Interval uncertainty analysis of a confined aquifer

Water inflow forecast is influenced by many factors and yields uncertain results. To more accurately predict the magnitude of water inflow and quantitatively define the corresponding response in the parameter change interval, this study combined a non-probabilistic set theory and uncertainty analysis to derive an equation for the confined water inflow. Using mining area data and comparing the calculation of upper and lower boundary limits obtained by a Monte Carlo method, results of the confined water inflow equation were calculated with relative errors of 5% and 10%. When corresponding to the rate of change of the variable parameter, the results showed that under the same error conditions, the allowable rate of change when calculating the minimum value using Eq. A was greater than when using Eq. B, and the maximum value using Eq. B yielded a greater allowable rate of change than the maximum value calculated by Eq. A. Thus, the obtained rate of change for Eq. A is indicative of the lower limit, and Eq. B is conducive to the calculation of the upper limit of mine water inflow.


Prediction equation for interval water inflow
The equation of confined water is the most commonly used mathematical equation for calculating mine water inflow, and defined using the following equations 23 : www.nature.com/scientificreports/ where Q is water inflow (m 3 ·day −1 ); K is hydraulic conductivity (m·day −1 ); a and b are working face length and width, respectively (m); η is the calculation factor (see Table 1); r 0 is the reference radius (m); R is the influence radius (m); R 0 is the large well reference radius (m); M is the thickness of the aquifer (m); and, S is the drawdown of the water table (m). The assumptions according to Eq. (4) are that the aquifer is almost horizontal, the distance between the top and bottom plates is relatively uniform, the water medium is relatively uniform, thus avoiding the fluctuation of water flow in the aquifer, and there is a certain range in the pumping well with a circular long radius head boundary. However, during the actual mining process, owing to the possible existence of faults, geological anomalies, and other extraneous factors, the formation permeability coefficient of the study area is locally variable; thus, the aquifer is characterized by non-uniformity, and it is difficult to achieve a circular or rule head boundary.
In the current prediction of mine water inflow, with a series of measurements such as mining area drainage, the groundwater level changes through a gradual process. Hence, the groundwater flow is similar to the steady flow, and the heterogeneous aquifer is regarded as stable. The flow is calculated using the traditional large-well method. Equation (1) is calculated under the assumption that the calculation object is approximated to a steady flow, and Q is nonlinear with K, M, S, and r 0 . The rate of change of the five variables K, M, R, S, and r 0 represents the variation interval of the variable Table 2. It is convenient to use Eq. (1) to calculate water inflow after considering the influences of the parameter changes.

Accuracy of the interval water forecasting equation
Based on a first-order Taylor series and the optimization theory, the obtained rate of change of the equation in Table 2 must be finite. The actual upper and lower limits of the response interval obtained by the Monte Carlo method were used to analyze the validity of the equation and the rate of change limit in Table 2, and the results are shown in Table 3.
The maximum relative error in "the allowable rate of change of the absolute value of the maximum relative error is smaller than the value of the variable" is the error between the upper limit value Q + of the water inflow according to the equation in Table 2, and the value derived using the Monte Carlo method 24 . In the calculation process shown in Table 2, the rate of change ranges from 0 to 0.5, with increments of 0.01.  Table 2. Interval equations of water inflow. Q 0 is the result when the corresponding variable takes the center of the interval. β K = K/ K 0 , β M = M/ M 0 ,β r 0 = r/ r 0,β R = R/ R 0 ,β S = S/ S 0 , β K , β M , β r 0 , β R , and β S are the rates of change of the corresponding variables. The "+" in "±" corresponds to the upper limit (maximum value) of the water influx change interval, and the "−" corresponds to the lower limit (minimum value). (2) was multiplied by 4, the water inflow amount was calculated using Eq. (1), and the other parameters were held constant. Table 3 shows the maximum rate of change of the corresponding variables of the five sets of test data for the two interval water inflow prediction equations at error levels β = 5% and 10%. Data 1, for example, used Eq. (1) to calculate the water inflow, and if the absolute value of the relative error of the calculated maximum value did not exceed 0.05, then the rate of change of the four parameters in Eq. (1) could not exceed 0.11. If the absolute value did not exceed 0.1, then the rate of change of the four parameters could not exceed 0.15.
It can be seen from Table 3 that under the same error requirement, the upper limit is greater than the lower limit when using Eqs. A and B. That is, when the rate of change of the variable is relatively large, the reliability of the calculated maximum water inflow using Eq. (1) is higher than the minimum water inflow. When it is necessary to obtain the upper and lower limits of the water inflow within the larger change interval and meet certain accuracy requirements, the interval can be divided into cells, and then the Eqs. A and B in Table 2 can be used between the cells.

