RUSSIAN JOURNAL OF EARTH SCIENCES VOL. 10, ES5004, doi:10.2205/2007ES000282, 2008

Modeling the Mechanism of Power Distribution of a Number of Deposits With Different Amount of Mineral Resources

[20]  Let us consider the formation of empirical distribution of a number of deposits from the amount of reserves. High concentration of reserves in a few largest deposits is known to be typical of ore and hydrocarbon deposits. Large hydrocarbon deposits [Burshtein, 2006; Kontorovich et al., 1985; Rodkin, 2006; Turcotte, 1997; and others] are known and the large ore deposits are expected [Burshtein, 2006; Laverov and Rundqvist, 2004, 2006; Turcotte, 1997; and others] to obey the power law. That means that the number of deposits N with the stocks amount of no less than V obeys the relation (8) similar to earthquake number distribution in relation to energy or seismic moment:

eq009.gif(8)

2007ES000282-fig03
Figure 3
where K is a coefficient; b is a power exponent of the distribution, in the most cases the meaning of b is close to one. For ore deposits, relation (8) is fulfilled for large and super large deposits only [Turcotte, 1997]. As an example of such approximation in Figure 3 the distribution of reserves of 8 lead-zinc deposits largest in the world is given using [Laverov and Rundqvist, 2006] data. Taking into account the problem of estimating of the total amount of reserves in case of multi-component deposits the reserve amounts are given in value terms (billion dollars).

2007ES000282-fig04
Figure 4
[21]  In Figure 4, data on ore reserves in carbonatite rocks are given using data [Frolov et al., 2005]. Data on reserves (million tons) for different kinds of mineral raw material are used, these values are standardized by median for the given type of reserves, and data for different ores are combined further to obtain a totality of normalized reserve amounts. The obtained total distribution in the area of large (normalized) values of mineral resources obeys the power distribution law.

[22]  For hydrocarbon deposits the relationship (8) is fulfilled reasonably well for mean size deposits besides the large size deposits. Therefore the power distribution law is often applied in practice to estimate the number of undiscovered deposits of a given size in the region under study [Burshtein, 2006; Kontorovich et al., 1985].

[23]  Let us consider the possibility of modeling the process of origin of power-law` relationship (8) for hydrocarbon deposits. Definite data testify for the geological "youth'' of the most oil and gas deposits [Muslimov, 2005, 2006; and others]. Moreover the effect of recent replacement of hydrocarbon reserves was revealed, and available data suggest a specific law of replacing of hydrocarbon reserves, i.e., the replacement rate is found to be approximately proportional to hydrocarbon total reserves in the deposit [Rodkin, 2002a, 2006]. It can be seen easily that such a regime of reserve replacement is in agreement with an avalanche-like process model (1). Such an agreement testifies for the usefulness of model (1) in description of processes of formation and replacement of oil and gas deposits.

[24]  In other cases and specifically for presumably long processes of formation of ore deposits the model of deposit formation in accordance with an avalanche-like scheme appears questionable. Actually, the process of ore deposit formation is treated generally as a long-term process and not avalanche-like. Note however, that this concept is not generally accepted; in paper [Romaniuk and Tkachev, 2007] geological short-term simultaneous processes of large endogenous ore deposits are supported. From this it follows that different models of ore deposit formation appear to be possible.

[25]  A likely variant of accommodating of a positive feedback (1) scheme for the case of ore deposits formation may be related to the model of deposit formation by transmagmatic fluid flows [Zotov, 1989; and others]. In this case the appearance of positive feedback can be associated with the fact that the current volume of ore deposit is determined by the volume of discharged transmagmatic fluid flow but the same flow (because of high heat content) traveling through the solidified magma sustains the channel suitable for transport of new portions of deep fluid and ore components contained in it. Thus a positive feedback can occur.

