Incorporating Hydrologic Uncertainty in Industrial Economic Models: Implications of Extreme Rainfall Variability on Metal Mining Investments
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Abstract
Water balance uncertainties have long been known to lead to potential environmental hazards, but their effect on economic profitability of mines is an understudied field of research. Historical rainfall data are analyzed using the extreme value theory (EVT) and the peak over threshold method (POT). The resulting distributions are used as inputs into a system dynamics technoeconomic metal mining investment profitability model, and simulation analysis is performed. The proposed methodology incorporates rainfall extremes and uncertainty into technoeconomic modeling of metal mining operations. A case study with reallife historical rainfall data was used to illustrate the relationship between hydrologic uncertainty and the economic value of a metal mining investment.
Keywords
Open pit flooding Temporary mine shutdown Investment analysis System dynamics modeling Extreme rainfall eventsIncorporación de la incertidumbre hidrológica en los modelos económicos industriales: implicancias de la variabilidad extrema de la lluvia en las inversiones en minería de metales
Resumen
Es ampliamente reconocido que las incertidumbres del balance hídrico conducen a peligros ambientales potenciales pero su efecto en la rentabilidad económica de las minas es un campo de investigación poco estudiado. Los datos históricos de lluvia son analizan utilizando la teoría del valor extremo (EVT) y el método de pico sobre umbral (POT). Las distribuciones resultantes se utilizan como entradas en un modelo técnicoeconómico de rentabilidad de la inversión en minería metalúrgica y se realiza un análisis de simulación. La metodología propuesta incorpora extremos de lluvia e incertidumbre en el modelado técnicoeconómico de las operaciones de minería de metales. Se utilizó un estudio de caso con datos históricos de lluvia de la vida real para ilustrar la relación entre la incertidumbre hidrológica y el valor económico de una inversión en minería de metales.
水文不确定性纳入工业经济模型:极端降雨变异性对金属采矿投资的影响
抽象
水量平衡不确定性的潜在环境灾害性早已为所知,但它们对采矿盈利的影响还未被充分研究。使用极值理论(EVT)和峰过阈值法(POT)分析历史降雨数据。计算所得降雨分布被用作输入项,融入技术经济的金属采矿投资盈利模型,实现模拟和分析。提议方法将极端降雨及不确定性纳入了金属采矿的技术经济模型。用真实的历史降雨案例说明了水文不确定性与金属采矿投资之间的关系。
Introduction
Large, irreversible industrial investments with heavy initial capital layouts, such as the metal mines that are the focus of this research, are designed to operate for decades in uncertain technical and market environments. Investors are faced with a tradeoff between constructing lower waterrelated risk designs and declining economic returns. This means that exante planning of these investments is very important from the point of view of being able to secure the longterm technical and economic sustainability of mining investments. It also implies that the planning models used should be able to capture the type and kind of uncertainty that surrounds the analyzed investments (Collan et al. 2016) and properly reflect their complexity (Ashby 1958).
In this vein, our research focuses on metal mining investments and specifically on the modeling and analysis of the relationships between water management risks and mining economics. The profitability of mining projects can be severely threatened by improper water management (ICMM 2012). Quantification of the cumulative effects of mine water management is, however, difficult, and it is common that water management policies and decision making include conflicting interests (Zhang et al. 2014).
Brown (2010) reviewed the current water management planning policies of mines and found that mine water evaluations are unreliable, and that the magnitude of errors in the estimation of economic losses due to improper water management are often significant. Some of the most important reasons highlighted by Brown (2010) leading to failures to successfully predict the effects of minewater management include (a) high uncertainty of initial data, (b) computational complexity of analysis (methods), and (c) perceiving water management as a secondary issue in a mining project. According to Gao et al. (2014), current water balance models are not sufficiently reliable for evaluating longterm water management strategies, particularly when dealing with climate change scenarios. Water management costs can constitute as much as 10% of the total investment in a mine (e.g. Fleming 2016), and the inability to estimate the consequences of an erroneous water balance in a mining project analysis can lead to severe economic losses.
Despite the importance of water management issues for mining project profitability, there seems to be little written on the economic effects of mine water management. This lack of research may be due to the lack of models that simultaneously embed economic and environmental variables in a suitable model. The purpose of this work was to demonstrate that a recently developed system dynamics (SD) metal mining investment model (Savolainen et al. 2017a, b) could be extended and successfully used to quantitatively explore the effects of water management on the value of a metal mining investments within a riskbased framework. We illustrate how dynamic variables that depend on probabilistic thresholds and characterize the distribution of historical rainfall events, when evaluated using the extreme value theory, EVT (e.g. Gilli and Këllezi 2006; Pedretti and Beckie 2016; Serinaldi and Kilsby 2014; Pedretti and Irannezhad 2018), can be used to assess the feasibility of mining investments. The SDbased model provides a completely new approach that circumvents some of the limitations of traditionally adopted static and/or deterministic investment analyses (e.g. Bartrop and White 1995; Bhappu and Guzman 1995; Eves 2013; Lawrence 2002; Smith 2002) used in the metal mining industry.
