GEOS 4430 Lecture Notes: Well Testing

Dr. T. Brikowski

Fall 2013

0file:well hydraulics.tex,v (1.32), printed November 11, 2013

Motivation

I aquifers (and oil/gas reservoirs) primarily valuable whentapped by wells

I typical well construction

I typical issues: how much pumping possible (well yield),contamination risks/cleanup, etc.

I all of these require quantitative analysis, and that usuallytakes the form of analytic solutions to the radial flow equation

Introduction

I Well hydraulics is a crucial topic in hydrology, since wells are ahydrologist’s primary means of studying the subsurface

I Lots 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.

I radial coordinates greatly simplify the geometry of wellproblems (Fig. 1)

I in such systems a cone of depression or drawdown cone isformed, the geometry of which depends on aquifer conditions(Fig. 2)

Geometry of Radial Flow

Figure 1: Geometry of radial flow to a well, after Freeze and Cherry(1979, Fig. 8.4).

Representative Drawdown Cones

Figure 2: Representative drawdown cones, after Freeze and Cherry (1979,Fig. 8.6). See Wikipedia animation for boundary effects.

Flow equation in radial coordinates

I Recall the transient, 2-D flow equation (the second form usesvector-calculus notation)(

∂2h

∂x2+

∂2h

∂y2

)=

S

T

∂h

∂t

∇2h =S

T

∂h

∂t(1)

I Equation (1) can be converted to cylindrical coordinatessimply by substituting the proper form of ∇:

∇2r =

∂2

∂r2+

1

r

∂

∂r(2)

Flow equation in radial coordinates (cont.)

I the extra 1r term accounts for the decreasing cross-sectional

area of radial flow toward a well (Fig. 3). Using (2) (1)becomes:

∂2h

∂r2+

1

r

∂h

∂r=

S

T

∂h

∂t(3)

I in the case of recharge, or leakage from an adjacent aquifer,an additional term appears:

∂2h

∂r2+

1

r

∂h

∂r+

R

T=

S

T

∂h

∂t(4)

Cross-Sectional Area in Radial Flow

θ

r*d θ

(r+dr)*dθ

dθr

dr

Figure 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 rdθ over adistance dr .

K Ranges

Figure 4: Relative ranges of hydraulic conductivity (after BLM HydrologyManual, 1987?).

T Ranges

Figure 5: Relative ranges of transmissivity and well yield (after BLMHydrology Manual, 1987?). The irrigation-domestic boundary lies at

∼ 0.214m2

sec .

Effect of Scale on Measured K

Figure 6: Effect of tested volume (i.e. heterogeneity) on measured K(Bradbury and Muldoon, 1990).

Theim Equation:Steady Confined Flow, No Leakage

I simplest analytic solution to (3), for steady confined flow, noleakage

I Assumptions: 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 aquifer

I then 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 the dh

dr , i.e.integrating directly

Theim Equation:Steady Confined Flow, No Leakage (cont.)

I for steady flow in homogeneous confined aquifer we can startwith Darcy’s Law (eqns. 5.41 to 5.44, Fetter, 2001)

Q = (2πrb)K dhdr = 2πrT dh

dr → dh =Q

2πT

1

rdr∫ h(r)

hw

dh =Q

2πT

∫ r

rw

dr

r

h(r) = hw +Q

2πTln

(r

rw

)(5)

I where h(r) is the head at distance r from the well, hw is headat the well, Q is the pumping rate (for a discharging well, i.e.water is removed from the aquifer), and rw is the well radius.More generally this equation applies for any two points r1 andr2 away from the well.

Theim: Obtaining Aquifer Parameters

I when 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 +Q

2πTln

(r2r1

)T =

Q

2π(h2 − h1)ln

(r2r1

)

K =Q

π(h22 − h2

1)ln

(r2r1

)(6)

I (6) is derived from unconfined version of Darcy’s Law, seeFetter (eqns. 5.45-49 2001)

I Advantages: T (or K ) determination quite accurate(compared to transient methods)

Theim: Obtaining Aquifer Parameters (cont.)

I Disadvantages: need 2 observation wells, can’t get storativityS , may require very long term pumping to reach steady-state

Theis Equation: Transient-Confined-No Leakage

I Assumptions: as in Theim equation (except transient), andthat no limit on water supply in aquifer (i.e. aquifer is ofinfinite extent in all directions)

I in this case, the solution of (1) is more difficult. Thirty yearsafter Theim equation was derived, Theis published thefollowing solution

s(r ,t) =Q

4πT

∫ ∞u

e−u

udu (7)

u =r2S

4tT(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.)