[26]  However, keeping in mind a vagueness in deposit formation regime, let us consider possible alternatives to avalanche-like model of formation of power-law distribution (1). Differences in the duration of process of deposit formation may be suggested as an alternative. Indeed, indications to relatively longer duration of process of formation of large and super large earthquakes were found [Laverov and Rundqvist, 2004]. However with such a mechanism to accommodate power distribution, the typical duration of formation of smaller, medium and large deposits should differ very strongly. With comparable intensity of deposit formation rates, differences in the duration of forming of smaller deposits and super large deposits would be 10 3 -10 4 as much, and to form largest deposits the whole time of existence of the Earth would not suffice. Empirical data in favor of so great differences in the formation time of deposits of different stock values are missing.

[27]  The only characteristic that in the context of the used simple model may lead to the required power distribution of a number of deposits in relation to stock values is a power distribution of rates of resource accumulation values. Such distribution seems possible if the formation of endogenous deposits is associated with fluid-magmatic flows. Intensities of these flows are determined by features of corresponding magmatic diapirs and intrusions as well as permeability of faults associated with magma and deep fluid discharge. The structure of tectonic dislocations and diapirs is suggested to be self-similar and hierarchic. Thus, it seems reasonable to suggest that fluid flows corresponding to such structures are self-similar and hierarchic also.

[28]  Let us consider a very simple stochastic model of such a process. Suppose that the process of reserve accumulation in an ore deposit might continue with probability of p or cease with probability of ( 1-p ) at any step. The average rate of resource accumulation for each deposit is suggested to correspond to the value of random Yi being member of a power-law distribution with typical parameter value b=1. We preset the growth of reserves of i -deposit Xi at each subsequent step as the product of mean intensity value of endogenous flow Yi fixed for each deposit and the random value r, for example evenly distributed in interval [0, 1]. It is reasonable to take the initial amount of reserves in deposit X0 equal to zero. Assuming the probability of continuing of process of reserve replacement is p=0.5, we obtain at each step

eq010.gif(9)

eq011.gif

If it appears that r1<0.5, then the formation of this deposit stops and value Xi-1 as it has been formed in the preceding time moment is taken as the amount of reserves of the given deposit.

[29]  In the used simple model, the power law distribution of the mean rates of filling of the given deposit simulates the well-known hierarchic character of the geological discontinuances, for example hierarchic character of fault system being channel for the deposit feeding deep fluid flows. And the random character of a filling process expressed by the product of mean rate and random value r simulates the stochastic character of operating of fluid-magmatic system feeding the deposit. Both these conditions are reasonable and necessary. Therefore model (9) seems to be extremely simple, realistic and structurally stable (from the viewpoint of [Arnold, 1998] that models are called stable if variation of characteristics involved changes the details of their behavior without a change in principal features of their behavior). Taking into an account such stability of the model and the reasonable character of assumptions underlying it, one can hope that the major features of behavior of the model are adequate to the major regularities in process of deposits formation.

2007ES000282-fig05
Figure 5
[30]  In Figure 5, curves are given of the distribution of obtained model values of reserves X for different number K of episodes of formation of deposits: K=100, 1000, 10,000, 100,000. Since the probability of continuing of process of deposit formation at each step is p=0.5, actually the process starts in a half of the cases only. It follows that the number of deposits having formed will be close to 50, 500, 5000 and 50,000 respectively. It can be seen that the obtained distributions are similar, and power law character of distribution (with exponent b close to 1) takes place for deposits of larger volume of reserves only. For deposits with smaller reserves volumes the curve deviates from the power-law distribution curve towards considerably less amount of deposits. It should be emphasized that this feature of distribution is a basic feature of the model and it cannot be removed by an insignificant change in parameters of the model.

[31]  Similar result of a relatively smaller number of relatively smaller events follows also from the Gnedenko-Pikands -Balkem-de Haan limit theorem of the theory of probability (it is described in more details in [Embrechts et al., 1997; Pisarenko and Rodkin, 2007]). Simplifying strict formulations, this theorem states that the distribution of maximum values of excess function of any empirical distribution corresponds to the generalized Pareto distribution (GPD) corresponding to the power law distribution in the area of maximum values and deviating from it in the area of moderate values. The general pattern of GPD distribution is similar to model curves presented in Figure 4, thus the general Gnedenko-Pikands-Balkem-de Haan limit theorem of the theory of probability also indicates a relatively smaller number of deposits with smaller stocks values than it should follow from the clear power-law distribution.

