SPE 110973

Understanding Stress Dependant Permeability of Matrix, Natural Fractures, and

Hydraulic Fractures in Carbonate Formations
H.H. Abass, I. Ortiz, M.R. Khan, J.K. Beresky, Saudi Aramco, and L. Sierra, Halliburton
Copyright 2007, Society of Petroleum Engineers
This paper was prepared for presentation at the 2007 SPE Saudi Arabia Technical Symposium
held in Dhahran, Saudi Arabia, 78 May 2007.
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Abstract
Most carbonate reservoirs behave as dual porositypermeability systems in which the rock matrix and both
natural and created hydraulic fractures contribute to the
hydrocarbon transport in a very complex manner.
Understanding the behavior of the permeability of the
matrix frame, natural fractures, and created hydraulic
fractures, as a function of reservoir depletion, is vital to
designing optimum stimulation treatments and to
maximize the carbonate formations exploitation.
Core samples were selected from a carbonate reservoir
and a testing procedure was applied to determine the
stress dependant permeability as a function of various
combinations of effective stresses. A tensile natural
fracture was simulated by splitting a whole core by failing
it under tension using a Brazilian test procedure. The
stress dependant permeability was evaluated under
varied effective stresses simulating a reservoir depletion
scenario. A shear fractured core was selected from a
given carbonate formation and a stress dependant
permeability was established. The tensile fractured core
was then propped with a low concentration of small
mesh proppants and the permeability of the simulated
propped fracture was determined. Using a new reservoir
simulator the testing results and selective functions were
used to predict the production performance of a
carbonate reservoir under the effect of the stress
dependant permeability.
The experimental results indicate that the tensile
fractures are much less conductive than shear fractures
and the shear fractures are less conductive than
propped fractures. The concept of effective stress within
the rock matrix is totally different than that of natural

fractures; therefore, the effective stress function for both

matrix and natural fractures should be separately
evaluated to obtain representative functions for any
simulation study. The tensile fractures lose conductivity
at very early stages of reservoir depletion.
Recommendations to manage these tensile fractures for
optimum hydrocarbon recovery are suggested.
Practical outputs of this work are: 1) Understand how
natural fractures are controlled to efficiently contribute to
well productivity, 2) Quantify the effective stress concept
in the matrix and fracture systems, 3) Provide stressdependant correlations for simulation studies.
Introduction
Tight
carbonate
reservoirs
behave
as
dual
porosity/permeability systems in which the rock matrix,
natural fractures network and created hydraulic fractures
contribute to the hydrocarbon transport in a very
complex manner. Permeability loss due to increasing
effective stress as a result of reservoir depletion can
result in substantial cumulative recovery loss1. The
permeability of fissured/fractured reservoirs has been
postulated to be highly sensitive to changing effective
stresses. An increased effective stress, which is the
combined effect of stress and pore-pressure, may
decrease the reservoir permeability considerably. This
permeability sensitivity to changing stresses is most
pronounced in tight, over pressured, naturally fractured
reservoirs where the apertures of natural fractures are
very sensitive to applied closure stress resulted from
reservoir depletion.
A working model consistent in both fluid flow and
geomechanical considerations is required to link various
fluid-rock information (e.g., flow/storage properties, rock
mechanical properties, reservoir fluid pressure and
stress level) measured by different techniques and to
forecast reservoir performance2.
Identifying natural fractures is incumbent in the
economic optimization of fracture stimulation designs.
Techniques designed specifically to stimulate natural
fractures include low gel loadings, energized fluids and
100 mesh proppant for natural fracture preservation.3-5
Pressure depletion during production will significantly

Understanding stress dependant permeability of matrix, natural fractures, and hydraulic fractures in carbonate formation