Application
The aquifer in the Jurassic era of one mine was primarily composed of coarse sandstone. The average elevation of the aquifer was 838.18 m. According to the drilling data of the working face, the average thickness of the Jurassic-era system was 108 m. For the mining area, some boreholes were laid and pumping tests were carried out, and the hydrogeological parameters of the area were obtained. The aquifer had a maximum permeability coefficient of 0.0654 m day −1 , a minimum of 0.00043 m day −1 , an average of 0.02265 m·day −1 , a single-hole water inflow of 0.05-3.85 L s −1 , and a unit water inflow of 0.0015-0.1171 L s −1 m −1 . The layer was water-invariant, an indirect water-filled aquifer mined by the 3-1 coal seam, and was also the main aquifer. There were 14 normal faults, one reverse fault, one fault with a drop of more than 10 m, two fault gaps from 5 to 10 m, two fault gaps from 3 to 5 m, and the remaining nine faults were < 3 m. After the working face of the mine was drained, the pressure outside of the funnel boundary was confined water, while inside of the funnel was unconfined water. There was a bounded difference in the amount of water inflow calculated using the maximum, minimum, and average values of variables, indicating that the variable interval has practical significance for the prediction of water inflow.
In order to further explore the relationship between the calculated results of Eq. A and the rate of change of each variable, a comparison analysis was performed Fig. 1. Analyzing the data, the value calculated by the formula is larger than the observed value. The calculated maximum value is closer to the observed value than the minimum value. Thus proved the reliability of the calculated maximum water inflow using Eq. (1) is higher than the minimum water inflow.
In Fig. 2, when the rate of change of the variable was 0.3, the error of the maximum value was substantially < 20%. Further, when the rate of change was 0.2, the minimum value error was substantially < 10%. The actual calculated values revealed that the interval water inflow calculated by the empirical equation was adaptive within a certain range of variables. It can be concluded from Fig. 2 that there is a nonlinear relationship between the relative error, which is calculated by the empirical equation and the rate of change of the variable.

Discussion and conclusion
Discussion. Most of the existing theories and methods for studying uncertain structures assume that the uncertainty of the structure is a random variable or a random process, and satisfies a certain probability distribution assumption. In this case, the rationality of structural analysis and design can be justified; however, the probability density of uncertain variables and experimental information about the structure are often lacking. If these probability distribution assumptions are not satisfied, the rationality of structural analysis and design lose much of their value. It is generally difficult to verify whether the variables of the actual structure satisfy a certain assumption, and this results in the following contradiction: Although the complexity of the actual structure is acknowledged, it is not always possible to use simple model systems; but alternatively, the assumptions of the analysis model due to the uncertain variables are artificial, so almost all uncertain variables can obtain estimates that are arbitrarily close to the real system through probability models. It is precisely because of this contradiction that the use of non-probabilistic models to study various uncertainties is gaining popularity. The equations proposed in this study take into account the uncertainty of the interval and is more applicable to areas with complicated hydrogeological conditions. When calculating the maximum water inflow in a mining area, the formula given in this study is more accurate under the same parameters.

Conclusion.
(1) An uncertainty analysis of the interval, combined with the theory of a non-probability set, allowed the prediction equation of confined water inflow to be derived. This equation considered the traditional largewell method, and as the parameters changed, the amount of water influx also changed within a certain interval. (2) The mathematical equation derivation and the calculation of the mining area example revealed that the predictive accuracy of the equation for the interval water inflow could be improved for magnitude, and provide some reference for the calculation of water inflow in mining areas.