SPE 96896
A New Experimental Method To Determine Interval of Confidence for Capillary
Pressure and Relative-Permeability Curves
P. Egermann, J.-M. Lombard, C. Fichen, E. Rosenberg, E. Tachet, and R. Lenormand, Inst. Français du Pétrole
Copyright 2005, Society of Petroleum Engineers
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Abstract
The saturation profiles are routinely measured during Special
Core Analysis (SCAL experiments). Several ways exist to use
this information in the inversion procedure of the relative
permeability data. To date, the local saturation profiles are
either included in a global objective function that is minimized
during the inversion process, or smoothed and used as input
data in the simulation, which leads to non smoothed simulated
pressure drop. None of these methods permit to reproduce the
local saturation fluctuations that are measured during
coreflooding experiments. Our approach has the advantage to
take benefit from these fluctuations to characterize the
petrophysical properties for a given rock-type.
In this paper, the methodology is first recalled and illustrated
on a synthetic case. It is based on upscaling techniques applied
in reservoir simulation. Both the stabilized and transient
saturation profiles are analyzed to evaluate the local
properties. The quality of the history matching is significantly
improved. In addition, both kr and Pc-curves are obtained
from only one experiment and the variability of these
parameters with the heterogeneity is also investigated. This
estimated interval of confidence for kr and Pc can be very
useful to perform uncertainty studies at the reservoir scale.
The Pc-curve interval of confidence determined with this
method on a carbonate sample, is in good agreement with the
variability observed with several centrifuge experiments
performed on sister plugs.
Finally an alternative approach derived of the heterogeneous
method is proposed, which allows a very quick acquisition of
all the information relative to kr and Pc during a multi-rate
flow experiments.
Introduction
Petrophysical parameters assessment
Both relative permeability (kr) and capillary pressure (Pc)
curves are key factors for assessment and prediction of the
hydrocarbon recovery of a field. Representative curves are
routinely obtained through SCAL (Special Core Analysis
Laboratory) studies, in operating conditions as close as
possible of reservoir conditions. These conditions mainly
concern
the
thermodynamic
conditions
(pressure,
temperature), the reservoir stress (overburden pressure), the
nature of the fluids (synthetic reservoir brine, live oil), the
initial saturation state (low Swi value) and the in-situ
wettability (preserved samples or restored wettability through
an aging process at Swi).
Relative permeabilities are usually determined from flow
experiments performed on core samples using either the
UnSteady-State (USS) or the Steady-State (SS) method, where
either one or two fluids respectively are injected. The
centrifuge technique can also be used but not at full reservoir
conditions (Ruth, 1997; Chardaire et al., 1992). Whatever the
method used, the experimental results must be interpreted
using capillary pressure curves. Virnovsky et al. (1998)
proposed an analytical method to correct the SS data for
capillary pressure. However, numerical simulations are needed
for both USS and SS because a complete analytical solution
does not exist. The kr curves can either be optimized in order
to history match the experimental data knowing the Pc curve
(from another experiment) or both kr and Pc curves can be
optimized simultaneously (Chardaire et al., 1992; Helset et al.,
1998).
More recently, it has been shown that both kr and Pc curves
can be determined at reservoir conditions using the semidynamic approach (Lombard et al., 2002). The main
advantage of this method is to establish several equilibrium
states between the viscous and the capillary forces within the
sample by injecting one fluid while the other circulates at the
outlet face. These equilibrium states enable the determination
of both the kr of the injected phase and the Pc curve. The kr
of the displaced phase can also be obtained by history
matching of transient evolution of the pressure drop.
In-situ saturation monitoring, which is now widely
implemented in SCAL studies, brings a significant added
value to the interpretation process because it enables the direct
identification of the influence of the capillary effects on the
experimental data. Several ways exist to use this information
in the inversion procedure of the kr data. To date, the local
saturation profiles are either included in a global objective
function (in addition to the production and the pressure drop
data) that is minimized during the inversion process, or
smoothed and used as input data in the simulation. Local
saturation fluctuations often take place in carbonate samples
2
during coreflood experiments. These saturation fluctuations
cannot be anticipated during the preliminary core selection
process because they are due to small-scale local variations of
the petrophysical properties. Therefore, none of the available
methods permit the accurate reproduction of both the
production data and the heterogeneous saturation profiles in an
automatic way.
Background on rock heterogeneities
Because the core heterogeneities can severely affect the
determination of the petrophysical parameters, a SCAL study
always begins by a preliminary selection and sampling process
in order to retain the best cores within a given rock type.
