HYDROGEOLOGY

GEOLOGICAL ENGINEERING

EART TECHNOLOGY AND SCIENCE FACULTY

HALU OLEO UNIVERSITY

2015

GROUNDWATER FLOW TO WELL5

SCOPE

Steady Radial Flow to Well1

2

3

Unsteady Radial Flow to Well

Step Pumping Test

Radial Flow to Well

There are two condition, which can be occurred during pumping of water in a well:

1. Steady Stage

2. Unsteady Stage

Steady Stage pumping with constant discharge rate until the drawdown of GWL or Piezometric Level Constant

Unsteady Stage pumping with constant discharge rate but not until the drawdown constant

Aquifer Characteristics & Radial Flow to Well

• Specific Capacity (Sc)

• Transmissivity (T)

• Storativity (S)

• Hydraulic Conductivity (K)

• Steady Radial Flow to Well Sc, T, K

• Unsteady Radial Flow to Well Sc, T, S, K

PUMPING TEST

Pumping Well Terminology

• Static Water Level [SWL] (ho) is theequilibrium water level before pumpingcommences

• Pumping Water Level [PWL] (h) is the waterlevel during pumping

• Drawdown (s = ho - h) is the differencebetween SWL and PWL

• Well Yield (Q) is the volume of water pumpedper unit time

• Specific Capacity (Q/s) is the yield per unitdrawdown

Cone of Depression

• A zone of low pressure is created centred on the pumping well

• Drawdown is a maximum at the well and reduces radially

• Head gradient decreases away from the well and the pattern resembles an inverted cone called the cone of depression

• The cone expands over time until the inflows (from various boundaries) match the well extraction

• The shape of the equilibrium cone is controlled by hydraulic conductivity

Low Kh aquifer

High Kh aquifer

Kh Kv

Steady Radial Flow to a Well – Confined Aquifer

• In a confined aquifer, the drawdown curve or cone of depression varies with distance from a pumping well.

• For horizontal flow, Q at any radius r equals, from Darcy’s law,

Q = -2π.r.b.K dh/dr

for steady radial flow to a well where Q,b,K are constant.

• Integrating after separation of variables, with h2 & h1 and r2 & r1

at the well, yields Thiem Equation:

Q = 2πKb[(h2-h1)/(ln(r2/r1 )]……. T = K.b

Note, h increases indefinitely with

increasing r, yet the maximum head is h0.

Steady Radial Flow to a Well- Unconfined Aquifer

• Using Dupuit’s assumptions and applying Darcy’s law for radial flow in

an unconfined, homogeneous, isotropic, and horizontal aquifer yields:

Q = -2πKh dh/dr

integrating,

Q = πK[(h22 - h1

2)/ln(r2/ r1)

solving for K,

K = [Q/π(h22 - h1

2)]ln (r2/ r1)

where heads h1 and h2 are observed at adjacent wells located distances

r1 and r2 from the pumping well respectively.

Unsteady Radial Flow to a Well

Aquifer Categories

• PDE 1 (rh ) = S hr r r T t

• Solution is more complex than steady-state

• Change the dependent variable by letting u = r2S

4Tt

• The ultimate solution is:

ho- h = Q exp(-u) du4pTu u

where the integral is called the exponential integral written as the well function W(u)

This is the Theis Equation

• Assumptions

Isotropic, homogeneous, infinite aquifer, 2-D radial flow

• Initial Conditions

h(r,0) = ho for all r

• Boundary Conditions

h(,t) = ho for all t

Unsteady Radial Flow – Confined Aquifer

The Theis Method

s' = (Q/4πT)W(u)

r2/t = (4T/S)u

For a known S and T, one can use Theis to compute s’ directly at a given r from the well as a function of time:

First compute u = r2S / (4T t)

Then W(u) from Table

Finally s' = (Q/4πT)W(u)

Theis Method - Graphical Solution

Well FunctionData Pts

W(u) vs u

s' vs r2/t

T =Q

4π s’W(u)

S =4T· u

r2/t

Cooper-Jacob Method

Cooper and Jacob noted that for small values of r and large values of t, the

parameter u = r2S/4Tt becomes very small so that the infinite series can be

approx. by:

W(u) = – 0.5772 – ln(u) (neglect higher terms)

Thus s' = (Q/4πT) [– 0.5772 – ln(r2S/4Tt)]

Further rearrangement and conversion to decimal logs yields:

s' = (2.3Q/4πT) log[(2.25Tt)/(r2S)]

A plot of drawdown s' vs.log of t forms a straightline as seen in adjacentfigure.

