Abstract n°2120

Anouck FERROUD, Romain CHESNAUX, Silvain RAFINIanouck.ferroud@gmail.com

Numerical investigations of the spherical flow

regimes induced by constant-rate pumping tests

25-29thSeptember, 2016

3

1.1 Terms and concepts

1. Introduction

Pressure front pulse (or pressure wave)

Cross flow area A(r) (equipotential surface)

Radial distance r

r

Pumping well

Aquifer

Constant rate pumping test

Flow lines

Fig. 1: Conceptuel model of the pressure during a pumping test

4

1. Introduction

1.2 Constant rate pumping tests : diagnostic tools

Fig. 3: Log-log plot of ds/dlogt

(Bourdet et al., 1989).Fig. 2: Semi-log plot of the s.

Drawdowns

Drawdown-log derivativeds/dlogt

Much more sensitive signalHelp to select the most adapted analytical model

Allows to investigate the flow regimes

+

" The new way " *

s (

m)

ds/d

log

tt (s)

t (s)

* Mattar (1997)

" The old way " *

…But the physical interpretation = still enigmatic

5

1.3 Hydraulic interpretation of ds/dlogt

1. Introduction

n = 1 ; A ~ r0

n = 2 ; A~ r

n = 3 ; A ~ r²

Generalized Radial flow (GRF) Model

(Barker, 1988)

A(r): cross-flow area

r: radial distance from the well (m)

n: flow dimension parameter (= flow regime)

n reflects the geometry of the

pressure wave when it diffused

through the aquifer

"Flow geometry combined to the

hydraulic properties of the aquifer"

(Beauheim et Roberts, 1998).

6

1.3 Hydraulic interpretation of ds/dlogt

1. Introduction

n = 3 ; A ~ r² spherical

flow regime

How to estimate n from transient well test data? n = 2.(1-v)

n: flow dimension

v: slope of ds/dlogt

8

2. Objectives and issue of the study

Objective

To develop advanced tools of intepretation of pumping tests

based on ds/dlogt signal and Barker’s theory

Issue

What are the conceptual models producing a spherical flow

regime?

9

1.4 Field data: observation of n in nature

Previous study:

Statistical occurrence of flow dimension in various geological

environments n distribution in nature.

2. Objectives and issue of the study

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 >3.6

Flow dimension

Interpretable n

10

1.4 Field data: spherical flow regime

2. Objectives and issue of the study

27 m3/s32 m3/s

27 m3/s

23 m3/s32 m3/s

p = - 0.57 ; n = 3.1

Prof : 76 m

Diamètre : 304 mm

12

3.1 The model: HydroGeoSphere (HGS)

3. Methodology

HydroGeoSphere (Therrien et al., 2010, Aquanty, 2015)

HGS = Spatially distributed and physically-based 3D modelling software.

Discretization :

• Finite element method,

• Control-volume finite element method.

Diffusivity equation

For sub-surface and saturated flow:

Numerical implementation:

K: Hydraulic conductivity (m/s)

Ss: Specific storage

h: hydraulic head (m)

13

3. Methodology

3.2 The conceptual models

0 1500 3000

xy

z

Well

Pumped

aquifer

Initial head H = 200 m

100

0y = 3000 m

50Const

ant hea

d b

oundary

H =

200

m

Constant h

ead b

oundary H

= 200 m

0 500 1000

xy

z

Well

Inclined substratum

Pumped

aquifer

Inactive zone

Initial head = 600 m

500

0ywell

= 1000 mα

Partially penetrated aquiferInclined substratum

15

4.1 Propagation of frontal the equipotential surface

4. Results

Pumpedaquifer

Inactive zone

Inclined substratumWell

16

4.1 Propagation of frontal the equipotential surface

4. Results

Inactive zone

Well

A ~ r²Spherical flow regime

A ~ rEarly cylindrical-radial flow

regime

n = 2

0 500 1000

xy

z

Well

Inclined substratum

Pumped

aquifer

Inactive zone

Initial head = 600 m

500

0ywell

= 1000 mα

17

4.2 The ds/dlogt signal: inclined substratum

4. Results

Inclined substratum

Interpretation of the hydraulic

signal and detection of the

boundary conditions = more

reliable

0 500 1000

xy

z

Well

Inclined substratum

Pumped

aquifer

Inactive zone

Initial head = 600 m

500

0ywell

= 1000 mα

Constant-head hydraulic boundary

18

4.2 The ds/dlogt signal: inclined substratum

4. Results

0,01

0,1

1

10

1,E-02 1,E-01 1,E+00 1,E+01 1,E+02 1,E+03 1,E+04 1,E+05 1,E+06 1,E+07 1,E+08 1,E+09

ds/d

log

t

α5

10

20

65

75

Spherical flow regime

α: slope of the inclined substratum

0 500 1000

xy

z

Well

Inclined substratum

Pumped

aquifer

Inactive zone

Initial head = 600 m

500

0ywell

= 1000 mα

α↘

19

4.2 The ds/dlogt signal: inclined susbtratum

4. Results

y = 4E-05x-0,487

y = 0,0013x-0,49

y = 0,0398x-0,49

y = 37,4x-0,489

0,000001

0,00001

0,0001

0,001

0,01

0,1

1

10

1,E-02 1,E-01 1,E+00 1,E+01 1,E+02 1,E+03 1,E+04 1,E+05 1,E+06 1,E+07 1,E+08 1,E+09

ds/d

logt

t (s)

