cydar theory overview exp 5

CYDARTheory Overview

Version 14 May 2006

R. Lenormand Conventional and Special Core Analysis

Conventional and Special Core Analysis

Theory Overview

Foreword ............................................................................................................................. 3

Rock properties................................................................................................................... 4 Grain density ......................................................................................................................................................................................4 Porosity................................................................................................................................................................................................6 Mercury pore size distribution .................................................................................................................................................... 10

Capillary effects........................................................................................................................................... 10 Experimental principles............................................................................................................................... 11 Pore size distribution (psd) .......................................................................................................................... 12 Smoothing the experimental data ................................................................................................................ 13 What do we measure?.................................................................................................................................. 14 Mercury imbibition...................................................................................................................................... 16

Estimation of permeability .......................................................................................................................................................... 19 Kozeny Carman........................................................................................................................................... 19

Leverett J Function......................................................................................................................................................................... 20 Reservoir transition zones............................................................................................................................................................ 20

Permeability...................................................................................................................... 22 Darcy’s law historic........................................................................................................................................................................ 22 Local Darcy’s law ........................................................................................................................................................................... 23 Compressible fluid (gas)............................................................................................................................................................... 24 Inertial effects - Forchheimer's law............................................................................................................................................ 25 Klinkenberg effect .......................................................................................................................................................................... 26 Steady state methods ................................................................................................................................................................... 27 Unsteady state : Pulse decay method ....................................................................................................................................... 28 Unsteady state : Luffel method................................................................................................................................................... 28 Darcylog method for cuttings .................................................................................................................................................... 28 From NMR T2 .................................................................................................................................................................................. 29 From Capillary Pressure ............................................................................................................................................................... 29

Two-phase flow................................................................................................................. 30 Introduction .................................................................................................................................................................................... 30 Two-phase flow model................................................................................................................................................................. 32 Capillary effects .............................................................................................................................................................................. 32 Nomenclatures and definitions.................................................................................................................................................. 34

Reference and non-reference fluids: ............................................................................................................ 34 Saturation: ................................................................................................................................................... 34 Drainage-imbibition: ................................................................................................................................... 34

Flow equations ............................................................................................................................................................................... 35 Pc and Kr curves ............................................................................................................................................................................. 36

Capillary pressure........................................................................................................................................ 36 Relative permeability................................................................................................................................... 38 Local properties ........................................................................................................................................... 39

Wettability ....................................................................................................................................................................................... 40

Version: 13 May 2006 1


USBM.......................................................................................................................................................... 40 Amott index................................................................................................................................................. 40

Boundary conditions..................................................................................................................................................................... 41 Pressure and flow rate in a fluid .................................................................................................................. 41 Free surface: ................................................................................................................................................ 41

Experiments in Core Analysis ...................................................................................................................................................... 43 Spontaneous displacement........................................................................................................................... 43 Porous Plate................................................................................................................................................. 44 Injection of one fluid ................................................................................................................................... 45 Simultaneous injection of two fluids ........................................................................................................... 46 Gravity displacement................................................................................................................................... 47 Centrifuge.................................................................................................................................................... 47 Semi-dynamique Method ............................................................................................................................ 48

Kr and Pc Measurements.................................................................................................. 50 Pc measurements........................................................................................................................................................................... 50

Porous Plate................................................................................................................................................. 50 Centrifugation.............................................................................................................................................. 51

Kr Measurement............................................................................................................................................................................. 53 Analytical steady-state method.................................................................................................................... 53 Kr JBN (or Jones and Roszelle) .................................................................................................................. 54 Kr by history matching................................................................................................................................ 60

Pc and Kr by SDM ........................................................................................................................................................................... 61 References ......................................................................................................................... 63



Foreword

This document has been written by Roland Lenormand and Fabrice Bauget and most of the illustrations showing flow in transparent models and micromodels were provided by César Zarcone.

The main objective of this document is to provide the basis for understanding core analysis and the relationship between core measurements and reservoir simulations.

For Special Core Analysis measurement (SCAL), we have tried to focus on some important features:

• any measurement must be "representative" of a specific displacement in the reservoir, either in terms of flow rates, fluids or rock-fluid properties. Especially, wettability that controls the displacements must be as close as possible to the reservoir wettability.