Understanding the hydraulic characteristics of the matrix

frame, natural fractures, and created hydraulic fractures
as a function of effective confining stress is vital to
design optimum stimulation treatments, to predict
reservoir performance via reservoir simulation and to
maximize the carbonate formation exploitation.
The objective of this study was to investigate
permeability reduction characteristics of natural fractures
in highly stressed reservoirs with lowered pore pressure.
An understanding of production from fractured carbonate
reservoirs that exhibit permeability reduction with lower
bottom hole pressure (BHP) is required.
A pore
pressure versus permeability relationship is essential for
optimizing field development scenarios, is also required.
This matrix-fracture flow transfer function has been used
to simulate fluid flow through fractured porous medium.
A shape factor is usually imbedded within the transfer
function. Considering the work presented in this paper, a
new perspective for the transfer function including the
shape factor should be considered to include the stress
dependant fracture aperture and its permeability, in
addition to the stress dependant matrix permeability.
Effective Stress Concept
The effective stress concept as introduced by Biot
(1941) suggests that pore pressure helps counteract the
mechanical stress carried through the grain-to-grain
contact. The efficiency of reservoir pressure , pr , in
supporting the earth stresses is measured by the
poroelastic factor ; the relationship is:

= a p .... (1)
Where is the effective stress, and is the total stress.
The poroelastic constant, , is given by:

= 1

cma
,
cb

0 1 (2)

With the bulk compressibility, cb, given by:

cb =

3(1 2 )
(3)
E

If the rock has no porosity, the rock matrix

compressibility, cma, is equal to cb, and becomes zero.
Conversely, with high porosity, the matrix compressibility
is small compared to the bulk compressibility, and
approaches unity. The poroelastic constant can be
evaluated in the laboratory. This study provides a new

SPE110973

technique to indirectly determine this important

coefficient. The role of effective stress concept near the
wellbore is more pronounced as the reservoir pressure
declines rather rapidly approaching the bottom hole
flowing pressure. The reservoir pressure as a function of
radial distance from a given well is given by:

r
ln
rw
p (r ) = Pw + ( Pe Pw )
r
ln e
rw

... (4)

For example, lets assume pe = 8500 psi, pw = 2500 psi,

re = 9000 ft, rw = 0.5 ft, 8500, =12000 psi, and = 0.65.
Then the pressure and effect stress gradients around the
flowing wellbore for the given example is shown in
Figure 1.
Reservoir pressure
effective stress
9,000

12,000

6,000

8,000

3,000

4,000

0
0

1,000

2,000

3,000

E ffe c tiv e s tre s s , ps i

change the productivity of a reservoir with similar natural

fractures6. Natural fractures impact on production has
been studied by various authors whom have proposed
that these fissures if kept open can contribute
substantially to production.7-15

P re s s ure , ps i

4,000

Distance (ft)
Figure 1: Reservoir pressure and effective stress gradients
around a given wellbore.

The greatest pressure drop occurs within a short

distance from the wellbore. Therefore the effective stress
will be the highest near the wellbore causing
permeability reduction additional to that caused by radial
flow convergence and skin.
Although we assumed to be constant, it is really a
function of pressure. The near wellbore permeability in
specific and the reservoir permeability in general
changes with pressure depletion thus affecting our well
test analysis, single-well reservoir simulation studies,
and the reservoir management strategies as a whole.
Therefore modeling this mechanism is critical to these
petroleum engineering aspects.
Changes in fracture permeability caused by changes in
effective stress, caused by changes in pore pressure,
have been observed at both laboratory and field studies.
Although the terminology Stress-sensitive reservoirs

Understanding stress dependant permeability of matrix, natural fractures, and hydraulic fractures in carbonate formation

Stress Dependant Matrix Conductivity

Selected samples such that they do not appear to have
microfractures were tested to determine the stressdependant matrix permeability. Various combinations of
net effective stresses were applied and the permeability
measured at each stress level. Table 1 presents all
combinations of applied confining stresses and the pore
pressure levels and gradients for a given flow test.
Recalling the definition of effective stress as given in
equation 1, it is required to assume a value for the Biots
coefficient (). Assuming is 1, and plotting matrix
permeability as a function of effective stress, we obtain
Figure 2. Close examination of Figure 2 suggests that
for a given effective stress multiple values of
permeabilities are measured. This is not an experimental
error rather the assumption of being one is not valid.
The next step is to change and replot the stressdependant permeability function until a meaningful trend
is obtained. Since is function of stress, then varying it
within the constraints from the first step would produce
the stress-dependant permeability presented in Figure 3
with estimated function (p).