Typically, system dynamics are modeled using stochastic processes, which treat input parameters as random variables. The randomness of these parameters is generally linked to the poor predictability of fluctuations in stock markets, societal and economic crises, and consumer habits and demand (e.g. Banerjee and Siebert 2017; Caballero and Pindyck 1992; Gilli and Këllezi 2006; Wernerfelt and Karnani 1987; Zachary 2014). When the performance of industrial activities depends directly on the variability in environmental conditions, longterm economic investments must also embed other sources of uncertainty that are linked to the endemic variability of natural phenomena. Because of geological, atmospheric, and oceanrelated processes, several potential extreme phenomena can occur, such as intense rainfall events or temperatures, leading to flooding, heat waves or droughts (e.g. Easterling et al. 2000; Kløve et al. 2014; Luoma et al. 2013; Merz et al. 2004; SchmidtThomé et al. 2013, 2015).
Incorporating environmental variables into economic models is not a straightforward task (e.g., Garrick et al. 2012). It requires a versatile dynamic simulation method that can predict the feasibility and profitability of an investment by simultaneously capturing (a) the technical and economic complexity of large industrial investments and (b) the random occurrence of natural events. SD models provide a promising approach in this sense, although they have not yet been widely applied to incorporate environmentally driven source of uncertainty. SD models are generally used for evaluating complex real investments by means of nonlinearity and feedback loop structures (Forrester 1994; Inthavongsa et al. 2016; Johnson et al. 2006; O’Regan and Moles 2006; Tan et al. 2010). Größler et al. (2008) discussed the use of SD models in the operations management context. They pointed out that the power of SD models lies in their ability to present how system components interact. Qureshi (2007) studied the value of a company using a SD approach, although they assumed no variability in the input parameters used (i.e. they used a purely deterministic approach).
SD models are potentially wellsuited to accounting for undesired economic losses due to the nonlinear nature of hydrologic processes that can lead to water security issues. Côte et al. (2007) evaluated mine water treatment using a SD approach. Garrick et al. (2012) described water security as the ability to deal with (or predict the impact of) the probability of occurrence of high impact hazardous events. This could, for instance, be the probability of flooding due to the occurrence of extreme rainfall events (e.g. Merz et al. 2004). Garrick et al. (2012) maintained that decision makers should focus on a system’s vulnerability and develop the ability to “anticipate, cope with, resist and recover” from water security risks. System dynamics investment models linked to the variability of hydrologic events would be ideal tools to develop such abilities and support the decisionmaking process. This work is among the first to illustrate how realworld environmental data can be used with a SD metal mining model to study the effects of water risks on metal mining project feasibility.
The paper presents the methodology used to describe hydrologic and the economic variability and an approach to combine them in a unique investment model. It describes how the approach can improve predictive outcomes and presents a case study of a longterm mining investment using realworld precipitation data.
Methodology
Modeling Metal Mines with System Dynamics
The SD model of metal mining investments used in this research was introduced in Savolainen et al. (2017a), where its key SD characteristics—namely modularity and feedback loops—were presented in detail. The model mimics the characteristics of realworld mining investments, where different aspects—most importantly production rates, cashflow (CF), and the balance sheet—simultaneously affect the overall economic profitability of the mining investments studied. These “aspects” are presented as standalone submodels tied together by immediate, or delayed, interactions that originate from feedback/feedforward loops and stocks.
As a simple example of the SD model, production during a long metal price recession results in a series of negative cashflow periods (feedforward). This in turn decreases the accumulated cash balance (stock), which in turn inhibits production (feedback) in the longterm. In other words, a permanent abandonment option would be exercised in the simulation if the cash runs out and the accumulation of losses to the owners is stopped. The SD model accounts not only for the variability in key parameters, but more importantly models the reactions to changes by altering the outcomes when the changes occur. For example, one can create a trigger for temporary mine closure to limit economic losses and save the remaining ore reserves for the future.
The importance of modeling operational changes in metal mines is supported by Moel and Tufano (2002) who studied North American gold mines. They showed that most existing mines have been closed for some period of time. During times of lower metal prices, it may be economically favorable to temporarily stop mining and wait for the prices to recover. However, as mine closing and opening decisions entail large capital and operational costs, metal prices should be well below or above breakeven profitability before making a decision to temporarily close or open, respectively, an operation. The hysteresis effect of real investments has been discussed, e.g. Dixit and Pindyck (1994) and Dixit (2004).