• The determination of two-phase properties such as capillary pressure or relative permeability are never simple measurements like porosity or permeability: they must be determined by numerical calculations generally with inverse methods. For many decades, approximate analytical solutions have been used due to the lack of numerical simulators (Hassler-Brunner method for capillary pressure for centrifugation or JBN for relative permeabilities). In most of cases, the analytical calculations lead large errors.

The content is based on the large experience of IFP concerning Core Analysis and Special Core Analysis, and all the recent publications on the subject, mainly in the literature of the Society of Core Analysis.

R. Lenormand

©Copyright CYDAREX 2006



Rock properties Any porous material, like rocks, are made of solid grains and void space that contains the fluids, water, oil or gas.

Vs

VpVt

Vs

VpVt

We will use the following notation (Figure 1): • : Pore volume Vp• : Volume of solid Vs• : total volume (also called bulk

volume) Vt

In this study, we will assume that the volumes of pore and grains do not change during an experiment (no plugging, no dissolution and non-deformable media).

We will first examine the properties linked to the solid (grain density) and to the pore space (porosity and pore size distribution). In the next

chapters we will study the flow properties when one or two fluids are flowing.

Figure 1- Notations for a porous medium: Pore volume Vp ; Volume of solid Vs and

total volume Vt (also called bulk volume).

Grain density "Grain density" is the density of the solid part of the rock. A few examples are listed in Table 1. This property is mainly used to determine the mineralogy of the rock and to calibrate density logs.

d

Equation Chapter (Next) Section 1Table 1- Examples of grain density

mineral d (g cm-3)

quartz 2.65

calcite 2.71

dolomite 2.87

kaolinite 2.59

anhydrite 2.96

The grain density is derived from the mass of the dry sample and the volume Vs of solid:

m

=mdVs

(1)



The volume of solid Vs can be measured by the buoyancy exerted when the sample is immersed in a liquid. However, most of laboratories are now using a method based on gas expansion in a vessel containing the dry sample to be studied. The principle of the method is shown in Figure 2.

The measurement is made on dry samples and is based on Boyle's law. The principle is that the gas molecules invade the pore space and not the solid. The volume of solid Vs is derived after calibration of the volumes of the apparatus. Any gas can be used, but for very small pores, the Kelvin condensation of helium is less than for nitrogen. For catalysts pellets, helium is necessary, but for reservoir cores that always contain residual traces of water when dried at low temperature (60°C), the condensation of nitrogen is negligible and this gas or dry air can be used. Examples of equipment are the "Accupyc" from the company Micromeritics or the "Helium Porosimeter" sold by Vinci Technology.

heliumVs

1V 2V

atmP initPfinalPfinalP

a) Initial: valve V closed

V V

a) final: valve V open

heliumVs

1V 2V

atmP initPfinalPfinalP

a) Initial: valve V closed

V V

a) final: valve V open

Figure 2- Principle of measurement of the volume of solid by gas expansion.

The volume of solid is derived from Boyles' law applied to helium before and after opening the valve V separating the two reservoirs.

(2) 1 2 2 1( ) (atm init finalP V Vs P V P V V Vs- + = + - )

Where the pressures are the absolute pressures, and the volumes and known after calibration using reference samples.

1V 2V



Porosity For any porous material (for instance cores or cuttings), porosity is defined by:

Vp VpVt Vp Vs

φ = =+

(3)

where is the volume of solid, Vt the total (or bulk) volume and Vp the pore volume. Porosity can be calculated using any combination of two of these three parameters:

Vs

• volume of solid, • volume of pores, • bulk volume.

Volume of solid

Gas expansion pycnometer

This method is described in the previous section for grain density measurement.

Archimedes force

The sample fully saturated with brine (saturation under vacuum, after flushing by CO2 or under high pressure) is immersed in brine under a balance. The difference of weight between the dry core and the core immersed in brine is equal to the volume of displaced liquid (Archimedes' theorem). The brine density can be measured separately or determined by comparison with a calibration volume.

0

10

20

30

40

0 10 20 30 4

volume of solid by Archimedes force (cc)

volu

me

of s

olid

by

heliu

m e

xpan

sion

0

Figure 3 - Volumes of solid for small plugs measured by the Archimedes and the gas expansion methods (Accupyc, Micromeritics)

Figure 3 shows the good agreement between Archimedes and gas expansion methods for small pieces of rocks of irregular shapes. The Archimedes method is less time consuming, especially if associated with pore volume measurement by the saturation method.