Various techniques can be used and combined to finalize the
selection of core samples: CT-scanner, NMR, HPMI, tracer
tests (Dauba et al., 1999). Petrophysical parameters (k, φ, kr,
Pc) are seldom uniform within the entire core.
Graue (1994) reported a study on an eolian sandstone
originally considered as homogeneous, which showed
unexpected saturation profiles during coreflood experiments.
The saturation fluctuations were successfully interpreted as a
result of permeability and capillary pressure heterogeneities.
Honarpour et al. (1994) showed through several displacement
experiments conducted on cross or parallel bedding cores, that
the structure of the heterogeneity itself played a major role in
the shape of relative permeability curves. These conclusions
were confirmed by Mannseth et al. (1998), who interpreted
synthetic experimental data with different heterogeneity
structures (tilted lamina, unstructured heterogeneity
distribution, and flow barriers). Hamon and Roy (2000)
explored the influence of the coreflood design as a function of
heterogeneity type, along or across bedding samples. They
generated numerical experiments through a finely well
characterized model, and the interpreted kr curves were
compared to the input curves. One of their conclusions was
that the USS experiment is very sensitive to along-axis
heterogeneities.
Saad et al. (1995) succeeded in history matching both the
production data and the heterogeneous saturation profiles
presented by Honarpour et al. (1994) by introducing a limited
number of kr, Pc curves in the simulation code. Fincham and
Gouth (2000) analysed a multi-rate coreflood experiment,
which showed heterogeneous saturation profiles. Because the
stabilized profiles result from an equilibrium between the
viscous and the capillary forces, they observed that the
saturation profiles tend to become more homogeneous at the
highest bump flow rates (higher viscous forces). It was
possible to reproduce all the experimental data by
implementing a single set of kr curves and two Pc curves
accounting for low and high permeability facies.
Outline
This paper presents an innovative methodology to fully
account for the local heterogeneity in the history matching
procedure to determine kr and Pc. The proposed approach
takes advantage of local saturation fluctuations to characterize
the petrophysical property variability within a rock-type. The
method is derived from upscaling techniques that are routinely
applied in reservoir simulation and that are based on the
pressure data, which are, contrary to the saturation data,
SPE 96896
already a smooth function even when the heterogeneity level
is significant. Both the stabilized and transient saturation
profiles are analyzed to evaluate the local kr and Pc.
The first part of the paper is devoted to the presentation of the
proposed approach. A literature review on the use of local
saturations in SCAL and the background about upscaling
techniques in reservoir engineering is first presented. The
principle of the approach is then illustrated through a synthetic
case and the general workflow to interpret an USS experiment
is described. The application of the method for the
interpretation of a multi-rate gas/brine experiment performed
on a carbonate sample is then detailed. The results are
compared to centrifuge data obtained on companion plugs.
Finally, a semi-analytical approach derived of this
heterogeneous method is proposed to speed up the
interpretation of any multi-rate experiments.
The proposed heterogeneous approach
As described in the introduction, the rock heterogeneity very
often affects the experimental saturation profiles. Several
experimental techniques exist to identify this heterogeneity,
and results from these techniques can be accounted for in the
numerical simulation. Nevertheless, the heterogeneity is
seldom taken into account in the interpretation process
although several studies demonstrated that significant
improvements can be made using local petrophysical
parameters. The mathematical approach based on numerical
inversion of the problem is too complex and time consuming
to be conducted. The trial-and-error method is too subjective
and only applicable on a limited number of petrophysical
parameters. Therefore there is a need to develop a physical
approach to assess these parameters directly from the
experimental data in a consistent manner.
Use of local saturations
Local saturation profiles can bring a significant improvement
in the interpretation of coreflood experiments by accounting
for capillary pressure effects during the relative permeabilities
determination.
The most conventional approach for including saturation data
in the history matching process is to use a homogeneous
numerical model, a single kr curve (for each fluid) and a single
Pc curves. The saturation data are added to the production data
to calculate the objective function. At each iteration step, this
objective function is minimized by adjusting kr and Pc curves.
Several papers showed that the inclusion of the saturation data
considerably improved the determination of kr, Pc curves
especially at low injected fluid saturation (Courtial and
Ghalimi, 2000). Nevertheless, the optimized simulated
saturation profiles pass through the experimental data in only
an average sense but cannot reproduce the local
heterogeneities. Mejia and Mohanty (1995) proposed to
improve this procedure by incorporating local end-point
saturation in the inversion procedure. They found a better
match of the saturation profiles but the large number of
parameters to be considered makes the optimization process
very time consuming.