Cp

Pout

Pin

Pav

psi

psi

psi

psi

psi

mlcc/m

2488

2004

1000

1502

1004

7.644

0.313

4519

4010

3000

3505

1010

6.359

0.2588

4506

2013

1000

1507

1013

4.224

0.1714

8555

8006

7000

7503

1006

5.124

0.2093

6495

4002

3000

3501

1002

3.616

0.1484

6487

1999

1000

1499

999

3.178

0.1308

8511

4006

3000

3503

1006

2.857

0.1167

8511

2005

1000

1502

1005

2.595

0.1061

10012

2005

1000

1502

1005

2.521

0.1032

Stress-dependant matrix permeability

0.4

0.3

0.2

0.1

0
0

2,000

4,000

6,000

8,000

10,000

Effective confining stress, psi

Figure 2: Stress-dependant matrix permeability ( =1).

Stress-dependant matrix permeability

0.9

0.5
-0.1236

y = 1.9989x

0.4

Perm
Biot Coeff.

0.8

0.3
0.7
0.2
-0.5937

y = 21.548x

0.1
0.0
0

2,000
4,000
6,000
8,000
Effective confining stress, psi

0.6

Biot Coefficient

Experimental Simulation
An experimental procedure was designed to simulate the
reservoir permeability (matrix, natural fractures and
induced fractures) reduction as a function of increasing
effective stress. Whole core samples were used with
dimensions of 4-in. diameter and various lengths (4 - 8
in). The sample is then tested for matrix conductivity. A
tensile fracture was created by failing a whole core
similar to a Brazilian test. The core would split into two
halves. A natural shear fracture from a cored formation
was used to study the hydraulic conductivity of a shear
fracture as a function of effective stress. A proppant bed
is sandwiched between the two splitted halves to form a
propped fracture. The sample (intact, with tensile
fracture, with a natural shear fracture, or propped
fracture) is then positioned inside the rock mechanics
loading frame. Then the confining pressure is applied
around the sample and a linear flow is established at a
given pore pressure to determine the hydraulic
conductivity of a given porous medium component
(matrix, tensile fracture, shear fracture, and propped
fracture).

Table 1: applied confining stresses for various flow

tests.

Matrix permeability, Micro-Darcy

has been widely used in the literature, all reservoirs

exhibit stress-sensitive permeability. The effective stress
magnitude and resulting deformation which produce
permeability changes, differs from one reservoir to
another. The effective stress is a function of pore
pressure, total stress, and Biots coefficient, and the
deformation is a function of the rock elastic and plastic
characteristics. Therefore the resulting permeability
change is a very complex function for an analytical
equation and should be experimentally evaluated for a
given rock formation at in-situ conditions.

Matrix perm, Micro-Darcy

SPE 110973

0.5
10,000

Figure 3: Stress-dependant matrix permeability with variable (p).

Stress dependant tensile fracture conductivity

When the reservoir pressure decreases, the elastic
displacement in response to the increase in effective
stress will cause natural fractures to close leading to a
decline in reservoir productivity. The matrix medium
feed the natural tensile fractures and the latter conduct
the fluids to the wellbore. The decline in conductivity with

Understanding stress dependant permeability of matrix, natural fractures, and hydraulic fractures in carbonate formation

increasing effective stress should follow a logical

declining rate to support a given production rate.

