# Exercise Set 2

If you have not already downloaded the spreadsheets for the exercises and their solutions that are presented in this book, you can do so at the gw-project.org website on the Groundwater Velocity book page by downloading the interactive Microsoft-Excel spreadsheets titled “**GWP_Velocity_Exercises.xlsm**” and “**KeyFile_GWP_Velocity_Exercises.xlsm**”.

**1.** Consider the conceptualized diagram of an aquifer shown in Figure Exercise 2-1. A lucky hydrogeologist placed two wells in the ground at this site and somehow aligned them perfectly with the flow direction. As a result, these two wells are all that are needed to estimate the Darcy flux and seepage velocity in the aquifer.

Open the spreadsheet “GWP_Velocity_Exercises.xlsm”. You may receive a message about enabling content, updating content, or circular references. Proceed by clicking enable content, not updating content, and clicking OK for circular references. Then click on the tab for ‘Exercise 2’. Note that the wells are 20 m apart (Dℓ) with a difference in water levels between them of 0.02 m (DH). Prior work in the wells led to an estimate of* K *= 20 m/d for the aquifer. The sediments are dominantly sand, and the effective porosity (*n*_{e}) is estimated to be 0.28.

*Given the equations for Darcy flux and seepage velocity given on the Exercise 2 sheet, calculate these quantities in the space provided on the sheet.*

***The solution for Exercise 2-1** begins on row 71 of the Solutions Tab of KeyFile_GWP_Velocity_Exercises.xlsm*****

**2.** Scroll down to row 31 and consider the conceptualized diagram of an aquifer as shown in Figure Exercise 2-2. In this case a more experienced hydrogeologist placed three wells in the ground, confident that no matter what the flow direction, it could be determined, so no luck would be required.

The scales (arbitrary units) provided on the map of Figure Exercise 2-2b and in the Exercise 2 tab of the “GWP_Velocity_Exercises.xlsm” spreadsheet can be used to determine the distance between the wells.

Prior work in the wells led to an estimate of* K *of 20 m/d for the aquifer. The sediments are dominantly sand, and the effective porosity (*n*_{e}) is estimated to be 0.28.

*Use** graphical** construction** to determine the flow direction, relative to North** (north is aligned with the positive direction of the y-axis)**, and the hydraulic gradient** by interpolating to find the location of 9.8 between the north and east well, drawing a line from that point to the west well and constructing a perpendicular to that line from the north well**.*

*Given the equations for Darcy flux and seepage velocity **provided** on the Exercise 2 sheet, calculate these quantities** for the system of Figure* *Exercise* *2**–**2** in the space provided on the sheet.*

***The graphical solution for Exercise 2-2** extends from row 97 to row 114 of the Solutions Tab of KeyFile_GWP_Velocity_Exercises.xlsm*****

Item 3 below describes a mathematical versus graphical approach to solving a three-point problem.

**3.** There are several ways to solve the three-point problem, but one way that lends itself to finding the gradient (and its direction) of a water table with three wells, or more, involves matrix algebra as shown in Figure Exercise 2-3.

Open the spreadsheet “GWP_Velocity_Exercises.xlsm”. You may receive a message about enabling content, updating content, or circular references. Proceed by clicking enable content, not updating content, and clicking OK for circular references. Then click on the tab for ‘Exercise 2’ and fill in the table starting near row 69 to verify the graphical solution you obtained for Exercise Set 2, #2.

*Given the equations for Darcy flux and seepage velocity given on the Exercise 2 sheet, calculate these quantities in the space provided on the sheet.*

***The solution for Exercise 2-3** extends from row 116 to row 164 of the Solutions Tab of KeyFile_GWP_Velocity_Exercises.xlsm*****