The SD model used is generic in the sense that the number of submodels attached to it is not limited. For the purposes of this study, we created a standalone SD water management submodel with a (water) stock and feedback loop interaction. This submodel is attached to the SD model. It should be noted that the water balance submodel may be replaced by any hydrogeological model. A schematic diagram of the SD model used is presented in Fig. 1, and a detailed function blockdiagram of the water management submodel is shown in Fig. 2a. The model is created in and runs on the Matlab^{®}/Matlab Simulink® software environment.
Modeling Rainfall Extremes
The probability of rainfall events is modeled in this work using the extreme value theory (EVT). The foundation of EVT stems from the need to estimate the probability of events that are less frequent but more intense than common lowintensity events (e.g. Coles 2001). EVT is used to describe the stochastic frequency (or probability of occurrence) of rainfall events using parametric analysis and distributions with heavy tails. Traditionally, daily rainfall events have been statistically described using heavytailed models, such as the Generalized Extreme Values (GEV) model (Papalexiou and Koutsoyiannis 2013), the Gamma model (Aksoy 2000), and the Generalized Pareto (GP) model (Solari et al. 2017; Van Montfort and Witter 1986). The modeling described in the paper is for precipitation falling as rain and is not relevant for precipitation falling as snow.
The question of which EVT model best describes the data is currently debated in the literature, in light of the potential implications of nonstationarity of the time series used, driven by changes in the climate and other timedependent factors (e.g. Milly et al. 2008; Serinaldi and Kilsby 2015; Pedretti and Irannezhad 2018). This debate falls outside the scope of this research, given that the main purpose of this work was to illustrate how to incorporate general hydrologic uncertainty in industrial economic models. We used an established approach based on the wellknown stationary peakoverthreshold (POT) analysis method (e.g. Smith 1990; Coles 2001; Pedretti and Beckie 2016; Pedretti and Irannezhad 2018), which evaluates the parametric distribution of the extremes of empirical rainfall distributions that exceed a given threshold, \(\theta\). The implication of the selection of a different modeling approach is left open for a future extension of the analyses in this paper.
Connecting the Rainfall Model to the Metal Mining Investment Model
The SD mining model can embed multiple parametric models to describe \(g(rk,\sigma ,\theta )\). The selected distribution is used to generate a set of Monte Carlo simulations of the investment model, as illustrated in Fig. 2b. The SD model uses a discrete monthly timestep, and monthly aggregated rainfall data were created and used as input to the model. Using a shorter step time would be unlikely to produce any additional insights to the cashflow (CF) modeling due to the joint uncertainty caused by several parameters. We believe that the monthly timestep is robust enough to allow for good control of the model and give accurate enough analysis for the purposes of this research. The total timeframe of SD simulations performed was set to 300 months.
A random monthly rainfall input to the investment model was created as follows:

31 random daily precipitation events were drawn from the modeled GP distribution and summed to generate the monthly precipitation for each of the 300 simulated months. We assumed that varying the length of individual months (from 27 to 31 days) would not play a significant role, considering the overall simulation accuracy.

The obtained sum of random draws was multiplied by 365/12/31 (≈ 0.98) to represent the average length (≈ 30.4 days) of a month in a year, so in fact every month was assumed to be ≈ 30.4 days long.
The other key parameters used in the SD metal mine model are listed in Table 1. We note that in Savolainen et al. (2017b), the water balance model assumed a Gaussian distribution with a mean cumulative annual precipitation R of 600 mm, a standard deviation of 150 mm, and an arbitrary water output limit of 200,000 m^{3}/month. In the present study, we increased the output to 300,000 m^{3}/month to cope with the overall larger water amounts derived from the empirical distribution fitting. The critical limit of water balance, \({l_{crit}}\), to avoid pit flooding was set to 1 Mm^{3}. No seasonal patterns of water flows (e.g. FernándezÁlvarez et al. 2016; Sahu et al. 2009) or possible discrete changes in water flows (Rapantová et al. 2012) were considered. In the case of an operating mine, these characteristics could be derived from measured precipitation data and a detailed hydrogeological model and added to the water balance submodel for additional insight.