Bulk volume

Geometrical volume

For cylindrical cores, the total volume is generally obtained by measuring the diameter and the length of the cylindrical sample. This method is not applicable for pieces of cores of irregular shape.

Mercury pycnometer

Mercury is a non-wetting fluid with respect to air for all the rocks. Consequently, mercury does not enter in a sample filled by air if no pressure is applied. The mercury pycnometer method consists in measuring the volume of mercury without and with the core immersed.

Due to safety reasons, this method is no longer used in most of laboratories.

Powder pycnometer

The principle is the same as for mercury but mercury is replaced by a fine powder. A commercial apparatus is the Geopyc from Micromeritics. The powder is first packed in a piston using a controlled vibration and force. The position of the piston is measured with high accuracy (Figure 4). Then, the sample is introduced in the cell, keeping the same volume of powder. The powder is packed again under the same vibrating process and the volume of the sample is derived from the difference in position of the piston, knowing the section of the cell.

V(powder)

V(powder+sample)

fine powder

Figure 4 Measurement of the bulk (or total) volume of a sample by the powder method:

principle and automatic apparatus (Micromeritics)

Pore volume

There are mainly three techniques to determine the pore volume of cores:

Saturation method

The pore volume is derived from the mass of the sample saturated with brine and after drying.

The complete saturation of the samples is performed by evacuating the sample under vacuum for around one hour, and then saturating the sample at elevated pressure (150



bars) during a few hours to dissolve the remaining air bubbles trapped in the sample (Figure 5).

Mercury injection (Purcell)

The pore volume can be derived from the total volume of mercury injected during invasion under pressure. This measurement is performed on a small piece of rock (a few cc). This measurement is time consuming and its main objective is not the porosity but the pore size distribution.

NMR relaxation

The NMR magnetization is proportional to the number of protons in the sample (water or oil). After calibration, NMR relaxation can be used to determine the amount of water in a core. This method gives accurate results, in agreement with the saturation method. However, all the water is measured, including water remaining in the smallest pores after drying at low temperature (for instance inside the clay). Pore volume is expected to be always higher

with NMR than with the saturation method based on a limited drying of the sample, as shown in Figure 6. This method is recommended when the samples cannot be dried (preserved cores in core analysis for instance).

Figure 5 – Example of equipment used to saturate cores (Vinci-

Technologies)

0

2

4

6

8

10

12

0 2 4 6 8 10

pore volume by weight (cc)

pore

vol

ume

by N

MR

12

Figure 6 - Comparison of the pore volumes determined by NMR and saturation methods on small carbonate plugs.

Remark:

1) As a short-cut in language, the gas expansion apparatus is sometimes call a "gas porosimeter"; but this is true only for cores of cylindrical shape. For cores with non regular



shaped, another parameter must be determined, either the pore volume or the bulk volume.

2) using the saturation method always introduce an error due to the film of water that remains at the surface of the core. This effect is quite important for small fragments where the surface/volume ratio is relatively large. However, even for a standard cylindrical sample (8 cm height and 4 cm diameter), a film of 0.1 mm of water covering its lateral surface corresponds to a volume of around 1 cc, to be compared to the 20 cc of pore volume if porosity is 0.2.



Mercury pore size distribution Capillary effects

The non miscibility of two fluids in contact leads to the existence of an interface. The equilibrium between two non-miscible fluids is governed by the Laplace’s equation that stipulates that the pressure difference, (P2-P1), between two fluids is controlled by the interfacial tension γ and the radius of curvature R as follow:

γ

1 22

cP P PR

= − = (4)

Where the interfacial tension γ characterizes the energy spent in the creation of the interface. R is the curvature radius which will be such as the energy is minimized. Then a fluid surrounded by another one will tend to form a sphere. The pressure is always higher in the fluid "inside" the meniscus.

