A New Method for Normalization of Capillary Pressure Curves
Oil & Gas Science and Technology - Rev. IFP, Vol.
A New Method for Normalization of Capillar y Pressure Cur ves
S.E.D.M. Desouky 0
0 King Saud University, College of Engineering, Petroleum Engineering Department , PO Box 800, Riyadh 11421 - Saudi Arabia
- A New Method for Normalization of Capillary Pressure Curves - A new method for normalization of capillary pressure data of a reservoir was developed that incorporates the effects of pore geometry (pore size distribution index and flow zone indicator index), lithology index and irreducible water saturation. The hydraulic flow unit's approach was used to classify the reservoir formation into constant pore geometry units. The Leverett J-function was then correlated with the normalized water saturation for each reservoir unit to group the capillary pressure data of that unit into one curve. The method requires the measurements of capillary pressure, irreducible water saturation, and routine core data such as permeability and porosity. To check the validity of the proposed method, capillary pressure data, irreducible water saturations, and routine core data of a shaley-sandstone reservoir in the Gulf of Suez (Egypt) were obtained and were used in determining the accuracy of the developed equations. The calculations showed that the studied reservoir has four constant pore geometry units. An excellent agreement existed between the measured data and the calculated ones with average relative errors of - 5.8, - 4.3, 3.7 and 4.4% for flow unit-1, unit-2, unit-3 and unit-4, respectively.
The capillary phenomenon occurs in porous media when two
or more immiscible fluids are present in the pore space. The
capillary pressure is defined as the difference in pressure
between the non-wetting and wetting phases .
For oil and water:
Pc = Pnw – Pw
Pc = Po – Pw
Since the gravity forces are balanced by the capillary
forces, capillary pressure at a point in the reservoir can be
estimated from the height above the oil-water contact and the
difference in fluid densities. For an oil-water system:
Pc = gh (ρo – ρw)
Capillary pressure curves are usually determined in the
laboratory by three methods: mercury injection, restored
state cell (porous plate) and centrifuge. The description of
these methods can be found in the reservoir text books [1, 2].
Reservoir calculations require a normalized curve for the
capillary pressure measurements which are obtained from
several plug samples that at different depths. Because of the
reservoir heterogeneity, no single capillary pressure curve
can be used for the entire reservoir. Attempts were made to
correlate capillary pressure curves with the petrophysical
properties of the reservoir rock. Leverett  was the first to
introduce a dimensionless capillary pressure correlation
function. This function is defined as:
J(Sw) = Pc √K/ϕ /(σ cos θ)
The above function accounts for change of permeability,
porosity and wettability of the reservoir as long as the general
pore geometry remains constant. This fact was confirmed by
Brown , and Rose and Bruce . Guthrie and Greenberger
 proposed a linear correlation of the water saturation with
porosity and permeability at a constant value of capillary
Sw = a1 ϕ + a2 Log K + C
Wright and Woody  applied this correlation to a group
of capillary pressure curves of different permeabilities, but
with constant porosity. Pletcher  suggested the use of
) to averaging capillary pressure curves
corresponding to average permeability and porosity. El-Khatib 
modified the J-function by introducing tortuosity and
irreducible water saturation for averaging capillary pressure data.
J*(Sw) = Pc √K/ϕ (1 – Swr)/(σ cos θ)
Pc = a/(Sw – Swr)b
) requires the equality of the saturation
exponent b in the different formations to obtain a unique
correlation. Khairy  proposed a power relationship between
capillary pressure and saturation:
Pc = A Sw–B
Although the above relation correlated the data with
acceptable degree of accuracy, it fails to correlate capillary
pressure data of another formation. Donaldson et al. found
that the least square solution of a hyperbolic function can be
used to average capillary pressure data. Thus the proposed
models, discussed above, apply locally only because there
may be large differences in depositional characteristics at
other locations. Most of the models explicitly ignore the
scatter of data about the normalized capillary pressure curve
and implicitly attribute any scatter to measurement errors,
fluctuations in reservoir characteristics, or absence of some
reservoir parameters. An improvement to J-function,
), can be achieved by first identifying lithological
categories of formation and then calculating regression curves for
measurements that belong to each lithology class. The proper
method to identify the different lithological categories is the
use of hydraulic flow unit technique.