Another approach consists in processing directly the local
saturation data (Goodfield et al., 2001; Element and Goodyear,
2002). The particularity of their approach is to eliminate the
SPE 96896
Numerical example
The main finding of these upscaling studies is that the pressure
profiles are less sensitive to the heterogeneity than the
saturation profiles. Therefore, it suggests that the pressure can
be reasonably approximated using the petrophysical properties
determined with a homogeneous model. Consequently, the
pressure can be calculated at every saturation measurement
location without the need for local pressure taps.
Synthetic data have been generated by simulation to illustrate
this point on a conventional USS coreflood imbibition
experiment. The IFP in-house simulation code has been used
with a 1D geometry and 30 cells. The core properties are
presented in Table 1. We consider a multi-rate brine injection
in a core originally at Swi (10, 50, 200, 400 cm3/hr). The same
viscosity was considered for both fluids (1 cP) and a unique
set of kr curves was used. The Corey exponents and kr end-
Table 1. Core properties
Length
Diameter
8 cm
5 cm
Phi
Kmin
Kmax
25 % 11 md 47 md
K average
Swi
Sorw
24.5 md
0.1
0.1
Two sets of data were generated using either a homogeneous
or a 1D heterogeneous core with non-uniform permeability
and Pc curves. The permeability values range between 11 and
47 md and distributed at random along the sample. The local
Pc curves were obtained using the Leverett function. The set
of Pc curves is shown in Figure 1 and corresponds to an oilwet case. The permeability of the homogeneous core is the
harmonic average. A unique Pc curve based on this average
permeability value is considered for the homogeneous core.
Sw (fraction)
0
0.2
0.4
0.6
0.8
1
1
0
-1
Pc w/o (bar)
Pressure based approach
Upscaling techniques in reservoir engineering
A tremendous amount of work has been published on flow in
heterogeneous porous media in the reservoir numerical
simulation community. The geological codes often provide
fine description models of millions of cells, which account for
the complexity and the heterogeneities of the reservoir.
Because the fluid flow simulations cannot be conducted on
these high resolution grid, an upscaling (or averaging)
technique is needed to reach reasonable calculation times. The
conventional method is to coarsen the grid by doing a static
upscaling before the flow simulation but it can lead to
significant errors due to the averaging of the saturation profile.
The principle of the Dual Mesh Method (DMM) is to solve the
pressure equation on the coarse grid and the saturation
equation on the fine grid. The pressure data are then
interpolated on the fine grid to better reproduce the saturation
profile evolutions (Guerillot and Verdière, 1995). From a
mathematical point of view, the DMM approach is justified by
the parabolical (or elliptical) nature of the pressure equation
(long range forces) compared with the hyperbolical nature of
the saturation equation. It makes the pressure profiles to be
smoother than the saturation profiles (Verdière et al., 1996).
point values are equal to 2 and 0.6 for the water phase, and, 4
and 0.9 for the oil phase respectively.
-2
-3
-4
-5
-6
Figure 1 : Synthetic case with small-scale heterogeneities: the
set of Pc curves implemented in the simulator at each location
5000
30
4500
25
4000
Pressure drop (mbar)
local heterogeneity effects by smoothing the saturation
profiles and to use them as input data for the numerical
simulator. This leads to simulated pressure profiles that are not
smooth, which is not consistent with the experimental pressure
drop responses that exhibit always a smooth, strictly
monotonous decreasing behavior after breakthrough, whatever
the level of heterogeneity inside the core. Therefore, this
approach permits the reproduction of the saturation data in all
the cases (input data in the model) but it fails to take into
account the heterogeneity and produces non-physical pressure
drop responses.
The latest approach is the one followed by Graue (1994),
Fincham and Gouth (2000) and Saad et al. (1995). It consists
in implementing several kr, Pc curves in the simulator to
account for the local heterogeneities. These existing studies
demonstrated that this approach permits significant
improvements in the quality of the history match but is only
applicable with a limited number of curves.
3
3500
Oil
pro
du
cti
15 on
(c
m3
10 )
20
3000
2500
2000
1500
Pressure drop
1000
Oil production
5
500
0
0
50
100
150
0
200
Time (h)
Figure 2 : Synthetic case with small-scale heterogeneities:
production data
The production data (Figure 2) are very similar to real
experiments performed in the laboratory in an oil-wet core.
Each increase of the injection rate (bump) leads to production
4
SPE 96896
1
Hete 10 cm3/hr
0.9
Hete 50 cm3/hr
0.8
Hete 200 cm3/hr
Hete 400 cm3/hr
Sw (fraction)
0.7
Homo 10 cm3/hr
0.6
Homo 50 cm3/hr
0.5
Homo 200 cm3/hr
Homo 400 cm3/hr
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
Core length (cm)
Figure 3 : Saturation profiles for the homogeneous (solid or
dotted lines) and the heterogeneous cores (points) descriptions
Principle of the "pressure based" method
First, the experiments are interpreted as if the sample were
homogeneous using standard methods used in SCAL, and
pressure profiles at each injection step are calculated using a
numerical simulator.