SPE110973

asperities. Compressive failure also generates rock

particles and fines that will further reduce fracture
conductivity. The fracture asperities in tensile and shear
fractures differ considerably, as the first one is not
accompanied with formation shifting while the rock in the
latter experience formation shifting which generate a
higher conductivity. As explained earlier, a 4-in sample
shown in Figure 4 is failed in tension following a
Brazilian test. The induced failure plain represents a
tensile fracture. The flow testing through the fracture was
performed and separated from the total permeability
using the following equation:
K t At = K f A f + K ma Ama ..... (6)
Stress dependant shear fracture conductivity
A natural shear fracture from a cored formation is shown
in Figure 5. Conductivity tests through the fracture were
performed similar to the tests conducted on the tensilefracture sample. The natural shear fracture as shown in
Figure 6 exhibits the shear fracture and its natural
surface.

Figure 4: Simulated tensile fracture; (Top) shows the failed whole

core and (Bottom) shows the resulting fracture surface.

The elastic closure response occurs when the net

effective horizontal stress increases as a result of
reservoir depletion. The elastic response to close the
fracture follows Hookes law of elasticity and it is
controlled by Youngs modulus of the formation:

e =

.. (5)

The aperture of the fracture will decrease causing a

corresponding reduction of fracture conductivity. If we
assume 50 ft of the rock perpendicular to the fracture will
contribute to fracture closure, then for a Youngs
modulus of 3 x 106 psi, the decrease in fracture width
corresponding to a decrease in reservoir pressure from
7,000 to 4,000 psi will be 0.05 inches. The fracture will
not close by 0.05 inches rather the contact points
(asperities) will carry the applied stress to prevent
fracture closure if they are strong enough to withstand
the stress. The compressive strength of the asperities
will determine the final fracture permeability. The
reduction in conductivity is due to a combined effect of
elastic response and compressive failure of the

Figure 5: Natural shear fracture from a retrieved core; (Top)

shows the naturally sheared fracture whole core and (Bottom)
shows the natural shear fracture surface.

SPE 110973

Understanding stress dependant permeability of matrix, natural fractures, and hydraulic fractures in carbonate formation

Stress
Dependant
Conductivity.

Proppant

hydraulic-fracture

Proppant hydraulic fracturing is used to create a

conductive fracture in the pay zone to enhance well
productivity. Proppant is used to keep a fracture open
during the life of the well.

Figure 6: A 100-mesh propped tensile fracture

A tensile-failed sample was propped with 100 mesh

Intermediate Strength Proppant (ISP) (Figure 6), or with
30 mesh Intermediate Strength Resin Coated Proppant
(ISRCP) (Figure 7), and flow testing was performed at
variable effective confining stress.
The permeability of a one layer 30 mesh RCP decreased
drastically and a lot of fines were generated (Figure 7)
at an effective closure of 4,000 psi. This is an important
criterion to consider when deciding on the type of
proppant to be used in the proppant fracturing treatment
of a given reservoir.
Figure 8 shows the stress-dependant permeability of the
porous media components considered in this study. The
best reservoir management will be produced if the
conductivity of all components declines at the same rate
assuming they start at similar original values. Figure 9
shows the stress dependant conductivity for the same
porous media components.
Normalized fracture conductivity defined as a
percentage of the initial conductivity is presented in
Figure 10. A 100-mesh proppant fracture will sustain
well productivity better than a 30-mesh RCP which
crushed at a closure stress of 6500 psi effective stress.
The 30-mesh proppant conductivity becomes less than
the ISP 100-mesh conductivity when

Figure 7: A tensile fracture propped with 30-mesh proppant (top),

and the re-sieved meshes observed after the test (bottom).

the effective stress becomes more than 7000 psi.

Although this high stress may not occur deep in the
reservoir, near the wellbore it may be very well
applicable. Low concentrations of proppant (Waterfracs)
have been very successful in stimulating tight gas
reservoirs17,18. A WaterFrac is a proppant fracturing;
however, proppant concentration used is very low
ranging from 0.5 to 2 Ib/gal. Therefore, it is possible that
a 100-mesh proppant perform better that a 30 mesh
resin-coated proppant if the later is exposed to crushing
stress as fines will be generated from the resin coat
resducing fracture conductivity below that of a 100 mesh
proppant.

