = y coordinate of pumping node [ft], and
zj = z coordinate of pumping node [ft].
Optimization of Dewatering System Using MINOS
Design of the dewatering system will be optimized using MINOS, a code developed at Stanford University to solve non-linear optimization problems (Murtagh and Saunders, 1987). Optimization of the example problem is non-linear in both the objective function and the constraint, but this optimization approach will linearize the constraint.
where
Δh = change in hydraulic head at a water level observation point due to changes in pumping rates [ft],
a = unit-response coefficient [s/ft²],
ΔQ = change in pumping rate of well [ft³/s],
i = index for water level observation point [ ],
j = index for pumping well [ ],
m = number of pumping wells [ ],
t = index for time-step interval of the water level observation point [ ], and
T = index for time-step interval of pumping [ ].
The unit-response coefficients of Equation 10 are derived from a finite-difference approximation. Using current values of the pumping rates for each well (Equation 9), the pumping in a particular well for a particular time-step interval is perturbed; and the resulting change in hydraulic heads is simulated using MINEDW.
Results
The results of optimizing the example mine dewatering problem are given for five different scenarios. The scenarios differ with respect to the anisotropy ratio of the ground-water flow system and the specified objectives, but in each scenario the same ten wells (representing 80 wells for a full 360-degree flow system) shown in Figure 3 were available for pumping.
As indicated in Tables 1 and 2, the cost and volume of water pumped from the optimized dewatering system under three significantly different anisotropy ratios are not very different, but the selection of wells, pumping rates, and pumping schedules is substantially different.
