We greatly appreciate the support from the

for this project

Interpreting Mechanical Displacements During Hydromechanical Well Tests in Fractured Rock

Hydromechanical well tests involve measuring and interpreting mechanical displacements of an aquifer that accompany the transient pressure signal during hydraulic well tests. The displacement signal can be interpreted along with the transient pressure record to estimate characteristics of the aquifer. We have used a precision extensometer between straddle packers to measure axial displacements during slug tests and pumping tests in fractured crystalline rock. The field data show an apparent normal compliance of hydraulically active fractures in the range of 1 to 5 microns of displacement/(m of head change in the wellbore).

However, during the more than 200 hydromechanical pumping and slug tests we have conducted, the displacement has always appeared as a hysteretic function of the well bore pressure; that is, displacements are smaller earlier in the test than they are at the same pressure late in the test. This hysteretic behavior is also typical of results from theoretical analyses using a discrete fracture model that considers coupled fluid flow and deformation. As a result, both field and theoretical data indicate that during well tests the apparent compliance of a formation can increase by a factor of 10 or more, and the theoretical analysis indicates the compliance approaches the inverse of the fracture normal stiffness at late times.

This hydromechanical effect occurs because opening displacements are relatively small when the pressure change is limited to a small region of the fracture early in a test. This causes the apparent compliance to be less than it is later in the test when the pressure change has spread over a larger area of the fracture. Compliance is closely related to storativity, a basic aquifer parameter commonly assumed to be constant. These results indicate that storativity may actually change during a well test. One consequence is that variations in drawdown curves that are interpreted as resulting from variations in aquifer properties or transient changes in flow dimension may instead result from transient changes in storativity produced by this hydromechanical

effect.

H13D-1437

•L.C. Murdoch (lmurdoc@clemson.edu) , Clemson University

T. Schweisinger (tschwei@ces.clemson.edu), Clemson University

L Germanovich (leonid @ce.gatech.edu) , Georgia Institute of Technology

Discrete fracture in biotite gneiss exposed in borehole at the field site.

Analytical

Pe

c

Consider a circular fracture of radius, a, and aperture, , internal fluid pressure, P. The effective stress on the asperities supporting the fracture is e. Youngs modulus of the wall rock is E. The total driving pressure causing displacement of the fracture walls is

(1)

The average aperture is

(2)

so

(3)

where the driving pressure is distributed over the fracture according to = w p1(r).

The aperture of the fracture is assumed to be a linear function of e over small changes in e, so

(4)

Where Cn is the normal compliance, inverse of the normal fracture stiffness. It follows from (3) an d(4) that the aquifer compressibility is

(5)

Example

Driving pressure during a well test will start out concentrated near the well, and will spread radially with time. The distribution will resemble

So, for this distribution of

(6)

2 12

1

0

16 1( ) 1

w

aC p a d

E

e cP

o e nC

12

1

0

16( ) 1

' w w

ap a d C

E

1 1 n

n

CC

P C C

1/ 22

1w

r

a

r0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

=500 50 103

1

0

Idealized fracture in the analysis

Following Doe et. al. (1982), storativity of a fracture is related to compliance

(7)

Using (5) and (6) we can get the storativity as a function of the distribution of driving pressure

(8)

From (6) it is apparent that will increase with time during a well test. This gives

Early time

Late time

FindingsStorativity changes with time due to changes in controls on aquifer compressibility.

216 (1 )

31

13

aC

E

SP

Doe, T.W., J.C.S. Long, H.K. Endo and C.R. Wilson. 1982. Approaches to evaluating the permeability and porosity of fractured rock masses. Proceedings of the 23rd U.S. Rock Mechanics Symposium, Berkeley, Calif.: pp. 31-37

16

3 ' 16 / n

aS

E a C

nS C

216(1 )

3

aS

E

early deformation of wall rock

late deformation of asperitiesn

a

EC

Field Observations

Portable borehole extensometer deployed between packers for hydromechanical well testing

Time (sec)

0 100 200 300 400

Axi

al C

ompl

ianc

e

(d(a

pert

ure)

/dh)

x 1

06 )

-10

0

10

20

30

40

50

22.3 m 25.3 m 27.2 m

Axial compliance as a function of time during slug tests at different depths.

Stage 2

Stag 1

Stage 3

Stage 4

Head (m)0 2 4 6 8

Dis

plac

emen

t (μ

m)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Head vs Disp.

Time(seconds)1 10 100 1000

Hea

d(m

)

0

1

2

3

4

5

6

7

Dis

pla

cem

ent( m

)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5HeadDisp.

Stage 1 Stage 2 Stage 3 Stage 4

Hydraulic head and axial wellbore displacements as function of time during a slug test

Displacement as a function of head during a slug test. Slope = axial compliance

h/ho

0.0 0.2 0.4 0.6 0.8 1.0

Axi

al C

ompl

ianc

e

(d(a

pert

ure)

/dh)

x 1

06 )

-10

0

10

20

30

40

50

22.3 m 25.3 m 27.2 m

Axial compliance as a function of h/ho during slug tests.