I The integral in (7) is often written as the “well function”

W (u) =

∫ ∞u

e−u

udu (9)

I Values are tabulated in many hydrology references (e.g. Table4.4.1, Todd and Mays, 2005)

Theis: Obtaining Aquifer Parameters

I type-curve fitting : Theis solution (popular before the adventof computers)

I Theis 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.

I given (7) written using the well function

s(r ,t) =Q

4πTW (u) (10)

and (8) rearrangedr2

t=

4T

Su (11)

Theis: Obtaining Aquifer Parameters (cont.)

I solve these simultaneously for S and T

T =QW (u)

4πs(12a)

S =4Tur2

t

(12b)

need values for u and W (u) to solve these.I Determining u and W (u):

I take the log of both sides of eqns. (10)–(11):

log s = log

(Q

4πT

)+ log[W (u)] (13a)

log

(r 2

t

)= log

(4T

S

)+ log u (13b)

Theis: Obtaining Aquifer Parameters (cont.)I solve (13) simultaneously by plotting W (u) vs. 1

u(Fig. 7) and

s vs. tr2 (or just t for a single observation well) at same scale

on 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)

I then a pin pushed through the papers will show the values of sand t

r2 corresponding to the selected W (u) vs. 1u

. This iscalled choosing a match point.

I once the curves are matched, the match point can be chosenanywhere on the diagrams, since it fixes the ratios u(

r2

t

) and

W (u)s

, which arise in (12)I the plot W (u) vs. 1

uis called a type curve, since its form

depends only on the “type” of aquifer involved (e.g. confined,no-leakage)

I modern software solves (12) directly using numerical methods.Results often graphically compared to type curve forfamiliarity.

Type Curve, Confined No-Leakage

Figure 7: Type curve for confined flow, no leakage, after Fetter (Fig. 5.6,2001).

Confined No-Leakage Data

Figure 8: Observed drawdowns for confined flow, no leakage, after Fetter(Fig. 5.7, 2001).

Curve Matching (Theis Soln)

Figure 9: Type curve matching, Theis Method, indicating W (u)s = 1

2.4 ,

and 1/ut = 1

4.1 . After Fetter (Fig. 5.8, 2001).

Multi-Observation Wells

Figure 10: Cone of depression with multiple observation wells, setting fordistance-drawdown solution Driscoll (Fig. 9.23, 1986).

Distance-Drawdown Solution

Figure 11: Distance-drawdown solution. Slope is determined by ∆s overone log cycle on the distance scale. Fit line can be used to predictdrawdown beyond observation wells Driscoll (e.g. point at 300 ft, Fig.9.23, 1986).

Semi-confined (Leaky) Aquifers, Transient Flow

I Introduction:I more complicated class of problems: Non-ideal aquifersI Theis solution assumes all pumped water comes from aquifer

storage (ideal aquifer)I 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 classicTheis curve (Fig. 12)

I Assumptions: as in Theis solution, plus vertical-only flow inthe aquitard (i.e. leakage only moves vertically), no drawdownin unpumped aquifer, no contribution from storage in aquitard

Variation in Drawdown vs. Time

10 100 1000 10000

1

10

Time (min)

Leaky

TheisBarrierD

raw

do

wn

(ft

)

Figure 12: Comparison of drawdown vs. time curves for confinedaquifers. Ideal (Theis), leaky, and barrier cases.

Leaky Confined Aquifer Type Curve

Figure 13: Type curves for leaky confined (artesian) aquifer, after Fetter(Fig. 5.11, 2001)

Impermeable barriers

I the principal effect is to reduce the water available for removalfrom the aquifer (i.e. storage reduced at some distance fromwell), increasing drawdown rate when the drawdown coneintersects the barrier (Fig. 14)

I analytic solutions are available for this case (using image welltheory, Ferris, 1959), allowing estimation of the distance tothe boundary/barrier as well as the standard aquiferparameters

Image Well Geometry

Figure 14: Image well configuration for aquifer with barrier. After Freezeand Cherry (1979, Fig. 8.15).