When stabilization is reached, the pressure in the displaced
phase is uniform along the sample and equal to the outlet
pressure (no flow rate implies no pressure gradient). The local
pressure of the injected phase corresponds to the capillary
pressure (plus the outlet pressure) as in a semi-dynamic
experiment. The saturation Sw is known at any location of the
in-situ measurement and the corresponding capillary pressure
Pc is derived from the calculated pressure profile. The
capillary curve is determined by all the points Pc(Sw)
measured at the various locations along the sample and the
various flow rates.
The above described approach will be referred as “pressure
based” approach in the following.
600000
Hete 10 cm3/hr
Hete 50 cm3/hr
Hete 200 cm3/hr
500000
Hete 400 cm3/hr
Water pressure (Pa)
of additional oil due to the increase of the viscous forces. As
expected, many fluctuations are observed in the saturation
profiles with the heterogeneous core (Figure 3) due to local
variations of oil trapping. The amplitude of the saturation
fluctuations (10-15 % saturation units) is in good agreement
with real experimental data. The fluctuations also tend to
minimize at higher rates because the viscous forces are higher,
leading to saturation close to the asymptotic value (= 1 – Sor).
This behavior is also observed in experimental data
(Egermann et al., 2003; Fincham and Gouth, 2000). The
saturation profiles obtained with the homogeneous core pass
on average through the heterogeneous saturation points but the
saturation differences can be locally as high as 10 % (Figure
3).
This is completely different for the pressure profiles. Some
small amplitude fluctuations can be detected on the profiles
derived from the heterogeneous core but they remain very
comparable to the homogeneous case (Figure 4). This example
illustrates and confirms that the pressure profiles are less
prone to fluctuations due to small-scale heterogeneity (on the
order of the centimeter scale) compared to the saturation
profiles.
Homo 10 cm3/hr
400000
Homo 50 cm3/hr
Homo 200 cm3/hr
Homo 400 cm3/hr
300000
200000
100000
0
0
2
4
6
8
10
Core length (cm)
Figure 4 : Pressure profiles for the homogeneous (solid or
dotted lines) and the heterogeneous core (points) descriptions
General workflow for interpretation
In this part, the general methodology to account progressively
for heterogeneities is described. The general workflow is
given in Figure 5 and includes several steps.
Determination of the "homogeneous" curves
As a "first guess" for the calculation, a first set of curves noted
<kr> and <Pc> are determined as if the sample were
homogeneous, using standard methods in SCAL.
For a multi-rate unsteady-state (USS) experiment, the <kr>
and <Pc> curves can be determined analytically using the
procedure followed in a semi-dynamic experiment and
described in a previous SCA paper (Lombard et al., 2002).
The following analytical expressions with full account for
capillary effects are used :
dS
(1)
S inlet (∆Pi ) = S + q
dq
µL dq
(2)
kr inj =
KA d∆Pi
The capillary pressure points correspond to the ∆Pi, Sinlet(∆Pi)
values because the stabilized pressure drop is measured at the
inlet. ∆Pi can be considered as the capillary pressure because
the displaced phase is no longer mobile when stabilization
state is reached, leading to an uniform profile in this phase.
Therefore, the pressure drop that is measured across the
sample corresponds to the pressure difference between the
injected and the displaced phases at the inlet, which is the
definition of the capillary pressure.
If there is only one or a limited number of flow rates, the
previous method cannot be used (need to calculate the
derivative of saturation and pressure versus flow rate). In this
case, the capillary pressure must be measured on a separate
experiment, and the relative permeability determined from the
displacement (JBN + numerical simulation accounting for
capillary pressure).
SPE 96896
5
Using the homogeneous curves <kr> and <Pc>, a pressure
profile is numerically calculated at each step (Figure 5).
Step 1: determination of the local Pc(x)
In this part the “pressure based” approach is applied by
combining the experimental saturation measurements with the
simulated pressure profiles at the equilibrium points. This
enables derivation of several values of Pc at all locations
where a saturation measurement was performed. These
calculated values are then used to obtain the local Pc(x) curves
for each position.
These different Pc(x) curves are introduced as inputs in the
simulator in addition to <kr> obtained through the
homogeneous approach (Figure 5). This procedure can be
repeated several times to improve also the simulated pressure
profiles, but our experience shows that one iteration is often
enough to considerably improve the match of all the data.