Typically a spherical liquid-gas meniscus with a radius μ1 m and a surface tension

gives a capillary pressure around 1 bar. That illustrates the importance of capillary effects. γ -10.05 mN.m∼

Now in a porous medium the solid phase has to be considered. The wettability illustrates the fluid’s affinity with the solid. It is characterized by the contact angle • which measure the angle of the contact line between a fluid and a solid. A fluid is wetting for

θ π0≤ < 2 and non-wetting for π θ2≤ (Figure 7). Mercury is typically a non-wetting fluid with a contact angle around 140°. Usually in a system gas-liquid-solid the gas is the non-wetting phase but it depends on the physico-chemical properties of the solid.

Figure 7- A liquid drop on a solid surface.

Figure 8 -Fluid at pressure P2 in equilibrium in a tube saturated with a fluid at pressure P1



The wettability will affect the capillary pressure. For example in a circular channel of radius r the curvature radius is θcosr R= then equation (4) becomes (Figure 8):

γ θ2 cos

cPr

= (5)

Because there are two fluids, there distribution in the porous medium will affect the flow. In an elementary pore volume , the local saturation is the ratio between the fluid volume dV and :

dVpr dVp

rr

p

dVS

dV= (6)

Obviously we have , and at the scale of the core, we can define a mean saturation by:

1r nS S+ =

1r

rp p V

VS

V V< >= = S dV∫ (7)

Experimental principles

Mercury porosimetry is often used to characterize a pore size distribution of porous materials (Purcell 1949). The experimental principle is the following (Figure 9):

• The sample is saturated under vacuum (or sometime with air) and surrounded by mercury.

• The liquid is forced to invade the pore space by increasing its pressure.

• Injected volume is measured for each pressure step. • A volumetric pore size distribution is extrapolated.

The determination of the pore size distribution is based on a porous medium modeled by a bundle of capillary tubes with different radii. The capillary pressure in a circular tube of radius r is :

γ θ2 cos

c n rP P Pr

= − = (8)

Then to penetrate a tube, the mercury pressure must be at least equal to Pc . That is the basis of the mercury porosimetry. To a pressure increase corresponds a radius step of invaded tubes and an injected volume . From this, a volumetric pore size distribution can be derived. But it

is important to remember that this method is based on the assumption of a pore space modeled by a bundle of parallel circular tubes. Obviously this is not the case in reality and so it is not the real pore size distribution. This point will be discussed below.

nPr nV

Figure 9 -Automated apparatus for mercury

injection (from Micromeritics)



In the following, the theoretical background to derive a pore size distribution (psd) is first detailed. Afterwards the importance of smoothing the experimental data is discussed.

Pore size distribution (psd)

We consider a porous medium fully saturated with air. During a mercury experiment, the pressure, , is increased by a step and the corresponding injected volume is measured (Figure 10). The total pore space is assumed to be filled at the end of the experiment, i.e. at the maximum pressure.

nP ndP

nV

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

log(pressure), P in bar

Hg

satu

ratio

n

Figure 10 - Mercury injection experiment, Hg saturation as a function of P.

As explained in the introduction, the Laplace’s equation controls the process and when the pressure varies from to , the threshold radius varies from r to . The derivative is then:

nP nP dP+ n r dr+

γ θ

22

ncos

dP drr

=− (9)

During this step, the injected volume is given by:

(10) ( )n pdV V f r dr=−

where f(r) is the psd. The negative sign comes from the relationship between pressure and radius: when the pressure is increased, the corresponding radius decreases. The previous equation is rewritten using the standard definition of saturation n nS V V= p :

(11) ( )ndS f r dr=−

and consequently



( ) n n

n

dS dS dPf r

dr dP dr=− =− n (12)

Using equation (9), this leads to the pore size distribution (psd):

γ θ

2( )

2n

n

P dSf r

cos dP= n (13)

Figure 11 shows the psd derived from the experimental curve displayed Figure 10.

0

5

10

15

20

25

30

35

40

45

1.E-03 1.E-02 1.E-01 1.E+00

radius (micron)

pore

siz

e di

strib

utio

n (1

/mic

ron)

Figure 11- psd corresponding to the experiment shown Figure 10.

By definition, is a probability distribution function and its integral over all the range of radius (and corresponding pressure) is unity:

( )f r

(14) 0 1

0( ) 1nf r dr dS

∞=∫ ∫ =

The dimension of is the inverse of a length (since its product by is dimensionless); ( )f r dr

Smoothing the experimental data

The more accurate way to determine a derivative is to smooth the experimental data with an analytical function (Lenormand 2003). The suitable method for this kind of curves is the "splines method". Figure 12 shows the difference between a simple linear interpolation and the splines. This method fits the curve by a polynomial inside a given interval limited



by two knots. Using splines of third degree with about 20 knots leads to a good approximation of the experimental curve but removes the high frequency noise due to experimental uncertainties. The coefficient of the polynomial of the splines are adjusted in order to assure continuity of derivatives at the knots.