Numerical values of key uncertainties (adopted from Savolainen et al. 2017b)
Variable  Unit  Pessimistic  Most likely  Optimistic  Volatility, % 

Reserve size  Tons  72,000  140,000  210,000  – 
Metal yield  Tons/month  1000  1200  1400  10 
Production ramp  Tons/month  50  100  200  – 
Unit cost  EUR/ton  4000  3500  3000  – 
Fixed cost (operating)  EUR/month  3,500,000  3,000,000  2,500,000  – 
Fixed cost (closed)  EUR/month  500,000  
Construction time  Months  36  24  12  – 
Construction cost  EUR  80,000,000  60,000,000  40,000,000  – 
Unit price  EUR/ton  14,000  16,000  18,000  5 
Exchange rate  USD/EUR  1.1  – 
Traditionally, metal mining investments have been valued using static net present value (NPV) calculations, or other discounted cashflow (DCF) based approaches. According to Brennan and Schwartz (1985a, b), a metal mine can be valued analogically to a financial option. That is, the owner of the metal mine holds an option to operate the mine when the metal sales price exceeds the cost of production, and the option to temporarily close the mine during metal market price recessions. In addition to the temporary closure option, other flexibilities to steer metal mine profitability exist, such as mine planning, contracting, and expanding production—these are typically referred to as real options (e.g. Savolainen 2016; Newman et al. 2010).
In Eq. 3 [Expanded NPV], refers to a case in which the value of real options has been taken into account (e.g. by considering the effect of temporary mine closure), and [Passive NPV] is the static NPV under a fixed operation mode.
Model Application: A Case Study
As an illustrative example of how SD models can be used to generate relevant and highquality decision support for mining operations in a way that combines economic and environmental analyses, we analyzed the effects of extreme rainfall events on a hypothetical metal mine operating in a location potentially exposed to flooding.
Modeling the Flooding of an Openpit Mine
The problem is conceptualized as presented in Fig. 3. An openpit metal mine is designed to optimally operate under normal (nonflooded) conditions (Fig. 3, top). The mined ore is fed into the concentration plant, where water is used for ore processing. Residual water from the concentrator is pumped to a waterstorage facility before final treatment and discharge. Under anomalous, or undesired conditions, such as a pit flooding event (Fig. 3, bottom), water accumulates in the pit after collapsing the drains. The pit flooding causes the mine to temporarily interrupt ore processing, resulting in a negative cash flow and economic losses. Given that the SD model combines the use of hydrologic and economic variables, the two major ingredients for estimation of economic losses associated with flooding events are:
 a.
the probability of occurrence of volumes of water that cause flooding of the open pit; and
 b.
the estimation of the economic damage caused by the ensuing interruption to ore production.
The probability of flood waters entering the open pit (a) can be calculated in several ways. It is reasonable to associate these volumes with excessive runoff resulting from extreme precipitation, expressed, for example, as daily precipitation rates \(r\) [mm/day]. During extreme events, rainfall rates tend to exceed evaporation and infiltration rates and the storage capacity of the subsurface, generating runoff (e.g. Boughton 2007). Because the occurrence of extreme rainfall events is random at a given site, the occurrence of runoff and pit flooding is also random and can be estimated in two general ways. The most correct yet unrealistic option (in many practical mining conditions) is through the direct measurement of flooding volumes. Very often, rainfall is sparsely distributed upgradient of open pits and results in diffuse sources of runoff entering the open pit that are not necessarily funneled in a single stream or water body (where a gauge station could be located). Due to this difficulty, an indirect approach is used to estimate the probability of flood events, based on a water balance model and measured rainfall from a weather station located at a mining site (e.g. Beven 2012; Chiew et al. 1993). Here, we used the water balance model approach. Since we are interested in evaluating the effects of hydrologic randomness on economic investments, we used a simplified stochastic modeling approach, in which we assigned the same probability of occurrence of rainfall extremes to the occurrence of pit flooding events characterized by a specific water volume. In reality, the two probabilities may be different, given that the amount of runoff generated from precipitation inputs can be highly nonlinear (e.g. Bo et al. 2018; Romanowicz et al. 2006). Nevertheless, the conclusions of this study from the method demonstration point of view are not limited by this assumption, as such nonlinearities, derived from the mine specific hydrologic model, can be included when dealing with a real world case.
A number of previous studies have identified precipitation extremes as one key variable limiting the global distribution of economic growth, because private investors prefer developed countries without water security related risks (e.g. Dadson et al. 2017; Khan et al. 2017). Therefore, embedding a realworld probabilitybased model with either realworld rainfall or runoff events is valuable in terms of obtaining more information on the profitability effects of these events and generally in making the model more holistic and realistic.
The estimation of the economic damages caused by interruptions in ore production (b) depends on the market situation at the time of the interruptions. Regardless of the water balance, it may sometimes be favorable to temporarily stop mining during the recession of prices. In our illustrative example, the cost to temporary stop the mine was set at 1.2 million Euros (M€) and the reopening cost was set at 0.5M€ (see Table 1). These costs effectively restrict the shutdownrestart flexibility of the investment only to situations caused by critical water imbalances.
Equations 4 and 5 dictate the behavior of the waterbalance submodel (see Fig. 2a). The water storage level depends directly on stochastic pit flooding volumes during rainfall events, as well as on the fixed water inflows from metal concentrate production and mine dewatering. Thus, the amount of water is directly linked to the probability of the extreme rainfall events, \(g(r)\). The monthly metal output in the production submodel is fixed by default, and the production on/off decision serves as a binary control variable that is based on the actual level of water storage. That is, a temporary stoppage decision will set the residual water from production–term (Eq. 4) to zero and steers the overall waterbalance.