This approach results in a very fast calculation since only a
direct simulation is involved.
Experimental data
multi rate injection
Analytical interpretation
JBN
Semi-dynamic method (Pc and kr
analytical values)
Pc curve on companion plug
Homogeneous approach
Inputs : Pc, first guess krs
Coreflood simulator
krs
Application case and comparison with conventional
measurements
In a previous publication, the proposed methodology has been
successfully applied to interpret USS experiments conducted
on heterogeneous samples with a gas-oil and a water-oil
systems (Egermann and Lenormand, 2004). The simulated
saturation profiles were in very good agreement with the
experimental profiles. The same approach was also adopted
for the interpretation of semi-dynamic water/oil experiments
performed on carbonate samples from several rock types
(Lombard et al, 2004). This section is focused on the
comparison between the variability of the Pc data that can be
obtained from only one experiment using the proposed
approach, and the spreading that can be obtained within the
same rock-type from Pc curves obtained on several companion
plugs (centrifuge technique).
Experimental procedures and results
The experiments were conducted at ambient conditions, on
carbonate samples cored at several locations and directions in
a quarry bloc: one sample for the multi-rate experiment, three
plugs for the centrifuge drainage experiments. Their main
properties are gathered in Table 2.
Table 2. Plug properties
Outputs : krs, Psim(x,t)
Stabilized
Simulated data
Gravity is neglected in the above expression.
Swexp(x,t)
Heterogeneous approach step1
Inputs : Pc(x), homogeneous krs
Samples
Multi-rate
centri 1 centri 2
centri 3
K (mD)
402
528
79
52
Porosity
0.31
0.35
0.29
0.34
Outputs : krs, all data matched
Heterogeneous approach step2
Inputs : Pc(x), Transient Swexp(x,t)
Output : local kr values
Figure 5. General workflow to interpret an USS multi-rate
experiment
Step 2: determination of the local kr(x)
The determination of local kr(x) does not require simulations.
The relative permeability of the injected fluid is derived by
using Darcy's law and assuming a uniform permeability. When
steady-state is reached, the pressure gradient in the injected
fluid is equal to the gradient in capillary pressure (derived
from step 1):
k kw ∂Pc
(3)
qw =
µ w ∂x
The relative permeability of the displaced fluid is derived from
the transient evolution of the saturation profiles. The displaced
phase relative permeability is derived from the general
expression of the fractional flow, calculated using the mass
balance (Craig, 1993):
k kro ∂Pc
1−
ut µo ∂x
(4)
fw =
µ w ko
1+
µo k w
For the multi-rate experiment, the sample was set in a core
holder transparent to X-ray, to enable local saturation
measurements with the CT-scanner. The experiment consisted
in injecting nitrogen (N2) in the core fully saturated with brine
(salinity: 20g/l). N2 was injected at a constant pressure using a
pressure regulator. Three steps of injection were considered :
115, 230 and 620 mbar. Pressure was changed from one step
to another when no water production was detected at the
outlet. The local saturation profiles were measured during the
whole experiment to follow the transient evolutions and also
to check the stabilization at the end of each step.
The three periods of injection can be clearly distinguished on
the production data, especially for water production curve
(Figure 6). Due to the heterogeneity of the studied sample, the
stabilized saturation profiles exhibit large fluctuation (Figure
11). The fluctuation can reach 6-7 saturation units, which is in
agreement with other experimental works reported in the
litterature (first part of the manuscript, example: Graue, 1994).
6
SPE 96896
18
100000
16
1400
12
10
8
6
Experimental
Numerical simulation
4
Pc centri 2
1200
Local Pc: step 1
1000
100
Experimental
10
Numerical simulation
2
0
0
100
200
300
1
400
0
Tim e (m in)
100
200
Tim e (m in)
300
800
600
400
Interpretation of the local Pc values
The interpretation of the USS experiment with the
heterogeneous approach led to the capillary pressure points
plotted in Figure 7. For each pressure, local Pc data were
calculated at the equilibrium. The connection between the
points calculated from three different stabilized saturation
profiles is satisfactory and, thanks to large capillary end
effects, a wide range of saturation was investigated (Sw varies
from 1 to 0.3)
1000
Local Pc: step 1
Local Pc: step 2
800
Local Pc: step 2
Local Pc: step 3
400
Figure 6 : Production data (gas-water experiment)
900
Pc centri 3
1000
Pc gaz/eau (mb
Gaz Production (cm3)
Water Production (cm3)
Pc centri 1
10000
14
Local Pc: step 3
200
0
0
0.2
0.4
0.6
0.8
1
Sw (fraction)
Figure 8 : Comparison between local Pc points and a set of Pc
curves from companion plugs
Alternative approach using the saturation profile
In the above-described method, the pressure profile is
numerically calculated using the homogeneous values <kr>
and <Pc>. However, when the homogeneous value <Pc> is not
known, a determination of local values is still possible using
the saturation profile. We will demonstrate this approach using
the example described in the previous section.
