Shu Long-cang, Liu Pei-gui, Dong Gui-ming, Basil Iro Ongor

(State Key Lab of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, China)

Abstract: Calculation of groundwater allowable withdrawal is the key task of evaluating groundwater resources. However, the precise amount of groundwater allowable withdrawal is under uncertainties associated with the real values of the parameters and simplification of mathematical model. The traditional deterministic method of water balance evaluation has a number of limitations, and the risk associated with the results is overwhelming. In this study, JC method which is an improvement method of first-order second-moment is used in determining the groundwater allowable withdrawal for the well field in Jining City, Shandong Province, China. First, this method transforms the non-normal distribution of random variables to the normal distribution, and then employs a First-Order Taylor series expansion to combine model sensitivity with uncertainty in the design check point. Based on the sensitivity analysis carried out, transmissivity, coefficient of leakage, and the gradient and head difference between deep confined aquifer and shallow phreatic aquifer are chosen as the random variables in the research area and their distributions are considered as lognormal and normal distribution, respectively. Using the JC method, the reliability of groundwater allowable withdrawal and different groundwater pumpage are analyzed with different probability density function. Results show that JC method in evaluation of allowable groundwater withdrawal (GWAW) has higher reliability due to uncertainties considerations. An attempt is made to explain the results and influence of variable parameters on the JC method.

Key words: groundwater allowable withdrawal; uncertainty analysis; JC method; traditional deterministic model; groundwater resources

Introduction

Groundwater is an important water supply source due to its wide distribution, good water quality and the fact that it is not easily polluted. It is virtually a nature resource and plays a vital role in the society, economy and ecology development. However, with the population increasing and the city size broadening coupled with agricultural and industrial growth, the over exploitation of groundwater has resulted in some critical problems, such as decline of water table level, land subsidence, cone of depression formation, seawater intrusion, etc. Other negative effects that may arise are depletion of stream flow and drying of wetlands (Shu & Chen, 2003). To combat these situations, much emphasis should be put on the groundwater resource evaluation, especially the estimation of groundwater allowable withdrawal (GWAW), which is traditionally evaluated by use of deterministic model. Like most other systems, groundwater system is uncertain in essence. Accordingly, there exist a lot of uncertain factors in the process of determining the GWAW. Based on the sources of influencing factors, uncertainties are divided into two parts: objective uncertainties and subjective uncertainties (Shu & Zhu, 2000). The former is mainly concerned with the values of parameters, source and sink items, through time and space. The groundwater level and groundwater quality are random variables changing with time and in space. This is well illustrated by the hydrogeological parameters, whose real values are virtually impossible to obtain at the research area by using pumping tests, and quite often the average value obtained from the limited number of pumping tests is considered as the true value of the whole research area.

Subjective uncertainties in groundwater resource evaluation are introduced: developing numerical simulation model of groundwater flow, and using the different mathematical models in the same study area, and collecting the data through observation or measurement. Quite often, due to lack of adequate data, groundwater flow model development entails simplification of the actual hydrogeological conditions. In some cases, most aquifers are heterogeneous and anisotropic, but are often simplified as heterogeneous and isotropic, or even homogeneous and isotropic, which introduces uncertainties into the GWAW determination inevitably. Moreover, adoption of uncorrected mathematical model in analyzing hydrogeological conditions with limited information also introduces more uncertainties. The following case study on a karstic aquifer is a good example of interpretation the uncertainties. Under the condition of limited data, and if the steady state model is applied to simulate groundwater flow, the calibration residual is normally big in the case of the mine drainage in the karsts aquifer. This real condition proves that unsteady flow model should be applied to predict the mine drainage (Lin & Liao, 1992). Observation and measurement during data acquisitions also immensely introduce uncertainties.

In general, there exist a number of uncertainties in groundwater systems, which directly or indirectly affect the reliability of GWAW and the sustainable management of well field. The traditional deterministic model can not take uncertainties into consideration very well; hence, it is greatly necessary to establish a new method to deal with uncertainties in order to design optimal and sustainable groundwater exploitation strategies.

Uncertainty is closely related to risk, and the reliability and credibility of the GWAW can be improved by combining the risk analysis method with uncertainties analysis. Presently First-Order Second-Moment (FOSM) method is the most widely used technique in engineering application, partly due to its simplicity (Jang Y S, et al.1994; Shreedhar & Vincent, 2003). However, this method suffers from a number of limitations. The main problem arises from the linearization in the mean values of the input variables and inconsideration of the different distributed model of random variable. An improvement of the FOSM is JC method which is illustrated in this paper in assessing the reliability of GWAW. First, the JC method is described, and then an application example is presented. Finally, discussion and conclusion are made.

JC METHOD

Principle of FOSM method

For the sake of generality, consider a function of several random variables:

                                      (1)

Whererepresents the random variable, the indices 1, 2, . , n respectively indicate the type of influencing factors.

According to the Taylor expansion, the function is expanded about the point of, and yields the following expressions in the first order (Zhao, 1996; Ma, 2004):

(2)

    (3)

                                          (4)

The reliability index is expressed as:

(5)

If the point of is chosen as the design check point, Eq. (5) is then transformed into:

                                                      (6)

Where is the sensitivity index, given by:

                                         (7)

With the iterative design method, the values of and reliability can be obtained from the equation:

                                                                 (8)

Principle of JC method

Despite the simplicity of FOSM, the method considers all of the random variables in the function as normally distributed, which is not the case. One possible way to improve the accuracy of the approximation is to use higher order from the Taylor series expansion. In practice, it is difficult and impossible to evaluate the 3^rd and higher order moments due to the lack of sufficient data.