Single-Well Tests: Introduction

I Use recovery data (Fig. 15)

I plot ho − h vs. log(

tt−t1

), where ho is the head in the well

prior to pumping, t is the time since pumping started, t1 isthe duration of pumping

I Note: for Theis or Jacob method: pumping rate must beconstant. Recovery data can be used if pumping rate variedconsiderably during the test. Well losses often important, sodrawdown in the pumping well often not useful duringpumping.

Recovery Data

Figure 15: Drawdown and recovery data. After Freeze and Cherry (1979,Fig. 8.14).

Slug (Injection) Tests

I useful for low to moderate permeability materials

I a volume of water (or metal bar called a “slug”) is added tothe well, and relaxation of the water levels to the regionalwater table is observed vs. time

I type curve solutions are available(Cooper-Papodopulos-Bredehoeft) , plotting the data as therelative slug height (ratio of current over initial slug height)vs. t

r2c

, where rc is the well casing radius

I for partially-penetrating wells or simple settings, the Hvorslevmethod is very popular approach, Eqn. 14. In this case a plotof relative slug height vs. log t is used (Fig. 16)

K =r2 ln

(LR

)2 L t37

(14)

Slug (Injection) Tests (cont.)

where r is the well casing radius, L is the length of thescreened interval, R is the radius of the casing plus gravelpack, t37 is the time required for water level to recover to 37%of the initial change (method can use withdrawal or injection)

Hvorslev Method

Figure 16: Hvorslev slug test analysis procedure (todd-mays-2005),after Fetter (Fig. 5.22, 2001).

Pump Test Sequence

Figure 17: Pump test sequence, after online notes. Surging is done to remove fines from and stablize gravelpack, step drawdown to measure well efficiency and observe non-linear effects (1 hr each), constant rate test atabout 120% of target rate (24 hr at least), subsequent recovery is often the most stable data.

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.

I Confined aquifersI steady-state: Theim solutionI transient: Theis solution (curve matching) or Jacob

straight-line method (ignores early data)

I Leaky confinedI Hantush (“Cooper”) curve matchingI Hantush-Jacob straight-line (ignores late data, same basic idea

as Jacob straight line)

I Unconfined: dual curve match

Single-Well Testing Summary

I slug/withdrawal testsI type-curve matching (Cooper-Papodopulos-Bredehoeft)I straight-line approximation (Hvorslev method)

I indirect tests: point dilution, specific capacity

Well-Testing Summary Table

Method Idea

l

Tra

nsi

ent

Co

nfi

ned

Lea

ky

CommentsTheim

√ √Steady state hard toreach in field

Theis√ √ √

Uses well functionW (u)

JacobStraight-Line

√ √ √Emphasizes late time(aquifer) data

Hantush-Jacob

√ √ √Uses leakywell-function W (u, r

B )HantushInflectionPoint

√ √ √Jacob straight line fortime before leakageappears

Unconfined√

Combined type curvesfor decompression andgravity drainage

References

Bradbury, K. R. and M. A. Muldoon (1990). Hydraulic conductivitydeterminations in unlithified glacial and fluvial materials. Special TechnicalPub. ASTM, pp. 138–151.

Dawson, K. J. and J. D. Istok (1991). Aquifer Testing. ISBN 0-87371-501-2.Chelsea, MI: Lewis, p. 344.

Driscoll, F. G. (1986). Groundwater and Wells. St. Paul, Minn. 55112: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. 598. isbn: 0-13-088239-9. url: http://vig.prenhall.com/catalog/academic/product/0,1144,0130882399,00.html.

Freeze, R. A. and J. A. Cherry (1979). Groundwater. Englewood Cliffs, NJ:Prentice-Hall, p. 604.

Hantush, M. S. (1964). “Hydraulics of wells”. In: Advances in Hydroscience 1.Ed. by V.T. Chow, pp. 281–442.

Kruseman, G. P. and N. A. de Ridder (1991). Analysis and Evaluation ofPumping Test Data. Publi. 47. Wageningen, The Netherlands: InternationalInst. Land Reclam. and Improvement, p. 377.

References (cont.)

Lohman, S. W. (1979). Ground-water hydraulics. Vol. 708. 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. 636. isbn: 978-0-471-05937-0. url: http://www.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000351.html.

Walton, W. C. (1984). Practical aspects of groundwater modeling. Nat. WaterWell Assn., p. 566.