1 GEOS 4430 Lecture Notes: Well Testing Dr. T. Brikowski Fall file:well hydraulics.tex,v (1.32), printed November 11, 2013
2 Motivation aquifers (and oil/gas reservoirs) primarily valuable when tapped by wells typical well construction typical issues: how much pumping possible (well yield), contamination risks/cleanup, etc. all of these require quantitative analysis, and that usually takes the form of analytic solutions to the radial flow equation
3 Introduction Well hydraulics is a crucial topic in hydrology, since wells are a hydrologist s primary means of studying the subsurface Lots of complicated math and analysis, the bottom line is that flow to/from a well in an extensive aquifer is radial, and can be approximated by analytic solutions to flow equation in radial coordinates. radial coordinates greatly simplify the geometry of well problems (Fig. 1) in such systems a cone of depression or drawdown cone is formed, the geometry of which depends on aquifer conditions (Fig. 2)
4 Geometry of Radial Flow Figure 1: Geometry of radial flow to a well, after Freeze and Cherry (1979, Fig. 8.4).
5 Representative Drawdown Cones Figure 2: Representative drawdown cones, after Freeze and Cherry (1979, Fig. 8.6). See Wikipedia animation for boundary effects.
6 Flow equation in radial coordinates Recall the transient, 2-D flow equation (the second form uses vector-calculus notation) ( 2 h x 2 ) + 2 h y 2 = S T h t 2 h = S T h t Equation (1) can be converted to cylindrical coordinates simply by substituting the proper form of : (1) 2 r = 2 r r r (2)
7 Flow equation in radial coordinates (cont.) the extra 1 r term accounts for the decreasing cross-sectional area of radial flow toward a well (Fig. 3). Using (2) (1) becomes: 2 h r r h r = S T h t in the case of recharge, or leakage from an adjacent aquifer, an additional term appears: (3) 2 h r r h r + R T = S T h t (4)
8 Cross-Sectional Area in Radial Flow dr (r+dr)*dθ r dθ θ r*d θ Figure 3: Cross-sectional area changes in radial flow. Water flowing toward a well at the origin passes through steadily decreasing cross-sectional area. Arc length decreases from (r + dr)dθ to rdθ over a distance dr.
9 K Ranges Figure 4: Relative ranges of hydraulic conductivity (after BLM Hydrology Manual, 1987?).
10 T Ranges Figure 5: Relative ranges of transmissivity and well yield (after BLM Hydrology Manual, 1987?). The irrigation-domestic boundary lies at m2 sec.
11 Effect of Scale on Measured K Figure 6: Effect of tested volume (i.e. heterogeneity) on measured K (Bradbury and Muldoon, 1990).
12 Theim Equation:Steady Confined Flow, No Leakage simplest analytic solution to (3), for steady confined flow, no leakage Assumptions: constant pump rate, fully-penetrating well, impermeable bottom boundary in aquifer, Darcy s Law applies, flow is strictly horizontal, steady-state (potentiometric surface is unchanging), isotropic homogeneous aquifer then an exact (analytic) solution to (3) can be obtained by rearranging to separate the variables in this differential equation, and to determine h(r) by adding up all the dh dr, i.e. integrating directly
13 Theim Equation:Steady Confined Flow, No Leakage (cont.) for steady flow in homogeneous confined aquifer we can start with Darcy s Law (eqns to 5.44, Fetter, 2001) h(r) Q = (2πrb)K dh dr h w dh = Q 2πT r r w dr r = 2πrT dh dr dh = Q 1 2πT r dr h(r) = h w + Q ( r 2πT ln r w where h(r) is the head at distance r from the well, h w is head at the well, Q is the pumping rate (for a discharging well, i.e. water is removed from the aquifer), and r w is the well radius. More generally this equation applies for any two points r 1 and r 2 away from the well. ) (5)
14 Theim: Obtaining Aquifer Parameters when two observation wells are available, (5) can be written as follows, then solved for transmissivity T, or for hydraulic conductivity K for unconfined flow (N.B. Q, h and T or K must have consistent units) h 2 = h 1 + Q 2πT ln T = K = Q 2π(h 2 h 1 ) ln Q π(h 2 2 h2 1 ) ln ( r2 r 1 ( r2 r 1 ( r2 (6) is derived from unconfined version of Darcy s Law, see Fetter (eqns ) Advantages: T (or K) determination quite accurate (compared to transient methods) r 1 ) ) ) (6)
15 Theim: Obtaining Aquifer Parameters (cont.) Disadvantages: need 2 observation wells, can t get storativity S, may require very long term pumping to reach steady-state
