Abstract: Drawdown data collected during a pumping test are used to assess hydrologic properties of the aquifer. The application of the Theis and Theis-Jacob non-steady state method are used to determine the coefficient of transmissibility (T) and the coefficient of storage (S). But, due to variation of specific capacity of wells with time during the earliest period of pumping test, the values of T and S not remain constant with the time of pumping. A simple method is proposed in this study using graphical method to estimate the transmissibility and storativity of aquifer. In the projected method which is based on Theis’formula, no assumptions for the hydrologic parameters are required and the values of T and S are determined simultaneously.
INTRODUCTION
The pumping test data analyses and interpretations are used as a method for estimating the hydrologic properties of the aquifer (Week, 2005; Stefan and Vitaly, 2005). The principle of existing formulas in this subject (Theis, 1935; Xaviez et al., 1999) is to determine the specific capacity after constant period of pumping assuming the value of the coefficient of storage to get the corresponding value of T and S (Singhal and Gupta, 1966; Vincent and Joshi, 1986).
The specific capacity of the well, which is the discharge per unit drawdown, decreases with time (Norman and James, 1971). If the discharge during the pumping test is constant, then the specific capacity should vary with time.
The estimation of the aquifer parameters (transmissibility, storativity) is of great importance for hydrogeological studies, as it constitutes an indicator of the hydrodynamic functioning. The pumping data set interpretations methods are developed taking in consideration time factor, to determine the coefficient of transmissibility, with the assumption of the value of storage coefficient.
In this study, a simple graphical method to estimate the transmissibility and storativity of aquifer is presented. The pumping data set are analyzed whilst the variation of specific capacity with time is taken in consideration and without assumption of the value of storage coefficient.
THEORY AND SIGNIFICANCE
In an artesian well that is pumped, the drawdown is equal to the difference between the pumping water level and the static water level (Forkasiewicz, 1972). It has two components. The first arising from the resistance of the formation and is proportional to the discharge. The second represent the loss of head that accompanies the flow; this well loss is proportional to the square of the discharge.
If Sw designs the drawdown inside the well in meter, F the formation resistance, c well loss and Q the discharge (m3/day) we can write:
(1) |
The ratio of discharge to the drawdown (specific capacity) is equal to:
(2) |
If the discharge Q is constant during pumping period, the specific capacity must decrease with time. Thus, due to the reality that the formation resistance F increases with time same as the everwinding area of influence of well expands.
For small values of well loss c which is usually ranges from 0.2x10-6 to 6x10-6, then cQ in Eq. 2 shall be very small if it is added to F and can be neglected without any serious error. Eq. 2 will be:
(3) |
For the non steady radial flow, Theis developed the following equation:
(4) |
W(u) is the well function and Sw is the drawdown inside the well (Stanley and Roger, 1966).
| S | = | Coefficient of storage |
| T | = | Coefficient of transmissibility in m2/day |
| t | = | Time in days since pumping began |
| rw | = | Effective radius of the pumped well in meter |
For sufficiently large values of t, the well function W(u) may be approximated by a simple logarithmic expression. Thus, the drawdown after sufficient time has onwards and values of u less then 0.02 is given by:
(5) |
When the well loss is appreciable the drawdown in the well is:
(6) |
(7) |
Comparing Eq. 7 with Eq. 1 therefore:
(8) |
in the other hand, according to Theis formula:
(9) |
In Eq. 9, the value of T/S is usually defined as the diffusivity. In this study, calculations of this parameter run in the range of 104 to 109. rw is considered to be 0.1 m which is a practical value for well diameter. We considered t as variable between 0.01 day to 100 days.
RESULTS AND DISCUSSION
Graphical method estimating transmissibility and the coefficient of storage. Table 1 shows some computed values of F for T/S equals to 0.5x106.
From Table 1 it is clear that the value of specific capacity is the arithmetic reciprocal for value of F. values of given transmissibility and the corresponding values of specific capacities are plotted on square grid. It gives a straight line for each time interval. It gives radial group of straight lines distributed from the origin. A set of graphs similar for Fig. 1 are plotted for values of T/S from 1x104 to 1x109. These graphics gave a clear variation in its divergent and slope. The divergence between two radiated lines decreases as time increase and as T/S value increase. The slope of the radiated group decreases as T/S values are increases.
The value of the well loss coefficient c is determined from step discharge test. During pumping test which is running with constant discharge test, the measured drawdown and time are tabulated. From each drawdown, we substract the value of CQ2 which equals the well loss constant during the test. Then, we can obtain the net Drawdown (DD) of the aquifer. After that we divide the value of the constant discharge Q by each value of net drawdown to obtain the variant values of specific capacity. We plot the obtained values and corresponding time where t in minutes as X axis and specific capacity values as Y axis. A straight line will be obtained.
| Table 1: | Calculated values of F and Specific Capacities (Sp.C.)
which is the reciprocal of F at T/S equal to 0.5x106 |
| Fig. 1: | Values of given transmissibility and calculated Sp.C. plotted at time in min |
| Table 2: | Well characteristics test data in PL1 well in Maknassy
basin |
| C: 0.10x10-6, CQ2: 2.4 | |
From this straight line, we determine the value of specific capacities at time 0.01 day (14.4 min) and at time 0.1 day (144 min), respectively. From the difference between these two values, the values of T and S can be obtained as shown in the example below.
The following table shows the observed data for one of wells in Maknassy basin (Central Tunisia).
Well loss is 0.2x10-6 determined from measured drawdown. Specific capacity values plotted against time as shown in Fig. 2.
At time 14.4 min the specific capacity is 632 m2 day-1. This value is 552 m2 day-1 at 144 min. Therefore, the difference equals 80 m2 day-1. It is the difference as shown in Fig. 3 between specific capacities. 632-552 = 80 is only found in the sheet of T/S = 4x106. Drop a vertical line on x axis, we will get the value of T = 800 m2 day-1. So, S = (800/4*106) = 2x10-4. Jacob method gives a value of transmissibility equal to 792 m2 day-1 and a coefficient of storage equal to 7.3x10-4.
Proposed method is based on the Thies non-equilibrium formula and its modification made by Jacob (Stanley and Roger, 1966).
| Fig. 2: | Specific capacity, time curve for well |
| Fig. 3: | The relation between Specific Capacity and transmissibility at T/S = 4x106 |
In the mathematical process to produce Theis`Formula, there are several assumptions. First, the aquifer is supposed homogenous, isotropic, with constant thickness and an infinite areal extent. Second, the well loss is negligible or must be taken in consideration if it is valuable. Third, the effective radius of the well has not been affected by the drilling and is equal to the nominal radius of the well. Fourth, the discharge, coefficient of transmissibility and storativity are supposed constant during the period of the pumping test.
CONCLUSION AND RECOMMENDATIONS
This graphical method as described and illustrated before is rapid and easy to assess T and S simultaneously, while being based on Theis`Fomula and its modification, without serious errors. For well diameter more then 0.1 meter other sets of calculations and T/S sheets must be prepared. Like Theis non-equilibrium formula, some assumptions, limit the field of application of this method. It is possible to correct values of transmissibility and storativity by using the well-known Hantush modification (Hantush, 1961) of the Theis method or Boonstra method that consists of a matching procedure between the drawdown observed in the field and the theoretical drawdown found from the analysis (Boonstra, 1992).