Introduction to Well Hydraulics Fritz R. Fiedler

Transcription

1 Introduction to Well Hydraulics Fritz R. Fiedler A well is a pipe placed in a drilled hole that has slots (screen) cut into it that allow water to enter the well, but keep the aquifer material out. A well is said to be fully penetrating when the screened section extends through the entire saturated thickness of the aquifer. Typically, a submersible pump is placed near the bottom of the well, and water is removed from the well via piping located within the well. When water is pumped from the well, the water level is drawn down below the static, or un-pumped water level within the well; because of the resulting pressure difference, water from the surrounding aquifer flows radially towards the pumped well, forming a cone of depression. After the well has been pumped for a long time at a constant rate, the water level equilibrates. The cone of depression represents the actual water surface in an unconfined aquifer, and the potentiometric (pressure) water surface in a confined aquifer. Figure 1 illustrates pumped wells in confined and unconfined aquifers. h 0 h 0 h h 1 Figure 1. Wells in confined and unconfined aquifers. (after Fetter, C. W., Applied Hydrogeology, Fourth Edition, Prentice Hall, Upper Saddle River, NJ, 001.) Aquifer properties (e.g., hydraulic conductivity, transmissivity, storativity) can be determined by pumping the well at a known, constant rate, and measuring the equilibrium (steady state) water level drawdown (static level, h 0, minus h) at various distances from

2 the pumped well, and/or measuring the water levels at various times and locations before equilibrium is reached. Smaller diameter monitoring wells are used to measure water levels at various distances from the pumped well. Confined Aquifer, Steady State The following derivation applies to a fully penetrating well in a confined, homogeneous, isotropic aquifer, pumped at a constant discharge rate until steady state is reached. The control volume is defined by a cylinder of radius r and height b, which is centered on the pumped well. dh ρwv da = Q = πrb (1) cs dr where ρ w = density of water v = velocity (vector quantity) A = area (vector quantity) Q = discharge from the well, L 3 /T r = distance from the center of the pumped well, L b = aquifer saturated thickness, L h = piezometric head, as defined in Figure 1, L Using the boundary conditions h = h r = r 1, and h = r = r, the differential equation can be solved for Q in the following steps: h h1 r Q dr dh = πkb () r r1 Q r h ln h1 = (3) πkb r1 h h1 Q = πkb (4) ln r1 Equation 4 is known as the Thiem Equation. Remember that the aquifer transmissivity is T = Kb. Also, if the drawdown s = h 0 h is used, the Equation 4 takes the more useful form s 1 s Q = πkb (5) ln r1

3 Unconfined Aquifer, Steady State The derivation for the unconfined situation is very similar to that for the confined case, but now the aquifer saturated thickness is variable. Again, the assumptions are a fully penetrating well in an unconfined, homogeneous, isotropic aquifer, pumped at a constant discharge rate until steady state is reached. Additionally, it is assumed that the vertical flow components are negligible. Starting from the control volume and using the variable h rather than the constant b dh ρwv da = Q = πrh (6) cs dr After integrating with the same boundary conditions as above, the Thiem-Dupuit formula is obtained Also as in the previous case, s can be substituted for h h h1 Q = πk (7) ln r1 s 1 s Q = πk (8) ln r1 Confined Aquifer, Transient Only confined aquifer transient well hydraulics are considered in this introduction. Using the same assumptions as above, plus the following: Start pumping at a rate of t = 0 Initial condition: h(r,0) = h 0 Boundary condition: h(,t) = h 0 Darcy s Law applies (laminar flow) The governing partial differential equation is h 1 h S h + = (9) r r r T t Theis developed an approximate analytical solution to this equation u Q e s = h = 0 h π du (10) 4 T u where r S u = 4Tt u

4 u e du W ( u) is called the well function u u A series expansion was used to approximate the well function integral u u W ( u) = lnu + u + L! 3 3! Values of W(u) as a function of u are given in tables and plotted as type curves. The latter are typically provided as a plot of log(u) versus log W(u), and are used in the graphical solution procedure known as type curve matching. 3 Equation 10 can be re-written in the form known as the Theis Equation Q s = W ( u) (11) 4πT This is sometimes expressed in U.S. customary units with s in feet, Q in gallons per minute, T in gallons per day per foot, r in feet, and t in days 114.6Q s = W ( u) (1) T 1.87r S u = (13) Tt Drawdown, s, can be predicted given Q, T, r, S, and t. Often pump tests, or aquifer discharge tests are performed to determine the aquifer properties (T and S). To do this, a well is pumped at a constant Q and s is recorded versus t at a monitoring well some distance r from the pumped well. As noted above, a graphical procedure can then be used to estimate T and S; this procedure is not described here. Instead, the Cooper-Jacob approach is presented. Cooper-Jacob Methods (for confined aquifer, transient) Cooper and Jacob (1946) found that for small r and large t the higher order terms in the series expansion are negligible and W(u) can be approximated W ( u) = ln( u) (14) This results in less than a 3% error when u is less than 0.01 (when this method is used, it is standard to have u be less than 0.01). Substituting Equation 14 into Equation 11 Q s = ( ln( u)) (15) 4πT In the time-drawdown method, this is solved by considering a change in drawdown, s = s s 1 over a time interval t 1 to t which are one log cycle apart

5 Q Tt Tt1 s = s s1 = ln ln πt (16) 4 r S r S Reducing this and switching to base-10 logarithms results in.3q t s = log (17) 4πT t1 Note that if t 1 and t are one log cycle apart, log (t /t 1 ) = 1. Using this and solving for T. 3Q T = 4 π s If it is assumed that s = 0 at t = t 0, an expression for S is obtained.5tt0 S = (19) r The following steps are taken to compute T and S from s versus t data: 1. Plot s versus t on semi-log scale as shown in Figure (18) Figure. Cooper-Jacob time-drawdown method. (after Fetter, C. W., Applied Hydrogeology, Fourth Edition, Prentice Hall, Upper Saddle River, NJ, 001.). Fit a straight line to the late-time data, as shown on Figure. 3. Extend the line to s = 0, and determine the value of t Find s for any (convenient) t 1 to t pair that spans one log cycle; in Figure 10 minutes to 100 minutes is used. 5. Compute T and S using Equations 18 and 19, respectively. 6. Check to make sure the value of u is less than If three or more monitoring wells are available, the distance-drawdown method can be used. Here, drawdown is measured simultaneously at various distances r from the pumped well. Drawdown is plotted as a function of distance (log) on a semi-log scale. The equations, derived in a manner similar to above, are

6 T. 3Q = π s (0).5Tt S = (1) r where r 0 is the distance where s = 0, and r 1 and r are taken over one log cycle. The solution steps are as follows: 1. Plot r versus s as shown in Figure 3. 0 Figure 3. Cooper-Jacob distance-drawdown method. (after Fetter, C. W., Applied Hydrogeology, Fourth Edition, Prentice Hall, Upper Saddle River, NJ, 001.). Fit a straight line to the data as shown in Figure Extend the straight line to s = 0 and determine the value of r Find s for any (convenient) r 1 to r pair that spans one log cycle (e.g., 10 to 100 feet). 5. Compute T and S using Equations 0 and 1, respectively. 6. Check to make sure the value of u is less than 0.01.