Pc g/w (mba
700
Linear interpolation of the pressure profile
The simplest approach consists in a linear interpolation of the
pressure between the inlet and outlet of the porous medium
(Figure 9). As the capillary end effects are not taken into
account with this approach, the local Pc points calculated are
not satisfactory, especially at the plateau level (Figure 10).
However, this curve can be used as a first guest for iteration.
600
500
400
300
200
100
0
700
0
0.2
0.4
0.6
0.8
1
Sw (fraction)
Figure 7 : Local Pc data derived from the USS experiment
500
Pressure (mbar)
Comparison with Pc data from companion plugs
In Figure 8, the Pc data obtained from the "heterogeneous"
interpretation of the USS experiment are compared to the data
measured on the three companion plugs using the centrifuge
technique. For this comparison, the centrifuge Pc curves were
corrected using Leverett function to account for the different
permeability and porosity of the samples and for IFT variation.
Each centrifuge curve is located in the area delimited by the
local Pc points of the USS experiment. The interval of
confidence drawn by the three centrifuge curves is comparable
to the one obtained from the USS experiment.
At ambient conditions, a centrifuge screening on several
samples as well as a USS experiment interpreted with the
heterogeneous approach can be used to study the variability of
Pc-curve within a given rock type. The main advantage of the
heterogeneous approach is its possible application for the
interpretation of experiments (USS or semi-dynamic)
performed at reservoir conditions, using only one sample.
step 1 : simulated
step 2 : simulated
step 3 : simulated
step 1 : linear
step 2 : linear
step 3 : linear
600
400
300
200
100
0
0
10
20
30
40
50
60
70
80
90
Length from the inlet (mm)
Figure 9 : Simulated (lines) and extrapolated (dots)
stabilized pressure profiles
SPE 96896
7
The local capillary pressure are then calculated using these
pressure profiles, and the results are very similar to the first
approach (Figure 13). The local capillary pressure points can
be accurately determined from these profiles and the
application of step 2 of the heterogeneous approach leads
finally to the local relative permeability values. Consequently
this method allows to obtain quickly most of the information
on capillary pressure and relative permeability variability.
1000
Pc centri 1
900
Pc centri 2
800
Pc centri 3
Local Pc: step 1
Pc gaz/eau (mb
700
Local Pc: step 2
600
Local Pc: step 3
500
400
300
200
700
100
step 1 : simulated
0
0
0.2
0.4
0.6
0.8
step 2 : simulated
600
1
step 3 : simulated
step 1 : calculated
Sw (fraction)
step 1 : experimental
0.8
step 2 : experimental
step 3 : experimental
0.7
step 1 : analytical
Sg (fraction
0.6
Pressure (mba
Analytical calculation of the pressure profile
A more accurate approach consists in using homogeneous
saturation profiles <Sw> together with homogeneous <kr> to
determine the pressure profiles when steady-state is reached.
The first step is the parametrization of the displacing fluid
relative permeability, using a standard analytical function
(Corey exponent for example). The saturation profiles are also
approximated by a smooth function such as:
λ
⎛ ⎛
L − x ⎞ ⎞⎟
(5)
S g ( x ) = S ginlet ⎜1 − ⎜ exp−
⎟
⎜ ⎝
H ⎠ ⎟⎠
⎝
where, Sginlet, L, H and λ are adjusting parameters to be fitted
according to the experimental data points. Sginlet nearly
corresponds to the gas saturation at the inlet, L is the total core
length and the other parameters control the shape of the
profile.
step 2 : calculated
step 3 : calculated
400
300
200
100
0
0
20
40
60
80
100
Length from the inlet (mm)
Figure 12 : Comparison between analytical and simulated
pressure profiles
1400
Local Pc: step 1
Local Pc: step 2
1200
Local Pc: step 3
Pc centri 1
1000
Pc gaz/eau (mb
Figure 10 : Local Pc data with linear interpolation of the
pressure – bad agreement especially in the plateau region
500
Pc centri 2
Pc centri 3
800
600
step 2 : analytical
step 3 : analytical
0.5
400
0.4
200
0.3
0
0
0.2
0.2
0.4
0.6
0.8
1
Sw (fraction)
0.1
0
0
20
40
60
80
100
Figure 13 : Comparison between local Pc points obtained
using the homogeneous saturation profiles and a set of Pc
curves from companion plugs
Length from the inlet (mm)
Figure 11 : Stabilized gas saturation profiles obtained
experimentally and with the analytical parametrization
Then the pressure profile in the displacing phase is
numerically calculated using an explicit scheme since
saturation is known. The pressure difference in each grid is
derived using either the liquid or gas Darcy's equation. The
total profile is then calculated by integrating from the exit
where the pressure is known. The result is in good agreement
with the profiles calculated by using the homogeneous
capillary pressure (Figure 12).