As a summary of the problem setting: the investment model dynamically tracks the development of the waterbalance at the mining site, and, if pit flooding occurs, stops the production of metal concentrate to ease the water situation. It is noted here that the water management continues “asis” and keeps on accruing operational costs to the owners. Using this logic, running n rounds of pseudorandom simulations (Monte Carlo) on the investment model with random rainfall realizations derived from historical data, the probability and magnitude of the economic effects of rainfall (events) can be estimated using Eq. 3. If the project runs out of cash, the metal mining operation is abandoned, and the accumulation of production and water management related costs stops. The costs related to permanent shutdown are assumed to be included in the prepaid environmental provision, which is (modeled here as) a part of the initial investment cost. For the purposes of this research, we have left out a detailed consideration of the full economic consequences of a permanent mine shutdown from our model and “only” focused on the normal state of the operations, including temporary shutdown periods.
Peakoverthreshold Evaluation of Rainfall Variability
We applied the SD model and the proposed method to a metal mining case with real rainfall observation data from the ÄänekoskiKalaniemi weather station in Finland. The precipitation data were used to estimate the rainfallmodel parameters for the water balance model (Figs. 1, 2). The collected data consist of daily rainfall measured between Jan 4, 1965 and Feb 22, 2016 by the Finnish Meteorological Institute, and can be considered an adequate set of data for the purposes of runoff modeling. However, as in many other rainfall series, the available dataset is incomplete. Here, approximately 50% of the data are missing, as 9,516 measurements for a total of 18,987 days exist. However, the selected POTmethod is robust enough to handle the presence of missing records. Indeed, the approach works even with strongly incomplete time series, where a minimal number of records exists, and is able to build an empirical distribution that reflects the overall variability of rainfall events.
The resulting empirical cumulative distributions (Eq. 1) of the estimated rainfall using the GP model are shown in Fig. 4a. Figure 4 also displays the other tested bestfitting heavily tailed models in the form of probability plots, namely Gamma, Gaussian, GEV, and LOG (logarithmic) models. Each panel shows a different result for increasing the rainfall threshold, θ, applied to the source data as a filter. We note that the observed rainfall data span from 0 to 70 mm/day, and most observations lie between 0 and 20 mm/day, with a probability of ≈ 99%. Put another way, rainfall events with intensities of up to 20 mm/day are expected at least once every 100 days. Naturally, more extreme events, grossly exceeding 20 mm/day, have an even lower probability of occurrence. For instance, a rainfall event of 40 mm/day is expected once every 1000 days, given that its probability lies somewhere around 0.1%. Although the frequency of highintensity events becomes many times lower than that of lowintensity events (e.g. the estimated 10fold difference), it is of fundamental importance to properly calculate the occurrence of the highly improbable, rare events, from the watersecurity point of view. That is, highvolume, lowprobability events are essentially the events that are associated with the potential extreme runoff volumes and consequently pit flooding risk and temporary mine closure.
The probability plots in Fig. 4a emphasize the extremes of the distributions. While all the tested models result in the same behavior for low probabilities (P < 10%), at the extremes, no model provides a good fit with the empirical data (marked with circles). The bestfitted Gaussian model (black line) seems to largely underestimate the extremes, as the calculated probability of occurrence of events with intensity r > 30 mm/day is virtually nonexistent. In contrast, the GEV model (dashed yellow line) tends to overestimate the extreme values, given that an event with an intensity of 70 mm/day has a probability of 5%, which corresponds to a theoretical frequency of one event every 20 days. The GP (solid red line) also overestimates the extremes of the data for θ = 0 mm/day. However, consistent with the theoretical behavior of this model, the GP tends to increase “matching of the extremes” with an increasing θ. While the other models still incorrectly estimate the probability of rare events, the GP provides a satisfactory fit with all the used empirical data for θ > 1 mm/day, including the very intense, least frequent events (e.g., r = 70 mm/day).
An optimal threshold for the empirical dataset, θ_{e}, can be found by using the KolmogorovSmirnoff test (Eq. 2). This is illustrated in Fig. 4b, where the cumulative distance D is plotted against θ. In this case, the distance goes to an asymptotic minimum value D\(\approx\) 10 for θ = 3 mm/day or larger values. As such, the threshold θ_{e} = 3 mm/day can be identified as an optimal value, when taking into account the tradeoff between realistic extreme events fitting (= increasing θ) and the preservation of the original data (= decreasing θ). We still considered and used a range of values for θ to study the sensitivity of the results with regard to θ to highlight the importance of the selection of the threshold, θ, and the corresponding GP parameters (k,σ). The bestfitting parameters for θ range between 0 and 9 mm/day (Table 2).