A projection of the lineback to s' = 0, where t = t0

yields the followingrelation :

0 = (2.3Q/4πT) log[(2.25Tt0)/ (r2S)]

Semi-log plot

The Final Result :

S = 2.25Tt0 /r2

Replacing s' by Δs', where Δs' is the drawdown

difference per unit log cycle of t:

T = 2.3Q/4π Δs'

The Cooper-Jacob method first solves for T and then for S and is

only applicable for small values of u < 0.01

For the data given in the Fig.

t0 = 1.6 min and Δs’ = 0.65 m

Q = 0.2 m3/sec and r = 100 m

Thus:

T = 2.3Q/4πΔs’ = 5.63 x 10-2 m2/sec

T = 4864 m2/sec

Finally, S = 2.25Tt0 /r2

and S = 1.22 x 10-3

Step Pumping Test

Well Loss

• Total drawdown sw at the well may be written for the steady state confined case:

sw = Q/ 2πT ln (ro/rw) + CQn

• Where C (well loss coefficient) is a constant governed by the radius, construction and condition of the well. For simplicity let:

B = ln (ro/rw) / 2πT

• So that

sw = BQ + CQn

Well Loss Coefficient

Well Loss Coefficient Well Condition

( C )

< 0.5 Properly designed and developed

0.5 to 1.0 Mild deterioration or clogging

1.0 to 4.0 Severe deterioration or clogging

> 4.0 Difficult to restore well to original

capacity

To evaluate well loss; a step drawdown test is required

Step-Drawdown Test

• Step-drawdown tests are tests at different pumping rates (Q) designed to determine well efficiency.

• Normally pumping at each successively greater rate Q1 < Q2 < Q3 < Q4 < Q5 takes place for 1-2 hours (Dt) and for 5 to 8 steps. The entire test usually takes place in one day.

• Equal pumping times (Dt) simplifies the analysis.

• At the end of each step, the pumping rate (Q) and drawdown (s) is recorded.

Time, t

Dra

wdow

n, ss1

s2

s3

s4

s5

Step-Drawdown Tests

Assumption – laminar flow exists in the aquifer during pumping

• Flow is directly proportional to pumping rate (e.g., Jacob approx.)

• B is almost constant after pumping is established

If turbulent flow exists, then linear relationship does not hold

Specific capacity (s/Q) is a linear function of discharge (Q), slope C and intercept B

QSr

Tt

TBQs

)

25.2log(

4

3.22

2CQBQsw

BCQQ

sw

Step-Drawdown Test Analysis

• Step-drawdown tests areanalysed by plotting thereciprocal of specific capacity(s/Q) against the pumping rate(Q).

Q (L/s)

s/Q

(m

/m3/d

)• The intercept of the graph at Q=0

is B = W(u)/4pT and the slope isthe well loss coefficient, C.

• B can also be obtainedindependently from a Theis orCooper-Jacob analysis of a pumptest.

B

C

Causes of Well Inefficiency

• Factors contributing to well inefficiency (excess head loss) fall into two groups:

• Design factors

• Insufficient open area of screen

• Poor distribution of open area

• Insufficient length of screen

• Improperly designed filter pack

• Construction factors

• Inadequate development, residual drilling fluids

• Improper placement of screen relative to aquifer interval

Well Efficiency Parameter

• Pumping efficiency

Ep = BQ / sw x 100 %, if Ep > 50% efficient

• Development Factor

Fd = C/B x 100

Development Factor Clasification

Development Factor (Fd) Class

< 0.1 Very good

0.1 – 0.5 Good

0.5 – 1.0 Moderate

> 1.0 Poor

Example – Step-Drawdown Test

Q (gpm)

s (ft)

s/Q (ft/gpm)

514 13 0.0253

1066 27 0.0253

1636 43.4 0.0265

1885 61.5 0.0326

2480 82.5 0.0333

3066 101.5 0.0331

3520 120.5 0.0342

6103 xSlope

0231.0Intercept

0231.0103 6 QxBCQQ

s

Fd = (3.10-6 / 0.0231) x 100 = 0.013

%74

100)2700(1030231.0

0231.0

100

100

6

2

x

gpmx

xCQB

B

xCQBQ

BQEp

Slug Test

Hvorslev Slug-Test Method