Conclusions
Within a same rock type, Pc and kr can vary significantly. The
study of the influence of heterogeneity on these petrophysical
parameters is of primary importance for the estimation of field
production.
A novel approach was successfully developed and tested to
take into account the rock heterogeneities that impact the
coreflood experimental data mainly in terms of saturation
profiles. The methodology is based on the inclusion of local
capillary pressure curves determined analytically by
combining experimental saturation profiles obtained during
stabilization states and simulated pressure profiles. The
8
SPE 96896
proposed approach takes advantage from previous works on
upscaling techniques for reservoir engineering, which showed
that pressure distribution is much less affected than saturation
distribution by the presence of heterogeneity.
Multi-rate injection experiments are recommended to get
analytically both Pc and kr data prior to the start of the
simulation process. The method enables the reproduction of
the saturation fluctuations by calculating analytically the local
Pc attributes using the stabilized saturation profiles. The local
values of kro can be obtained using the transient profiles
between two consecutive rates. The overall method was
successfully applied on experimental data related to
intermediate permeability carbonate samples (composite or
single plug) for a gas/oil system. The interpretation of the
local Pc attributes was also successfully applied on several
carbonate rock types, for a water/oil system.
This method can be used to reinterpret existing experiments
that were affected by small-scale heterogeneties. The
spreading of the kr, Pc data due to the heterogeneity within a
given rock-type can be assessed from only one experiment and
can be very helpful to prove the coherency between different
experiments performed on different samples, with different
methods in a SCAL program.
An alternative approach derived of this heterogeneous method
and based on the explicite determination of the pressure
profiles can also be applied to speed up the interpretation of
any multi-rate experiments.
in SPE Annual Technical Conference: Proceedings,
Washington, DC.
3.
Courtial, R., and Ghalimi, S., 2000, Techniques for
Relative Permeability Calculations : A Closer Look, SCA
2000-47, in the Society of Core Analysts Symposium:
Proceedings, Abu-Dhabi.
4.
Craig, F.F. :"The reservoir engineering aspects of
waterflooding", Monograph Series, SPE, Richardson,
Texas (1993), vol. 3.
5.
Dauba, C., Hamon, G., Quintard, M., and Cherblanc, F.,
1999, Identification of Parallel Heterogeneities with
Miscible Displacement, SCA 9933, in the Society of Core
Analysts Symposium: Proceedings, Denver, Colorado.
6.
Egermann, P., Robin, M., Lombard, J.-M., Modavi, A.,
and Kalam, M. Z., 2003, Gas Process Displacement
Efficiency Comparisons on a Carbonate Reservoir, SPE
81577 in the SPE Middle East Oil Show: Proceedings,
Bahrain.
7.
Egermann, P., and Lenormand, R., 2004, A New
Methodology to evaluate the impact of local heterogeneity
on Petrophysical Parameters (Kr, Pc): Application on
Carbonate Rocks, SCA 2004-18, in the Society of Core
Analysts Symposium: Proceedings, Abu Dhabi, UAE.
8.
Element, D. J., and Goodyear, S. G., 2002, New
Coreflood Simulator Based on Independent Treatment of
In-Situ Saturation and Pressure Data, SCA 2002-07, in the
Society of Core Analysts Symposium: Proceedings,
Monterey, CA.
9.
Fincham, A., and Gouth, F., 2000, Improvements of
Coreflood Design and Interpretation using a New
Software, SCA 2000-19, in the Society of Core Analysts
Symposium: Proceedings, Abu-Dhabi.
Acknowlegments
The author acknowledge O. Vizika and F. Kalaydjian for their
useful comments and fruitful discussions.
Nomenclature
A
:
∆Pi
:
fw
:
K
:
:
kri
krinj
:
L
:
µi
:
Pc
:
Q
:
:
Swi
:
S
:
Si
:
Sinlet
ut
:
lateral core area
total pressure drop measured at the inlet
watercut
absolute permeability
relative permeability of phase i
relative permeability of the injected phase
core length
viscosity of phase i
capillary pressure
flow rate
irreducible water saturation
average saturation
saturation of phase i
saturation at the inlet face of the core
total fluid velocity (Q/A)
References
1.