Rainfall distribution parameters from distribution fitting and comparison with the actual data
POTlimit  Distribution parameters  Sample size  Avg. yearly rainfall  0.999 perc of daily rainfall  

mm/day  k  sigma  n  vs. Act  mm/year  vs. Act  mm/day  vs. Act 
Actual  9082  –  1150  –  41.00  –  
Naïve^{a}  149  1188 ^{b}  38  4.52  − 36.48  
0.0  0.347  2.289  9082  0  1249  99  68.76  27.76 
0.1  0.265  2.655  8614  − 468  1248  98  55.65  14.65 
0.3  0.126  3.518  7563  − 1519  1251  101  38.63  − 2.37 
0.5  0.044  4.268  6737  − 2345  1236  86  35.66  − 5.34 
1.0  − 0.049  5.507  5547  − 3535  1249  99  31.73  − 9.27 
2.0  − 0.139  7.355  4140  − 4942  1224  74  33.41  − 7.59 
3.0  − 0.197  9.010  3204  − 5878  1188  38  34.60  − 6.39 
4.0  − 0.243  10.569  2535  − 6547  1210  60  35.96  − 5.04 
5.0  − 0.291  12.387  1955  − 7127  1182  32  36.65  − 4.35 
6.0  − 0.332  13.978  1582  − 7500  1188  38  38.16  − 2.84 
7.0  − 0.379  15.843  1255  − 7827  1197  47  38.88  − 2.12 
8.0  − 0.429  17.851  998  − 8084  1191  41  39.40  − 1.60 
9.0  − 0.483  20.021  798  − 8284  1197  47  39.99  − 1.01 
Results and Discussion
The SD model is used to evaluate the implications of the hydrologic modeling parameters discussed above on the overall profitability of the mine. Figure 5 shows the mean net present value (NPV, circles) resulting from over 1000 Monte Carlo simulation runs that in the context of this case correspond to over one thousand simulated rainfall accumulations and their effect on mine profitability. The red line is the reference NPV (23.1 M€) that represents the attainable “benchmark” value of the project excluding waterbalance issues. The results show that not taking water balance issues into account will result in an unrealistic overestimation of project value if a rainfall event causes the temporary shutdown of the open pit. It can be noted that all results generated with the SD model using empirical databased rainfall distributions are below the benchmark NPV. Table 3 shows the results, which indicate a clear nonlinear loss of NPV as total rainfall increases.
Monte Carlo simulation results (n = 1000 rounds/row) of a feasibility analysis of metal mining investment using different POTthreshold assumptions in creating GPdistribution for the investment model’s input
GPmodel’s  Net present value  ROV(*)  Abandons  Value lost  Diff  

POTthresh, mm  Mean M€  < 0, n  > 0, M€  M€  n  M€  vs. Naïve 
Unconstr.  23.1  494  129.0  65.2  146  –  19.2 
Naïve  3.9  568  124.4  53.7  187  − 19.2  – 
0.0  − 26.5  656  100.5  34.6  254  − 49.6  − 30.3 
0.1  − 14.0  599  101.2  40.6  217  − 37.0  − 17.8 
0.3  − 15.1  637  112.0  40.7  192  − 38.2  − 19.0 
0.5  − 5.7  590  114.4  46.9  214  − 28.8  − 9.5 
1.0  − 5.8  592  110.6  45.1  199  − 28.9  − 9.6 
2.0  − 8.4  600  113.8  45.5  204  − 31.5  − 12.2 
3.0  − 0.2  575  119.7  50.9  186  − 23.3  − 4.1 
4.0  − 2.5  569  107.4  46.3  168  − 25.6  − 6.4 
5.0  6.9  539  115.2  53.1  178  − 16.2  3.0 
6.0  6.1  546  114.1  51.8  174  − 17.0  2.3 
7.0  2.2  555  111.4  49.6  177  − 20.9  − 1.6 
8.0  − 1.1  577  118.0  49.9  196  − 24.2  − 5.0 
9.0  1.8  571  119.7  51.4  170  − 21.3  − 2.1 
The mean NPV results attained (Fig. 5, crossmarks) imply that there is a certain limit of annual rainfall (R < 1086 mm/year) that has no effect on the economic value of the mining operation. We refer to this condition as an “economically safe zone.” An “uncertainty zone” is found when R is between 1086 and 1609 mm/year, during which the NPV moves from positive values to strongly negative values. This suggests that the value of the project declines dramatically as capacity constraints in water management seriously inhibit the operational efficiency. That is, the mine suffers from temporary shutdowns due to exceedance of critical water storage levels. A “failure zone” can be defined as R > 1609 mm/year, where the NPV becomes and remains negative, independent of any additional increase in pit flooding volumes.
