GEOS 4430 Lecture Notes: WellTesting

Dr. T. Brikowski

Fall 2012

Vers. 1.32, November 13, 2012

Radial Flow

• 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

1

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)2

Geometry of Radial Flow

Figure 1: Geometry of radial flow to a well, after Freeze and

Cherry [1979, Fig. 8.4].3

Representative Drawdown Cones

Figure 2: Representative drawdown cones, after Freeze and

Cherry [1979, Fig. 8.6]. See Wikipedia animation for boundary

effects.

4

Flow equation in radial coordinates

• Recall the transient, 2-D flow equation (the second form uses

vector-calculus notation)(∂2h

∂x2+

∂2h

∂y2

)=

S

T

∂h

∂t

∇2h =S

T

∂h

∂t(1)

• Equation (1) can be converted to cylindrical coordinates

simply by substituting the proper form of ∇:

∇2r =

∂2

∂r2+

1r

∂

∂r(2)

5

• 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+

1r

∂h

∂r=

S

T

∂h

∂t(3)

• in the case of recharge, or leakage from an adjacent aquifer,

an additional term appears:

∂2h

∂r2+

1r

∂h

∂r+

R

T=

S

T

∂h

∂t(4)

6

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

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.7

Relative Aquifer Properties

8

K Ranges

Figure 4: Relative ranges of hydraulic conductivity (after BLM

Hydrology Manual, 1987?).9

T Ranges

Figure 5: Relative ranges of transmissivity and well yield (after BLM

Hydrology Manual, 1987?). The irrigation-domestic boundary lies at

∼ 0.214m2

sec.

10

Effect of Scale on Measured K

Figure 6: Effect of tested volume (i.e. heterogeneity) on

measured K [Bradbury and Muldoon, 1990].11

Steady Confined Flow

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 dhdr , i.e.

integrating directly13

• for steady flow in homogeneous confined aquifer we can start

with Darcy’s Law [eqns. 5.41 to 5.44, Fetter, 2001]

Q = (2πrb)Kdhdr = 2πrT dh

dr → dh =Q

2πT1rdr∫ h(r)

hw

dh =Q

2πT

∫ r

rw

dr

r

h(r) = hw +Q

2πTln(r

rw

)(5)

• where h(r) is the head at distance r from the well, hw is

head at 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 and r2 away from the well.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)

h2 = h1 +Q

2πTln(r2r1

)T =

Q

2π(h2 − h1)ln(r2r1

)

K =Q

π(h22 − h2

1)ln(r2r1

)(6)

• (6) is derived from unconfined version of Darcy’s Law, see15

Fetter [eqns. 5.45-49 2001]

• Advantages: T (or K) determination quite accurate

(compared to transient methods)

• Disadvantages: need 2 observation wells, can’t get storativity

S, may require very long term pumping to reach steady-state

16

Transient Confined Flow

17

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

udu (7)

u =r2S

4tT(8)

where s(r,t) = h(r,t) − h(r,0) is the drawdown at distance r

from the well.18

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

W (u) =∫ ∞

u

e−u

udu (9)

• Values are tabulated in many hydrology references [e.g.

Appendix 1, Fetter, 2001]

19

Theis: Obtaining Aquifer Parameters

• type-curve fitting : Theis solution (popular before the advent

of computers)

• Theis devised a graphical solution method for obtaining S&Tfrom (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

4πTW (u) (10)

20

and (8) rearranged

r2

t=

4TSu (11)

– solve these simultaneously for S and T

T =QW (u)

4πs(12a)

S =4Tu

r2

t

(12b)

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):21

log s = log(

Q

4πT

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

log(r2

t

)= log

(4TS

)+ log u (13b)

∗ solve (13) simultaneously by plotting W (u) vs. 1u (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 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)

∗ then a pin pushed through the papers will show the

values of s and tr2 corresponding to the selected W (u)

vs. 1u. This is called choosing a match point.

22

∗ once the curves are matched, the match point can be

chosen anywhere on the diagrams, since it fixes the ratiosu“r2t

” and W (u)s , which arise in (12)

∗ the plot W (u) vs. 1u is called a type curve, since its form

depends only on the “type” of aquifer involved (e.g.

confined, no-leakage)

• modern software solves (12) directly using numerical

methods. Results often graphically compared to type curve

for familiarity.