0

0

0

0

0

1

1

1

1

1.E-03 1.E-02 1.E-01 1.E+00

radius (micron)

dS/d

log(

P)

Figure 12 - Difference between smoothed and raw data.

What do we measure?

The model of parallel tubes

As described in the introduction, this method is based on an oversimplified model. The porous material is modeled as a bundle of circular tube, but the real pore space has a very complicated geometry.

Let’s take a simple model of tubes with two diameters (Figure 13.b). A step of pressure

γ θΔ 2 cosnP = r is needed to invade the pore b as well as to invade the pore a. So in both cases the psd will give a single value at a radius r, when in reality there are two radii in case b. Then the derived psd has to be seen a pore threshold distribution rather than a real pore size distribution (Dullien 1992). With a porous material modeled by a network of capillary tubes Figure 13.b, if the large part is called “pore” and the constriction is called “throat”, the “pdf” is more like a throat size distribution

Figure 13 - Capillary models for mercury injection

Which information?



Due to the factor in the formula (13), the probability distribution function f(r) exhibits a large contribution for low values of radius corresponding to high pressures. Then f(r) can be useful to characterize the structure of the roughness of the solid walls of the medium. For instance, it can be used to derive a fractal dimension by fitting it with a power law. However, this function is not adequate for estimating a mean pore radius or to found a double porosity. For the later one prefers to use the following:

2nP

( )( )

nn

n

dS dSg r P

d LnP dP= = n (15)

If we call ε Δ(ln )nP= the constant spacing in log scale, the incremental saturation is equal to εΔ ( )nS g= r . However, is not a probability distribution and its integral over all the values of the radius is not equal to unity. This point must be considered when comparing the results to other sources of information, such as NMR relaxation or image analysis. Compared to the standard psd, both functions have in common the derivative

( )g r

/n ndS dP

but multiplied only by for the logarithmic derivative instead of for the standard psd. That explains why there is more information contained in the intermediate values of pressure.

nP 2nP

0

5

10

15

20

25

30

35

40

45

50

1.E-03 1.E-02 1.E-01 1.E+00

radius (micron)

dist

ribut

ion

dS/dlog(P)

dS/dPpsd

0

5

10

15

20

25

30

35

40

45

50

1.E-03 1.E-02 1.E-01 1.E+00

radius (micron)

dist

ribut

ion

dS/dlog(P)

dS/dPpsd

Figure 14 – Example of 3 different "pore size distribution" calculated from the same injection curve.

Another possibility is to plot only the derivative /ndS dPn , that corresponds to saturation increments with equally-spaced pressures. Compared to the two other functions, this simple derivative leads to more weight for small values of pressure (large radius).

The function must be chosen depending on the information needed, as summarize in



Table 1.

Mercury imbibition

For the moment only mercury injection has been discussed. But the opposite experiment can be performed as well:

The sample is saturated with mercury at a maximum pressure . maxP

The mercury pressure is decreased to extract it from the porous medium.

It is no longer a drainage process but an imbibition since the saturation of mercury (non-reference fluid) is decreasing.

Table 1 - The type of information depends on the function used.

the pore size distribution equation (13) characterization of the distribution of the smallest pores,

roughness of the walls (fractal dimension).

the logarithmic derivative / (ln )dS d P pore topology,

mean diameter (to derive permeability),

presence of a double porosity.

information for large radius and low capillary pressures.

the derivative /dS dP

The interpretation of the imbibition curve is based on the previous approach. The Laplace’s equation (equation (8)) links the mercury pressure to a radius. In the injection case this radius was a threshold value. In a pore modeled by a large cavity (pore itself) and throats separating pores to each others (Figure 13.b), the injection is assumed to give a distribution of throat radius. With the same model the imbibition curve is supposed to give a distribution of pore radius (large diameter).