From the point of view of the annual rainfall events, the NPV of the case study is in the zone of uncertainty. This result suggests that the longterm profitability of the mining operation will likely suffer from waterbalance issues, given the uncertainty in rainfall variability derived from the historical data and the simulated water management measures onsite. From the project management perspective, it makes sense to study the feasibility of either adding water treatment capacity to allow greater discharge (which may be constrained by the environmental permit), or to increase water storage.
Figure 6a presents a threedimensional plot of the sensitivity of the mean NPV to a set of extreme rainfall parameters. The selected range of parameters reflects typical values reported in the literature (e.g. Serinaldi and Kilsby 2014) and is in line with the bestfitting parameters, empirically obtained from the MLE analysis from weather data for the ÄänekoskiKalaniemi case study. The sensitivity analysis indicates that there is a clear relationship between project value and the GP parameters. Rainfall distributions are predominantly controlled by the k (shape) and σ (scale), although the parameter σ has a larger effect on the NPV than does k. For example, when k = 0.01 or k = 0.3, all scenarios with σ = 1 are in the safe zone economically and the NPV remains positive, while all scenarios with σ = 5 fall in the failure zone, regardless of the k value (see also Fig. 6b). The extension of the uncertainty zone, on the other hand, depends on the bivariate combination of the parameters k and σ. For σ = 3, a heavy distribution tail (k = 0.3) implies a shutdown of the mining operation, while a light tail (k = 0.01) implies a safe condition. The sensitivity analysis shows that the SD model is able to handle extreme parameters and capture the nonlinear complex relationship between waterbalance issues and profitability, via mine production and cash flows.
As shown in Fig. 6b, there is strong variability in project value depending on the selected rainfall modeling parameters. For low k scenarios (more symmetric distributions), the variability is much higher than for high k scenarios (more heavily tailed distributions). At the same time, the variability increases with higher σ values. However, while for low σ values all project outcomes are positive (including the statistical outliers, marked with red crosses), for large σ the median of all project outcomes tends to become negative, although a number of realizations behave differently. The reason for this observed model behavior stems from the fact that the investment model uses an aggregate of ≈ 31 daily observations with no seasonal patterns as a model input: it is statistically unlikely that a tailless distribution with high sigmaparameter would yield tens of extreme events in successive months. Therefore, the lowest project values are found in simulations with tailed distributions (large k). For example, for k = 0.01 and σ = 3, more than 42% of project value realizations are equal or greater than the reference NPV (23.1 M€), whereas for k = 0.2275 and σ = 3 the corresponding figure is only 9.3%. As such, while statistically the stochastic model suggests that on average the mining operations would fail economically (i.e. NPV < 0), if the rainfall distribution shows k = 0.3 and σ = 0.2275, it is possible that in approximately 1 out of 10 mining cases, the NPV is still unaffected by the adverse flooding conditions that limit production. A sensitivity analysis is provided in Table 4.
Sensitivity analysis of rainfall modeling parameters
1  2  3  4  5  6  7  8  9  10 

k  σ  NPV, M€  < 0, n  > 0, M€  ROV, M€  Ab., n  mm/year  mm/day  mm/month 
0.0100  1.0  35  467  137  73  139  379  7.1  46.6 
0.0100  2.0  25  479  129  67  145  771  14.0  97.6 
0.0100  3.0  17  514  125  61  157  1155  21.9  144.8 
0.0100  4.0  − 108  930  64  4  566  1555  28.4  190.8 
0.0100  5.0  − 133  1000  NaN  NaN  1000  1913  36.3  225.9 
0.0825  1.0  21  485  120  62  156  410  9.3  50.9 
0.0825  2.0  29  464  128  69  148  823  18.5  104.1 
0.0825  3.0  − 9  597  113  46  202  1226  28.4  157.8 
0.0825  4.0  − 129  989  36  0  737  1651  38.9  208.5 
0.0825  5.0  − 131  1000  NaN  NaN  1000  2088  44.1  268.5 
0.1550  1.0  24  491  132  67  159  447  11.8  58.6 
0.1550  2.0  27  498  132  66  139  900  24.2  121.4 
0.1550  3.0  − 36  712  104  30  280  1361  36.9  184.5 
0.1550  4.0  − 131  999  14  0  990  1784  48.6  244.0 
0.1550  5.0  − 131  1000  NaN  NaN  1000  2230  60.9  290.3 
0.2275  1.0  32  481  139  72  132  490  18.3  69.8 
0.2275  2.0  30  487  140  72  147  988  37.3  136.0 
0.2275  3.0  − 90  873  67  8  453  1463  51.6  221.5 
0.2275  4.0  − 132  1000  NaN  NaN  1000  1979  70.4  278.8 
0.2275  5.0  − 131  1000  NaN  NaN  1000  2467  91.3  370.0 
0.3000  1.0  32  481  137  71  143  541  21.8  88.9 
0.3000  2.0  27  484  132  68  159  1086  46.5  173.8 
0.3000  3.0  − 122  973  47  1  643  1609  68.8  244.2 
0.3000  4.0  − 131  1000  NaN  NaN  1000  2196  94.8  379.6 
0.3000  5.0  − 131  1000  NaN  NaN  1000  2747  118.6  429.1 
The above discussion shows that initial estimates of rainfall uncertainty can have a crucial effect on investment profitability. In real life, when estimating parameters from empirical rainfall distribution, an uncertainty in σ of one unit may not be an unusual result of analysis when the effects of subsampling complicate the estimation of GP parameters (e.g. Pedretti and Beckie 2016; Pedretti and Irannezhad 2018). While these effects reduce as the length of the analyzed time series increases, sometimes the only useful data available for a remote greenfield mining site are based on measurements from a recentlyinstalled weather station on the site. As such, extra care is recommended when estimating longterm predictions for water balance requirements using EVTbased stochastic modeling approaches.