16 Theis Equation: Transient-Confined-No Leakage Assumptions: as in Theim equation (except transient), and that no limit on water supply in aquifer (i.e. aquifer is of infinite extent in all directions) in this case, the solution of (1) is more difficult. Thirty years after Theim equation was derived, Theis published the following solution s(r,t) = Q 4πT u e u u du (7) u = r 2 S 4tT where s(r,t) = h(r,t) h(r,0) is the drawdown at distance r from the well. (8)
17 Theis Equation: Transient-Confined-No Leakage (cont.) The integral in (7) is often written as the well function W (u) = e u u u du (9) Values are tabulated in many hydrology references (e.g. Table 4.4.1, Todd and Mays, 2005)
18 Theis: Obtaining Aquifer Parameters type-curve fitting: Theis solution (popular before the advent of computers) Theis devised a graphical solution method for obtaining S&T from (7), known as the Theis solution method. This method obtains values for u, given measurements of s vs. t. From this, S&T can be determined. given (7) written using the well function s(r,t) = Q W (u) (10) 4πT and (8) rearranged r 2 t = 4T S u (11)
19 Theis: Obtaining Aquifer Parameters (cont.) solve these simultaneously for S and T QW (u) T = 4πs S = 4Tu r 2 t need values for u and W (u) to solve these. Determining u and W (u): take the log of both sides of eqns. (10) (11): (12a) (12b) ( ) Q log s = log + log[w (u)] (13a) 4πT ( ) ( ) r 2 4T log = log + log u (13b) t S
20 Theis: Obtaining Aquifer Parameters (cont.) solve (13) simultaneously by plotting W (u) vs. (Fig. 7) and t s vs. (or just t for a single observation well) at same scale r 2 on log log paper (one curve per sheet, Fig. 8) and curve matching (sliding the papers around until the curves exactly overlie one another, keep the axis lines on each sheet parallel to the axes on the other! Fig. 9) 1 u then a pin pushed through the papers will show the values of s and t 1 corresponding to the selected W r 2 (u) vs.. This is u called choosing a match point. once the curves are matched, the match point can be chosen u anywhere on the diagrams, since it fixes the ratios ) and W (u), which arise in (12) s 1 the plot W (u) vs. is called a type curve, since its form u depends only on the type of aquifer involved (e.g. confined, no-leakage) ( r 2 t modern software solves (12) directly using numerical methods. Results often graphically compared to type curve for familiarity.
21 Type Curve, Confined No-Leakage Figure 7: Type curve for confined flow, no leakage, after Fetter (Fig. 5.6, 2001).
22 Confined No-Leakage Data Figure 8: Observed drawdowns for confined flow, no leakage, after Fetter (Fig. 5.7, 2001).
23 Curve Matching (Theis Soln) Figure 9: Type curve matching, Theis Method, indicating W (u) s = 1 and 1/u t = After Fetter (Fig. 5.8, 2001). 2.4,
24 Multi-Observation Wells Figure 10: Cone of depression with multiple observation wells, setting for distance-drawdown solution Driscoll (Fig. 9.23, 1986).
25 Distance-Drawdown Solution Figure 11: Distance-drawdown solution. Slope is determined by s over one log cycle on the distance scale. Fit line can be used to predict drawdown beyond observation wells Driscoll (e.g. point at 300 ft, Fig. 9.23, 1986).
26 Semi-confined (Leaky) Aquifers, Transient Flow Introduction: more complicated class of problems: Non-ideal aquifers Theis solution assumes all pumped water comes from aquifer storage (ideal aquifer) additional water can enter such systems via leakage from lower-permeability bounding materials or surface water bodies. This lowers the drawdown vs. time curve below the classic Theis curve (Fig. 12) Assumptions: as in Theis solution, plus vertical-only flow in the aquitard (i.e. leakage only moves vertically), no drawdown in unpumped aquifer, no contribution from storage in aquitard
27 Variation in Drawdown vs. Time Barrier Theis Drawdown (ft) 10 1 Leaky Time (min) Figure 12: Comparison of drawdown vs. time curves for confined aquifers. Ideal (Theis), leaky, and barrier cases.
28 Leaky Confined Aquifer Type Curve Figure 13: Type curves for leaky confined (artesian) aquifer, after Fetter (Fig. 5.11, 2001)
29 Impermeable barriers the principal effect is to reduce the water available for removal from the aquifer (i.e. storage reduced at some distance from well), increasing drawdown rate when the drawdown cone intersects the barrier (Fig. 14) analytic solutions are available for this case (using image well theory, Ferris, 1959), allowing estimation of the distance to the boundary/barrier as well as the standard aquifer parameters
30 Image Well Geometry Figure 14: Image well configuration for aquifer with barrier. After Freeze and Cherry (1979, Fig. 8.15).