At the beginning the mercury pressure is very high. By decreasing it the smallest throats are first emptied, but pressure will still be high enough to fill bigger channels or pores. Then the radius threshold is no more a lower bound but rather an upper bound. In other word:

The system is at a given pressure where the mercury meniscus is at equilibrium corresponding to a radius (Figure 15.a). pr

Injection (Figure 15.b): to fill a smaller channel of radius you need to increase the pressure . Then it gives a distribution of lower bound (i.e. throat).

tr

tP

Extrusion (Figure 15.c): to empty a bigger channel of radius you need to decrease the

pressure by at least . Then it gives a distribution of upper bound (i.e. pore). pr

pP



Figure 15 - Mercury intrusion-extrusion process.

Between drainage and imbibition hysteresis is typically observed. It illustrates the fact that the energy provided to inject the mercury is not restituted when the liquid is extracted. For example compressing and decompressing a spring between hands is almost a reversible process. The same spring in a tube with contacts along the solid wall will not be reversible due to friction forces. Indeed by pushing the spring inside the tube the hands will furnish all the energy but a part of it will be dissipated due to friction. The spring released will not be able to provide the same amount of energy, and it won’t recover its full length. Then in a graph representing the work versus the spring length, the first compression will be above the first decompression. In our system, in a graph (Figure 16) representing the mercury pressure (work) versus the saturation (length) the first drainage curve (compression) will be above the first imbibition (decompression).

All this have several origins: • In mercury injection “small pores” are invaded (Figure 17), in imbibition “big

pores” are emptied (Figure 18), then nV P∂ ∂ n won’t be the same.

• Contact angle hysteresis: In one case the non-reference fluid “pushes” the reference one, in the second it is the opposite. This leads to a change in the contact angle as fairly sketched in Figure 15.b and c.

• Trapping fluids. • Change in the rock wettability. This is not the case with mercury unless a very

specific physico-chemistry of the solid.



1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Hg volume

pres

sure

(bar

)

intrusion

extrusion2nd intrusion

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Hg volume

pres

sure

(bar

)

intrusion

extrusion2nd intrusion

Figure 16 – Cycle of intrusion, extrusion and second intrusion.

Figure 17 and Figure 18, representing a drainage and an imbibition in micromodel, illustrate perfectly the processes difference and the hysteresis.

Figure 17 - Drainage micromodel experiment.



Figure 18 - Imbibition micromodel experiment.

Estimation of permeability (to be completed)

The principle is that permeability is proportional to the square of the size of the thresholds of pores that controls the flow (see chapter on permeability). Several methods have been proposed to estimate the permeability from the capillary pressure curve derived from mercury injection.

Kozeny Carman

Permeability can be derived from porosity and tortuosity :

3

2 22

; 1/88

K r T KT

φ φφ= →∼ r= (16)

As a first approximation, the pore radius is taken as the value corresponding to the maximum of the distribution in

r/ logdS d P

Many other methods have been developed (to be developed in the future): • Purcell • Thomeer • Swanson • Kamath



Leverett J Function Mercury capillary pressure can be used to estimate a capillary pressure curve with other fluids in a similar rock sample, having different permeability and porosity.

The method is based on the Leverett J function, which corresponds to the following assumptions:

• The two rock samples or the rock sample used for mercury injection and the reservoir belong to the same rock type, that means that the pore size distribution are “similar”

• We can identify a wetting and non-wetting fluid. For reservoir applications this assumption is only valid to calculate the transition zones, since the oil migration is assumed to take place in a water wet medium. The non-wetting fluid is associated to mercury.

• The capillary pressure is proportional to the surface tension (multiplied by cos( )θ if the contact angle is known

• The pore sizes (or to be more exact, the threshold sizes) are related to permeability using a Cozeny-Carman relationship. In the J Leverett function, the following relationship is assumed :

K

Pcφ

∝ (17)

The dimensionless J Leverett function is then defined as

cosPc K

Jγ θ φ

= (18)

Then, it is assumed that two samples belonging to the same rock type are characterized by the same function J. Especially, for use of a mercury experiment for reservoir applications :

( ) ( )J reservoir J mercury= (19)

Reservoir transition zones Mercury capillary pressure is also used to calculated oil and water saturations in reservoirs,.