K (m/s)10-5

10-3

10-2

10-1

0 500 1000

xy

z

Well

Inclined substratum

Pumped

aquifer

Inactive zone

Initial head = 600 m

500

0ywell

= 1000 mα

m: y intercept

T↗

y = 1E-06x-1,491

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

1,E+00

1,E+01

1,E+02

1,00E-05 1,00E-04 1,00E-03 1,00E-02 1,00E-01 1,00E+00

m

K (m/s)

log 𝑚 ≈ −1,5. log(𝐾)

20

4. Results

0,001

0,01

0,1

1

10

1,E+00 1,E+02 1,E+04 1,E+06 1,E+08 1,E+10

ds/d

log

(t)

t (s)

bwell

1%

2%

3%

4%

5%

10%

25%

50%

75%

90%

100%

0 1500 3000

xy

z

Well

Pumped

aquifer

Initial head H = 200 m

100

0y = 3000 m

50

Const

ant hea

d b

oundary

H =

200

m

Constant h

ead b

oundary H

= 200 m

bwell↘

bwell

22

5. Discussion and conclusion

The derivative approach doesn’t need more data, but it gives much more

information!

- qualititative: analysis of flow geometry

- quantitative: estimation of hydraulic properties with more reliability

Non-uniqueness of the flow regimes!!!

Partially penetrated aquifer n = 3

An inclined substratum n = 3

Geological settings = essential to inteprret a pumping test!

Sequences of n = help to reduce the non uniqueness!

Sensistivity analysis empirical equation estimation of K from a ds/dlogt

curve fitting

02/11/2013Application et développement d’outils diagnostiques hydro-géo-physiques et géochimiques

pour la détermination des propriétés hydrauliques et physiques des aquifères 23

Marie-Claire Houmeau painting

Anouck FERROUD, Romain CHESNAUX, Silvain RAFINIanouck.ferroud@gmail.com

Quebec (CANADA)

24

Well

25

Algorithm of Bourdet et al. (1989)

26Bonus

Estimation de n

General solution

Generalized difffusivity equation

Laplace transformation

Hyp:

Constant head head at the boundaries

Initial head = 0 m at the begining

27Bonus

Estimation de n

For a little u,

𝑑ℎ

𝑑𝑙𝑜𝑔𝑡= 𝑡

𝑑ℎ

𝑑𝑡= 2,3. 𝐶. 𝑣 𝑡𝑣

Slope of ds/dlogt

28

Rafini, S., & Larocque, M. (2012). Numerical modeling of the hydraulic

signatures of horizontal and inclined faults. Hydrogeology Journal, 20(2),

337-350

29

1.4 Field data: spherical flow regime

1. Introduction

27 m3/s32 m3/s

27 m3/s

23 m3/s32 m3/s

p = - 0.57 ; n = 3.1

Prof : 76 m

Diamètre : 304 mm

Fig.: Hydraulic signal of s and ds/dlogt of the Mout_3 well,

Vendée, Pays-de-la-Loire, France.

Hyp 1 : Partial penetration

Geology: Granitic sand

Case study 2

Altered granitic aquifer

30

Water supply from the weathered horizon to the fissured

rock (Dewandel et al. 2006).

Fig. 13: Conceptual model of a partly eroded palaeo-weathering profile on hard rock

(Dewandel et al. 2006). Partial penetration or

Partial completion

Hydrogeological interpretationSpherical flow

and/orLeaky effects Wellbore effects

(Dewandel et al. 2006)

Fig. 14: Illustrations of a partial penetration /completion.http://www.fekete.com/

http://nptel.ac.in/courses/105103026/20

http://petrowiki.org/

0,001

0,01

0,1

1

10

1,E+00 1,E+02 1,E+04 1,E+06 1,E+08 1,E+10

s(m

)

t (s)

Boundaries and

hydraulic properties

Tested parameter

3D grid

0,001

0,01

0,1

1

10

1,E+00 1,E+02 1,E+04 1,E+06 1,E+08 1,E+10

s(m

)

t (s)

0,001

0,01

0,1

1

10

1,E+00 1,E+02 1,E+04 1,E+06 1,E+08 1,E+10

ds/d

log

(t)

t (s)

0,001

0,01

0,1

1

10

1,E+00 1,E+02 1,E+04 1,E+06 1,E+08 1,E+10

ds/d

log

(t)

t (s)

bs

100%

bs2%

100%

bs

100%

bs2%

100%

Confined aquifer

Well

5. Discussion

3.2 The conceptual model 1: inclined substratum

Screen thickness bs (m)

0,001

0,01

0,1

1

10

1,E+00 1,E+02 1,E+04 1,E+06 1,E+08 1,E+10

ds/d

log

(t)

t (s)

bs

1%

2%

3%

4%

5%

10%

25%

50%

75%

90%

100%

Boundaries and

hydraulic properties

Tested parameter

3D grid

Confined aquifer

5. Discussion

33

1.4 Field data: spherical flow regime

2. Objectives and issue of the study

Geology: clay; sand; rock