31 Single-Well Tests: Introduction Use recovery data (Fig. 15) plot h o h vs. log ( t t t 1 ), where h o is the head in the well prior to pumping, t is the time since pumping started, t 1 is the duration of pumping Note: for Theis or Jacob method: pumping rate must be constant. Recovery data can be used if pumping rate varied considerably during the test. Well losses often important, so drawdown in the pumping well often not useful during pumping.
32 Recovery Data Figure 15: Drawdown and recovery data. After Freeze and Cherry (1979, Fig. 8.14).
33 Slug (Injection) Tests useful for low to moderate permeability materials a volume of water (or metal bar called a slug ) is added to the well, and relaxation of the water levels to the regional water table is observed vs. time type curve solutions are available (Cooper-Papodopulos-Bredehoeft), plotting the data as the relative slug height (ratio of current over initial slug height) vs. t r 2 c, where r c is the well casing radius for partially-penetrating wells or simple settings, the Hvorslev method is very popular approach, Eqn. 14. In this case a plot of relative slug height vs. log t is used (Fig. 16) K = r 2 ln ( L R ) 2 L t 37 (14)
34 Slug (Injection) Tests (cont.) where r is the well casing radius, L is the length of the screened interval, R is the radius of the casing plus gravel pack, t 37 is the time required for water level to recover to 37% of the initial change (method can use withdrawal or injection)
35 Hvorslev Method Figure 16: Hvorslev slug test analysis procedure (todd-mays-2005), after Fetter (Fig. 5.22, 2001).
36 Pump Test Sequence Figure 17: Pump test sequence, after online notes. Surging is done to remove fines from and stablize gravel pack, step drawdown to measure well efficiency and observe non-linear effects (1 hr each), constant rate test at about 120% of target rate (24 hr at least), subsequent recovery is often the most stable data.
37 Multi-Well Testing Summary All these methods utilize data from one or more observation wells. Storage parameters can only be obtained from multi-well tests. Confined aquifers steady-state: Theim solution transient: Theis solution (curve matching) or Jacob straight-line method (ignores early data) Leaky confined Hantush ( Cooper ) curve matching Hantush-Jacob straight-line (ignores late data, same basic idea as Jacob straight line) Unconfined: dual curve match
38 Single-Well Testing Summary slug/withdrawal tests type-curve matching (Cooper-Papodopulos-Bredehoeft) straight-line approximation (Hvorslev method) indirect tests: point dilution, specific capacity
39 Well-Testing Summary Table Method Theim Theis Jacob Straight-Line Hantush- Jacob Hantush Inflection Point Unconfined Ideal Transient Confined Leaky Comments Steady state hard to reach in field Uses well function W (u) Emphasizes late time (aquifer) data Uses leaky well-function W (u, r B ) Jacob straight line for time before leakage appears Combined type curves for decompression and gravity drainage
40 References Bradbury, K. R. and M. A. Muldoon (1990). Hydraulic conductivity determinations in unlithified glacial and fluvial materials. Special Technical Pub. ASTM, pp Dawson, K. J. and J. D. Istok (1991). Aquifer Testing. ISBN Chelsea, MI: Lewis, p Driscoll, F. G. (1986). Groundwater and Wells. St. Paul, Minn : Johnson Division. Ferris, J. G. (1959). Groundwater Hydrology. In: ed. by C. O. Wisler & E. F. Brater. New York: John Wiley. Fetter, C. W. (2001). Applied Hydrogeology. 4th. Upper Saddle River, NJ: Prentice Hall, p isbn: url: com/catalog/academic/product/0,1144, ,00.html. Freeze, R. A. and J. A. Cherry (1979). Groundwater. Englewood Cliffs, NJ: Prentice-Hall, p Hantush, M. S. (1964). Hydraulics of wells. In: Advances in Hydroscience 1. Ed. by V.T. Chow, pp Kruseman, G. P. and N. A. de Ridder (1991). Analysis and Evaluation of Pumping Test Data. Publi. 47. Wageningen, The Netherlands: International Inst. Land Reclam. and Improvement, p. 377.
41 References (cont.) Lohman, S. W. (1979). Ground-water hydraulics. Vol Prof. Paper. Washington, D.C.: U.S. Geol. Survey, p. 70. Todd, D. K. and L. W. Mays (2005). Groundwater Hydrology. 3rd. Hoboken, NJ: John Wiley & Sons, p isbn: url: http: // Walton, W. C. (1984). Practical aspects of groundwater modeling. Nat. Water Well Assn., p. 566.
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