1 THEORETICAL ANALYSIS
A more reasonable approach for normalizing capillary
pressure data is to attribute the nature of interdependency
between capillary pressure and saturation to geological
variations in the reservoir rock and seek functional relationships
for capillary pressure that capture geological controls on flow
properties. Such relationships are best achieved if rocks of
similar fluid conductivity are identified and grouped together.
Each group is referred to as a hydraulic flow unit. The
hydraulic flow unit is a reservoir zone that is continuous
laterally and vertically and has similar flow and bedding
characteristics. The hydraulic flow unit that characterized a
specific reservoir zone is mathematically expressed by .
Where the reservoir quality index (RQI) is given by:
And the normalized porosity (Φn) is defined as:
RQI = FZI Φn
RQI = √K /Φ
Φ = Φ /(1 – Φ)
Substituting Equation (
) in Equation (13), one obtains:
Pc = σ cos θ J* Swn–1/λ / (FZI Φn)
Equation (16) can be rewritten as follows:
Pc = Ψ Swn–1/λ /Φn
Where the term Ψ is constant for each hydraulic flow unit,
and is defined by:
Ψ = σ cos θ J* /FZI
Equation (17) is the normalized capillary pressure
equation for a flow unit of a reservoir, and the number of the
normalized curves is equal to that of the hydraulic flow units.
Given the routine core data (k and Φ), capillary pressure data
(Pc – Sw), and irreducible water saturation (Swr), the following
steps are proposed for normalizing the capillary pressure
– Calculate the values of RQI and Φn from core data using
) and (
– Plot RQI versus Φn on log-log plot, determine the
optimum number of hydraulic flow units using the iterative
multi-linear regression clustering technique (
), and then
determine the values of FZI for each unit.
– Identify the capillary pressure data of each flow unit,
noting that the capillary pressure data of a flow unit are
corresponding to the core data of that unit.
– Use the capillary pressure data and core data to calculate
the values of J-function and normalized water saturation
from Equations (
) and (15), respectively.
– Plot the values of J-function against normalized water
saturation on log-log scale, and then determine the values of
lithology index (J*) and pore size distribution index (λ),
for each flow unit.
– Determine the normalized curve equation for each unit,
using Equations (17) and (18).
In this study, routine core data (k and Φ), capillary pressure
data (Pc – Sw), and irreducible water saturation (Swr) were
obtained from a shaley-sandstone reservoir in the Gulf of
Suez, Egypt. The reservoir thickness is 70 m (229.66 ft),
including 56 m (183.73 ft) shaley-sandstone and 14 m
(45.93 ft) shale. The routine core data were 344
measurements and 44 data-sets of capillary pressure data and
irreducible water saturation. The method used for measuring
capillary pressure and irreducible water saturation is the
restored state cell (porous plate), which relies on the selection
of a suitable porous plate (porcelains) to provide a barrier that
The value of flow zone indicator (FZI) is the intercept of a
unit-slope line with the coordinate Φn = 1, on a log-log plot
of RQI versus Φn. The basic idea of hydraulic flow unit
classification is to identify groupings of data classes that
form straight lines with unit slope on log-log plot of RQI
versus Φn. From Equations (
) and (
), the term √K/Φ is
Substituting Equation (12) in Equation (
), one gets:
2 APPLICATION TO FIELD DATA
√K/ Φ = FZI Φn
J(Sw) = Pc FZI Φn /(σ cos θ)
For capillary pressure data of a constant pore geometry
(i.e. a fixed value of FZI), the relationship between the values
of J(Sw) and normalized water saturation (Swn) is given by:
J(Sw) = J* Swn–1/λ (14)
Swn = (Sw – Swr)/(1 – Swr)
The term J* is known as the lithology index and its value
is equal to J(Sw) at Swn = 1. The term λ is the pore size
distribution index which is equal to the reciprocal of the slope.
permits the passage of the wetting phase which is the
simulated formation brine (i.e drainage process). The following
procedure was proposed by International Core Laboratories
 for measuring the capillary pressure by restored state
cell (porous plate).
