EAS 44600 Goundwate Hydology Lectue 3: Well Hydaulics D. Pengfei Zhang Detemining Aquife Paametes fom Time-Dawdown Data In the past lectue we discussed how to calculate dawdown if we know the hydologic popeties of the aquife. These hydologic popeties ae usually detemined by means of aquife test. In an aquife test, a well is pumped and the ate of decline of the wate level in neaby obsevation wells is ecoded. In the next two lectues we will discuss how to use the time-dawdown data to deive hydaulic paametes of the aquife. We will use the following assumptions in ou discussion:. The pumping well and all obsevation wells ae sceened only in the aquife being tested.. The pumping well and the obsevation wells ae sceened thoughout the entie thickness of the aquife. A B Figue 3-. Equilibium dawdown: A. confined aquife; B. unconfined aquife (Fette). 3-
Steady-State Conditions If a well pumps fo vey long time, the wate level may each a steady state, i.e., thee is no futhe dawdown ove time. The cone of depession will not gow unde steady-state conditions since the echage ate equates pumpage. In the case of steady adial flow in a confined aquife (Figue 3-A), the adial flow is descibed by Q = πt ( dh / d) (equation -7). Reaanging equation -7 yields: Q d dh = (3-) πt If we have two obsevation wells (hydaulic head h and distance fo the fist well, and head h and distance fo the second well), we can integate both sides of equation 3- with these bounday conditions: The esult is: h dh = (3-) h Q πt d Q h = h ln (3-3) πt Reaanging equation 3-3 gives the Thiem equation fo a confined aquife: Q T = ln (3-4) π ( h h ) whee T is the tansmissivity, Q is the pumping ate, and h and h ae the hydaulic heads at distances and fom the pumping well, espectively. Simila to the case of steady adial flow in a confined aquife (equation -7), the steady adial flow in an unconfined aquife is descibed by db Q = ( πb)k (3-5) d whee Q is the pumping ate, is the adial distance fom the cicula section to the well, b is the satuated thickness of the aquife, K is the hydaulic conductivity, and db/d is the hydaulic gadient. 3-
Reaanging equation 3-5 gives Q d bdb = (3-6) πk If we have two obsevation wells (hydaulic head h and distance fo the fist well, and head h and distance fo the second well), we can integate both sides of equation 3-6 with these bounday conditions: b bdb = b Q d πk (3-7) The esult is: Q b = b ln πk (3-8) Reaanging equation 3-8 gives the Thiem equation fo an unconfined aquife: Q K = ln (3-9) π ( b b ) whee K is the hydaulic conductivity, Q is the pumping ate, and b and b ae the satuated thickness at distance and fom the pumping well, espectively (Figue 3-B). Nonequilibium Flow Conditions In pevious section we discussed the methods of detemining hydologic paametes using timedawdown data unde steady-state flow conditions. In eality, howeve, many aquife tests will neve each the steady state (i.e., the cone of depession will continue to gow ove time). These conditions ae efeed to as nonequilibium o tansient flow conditions. Hee we will only discuss the methods of detemining tansmissivity and stoativity in a confined aquife unde nonequilibium adial flow conditions. Theis Method The Theis equation -0 can be eaanged as follows: T Q = W ( u) (3-0) 4 π ( h h) o whee T is the aquife tansmissivity, Q is the steady pumping ate, h o -h is the dawdown, and W(u) is the well function. Likewise, equation -9 can be eaanged as: 3-3
4Tut S = (3-) whee S is the aquife stoativity, T is the tansmissivity, u is a dimensionless constant, t is the time since pumping stats, and is the adial distance fom the pumping well. Duing an aquife test, wate is pumped out at a well fo a peiod of time; the dawdown is then measued as a function of time in one o moe obsevation wells. The data ae analyzed using diffeent methods to detemine aquife tansmissivity and stoativity. The Theis method is a gaphical method that involves the following steps:. Make a plot of W(u) vesus /u on full logaithmic pape, o using a speadsheet. This gaph has the shape of the cone of depession nea the pumping well and is efeed to as the Theis type cuve, o the nonequilibium type cuve (Figue 3-).. Plot the field dawdown at the obsevation well, (h o -h), vesus t using the same logaithmic scale as the type cuve (Figue 3-3). Since time is often ecoded in minutes in the field, you need to plot time in minutes on you field-data plot and covet minutes to days (equied in the Theis equation) late. 3. Lay the type cuve ove the field-data gaph and adjust the two gaphs until the data points match the type cuve, with the axes of both gaphs paallel (Figue 3-4). Select the intesection of the line W(u) = and the line /u = as you match point. Find the values of (h o -h) and t coesponding to the match point on the field-data gaph. You may use a pin to push though the two pieces of pape to locate the exact match point. 4. Calculate tansmissivity (T) value by substituting the values of Q, (h o -h), and W(u) fom the match point into equation 3-0. Once T is known, its value along with the values of, t, and u fom the match point can be substituted into equation 3- to find aquife stoativity (S). Figue 3-. Theis type cuve fo a fully confined aquife (Fette). 3-4
Figue 3-3. Field-data plot on logaithmic pape fo Theis cuve-matching technique (Fette). Figue 3-4. Match of field-data plot to Theis type cuve (Fette). Coope-Jacob Staight-Line Time-Dawdown Method This method is an appoximation to Theis method and is only valid fo u < 0.05. In this method a semi-log plot of the field dawdown data (linea scale) vesus time (nomal log scale) is made (Figue 3-5). A staight is then dawn though the field-data points and extended backwad to the zeo-dawdown axis (Figue 3-5). The time at the intecept of the staight line and the zeodawdown axis is designated t o. The value of the dawdown pe log cycle of time, (h o -h), is obtained fom the slope of the gaph. The values of tansmissivity and stoativity can be calculated fom the following equations: T.3Q = 4 (3-) π ( h h) o 3-5
S.5Tto = (3-3) whee T is the tansmissivity, Q is the pumping ate, (h o -h) is the dawdown pe log cycle of time, S is the stoativity, is the adial distance to the pumping well, and t o is the time whee the staight line intesects the zeo-dawdown axis (Figue 3-5). Figue 3-5. Coope-Jacob staight-line time-dawdown method fo a fully confined aquife (Fette). Notice that the time used in the time-dawdown plot is often in minutes and it must be conveted to days befoe it is used in equation 3-3. Jacob Staight-Line Distance-Dawdown Method If moe than thee obsevation wells ae used in an aquife test, and dawdowns ae measued at the same time in these wells, the Jacob staight-line distance-dawdown method can be used to detemine aquife tansmissivity and stoativity. In this method dawdown is plotted on aithmetic scale as a function of the distance fom the pumping well on the log scale (Figue 3-6). A staight line is then dawn though the data points and extended to the zeo-dawdown axis. The intecept is the distance at which the pumping well is not affecting the wate level and is designated o (Figue 3-6). The dawdown pe log cycle of distance is designated (h o -h) as befoe (Figue 3-6). The aquife tansmissivity (T) and stoativity (S) ae calculated as follows: T.3Q = (3-4) π ( h h) o.5tt S = (3-5) o 3-6
whee Q is the pumping ate, t is the time when dawdown is measued, and o is the distance at which the staight line intecepts the zeo-dawdown axis (Figue 3-6). Figue 3-6. Jacob staight-line distance-dawdown method fo a fully confined aquife (Fette). 3-7