23

Type Curve, Confined No-Leakage

Figure 7: Type curve for confined flow, no leakage, after Fetter

[Fig. 5.6, 2001].24

Confined No-Leakage Data

Figure 8: Observed drawdowns for confined flow, no leakage,

after Fetter [Fig. 5.7, 2001].25

Curve Matching (Theis Soln)

Figure 9: Type curve matching, Theis Method, after Fetter

[Fig. 5.8, 2001].26

Multi-Observation Wells

Figure 10: Cone of depression with multiple observation wells,

setting for distance-drawdown solution Driscoll [Fig. 9.23,

1986].27

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].

28

Non-Ideal Aquifers

29

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 aquitard30

Variation in Drawdown vs. Time

10 100 1000 10000

1

10

Time (min)

Leaky

TheisBarrier

Dra

wd

ow

n (

ft)

Figure 12: Comparison of drawdown vs. time curves for

confined aquifers. Ideal (Theis), leaky, and barrier cases.31

Leaky Confined Aquifer Type Curve

Figure 13: Type curves for leaky confined (artesian) aquifer,

after Fetter [Fig. 5.11, 2001]32

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

33

Image Well Geometry

Figure 14: Image well configuration for aquifer with barrier.

After Freeze and Cherry [1979, Fig. 8.15].34

Single Well Tests

35

Single-Well Tests: Introduction

• Use recovery data (Fig. 15)

• 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 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.

36

Recovery Data

Figure 15: Drawdown and recovery data. After Freeze and

Cherry [1979, Fig. 8.14].37

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. tr2c, where rc is

the well casing radius

• for partially-penetrating wells or simple settings, the Hvorslev

method is very popular approach, Eqn. 14. In this case a38

plot of relative slug height vs. log t is used (Fig. 16)

K =r2 ln

(LR

)2 L t37

(14)

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, t37 is the time required for water level to recover to

37% of the initial change (method can use withdrawal or

injection)

39

Hvorslev Method

Figure 16: Hvorslev slug test analysis procedure, after Fetter

[Fig. 5.22, 2001].40

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.

41

Summary

42

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)43

• Unconfined: dual curve match

44

Single-Well Testing Summary

• slug/withdrawal tests

– type-curve matching (Cooper-Papodopulos-Bredehoeft)

– straight-line approximation (Hvorslev method)

• indirect tests: point dilution, specific capacity

45

Well-Testing Summary Table

Method Idea

l

Tra

nsi

ent

Co

nfi

ned

Lea

ky

Comments

Theim√ √

Steady state hard to

reach in field

Theis√ √ √

Uses well function W (u)

Jacob Straight-

Line

√ √ √Emphasizes late time

(aquifer) data

Hantush-Jacob√ √ √

Uses leaky well-function

W (u, rB)

46

Method Idea

l

Tra

nsi

ent

Co

nfi

ned

Lea

ky

Comments

Hantush

Inflection Point

√ √ √Jacob straight line for

time before leakage

appears

Unconfined√

Combined type curves

for decompression and

gravity drainage

47

Bibliography

48

K. R. Bradbury and M. A. Muldoon. Hydraulic conductivity determinations in unlithified glacialand fluvial materials. Special technical pub., ASTM, 1990.

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

F. G. Driscoll. Groundwater and Wells. Johnson Division, St. Paul, Minn. 55112, 1986.

J. G. Ferris. Groundwater hydrology. John Wiley, New York, 1959.

C. W. Fetter. Applied Hydrogeology. Prentice Hall, Upper Saddle River, NJ, 4th edition, 2001.ISBN 0-13-088239-9.

R. A. Freeze and J. A. Cherry. Groundwater. Prentice-Hall, Englewood Cliffs, NJ, 1979.

M. S. Hantush. Hydraulics of wells. Advances in Hydroscience, 1:281–442, 1964.

G. P. Kruseman and N. A. de Ridder. Analysis and evaluation of pumping test data. Publi. 47,International Inst. Land Reclam. and Improvement, Wageningen, The Netherlands, 1991.

S. W. Lohman. Ground-water hydraulics, volume 708 of Prof. Paper. U.S. Geol. Survey,Washington, D.C., 1979.

W. C. Walton. Practical aspects of groundwater modeling. Nat. Water Well Assn., 1984.

49