# Cumulative Drawdown Analysis - Theoretical Method

The formulae described in the previous section are used to analyse systems of closely spaced wells, modelled as equivalent wells or slots. Such an approach is less satisfactory if the wells are widely spaced; in those cases, a cumulative drawdown (or superposition) method may be more suitable.

This method takes advantage of the mathematical property of superposition applied to drawdowns in confined aquifers. In essence, the total (or cumulative) drawdown at a given point in the aquifer, resulting from the action of several pumped wells, is obtained by adding together (or superimposing) the drawdown from each well taken individually (Figure 13.12). This approach is theoretically correct in confined aquifers but is invalid in unconfined aquifers, where the changes in saturated thickness that occur during drawdown complicate the interaction of drawdowns.

*Figure 13.12* Superposition of drawdown from multiple wells.

Expressed mathematically, the superposition principle means that the cumulative drawdown *(H - b)* at a given point as a result of *n* wells pumping from a confined aquifer is the sum of the drawdown contribution from each well:

Established mathematical expressions for the drawdown from an individual well can be applied to Equation 13.17 to estimate the drawdown at a given point. For example, using the method of Theis (1935) in a homogeneous and isotropic confined aquifer of permeability *k,* thickness D and storage coefficient *S,* the cumulative drawdown from *n* fully penetrating wells, each pumped at a constant rate <7,, at time *t* after pumping commenced is:

where

W*(u)* is the Theis well function, values of which are tabulated in Kruseman and De Ridder (1990) *u = (r ^{2}S)/(4kDt)*

*r* is the distance from each well to the point under consideration

For values of *и* less than about 0.05, the simplification of Cooper and Jacob (1946) can be applied, giving

This can also be expressed in decimal logarithms as

In many aquifers, the condition of w<0.05 is satisfied after only a few hours’ pumping, which means that Equations 13.19 and 13.20 can generally be used for the analysis of groundwater control systems in confined aquifers.

If the target drawdown (H - *b)* in the excavation area is known, these equations can be solved to determine the number, location and yield of wells necessary to achieve the required drawdown. This also allows the total discharge flow rate (the sum of flow from all the wells) to be determined. This method is most suitable for systems of relatively widely spaced wells. It is mainly used for deep wells and occasionally for ejector well systems; it is rarely used for wellpoint systems.

The following points should be considered when applying the method:

- (i) The method has been reliably applied to the estimation of drawdown within the area of excavation, away from the pumped wells themselves. Estimating the cumulative drawdown inside each well is more difficult, because well losses may not be accurately known. If large well losses occur, the method is less reliable, because the drawdown contribution becomes uncertain.
- (ii) Application of the method requires that the aquifer parameters and well yields be estimated. In practice, the most reliable way to obtain suitable estimates is from analysis of a pumping test. If pumping test data are not available, the estimated cumulative drawdowns should be treated with caution unless there is a high degree of confidence in the parameter values used in calculations. If a pumping test has been carried out, the graphical cumulative drawdown method (described in the subsequent section) may be a more appropriate method of analysis.
- (iii) It may be possible to obtain the required drawdowns in the proposed excavation using a few wells pumped at high flow rates or a larger number of wells of lower yield. Similarly, varying the well locations around (or within) the excavation may produce significantly different drawdowns in the area of interest. In years gone by, investigating the effect of the various options was a tedious process. However, using personal computers, it is possible to write routines or macros for spreadsheet programs to evaluate Equation 13.20, allowing many options to be rapidly considered. When evaluating the various options, it is vital that realistic well yields are used (see Section 13.9); otherwise, too many or too few wells will be specified.
- (iv) Equations 13.18 through 13.20 include a term for the time since pumping began, so each cumulative drawdown calculation is for a discrete time
*t.*The time used in calculation will depend on the construction programme. If the programme shows that a 2 week period is available for drawdown (between the installation of the dewatering system and the commencement of excavation below the original groundwater level), then that case should be analysed. However, in reality, there may be problems with the installation of a few of the wells and pumps, so not all*n*wells will be pumping for the full 2 week period. It may be prudent to design with the objective of obtaining the target drawdown in a rather shorter time. - (v) The assumptions inherent in Equations 13.18 through 13.20 (isotropic confined aquifer, fully penetrating wells and constant flow rate from each well) obviously will not apply in all cases. Provided that the basic aquifer conditions are confined or leaky, it may be possible to use Equation 13.18 for other conditions by substituting an alternative expression in place of the Theis well function W
*(u).*Kruseman and De Ridder (1990) give well functions for a number of cases, including leaky aquifers, anisotropic permeability, partially penetrating wells and variable pumping rates.