Chardaire-Rivière, C., Chavent, G., Jaffré, J., Liu, J.,
Bourbiaux, B., 1992, Simultaneous Estimation of Relative
Permeabilities and Capillary Pressure, SPE Formation
Evaluation, December, p. 283-289.
2.
Chardaire-Rivière, C., Forbes, P., Zhang, J. F., Chavent,
G., and Lenormand, R., 1992, Improving the Centrifuge
Technique by Measuring Local Saturations, SPE 24882,
10. Goodfield, M., Goodyear, S. G., and Townsley, P. H.,
2001, New Coreflood Interpretation Method for Relative
Permeabilities Based on Direct Processing of In-Situ
Saturation Data, SPE 71490 in SPE Annual Technical
Conference: Proceedings, New Orleans, LO.
11. Graue, A., 1994, Imaging the Effects of Capillary
Heterogeneities on Local Saturation Development in
Long Corefloods, SPE Drilling & Completion, March, p.
57-64.
12. Guerillot, D., and Verdière, S., 1995, Different Pressure
Grids for Reservoir Simulation in Heterogeneous
Reservoirs, SPE 29148 in the 13th SPE Symposium on
Reservoir Simulation: Proceedings, San Antonio, TX.
13. Hamon, G., and Roy, C., 2004, Influence of Heterogeneity,
Wettability and Coreflood Design on Relative
Permeability Curves, SCA 2000-23 in the Society of Core
Analysts Symposium: Proceedings, Abu Dhabi.
14. Helset, H.M., Nordtvedt, J.E., Skoeveland, S.M.,
Virnovsky, G.A., 1998, Relative Permeabilities From
Displacement Experiments With Full Account for
SPE 96896
Capillary Pressure, SPE Reservoir
Engineering, April, p. 92-98.
9
Evaluation
&
15. Honarpour, M.M., Cullick, A. S., and Saad, N., 1994,
Influence of Small-Scale Rock Laminations on Core Plug
Oil/Water Relative Permeability and Capillary Pressure,
SPE 27968, in SPE Centennial Petroleum Engineering
Symposium: Proceedings, Tulsa, OK.
16. Lombard, J.-M., Egermann, P., and Lenormand, R., 2002,
Measurement of Capillary Pressure Curves at Reservoir
Conditions, SCA 2002-09, in the Society of Core
Analysts Symposium: Proceedings, Monterey, CA.
17. Lombard, J.-M., Egermann, P., and Lenormand, R., Bekri,
S., Hajizadeh, M., Hafez, H., Modavi, A., and Kalam,
M.Z., 2004, Heterogeneity Study Through Representative
Capillary Pressure Measurements - Impact on Reservoir
Simulation and Field Predictions, SCA 2004-32, in the
Society of Core Analysts Symposium: Proceedings, Abu
Dhabi, UAE.
18. Mannseth, T., Mykkeltveit, J., Nordtvedt, J. E., and Sylte,
A., 1998, Effects of Rock Heterogeneities on Capillary
Pressure and Relative Permeabilities, SPE 50574, in the
SPE European Petroleum Conference: Proceedings, The
Hague, The Netherlands.
19. Mejia, G.M., Mohanty, K.K., Watson, A.T., 1995, Use of
In-Situ Saturation Data in Estimation of Two-Phase Flow
Functions in Porous Media, Journal of Petroleum Science
and Engineering, vol. 12, p. 233-245.
20. Ruth, D., 1997, Analysis of centrifuge relative
permeability data, SCA 9711, in the Society of Core
Analysts Symposium: Proceedings, Calgary, Canada.
21. Saad, N., Cullick, A. S., and Honarpour, M. M., 1995,
Immiscible Displacement Mechanisms and Scale-up in
the Presence of Small-Scale Heterogeneities, SPE 30779,
in the SPE Annual Technical Conference: Proceedings,
Dallas, TX..
22. Verdière, S., Guérillot, D., and Thomas, J.-M., 1996, Dual
Mesh Method for Multiphase Flows in Heterogeneous
Porous Media, in the 5th European Congress on the
Mathematical of Oil Recovery: Proceedings, Leoben,
Austria.
23. Virnovsky, G. A., Vatne, K. O., Skoeveland, S. M., and
Lohne, A., 1998, Implementation of Multirate Technique
to Measure Relative Permeabilities Accounting for
Capillary Effects, SPE 49321, in the SPE Annual
Technical Conference: Proceedings, New Orleans, LO.