Understanding stress dependant permeability of matrix, natural fractures, and hydraulic fractures in carbonate formation

Permeability, md

Tensile Frac
100 mesh frac

100000
1000
10

0.8
0.6
0.4
0.2
0

0.1

0.001

2,000
4,000
6,000
8,000
10,000
Effective Confining Pressure, psi

Figure 10: Normalized conductivity of various porous media

components associated with carbonate formation.

0.00001

2,000 4,000 6,000 8,000 10,000

Effective Confining Pressure, psi

Figure 8: Permeability of various porous media components

associated with carbonate formation.

Observing Figure 10, it be concluded that filling natural

fractures (tensile and shear) will improve productivity of
the given stress-sensitive reservoir under study. This is
not necessarily true in non-stressed reservoirs as it is
shown when the effective stress < 2000 psi, the shear
fracture is providing better conductivity than a 100 mesh
ISP propped fracture.
Matrix
Shear Frac
30 mesh RCP Frac

Tensile Frac
100 mesh Frac

10000

100

Mathematical Simulation
A mathematical model16 was used to achieve two
objectives; 1) Compare stress-dependant permeability
vs. constant permeability, and 2) Compare the effect of
100-mesh propped, a tensile, and shear hydraulic
fractures. The reservoir properties used in all simulation
cases were; a gas reservoir with a net pay of 46 feet,
reservoir pressure of 7620 psi, permeability is 0.5 md at
initial conditions and for fractured-well simulations; the
fracture is 150 feet long. The stress-dependant
permeabilities for all conductive components presented
above were used in the simulations. Two cases were
considered in the simulation runs; a single porosity
reservoir with constant matrix permeability and the other
case was considering the stress-dependant matrix
permeability. Figure 11 shows the results from these
two cases labeled as w/o stress and w/stress. In the
w/o stress case, the initial matrix permeability was kept
constant as the reservoir pressure decreases, while in
the w/stress case the initial matrix permeability
decreases as the reservoir pressure decreases following
the matrix function presented in Figure 11. The stress
effect is responsible for 50% loss of the PI, which can be
even more at higher production rate.
2.5

0.01

w/o stress
w/ stress

0.0001
0

2,000 4,000 6,000 8,000 10,000

Effective Confining Pressure, psi

Figure 9: Conductivity of various porous media components
associated with carbonate formation.

Figure 10 suggests that the tensile and shear fractures

losses more than 70% of its conductivity when the pore
pressure decreases by 1000 psi. This confirms rapid
decline in productivity in naturally fractured reservoirs
during early production. Propped fractures decline rather
much slower than natural fractures.

PI, Mscf/d/psi

Conductivity, md-cm

Matrix
Shear Frac
30 mesh RCP Frac

Tensile Frac
30 mesh RCP Frac
Normalized Conductivity

Matrix
Shear Frac
100 mesh Frac

SPE110973

1.5
1
0.5
0

50

100

150

200

250

Time (days)
Figure 11: PI as a function of time for single porosity model with
and without the effect of stress on matrix permeability.

Understanding stress dependant permeability of matrix, natural fractures, and hydraulic fractures in carbonate formation

Additional three simulation runs were made for

hydraulically fractured wells. The hydraulic fractures
considered followed the same characteristics of the
tensile, shear, and 100 mesh ISP propped fractures. The
initial conductivities of these fractures were kept
constant as the reservoir pressure decreases. The
results, represented by the well productivity index (PI)
behavior, for all fractured-well cases are presented in
Figure 12. The results show the benefit of the tensile,
shear and 100 mesh ISP propped fractures compared to
a single porosity non-fractured well. The 100-mesh
hydraulic fracture performed better than the hydraulic
fracture that has natural shear fracture conductivity or
the hydraulic fracture that has the tensile fracture
conductivity. This observation suggest that filling the
existing natural tensile and shear fractures with 100
mesh ISP proppant will lead to more production than
depleting the reservoir as a naturally fractured reservoir.
This conclusion does not carry a general consensus
rather it depends on the stress level and mechanical
characteristics of the reservoir formation. In shallow
reservoirs, the natural fractures may behave as infinite
conductivity fractures and filling them with 100-mesh
proppant will only reduce their contribution to the overall
reservoir flow efficiency.