The cumulative drawdown method assumes that individual wells do not interfere significantly with each other’s yield. For wells at wide spacings (greater than around 20 m) in confined aquifers (where the aquifer thickness does not change with drawdown), interference is usually low. In such cases, the observed drawdowns are likely to be close to those predicted directly from the cumulative drawdown method. However, in general, observed drawdowns will be slightly lower than predicted. It is not unusual for observed drawdowns to be between 80 and 95 per cent of the calculated values when applied using reliable parameters derived from pumping tests. To allow for this, the total well yield (or the number of wells to be installed) should be increased (by dividing by an empirical superposition factor *J *of 0.8 to 0.95). For example, the total system flow rate Q is determined from the sum of the individual flow rates g, from *n* wells:

The cumulative drawdown method is invalid in unconfined aquifers (or confined aquifers where the drawdown is so large that local unconfined conditions develop). This is because the saturated thickness decreases as drawdown increases, making each additional well less effective compared with the initial wells. Although the method is theoretically invalid in unconfined conditions, where drawdowns are small (less than 20 per cent of the initial saturated aquifer thickness), the method has been successfully applied using an empirical superposition factor / of 0.8 to 0.95. For greater drawdowns in unconfined aquifers, the cumulative drawdown method has been applied using empirical superposition factors of 0.6 to 0.8.

# Cumulative Drawdown Analysis - Graphical Method

If distance-drawdown data are available describing the aquifer response to the pumping of a single well, a graphical cumulative drawdown method can be used. This approach is based on the Cooper-Jacob straight line method of pumping test analysis (see Section 12.8.5), which uses Equation 13.19 expressed as

where all terms are as described previously, apart from R„, which is the radius of influence at time *t.* The equation is evaluated graphically and is used to obtain the total drawdown (H - *b)* at the selected location, resulting from a given array of wells, without the need to evaluate the aquifer parameters.

The method is described in detail by Preene and Roberts (1994) and involves the following steps: ^{1 2}

- 1. Determine the target drawdown level in critical points of the excavation. Typically, critical points where drawdown is checked include the centre and corners of the excavation. Normally, the target drawdown level is a short distance (0.5 to 1 m) below the excavation formation level.
- 2. From the pumping test data, construct a drawdown-distance plot on semi-logarithmic axes. Drawdowns recorded in monitoring wells at a given time after pumping commenced are plotted, and a best straight line is drawn through the data (Figure 13.13a). For short- duration pumping tests, the data used are normally from the end of the test. The drawdown in the pumped well is typically ignored, as it may be affected by well losses.

*Figure 13.13* Cumulative drawdown analysis: graphical method, (a) Distance-drawdown plot, (b) Specific drawdown plot.

- 3. Convert each drawdown data point to specific drawdown by dividing by the discharge flow rate recorded during the test. A straight line is then drawn through the monitoring well data to obtain the design-specific drawdown plot (Figure 13.13b); this plot shows the drawdown that results from a well pumped at a unit flow rate.
- 4. Draw a plan of the excavation and groundwater lowering system, marking on the well locations and the points where drawdown is to be checked. Measure and record the distances from each well to each drawdown checking point.

- 5. Estimate the yield of each well in the system. This may be based on the pumping test results (based on step-drawdown data) or may involve the guidelines of Section 13.9.
- 6. At each drawdown checking location, calculate the drawdown that will result from the assumed set of well locations and yields. This is done using the specific drawdown plot (Figure 13.13b). The drawdown contribution (H - />), from each well is calculated by reading the specific drawdown at the appropriate distance and then multiplying by the assumed well yield. The total calculated drawdown
*(H*-*b)*is the sum of the contribution from each well multiplied by an empirical superposition factor*J*:

- 7.
*J*is normally taken to be between 0.8 and 0.95 in confined aquifers. As with the theoretical cumulative drawdown method, the graphical method has been applied in unconfined aquifers where drawdowns are less than 20 per cent of the initial saturated aquifer thickness. An example calculation in a confined aquifer (from Preene and Roberts, 1994) is shown in Figure 13.14; in that case, /was back-calculated to be 0.92. - 8. The calculated drawdown at each checking location is compared with the target drawdowns from Step 1. If the drawdown is insufficient, the calculation is repeated after having either changed well locations; increased the number of wells; or increased individual well yields. It is vital that the well yields assumed are achievable in the field. If the assumed yields are too large, the system will not achieve its target drawdowns.

While the graphical method is most commonly used where site investigation pumping test data are available, the technique can also be used with the observational method. In this

*Figure 13 14* Case history of cumulative drawdown calculation. (Data from Preene, M and Roberts, T О L, The application of pumping tests to the design of construction dewatering systems, in *Groundwater Problems in Urban Areas* (Wilkinson, W B, ed.), Thomas Telford, London, pp. 121-133, 1994. With permission.) approach, one of the first wells in the groundwater lowering system is pumped on its own in a crude form of pumping test. Drawdowns are observed in the other dewatering wells (which are unpumped at that time); these data allow distance-drawdown plots to be produced. The cumulative drawdown calculations are then used to help with the decision-making progress to decide whether to install additional wells. As each new well is installed and pumped, further drawdown data are collected and the predicted drawdowns compared with the actual. In these cases, it is often found that as each additional well becomes operational, the empirical superposition factor *J* reduces further as interference between wells increases.