An improvement method of JC method aims at overcoming the problem based on the following premises. First, it transforms the non-normal distribution of random variables to the normal distribution in the design check point instead of mean values, which meets the two conditions simultaneously: the values of the probability density function (PDF) should be the same as the equivalent normal distribution in the point of failure. Cumulative density function (CDF) should also satisfy the limiting condition (Fig.1). The two equations are written as:

                            (9)

                     (10)

Where is the given CDF of the random variable, is the given PDF of the random variable, is the equivalent mean value, is the equivalent standard deviation.

Then the given means and standard deviations are replaced with the means and standard deviations of the equivalent normal distribution. Finally, it employs a First-Order Taylor series expansion to combine model sensitivity with uncertainty and use the FOSM method to compute the reliability index.

Application example

Jining City is located in the southwest of Shandong Province, China. The study area is around the city center, and spreads over an area of about 36 km². It is a part of Huaihe River watershed, and drained by several moderate rivers. There are two aquifers, one shallow and unconfined, the other deep and confined, with an aquitard between them. Prior to 1970, groundwater was pumped from shallow aquifer. Presently, water quantity and quality from shallow unconfined aquifers can not meet the domestic and industrial water demands; hence groundwater is pumped from deep confined aquifer (Shu & Sun, 2001). The deep confined aquifer recharges are leakage from shallow aquifer and the lateral inflow. The equation describing water balance can be expressed as:

          (11) Where represents the head difference between deep confined aquifer and shallow aquifer, is the coefficient of leakage, is the gradient, is the transmissivity, is the groundwater pumping rate, is the storage coefficient of the aquifer, F and are the distribution area and the perimeter of the aquifer respectively, is the average variation of the water level within the time of .

Determination of random variables

Sensitivity analysis (SA) is closely linked to uncertainty analysis, which aims at quantitative assessment of the overall uncertainty associated with the response as a result of uncertainties in the model input. The sensitivity analysis results can provide a better understanding of how the model response to variables changes in the inputs. The four variables are chosen as the main random variables in the study area (Table 1). The coefficient of leakage and the transmissivity are both considered as lognormal distribution (Law, 1944; Li & Shu, 2005). Bothand are assumed to be normal distribution. Using the statistical analysis, the mean and standard deviation of each random variable can be obtained from pumping tests (Table 2).

Results and discussions

The groundwater level above the top of aquifer is forty-five meters on average, and assuming that the well field is to be run for about twenty years, then according to the definition of GWAW, the annual average allowable drawdown is 2.25 meters in the study area. Owing to the limitations of traditional method, the effect of uncertainties on the calculation of GWAW in Jining City is assessed using the JC method on the basis of water balance equation. The reliabilities of different GWAW with different types of PDF for random variables are calculated and results are discussed below.

GWAW obtained from the traditional deterministic model is 14.68×10⁴ m³/d, but its corresponding reliability based on JC method is only 64.80%, hence the occurrence of risk as hydrogeological problems is still 35.20%. Presently, the well field reliability under traditional method of GWAW shows that whereas the groundwater abstraction was 14.35×10⁴ m³/d, GWAW was 14.68×10⁴ m³/d in 1987, yet the groundwater level was still decreasing. This reflects the omission of uncertainties effects, and proves the great influence of uncertainties in the groundwater evaluation results obtained from the traditional deterministic method. The JC method provides an effective method to deal with uncertainties in the process of determining GWAW and makes up the shortcomings of the deterministic model. With the JC method, the different GWAW and corresponding reliabilities can be achieved, with the GWAW of 100% reliability being 8.26×10⁴ m³/d (Table3).

Effects of the distributions of random variables

For a given mean and standard deviation, two different PDF are assumed for the transmissivity and the coefficient of leakage i.e. logarithmic normal and normal. At higher reliabilities, the computed results based on JC method become closer for these two distributions (Fig.2).

Effects of the means of random variables

Different means are obtained from different number of samples, and by changing the means of transmissivity and the coefficient of leakage by 10%, the effect of means is assessed (Fig.3).

Figure 2 and Figure 3 illustrate that the type of PDF for the transmissivity and the coefficient of leakage is not the most influential factor in the study area; hence a normal distribution can be replaced with lognormal distributions in this well field. However more work on groundwater exploration and investigation in the study area should be done in order to improve the reliability of groundwater resource evaluation.

Conclusions

In this paper, JC method is presented to deal with uncertainties that exist in the groundwater system and assess the reliability of GWAW, which the widely used traditional deterministic method can not effectively deal with, given the spatial distribution of main parameters. Compared with the GWAW of 14.68×10⁴ m³/d determined by the traditional deterministic method, the result obtained from the JC method has higher reliability due to uncertainties considerations. Hence, it minimizes the risk of ecological, environmental and hydrogeological problems generated by the overdraft of groundwater resources to a certain degree. It also makes up for the limitations of the deterministic model by establishing random mathematical model. Results from the study area illustrate that the means of random variables are the most influencing factors, as such more hydrogeological studies should emphasis in them

Uncertainty analysis and risk analysis present an essential new research field in groundwater resources evaluation, management and sustainable development.

Acknowledgement

The study is supported by the State Key Lab of Hydrology-Water Resources and Hydraulic Engineering of Hohai University. Thanks also to Jining Bureau of Water Resource, Shandong Province of China for their collaboration.

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Figure Caption:

Figure 1. Equivalent normal distribution of JC method

Figure 2. Effect of different PDF: log normal and normal

Figure 3. Effect of the given means