LECTURE SERIES: 3 Well Hydraulics and Design Radial Flows Consideringthe general partial differential equation for unsteady flow of groundwater in the horizontal direction - For axisymmetric groundwater flow to wells normally radial coordinates are preferable and for homogeneous & isotropic aquifer it can shown that the above equation is equivalent to - - Where r is the radial coordinate of the well ) 1 (2222 cc=cc+ccthSyhTxhTy x) 2 (122 cc=cc+ccthTSrhr rhRadial Flow in a Confined Aquifer Figure (a) shows a pumped well with discharge Q which fully penetrates a confined aquifer of constant saturated thickness m & uniform radial hydraulic conductivity Kr & Figure (b) shows a cylindrical element at radial distance r with width dr. From Darcys Law - Q = Area x hydraulic conductivity x head gradient - - rearranging) 3 ( 2 2 = =drdhrTdrdhK rm Qrt trdrTQdht 2=Radial Flow in a Confined Aquifer Integrating gives - The boundary condition is such that the groundwater head at radius R is H hence, constant of integration A is evaluated as - Substituting back the groundwater head equation is given as -( ) ) 4 ( ln2 + = A rTQht( ) RTQH A ln2t =( ) () ( ) ) 5 ( ln2ln ln2 |.|

\| = + =rRTQH R rTQH ht tRadial Flow in a Confined Aquifer Alternatively, the derivation can be made from the governing differential equation for radial flow which for the case of the given confined aquifer problem can be expressed as - For the given confined aquifer problem q = 0 and since the aquifer thickness is constant, equation (6) reduces to - Integrating twice and substituting the boundary conditions that (1) for any radius the total flow Q=2rT(dh/dr) and that (2) at any outer radius R the groundwater head is H, should lead to equation (5). ) 6 (1 = +|.|

\|=|.|

\|qdrdhKrmdrdhmKdrddrdhr mKdrdrr r r) 7 ( 01 =|.|

\|drdhrdrdrRadial Flow in a Confined Aquifer Instead of working in terms of h, it is more convenient to work in drawdown terms s below the rest water level s=H h Therefore a more general form of equation (5) can be writtenin terms of drawdowns as - Which is called the Thiem equation used for preliminary analysis of groundwater flow to wells. The equation can be used to determine the drawdown sw in a pumped well of radius rw when the drawdown at radius r1 is s1 -) 8 ( ln2122 1 ||.|

\|= rrTQs st) 9 ( ln211 ||.|

\|+ =wwrrTQs stRadial Flow in a Confined Aquifer The thiem equation enables the hydraulic conductivity to be determined from a pumped well The method consist of measuring drawdowns in two observation wells at different distances from well pumped at a constant rate When this approach is used to determine the hydraulic conductivity, pumping must continue at a uniform rate for sufficient time approach a steady state condition The derivation assumes that the aquifer in homogeneous, isotropic, of uniform thickness and that the initial piezometric surface/water table is horizontal Radial Flow in a Confined Aquifer Radial Flow in an Unconfined Aquifer with Recharge Assuming that the saturated depth remains approximately uniform so that a constant transmissivity can be used. The analysis of steady-state unconfined flow with uniform recharge q uses a simplified form of equation (6) given as - Equation (10) is a second order differential equation and hence necessary to perform two integration As indicated the figure below the two boundary conditions apply at the outer boundary r=R Discharge Q needs to be substituted by Q = R2q to arrive at the final expression ) 10 (1 =|.|

\|TqdrdhrdrdrRadial Flow in an Unconfined Aquifer with Recharge derivation with a pumped well Radial Flow in an Unconfined Aquifer with Recharge In equation (11) the first term in the bracket is the same as for confined flow while the second termis a correction which results from increasing flow towards the well as recharge enters the well Results for a typical problem in which Q/2T = 1.0 with a maximum radius R = 1000 m are given in the table below The third column lists the correction term which tends to a maximum radius 0.5 Drawdowns at radial distances of 900 m & 300 m are smaller for the unconfined situation because the flow through the aquifer at these radii is smaller than in the confined aquifer for which all the flow enters at the outer boundary For radial distances of 100 m or less, the drawdowns for the unconfined aquifer with uniform recharge is consistently 0.5m less than in the confined situation Radial Flow in an Unconfined Aquifer with Recharge comparison of drawdowns Radial Flow in Unconfined aquifer with varying saturated depth From Dupuit approx. soln forradial flow in an unconfined aquifer can be obtained Consider the pumping rate from the unconfined aquifer - Q = K2rh(dh/dr) Rearranging gives - Integrating gives - ) 12 ( 2 =rdrKQhdht( ) ) 13 ( ln2 + = A rKQhtRadial Flow in Unconfined aquifer with varying saturated depth Boundary conditions - At r = R, h = H; therefore A = H2 (Q/K)ln(R), thus - At r = rw, h = hw hence - Which can be rearranged as -) 14 ( ln2 2 |.|

\| =rRKQH ht) 15 ( ln2 2 ||.|

\| =wwrRKQH ht( )( )) 16 (/ ln2 2 +=wwr Rh H KQtRadial Flow in Unconfined aquifer with varying saturated depth As is the case with one dimensional flow (x direction) the flow equation (16) is correct but the expression for groundwater head is not due to Dupuit approximation Comparison between the flow equations for constant and variable saturated depths allow a better judgement to be made about the validity of constant saturated depth approx. Equation (16) can be rewritten as - Eq. (5) for radial flow in a confined aquifer is rewritten as -( )( )( )) 17 (/ ln + =ww wr Rh H h H KQt( )( )) 18 (/ ln2 =wwr Rm h H KQtRadial Flow in Unconfined aquifer with varying saturated depth The difference between the two equation is the last term in the numerators (H + hw) and 2m. If hw which is the elevation of waterlevel in the well is close to full saturated thickness H (i.e. small drawdowns) then H + hw 2m, where m is the constant saturated depth. Therefore if hw > 0.9H (the pumped well drawdown is less than 10 percent of the maximum saturated thickness) the error in the flows will be less than 5% when a constant saturated thickness approach is adopted Otherwise for hw < 0.9H the errors become significant Time Variant (Unsteady) Radial Flow Considering unsteady groundwater flow equation in two dimension - In polar coordinates and in terms of drawdowns s - Which applies to both confined and unconfined flow under conditions that S = Storage coefficient for confined flow and Specific yield for unconfined flow. For unconfined flow theequation is a linearized form of the Boussinesq equation which holds when drawdown s