[32]  The described above model feature - relatively smaller number of deposits with relatively smaller amount of reserves than it may be expected from the power law distribution - is a feature typical of empirical data [Burshtein, 2006; Kontorovich et al., 1985; Laverov and Rundqvist, 2004]. This feature is suggested ordinary to arise from the fact that it is more difficult to reveal deposits of smaller amount of resources. The assumption of lower probability of revealing of smaller deposits is feasible but the model used above suggests also another explanation of this phenomenon. The actual number of smaller deposits may be smaller than it should be assumed from the pure distribution power law.

[33]  To check this assumption we use data sampling from [Burshtein, 2006] on 18 oil basins of North America with a large number of oil deposits. In this paper the actual numbers of deposits with reserve amount of more than and less than 5 million tons and the expected numbers of deposits obtained by the extrapolation of the power law distribution are given. Besides the data on total reserves, age of sedimentary basins and the size of the basins (the areas of sedimentary basins and sedimentary rocks volumes) are presented.

[34]  It is reasonable to assume that the relation of actual numbers of oil deposits to the suggested numbers of deposits (theoretically expected number) will increase as the studies in the basin progress and this regularity will be manifested for both larger and smaller deposits (and even somewhat better for smaller deposits owing to a large number of such deposits and statistically more sufficient data).

2007ES000282-fig06
Figure 6
[35]  Having no direct data on the extent of previous studies, we use indirect assessments. Suppose that the extent of previous studies is better in basins with larger number of revealed deposits per area of the basin ( N/S ) and in basins with larger value of revealed reserves (  W ). Let us characterize the extent of coverage by value WN/S. Let us compare the amount of revealed deposits of volume greater than 5 million tons (points) and of volume smaller than 5 million tons (circles) to the expected number of such deposits from the model of pure power-like distribution (Figure 6). It can be seen that the share of revealed large deposits shows a tendency of increase with WN/S increasing. However for deposits of smaller volumes such regularity is not noted. The lack of it is corroborated with formal calculations of regression parameters and results obtained with the use of other ways of estimating of comparative extent of coverage in different sedimentary basins. Thus the studies of hydrocarbon deposit data of North America corroborate the conclusion inferred from the model that the number of deposits with relatively smaller volumes of reserves is actually relatively smaller as compared to a power law distribution.

2007ES000282-fig07
Figure 7
[36]  Let us consider statistical behavior of deposits at the stage of "ageing'' that corresponds to a stage of failure and disintegration of deposit. Let us consider two variants of disintegration of hydrocarbon deposits with time. The rate of disintegration can be determined by the parameters of feeding fault system Yi (similar to process of deposit formation; in this case faults by which the stocks are carried out from the deposit are considered to be a continuation of faults that feed it). According to the second variant the rate of deposit disintegration is determined by the amount of its current stock values Xi. Suppose the mechanism of useful raw material loss at each stage is proportional to either Xi or Yi. We take the rate of deposit disintegration 100 times as slow as the rate of its accumulation. This assumption seems likely and it is intended to model the presence of isolating seals in the near-surface sedimentary layers and the decrease in solubility at lower temperatures in near-surface horizons. Modeling typical result for the case of disintegration rate dependence of effective permeability of feeding system Yi is given in Figure 7a for times exceeding the deposit formation time by factor of 3, 50 and 500. Similar plot for the case when rate of deposit disintegration is assumed to be proportional to current stock values is shown in Figure 7b.

[37]  As it can be seen in Figure 7, in the model used the power character of deposits number distribution against the amount of reserves preserves in a process of deposits disintegration. Distribution plot retains its form and the slope of the plot changes insignificantly and irregularly and the number of deposits of different size linearly decreases with time. Thus within the model under discussion, the power law distribution of a number of deposits from their stock values persists at a very long stage of deposit degradation after the process of deposit formation was completed.


RJES

Citation: Rodkin, M. V., A. D. Gvishiani, and L. M. Labuntsova (2008), Models of generation of power laws of distribution in the processes of seismicity and in formation of oil fields and ore deposits, Russ. J. Earth Sci., 10, ES5004, doi:10.2205/2007ES000282.

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