In a reservoir, in static conditions, the pressure in the oil phase at any point is given by :

oil ref oilP P ghρ−= (20)

where the interface between oil and water in the well (free water level) is taken as reference for the pressures ( ) and the distance ( h ). The pressure in water is similar refP

w ref wP P ghρ−= (21)

The capillary pressure is equal to the difference between these two pressures:

(oil w w oilPc P P gh)ρ ρ= − = − (22)



Saturation at any distance from the free water level can be calculated from the balance between capillary and gravity forces (22). The reservoir capillary pressure is derived from mercury injection using the Leverett J function.

h



Permeability The porous material is fully saturated with a fluid which is displaced by the application of a pressure drop along the sample. The intrinsic property characterizing the resistance of the medium to the flow is the permeability. In the following, the notion of permeability is introduced with the Darcy’s law for a liquid:

( )

ρ θμ

cosout inP PQ Kg

A L

⎛ ⎞− ⎟⎜ ⎟=− +⎜ ⎟⎜ ⎟⎜⎝ ⎠ (23)

and for a gas:

( )

ρ θμ

2 21

cos2

out in

q

P PQ Kg

A P L

⎛ ⎞⎟−⎜ ⎟⎜ ⎟⎜=− + ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠

(24)

Different corrections of these laws that could be necessary are then discussed as: the inertial correction, the Klinkenberg effect and tubing correction. And finally some experimental remarks are given.

Figure 19 - Darcy’s experiment, after (Darcy 1856) Darcy’s law historic

In 1856 Henry Darcy (Darcy 1856), studying water injection trough a column of sand (Figure 19), found a relationship between the flow rate and the pressure gradient along the sample:

in outh

h hQ AK

L

−= (25)

with the flow rate, Q A the sectional area, the column length, the hydraulic conductivity. h is the hydraulic load defined as:

L hK

ρP

hg

= + z (26)

with P the pressure, • the density, g the gravity constant and z the height. The hydraulic conductivity depends on the fluid viscosity μ , which is avoided by defining an intrinsic permeability of the medium as K ρ μ( ) ( hK g K= ) .



Using the intrinsic permeability, the Darcy’s law becomes:

ρμ

ΔK PQ A g

L

⎛ ⎞⎟⎜= + ⎟⎜ ⎟⎜⎝ ⎠ (27)

where , and are respectively the inlet and outlet pressures. Equation (27) is valid for a vertical sample as shown in Figure 20.

Δ in outP P P= − inP outP

The Darcy’s approach is macroscopic, at one dimension, and requires some assumptions: • The porous medium and the fluid are homogeneous, i.e. the properties are the

same on each point of the system. • The variables are considered at the two extremities, so the flow is characterized in

one dimension along the sample axe. This assumes the porous medium isotropic.

Local Darcy’s law The previous assumptions are oversimplified in most cases. To take account of heterogeneities, anisotropy, fluid compressibility (like a gas, see the following paragraph), etc., one need a local formulation:

(Q KU gradP

A)gρ

μ= =− + (28)

where is the Darcy’s velocity, the gravity vector, the pressure gradient and K the permeability. This formulation is written, at the scale of the Representative Elementary Volume (REV), a very small volume f porous medium, but that contains a large number of pores. The permeability could be compared to the viscosity for a fluid which is relevant if the volume considered is larger than the molecules’ mean free path.

U g gradP

With this assumption (REV), equation (28) allows us to account for the spatial variation of the different variables. From now, we will assume the medium homogeneous, isotropic and the flow unidirectional along the core’s axe z , then we have:

ρ θμ

coszK P

Uz

⎛ ⎞∂ ⎟⎜=− + ⎟⎜ ⎟⎜⎝ ⎠∂g (29)

where zU is the velocity component in the flow direction, P

z

∂∂

is the pressure gradient

along the sample and θ is the angle between the vertical and the flow direction (Figure 20).

Let’s consider a core of length L and sectional area A saturated with a liquid of viscosity μ. If we integrate the local Darcy’s law between the inlet and the outlet along z:

ρ θμ

cosout out out

in in in

z z z

z z z

Q K Pdz dz g dz

A z

⎛ ⎞∂ ⎟⎜=− + ⎟⎜ ⎟⎜⎝ ⎠∂∫ ∫ ∫ (30)

We obtain the macroscopic formulation:



( )

ρ θμ

cosout inP PQ KU

A L

⎛ ⎞− ⎟⎜ ⎟= =− +⎜ ⎟⎜ ⎟⎜⎝ ⎠g (31)

Equation (31) is the Darcy’s law presented above (Equation (27)) with a cosine equal to one (vertical). The gravity term is dropped off for simplicity purpose in the following. We will see later it can be incorporated trough an offset term.