Conclusions
Adequately linking the randomness of environmental events to economic profitability is usually complicated by the large number of variables and processes involved, and a versatile modeling approach is required. In this work, we presented an approach that can integrate hydrologic uncertainty into a SD model of a metal mining operation. We focused in particular on a synthetic case study using real rainfall data, where we demonstrated the effect of extreme rainfall events on the resulting economic profitability of a mine. The model allows an examination of the effects of water balance issues on the longterm profitability of mining investments in a quantitative manner. A numerical example showed how the proposed method can make use of historical hydrologic data (in this case, daily precipitation) to create insights on the feasibility of an investment.
The modeled case study was found to be very sensitive to rainfall extreme variability. By using a GP model to describe the peakoverthreshold distribution of values exceeding predetermined thresholds, we found that the investment value strongly depends on both the shape (\(k\)) and scale (\(\sigma )\) rainfall parameters used in the GP model. For example, a change in scale parameter from \(\sigma =3\) to \(~\sigma =4\) in the case study turned the expected project value from nearly positive to grossly negative. Given the difficulties in estimating statistical parameters describing variability in extreme rainfall events, sufficient safety margins must be considered when making longterm investments involving rainfall distribution uncertainty.
Water management investments that could prevent pit flooding and a temporary mine shutdown include adding water treatment and water storage. Based on the results from the illustrative case study, some general implications for decision making regarding water management investments may be suggested:
 1.
The results suggest that the relationship between hydrologic uncertainty and investment value distribution is nonlinear and, in this case, negatively skewed; that is, higher hydrologic uncertainty leads to lower project value. Inadequate water management investments are as likely to destroy investment value as overinvesting. Therefore, from a rational decisionmaking perspective, and assuming uncertainty of rainfall modeling variables such as incomplete historical data, the strategy for determining the size of an initial water management investment would be in the borderline area between safe and uncertain zones.
 2.
A gradual increase in water management capacity (storage and/or treatment) would likely result in the highest overall returns on investment (ROI) in cases where more capacity is needed.
 3.
The view that water management investments are real options has been implicitly and intuitively applied in the mining industry (see point 2 above), even if the decisionmaking tools used may have been unable to handle the value of real options in the past.
The optimal level of water management investments will be left for future research efforts. Furthermore, the attained results could be refined by including trends into the simulation of future rainfall, e.g. to account for the effects of climate change on extreme rainfall distributions. Indeed, stochastic parameters associated with rainfall variability could be conditioned by nonstationary distributions, although uncertainty affects the actual observation of trends for limited empirical datasets (e.g. Serinaldi and Kilsby 2015; Pedretti and Irannezhad 2018).
The limitations of the study are related to the nature and availability of data. In this study, we assumed that all the relevant technical data for model building and uncertainties are parametric, whereas in real life industrial investments on the planning table often entail “nonnumeric” uncertainties (such as environmental permits, technical development or changes in legal environment). In such cases, less detailed models should be considered.
With the approach proposed in this paper, a simulationbased real option study for staged investment of additional water management capacity could be performed by providing the SD model with an optional water management investment, which is only triggered (automatically) if the water balance is trending towards a critical level. This would radically decrease the water security risks without increasing the initial capital investment. However, the substantially longer time need to install water management capacity vs. the potentially rapid trend toward a critical water balance level must also be considered.
Notes
Acknowledgements
Open access funding provided by LUT University. This research has been funded by the Finnish Cultural Foundation/EteläKarjalan rahasto (grant number 05172166) and the Academy of Finland project “Manufacturing 4.0”.
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