Matrix
100 mesh
Shear fracture
Tensile fracture

P I, M scf/d/psi

6
5
4
3
2
1
0
0

50

100
150
Time, days

200

250

Figure 12: PI performance for various scenarios without the effect

of stress.

Considering the stress effect all the simulated cases

change drastically, as it is showed in Figure 13. The
effect of the tensile fracture stands only for a shorter
time due mainly to the high effect of the stress. This is a
scenario of injecting water only to create tensile fracture;
however this type of fractures will close upon producing
the reservoir and the effective stress increases. The
effect of shear fracture stands for a longer time for the
simulated case. The effect of the 100 mesh ISP propped
fracture is superior to all other cases with the effect of
stress dependant permeabilities.

8
Matrix

100 mesh
Shear fracture

6
PI, Mscf/d/psi

SPE 110973

Tensile fracture

5
4
3
2
1
0
0

50

100
150
Tim e (days)

200

250

Figure 13: PI for various completion scenarios with stress effect .

Conclusions
1. This study has uncovered an important phenomenon
related to the stress dependant poroelastic effect
during production of naturally fractured reservoirs.
The poroelastic coefficient in the matrix domain is
considerably different than that of the natural
fractures system. A new laboratory procedure to
determine the two different poroelastic functions is
presented. These functions should be implemented
in a dual porosity/dual permeability reservoir
simulation study to obtain reasonable prediction of
reservoir performance.
2. In stressed reservoirs or stress-sensitive reservoirs
where permeability loss is substantial, keeping the
natural fractures open should be the primary
objective. Propping these fractures with small
proppant mesh at early time should be considered
as an effective reservoir management strategy for
these reservoirs.
3. Many wells in naturally fractured reservoirs are
initially good producers but after a short period of
time a sharp decline in productions is observed.
This is frequently interpreted as a flush production
which is a rapid drainage of the fracture network,
whereas fluid bleed-off from the lower permeability
matrix rock occurs at much lower rates. This study
suggests
a
new
explanation
related
to
unsynchronized permeability reduction rate in the
matrix and fissures media. Effort to synchronize the
permeability decline rates of the matrix and fractures
systems should be carefully designed.
4. The contribution of matrix, tensile fractures and
shear fractures to the overall reservoir productivity
follow different stress-dependant permeability
functions. The permeability functions of these
porous components should be carefully determined
for any reservoir simulation study.
5. The stress-dependant proppant conductivity should
be evaluated in these highly stressed reservoirs
before any proppant fracturing is considered as a
stimulation treatment. A crushed large size proppant
may perform poorer than 100-mesh proppant.
6. In low permeability gas reservoirs, the abnormal
production lost is normally attributed to the

Understanding stress dependant permeability of matrix, natural fractures, and hydraulic fractures in carbonate formation

condensate banking phenomenon; however this

paper offers another alternative explanation related
to lost permeability.
7. This study provides an explination of why slickwater
hydraulic fractures work in certain reservoirs. The
tensile fractures in low stress reservoirs may
continue to contribute to the flow system before their
permeability becomes negligible. Additionally the
waterfrac may perform better than the natural
fractures which are assumed to have infinite
conductivity.
Nomenclature
A
: Fracture area
E
: Youngs modulus
K
: Bulk modulus, psi
K
: Permeability, md
Pe
: External pressure, psi
Pw
: Wellbore pressure, psi
PI
: Productivity Index, Mscf/d/psi
P
: Pressure drawdown (Pe-Pw), psi
re
: External radius, inch
rw
: Wellbore radius, inch
w
:
Fracture
displacement
during
development

: Biots coefficient
: Displacement due to elastic response
e

: Effective grain-to-grain stress.

t
: Total minimum horizontal stress

width

Subscript
t
: total
ma
: matrix
f
: fracture
b
: bulk
e
: external
e
: elastic
w
: well
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SPE110973

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