We conducted pumping and slug tests between packers using a borehole extensometer. This resulted in pressure head and displacements as functions of time.

1. Displacement is a non-linear, hysteretic function of head during well tests. 2. Displacement vs head varies for each fracture—unique signal with information about the fracture.3. Compliance changes as a function of time, or head.

Findings

Exten

someter

Pac

ker

Pac

ker

-12

-10

-8

-6

-4

-2

0

2

0 2 4 6

Drawdown (m)

dis

pla

ce

me

nt

(mic

ron

s)

-12

-8

-4

0

4

8

0 200 400 600 800 1000

time (s)

Dis

p (

mic

ron

), D

raw

do

wn

(m

)

Slug Tests

Pumping Tests

0

5

10

0 200 400 600time (sec)

Co

mp

lian

ce

0

5

10

0 2 4 6drawdown (m)

Co

mp

lian

ce

pumpingpumping

recovery

recovery

Conclusions1. Field data: Displacement in the well bore during a well test is a non-

linear function of head.

2. Theoretical: This can be explained because compliance is proportional to (fracture length/Young’s modulus) early in a well test, and proportional to asperity compliance (inverse fracture normal stiffness) late in a test.

3. Storativity of fractured rock aquifers is related to compliance. So, Storativity increases with time during both pumping and slug tests as a result of this effectStorativity is not constant.

4. Interpreting displacement signals increases the resolution of well tests in aquifer with discrete fractures.

Numerical

Head(m)0.0 0.5 1.0 1.5 2.0 2.5 3.0

Dis

plac

emen

t ( m

)

0

2

4

6

8

10

Time (sec)1 10 100 1000

Hea

d (m

)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Dis

plac

emen

t ( m

)

0

2

4

6

8

10

Time (sec)1 10 100 1000

Hea

d (m

)

0

2

4

6

Dis

plac

emen

t ( m

)

0

1

2

3

4

5

6

Head(m)0 2 4 6

Dis

plac

emen

t ( m

)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Head(m)0.0 0.5 1.0 1.5 2.0 2.5 3.0

Dis

plac

emen

t ( m

)

0.00.20.40.60.81.01.21.41.61.82.0

Time (sec)1 10 100 1000 10000

Hea

d (m

)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Dis

plac

emen

t ( m

)

0

1

2

3

4

5

6

22.3m

25.3m

27.2m

r2=0.9893

r2=0.9830

r2=0.9974

The analysis considers radial fluid flow in a circular, deformable fracture of finite radius. The flow problem is treated in 1-D radial coordinates, and it is coupled to a solution from elasticity theory that gives the aperture of a circular crack loaded by an arbitrary pressure. Details of analysis in Murdoch and Germanovich (2006) (Int. Jour. Analyt. Numer. Methods in Geomech)

Parameter estimation methods were used to fit the model to displacements and head or drawdown. Unknown parameters were E, Kn (normal stiffness), aperture, and heterogeneities representing crossing fractures or cemented zones within the fracture.

The model (solid line) predicts the field data (points) quite well for both pumping and slug tests, giving estimates of fractured rock properties and heterogeneities.

-12

-7

-2

3

10 100 1000time (sec)

Dra

wdo

wn

479153910601275Aperture δo (μm)

6.5169.512.5Blockage (m)

2.7 x 1083.4 x 1086.8 x 1074.86 x 108Stiffness Kni (Pa/m)

1.92.70.82Leakage (m)

UpperLower

95% Confidence Interval

95 % Confidence LimitsEstimateEvaluated Parameters

479153910601275Aperture δo (μm)

6.5169.512.5Blockage (m)

2.7 x 1083.4 x 1086.8 x 1074.86 x 108Stiffness Kni (Pa/m)

1.92.70.82Leakage (m)

UpperLower

95% Confidence Interval

95 % Confidence LimitsEstimateEvaluated Parameters

Dis

plac

emen

t

td

10-1 100 101 102 103 104 105 106

Pd

0

2

4

6

8

10

12

14

h/ho0.01 0.1 1

Com

plia

nce

(m

/m)

0.1

1

Asperity compliance

Type curve from pumping a deformable fracture (red line), line source solution (Theis) (heavy black), finite diameter well (dashed).

Difference between red and black lines occurs because apparent storativity of a fracture changes with time.

Results from the numerical analysis show that the apparent compliance during a slug test increases and approaches the asperity compliance Cn (inverse fracture normal stiffness). This confirms findings from analytical solution in adjacent panel.

Axial Compliance

displacement

wellbore head

Field Methods

a: fracture half-lengthE: Elastic modulus of wall rockCn: Compliance of asperties

Some other model results