Radial FlowRelative Aquifer PropertiesSteady Confined FlowTransient Confined FlowNon-Ideal AquifersSingle Well TestsSummaryGEOS 4430 Lecture Notes: Well TestingDr. T. BrikowskiFall 20130file:well hydraulics.tex,v (1.32), printed November 11, 2013MotivationIaquifers (and oil/gas reservoirs) primarily valuable whentapped by wellsItypical well constructionItypical issues: how much pumping possible (well yield),contamination risks/cleanup, etc.Iall of these require quantitative analysis, and that usuallytakes the form of analytic solutions to the radial flow equationIntroductionIWell hydraulics is a crucial topic in hydrology, since wells are ahydrologist’s primary means of studying the subsurfaceILots of complicated math and analysis, the bottom line is thatflow to/from a well in an extensive aquifer is radial, and canbe approximated by analytic solutions to flow equation inradial coordinates.Iradial coordinates greatly simplify the geometry of wellproblems (Fig. 1)Iin such systems a cone of depression or drawdown cone isformed, the geometry of which depends on aquifer conditions(Fig. 2)Geometry of Radial FlowFigure 1: Geometry of radial flow to a well, after Freeze and Cherry(1979, Fig. 8.4).Representative Drawdown ConesFigure 2: Representative drawdown cones, after Freeze and Cherry (1979,Fig. 8.6). See Wikipedia animation for boundary effects.Flow equation in radial coordinatesIRecall the transient, 2-D flow equation (the second form usesvector-calculus notation)∂2h∂x2+∂2h∂y2=ST∂h∂t∇2h =ST∂h∂t(1)IEquation (1) can be converted to cylindrical coordinatessimply by substituting the proper form of ∇:∇2r=∂2∂r2+1r∂∂r(2)Flow equation in radial coordinates (cont.)Ithe extra1rterm accounts for the decreasing cross-sectionalarea of radial flow toward a well (Fig. 3). Using (2) (1)becomes:∂2h∂r2+1r∂h∂r=ST∂h∂t(3)Iin the case of recharge, or leakage from an adjacent aquifer,an additional term appears:∂2h∂r2+1r∂h∂r+RT=ST∂h∂t(4)Cross-Sectional Area in Radial Flowθr*dθ(r+dr)*dθdθrdrFigure 3: Cross-sectional area changes in radial flow. Water flowingtoward a well at the origin passes through steadily decreasingcross-sectional area. Arc length decreases from (r + dr)dθ to rd θ over adistance dr .K RangesFigure 4: Relative ranges of hydraulic conductivity (after BLM HydrologyManual, 1987?).T RangesFigure 5: Relative ranges of transmissivity and well yield (after BLMHydrology Manual, 1987?). The irrigation-domestic boundary lies at∼ 0.214m2sec.Effect of Scale on Measured KFigure 6: Effect of tested volume (i.e. heterogeneity) on measured K(Bradbury and Muldoon, 1990).Theim Equation:Steady Confined Flow, No LeakageIsimplest analytic solution to (3), for steady confined flow, noleakageIAssumptions: constant pump rate, fully-penetrating well,impermeable bottom boundary in aquifer, Darcy’s Lawapplies, flow is strictly horizontal, steady-state (potentiometricsurface is unchanging), isotropic homogeneous aquiferIthen an exact (analytic) solution to (3) can be obtained byrearranging to separate the variables in this differentialequation, and to determine h(r ) by adding up all thedhdr, i.e.integrating directlyTheim Equation:Steady Confined Flow, No Leakage (cont.)Ifor steady flow in homogeneous confined aquifer we can startwith Darcy’s Law (eqns. 5.41 to 5.44, Fetter, 2001)Q = (2πrb)Kdhdr= 2πrTdhdr→ dh =Q2πT1rdrZh(r )hwdh =Q2πTZrrwdrrh(r ) = hw+Q2πTlnrrw(5)Iwhere h(r ) is the head at distance r from the well, hwis headat the well, Q is the pumping rate (for a discharging well, i.e.water is removed from the aquifer), and rwis the well radius.More generally this equation applies for any two points r1andr2away from the well.Theim: Obtaining Aquifer ParametersIwhen two observation wells are available, (5) can be writtenas follows, then solved for transmissivity T , or for hydraulicconductivity K for unconfined flow (N.B. Q, h and T or Kmust have consistent units)h2= h1+Q2πTlnr2r1T =Q2π(h2− h1)lnr2r1K =Qπ(h22− h21)lnr2r1(6)I(6) is derived from unconfined version of Darcy’s Law, seeFetter (eqns. 5.45-49 2001)IAdvantages: T (or K) determination quite accurate(compared to transient methods)Theim: Obtaining Aquifer Parameters (cont.)IDisadvantages: need 2 observation wells, can’t get storativityS, may require very long term pumping to reach steady-stateTheis Equation: Transient-Confined-No LeakageIAssumptions: as in Theim equation (except transient), andthat no limit on water supply in aquifer (i.e. aquifer is ofinfinite extent in all directions)Iin this case, the solution of (1) is more difficult. Thirty yearsafter Theim equation was derived, Theis published thefollowing solutions(r ,t) =Q4πTZ∞ue−uudu (7)u =r2S4tT(8)where s(r ,t) = h(r ,t) − h(r ,0) is the drawdown at distance rfrom the well.Theis Equation: Transient-Confined-No Leakage (cont.)IThe integral in (7) is often written as the “well function”W (u) =Z∞ue−uudu (9)IValues are tabulated in many hydrology references (e.g. Table4.4.1, Todd and Mays, 2005)Theis: Obtaining Aquifer ParametersItype-curve fitting: Theis solution (popular before the adventof computers)ITheis devised a graphical solution method for obtaining S&Tfrom (7), known as the Theis solution method. This methodobtains values for u, given measurements of s vs. t. Fromthis, S&T can be determined.Igiven (7) written using the well functions(r ,t) =Q4πTW (u ) (10)and (8) rearrangedr2t=4TSu (11)Theis: Obtaining Aquifer Parameters (cont.)Isolve these simultaneously for S and TT =QW (u)4πs(12a)S =4Tur2t(12b)need values for u and W (u) to solve these.IDetermining u and W (u):Itake the log of both sides of eqns. (10)–(11):log s = logQ4πT+ log[W (u)] (13a)logr2t= log4TS+ log u (13b)Theis: Obtaining Aquifer Parameters (cont.)Isolve (13) simultaneously by plotting W (u) vs.1u(Fig. 7) ands vs.tr2(or just t for a single observation well) at same scaleon log–log paper (one curve per sheet, Fig. 8) and curvematching (sliding the papers around until the curves exactlyoverlie one another, keep the axis lines on each sheet parallelto the axes on the other! Fig. 9)Ithen a pin pushed through the papers will show the values of sandtr2corresponding to the selected W (u) vs.1u. This iscalled choosing a match point.Ionce the curves are matched, the match point can be chosenanywhere on the diagrams, since it fixes the ratiosur2tandW (u)s, which arise in