– Clean, dry core samples are evacuated and pressure
saturated with the simulated formation brine.
– The porous plate is saturated with the simulated formation
– The saturated core samples are placed on the porous plate
with a suitable material (tissue paper and diatomaceous
earth) to aid in establishing a good capillary contact
between the rock surface and porous plate.
– The pressure applied to the assembly is increased by small
increments, and each sample is allowed to approach a
state of static equilibrium at each pressure level. This aids
in controlling the wettability of the rock sample.
– The saturation of the core is calculated at each point
The routine core data (k and Φ) and irreducible water
saturation (Swr) were statistically analyzed and the results are
given in Table 1. The capillary pressure data are plotted in
Figure 1. It shows a lot of scatter which clearly indicates the
existence of more than one rock type. A normalization of
these data was attempted using J-function (Eq. (
normalized curve is overlaid on the capillary pressure data in
Figure 1, and the average relative error between the measured
data and those calculated from the normalized curve equation
was found be 37.6%. The existence of more than one rock
type or simply pore geometry in the studied reservoir can
also be confirmed by plotting log k versus Φ, as shown in
3 RESULTS AND DISCUSSION
By applying iterative multi-linear regression clustering 
to the core data given in Figure 2, the optimum number of
hydraulic flow units is four. Core data of each flow unit is
assigned to a regression line with a fixed value of flow
zone indicator, and is also characterized by a constant pore
geometry, as shown in Figure 3. The flow zone indicators are
3.25, 5.6, 9.2 and 24.3 micron for unit-1, unit-2, unit-3 unit-4,
respectively. Based on the core data of each flow unit and
plug-depths, the capillary pressure data and irreducible water
saturations are sub-grouped into four sets of data. These four
sets of data are plotted in Figure 4. To determine the values
of lithology index (J*) and pore size distribution index (λ)
from capillary pressure data and irreducible water
saturations, the values of J-function (Eq. 4) and normalized water
saturation (Eq. 15) are calculated and plotted in Figure 5.
This figure ensures the existence of four different pore
geometry rocks and the values of J* and λ are given in Table
2. The table shows that the results of pore size distribution
and lithology index are consistent with those plotted in
Characteristics of the reservoir four flow units
Flow zone indicator (FZI) (micron)
Lithology index (J*)
Pore distribution index (λ)
Normalized equations of the capillary pressure data
Normalized curve equation, psi (0.145 kPa)
*The value of (σ cos θ) is equal to 72 dyne/cm for air/brine system.
A crossplot between measured capillary pressures and
A new method for normalization of capillary pressure data,
using Leverett J-function and hydraulic flow unit approach,
was developed. The method incorporates the effects of pore
geometry, lithology variation and fluid saturations.
Pc* = 0.223/(Swn3.878Φn)
Pc = 0.139/(Swn2.574 Φn)
Pc = 0.046/(Swn2.141 Φn)
Pc = 0.013/(Swn1.687 Φn)
The proposed technique is successfully applied to field
data of a shaley-sandstone reservoir in the Gulf of Suez,
Egypt. The average relative errors between measured data
and the calculated ones were –5.8, –4, 3.7 and 4.4% for flow
unit-1, unit-2, unit-3 and unit-4, respectively.
The developed equations can be used to correlate capillary
pressure data of different zones in a formation in the different
wells according to the value of flow zone indicator (FZI) as
determined from core data measured on plug samples taken
from different wells.
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