Figure 20 - Experimental setup.

Compressible fluid (gas) With a compressible fluid, as a gas, the density variation with the pressure has to be considered. At constant temperature the perfect gas law is given by:

( )ρ ρ00

PP

P= (32)

And the mass conservation gives:

( )ρ

0d U

dz= (33)

Then by using the local formulation of the Darcy’s law (equation (29)) we obtain the following pressure equation:

0d dP

Pdz dz

⎛ ⎞⎟⎜ =⎟⎜ ⎟⎜⎝ ⎠ (34)

By integration along z this leads to ( )2 2 2 2in in out

zP P P P

L= − − , which gives:



( )

( )( )

μ

2 21

2

out inP PQ z K

A P z L

⎛ ⎞⎟−⎜ ⎟⎜ ⎟⎜=− ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠

(35)

Because •Q is constant in steady-state (equation (33)), we also have PQ constant along the experimental setup (equation (32) in (33)) and then we can write:

Q

PoutPin Pq

DP

Patm

Q

PoutPin Pq

DPDP

Patm

( )

μ

2 21

2

out in

q

P PQ K

A P L

⎛ ⎞⎟−⎜ ⎟⎜ ⎟⎜=− ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠

(36)

where is the flow rate measured at the pressure as shown in Figure 21. It is important to know at which pressure is

the flow meter. One can see the gas compressibility effect through the pressure squares.

QPq

Figure 21- Experimental setup for gas permeability.

Inertial effects - Forchheimer's law All the above equations show a linear relation between the velocity and the pressure gradient. It has been shown that a gap between measurements and the Darcy’s law appears and increases when increasing the velocity (Figure 22).

The most popular empirical law to model experimental data was established by Forchheimer as follow (Forchheimer 1901):

( ) μ βρ 2in outP P U

UL K

−= + (37)

where • is called the inertial coefficient and is of the unit of the inverse of a length. This is valid for an incompressible fluid. For gas the right term in equation (37) will be

( ) ( )2 2 2in out qP P P− L . The Forchheimer’s law assumes that the pressure gradient will

depend on a viscosity term (Darcy’s law) and on an inertial one in U2. We can see in the above equation the dependence of the inertial coefficient on the density, which is problematic for compressible fluids for example. One prefers to rewrite it in term of the Reynolds number:

ρμ

ReVd

= (38)

d is a characteristic length defined by dd in porous media, where is the

permeability at low velocity. V is the interstitial velocity defined as

K= dK

φ φ( )Q A= =V U . Then we can write:



ρμφ

Re dQ K

A= (39)

By introducing a dimensionless coefficient βφ dB = K we obtain:

( )

(μ1 Rein outP P U

BL K

−= + ) (40)

Typically the inertial effect becomes significant when R as shown in Figure 22. e 1≥

flow rate Reynolds6543210

pres

sure

diff

eren

ce m

bar

35

30

25

20

15

10

5

0

Figure 22 - Experimental results on an unconsolidated sand of3 mm quartz grains.

Klinkenberg effect The Darcy’s approach is based on the continuous assumption of the system which is not always the case. Two extremes cases and an intermediate one can be distinguished when a gas flows in porous medium:

• Poiseuille's flow (Figure 23.a): the collisions between molecules lead to the notion of viscosity. The classic equations of fluid mechanics may be used at the pore scale. The medium may be assumed to be continuous and the Darcy’s approach is valid.

• Knudsen's flow (Figure 23.b): the characteristic pore size is negligible in comparison with the mean free path of the molecules. There are no collisions between molecules but only between solid walls and molecules. Then the



medium can not be considered continuous and the classic approach is no longer valid.

• Intermediate case: due to the pressure gradient the number of collisions will change locally, and then the apparent viscosity will not be constant in the pore space.

Figure 23: Gas flow trough a pore constriction, a) Poiseuille’s flow, b) Knudsen’s flow.

The Klinkenberg effect corresponds to the intermediate case where the fluid is assumed continuous with a given viscosity and slip conditions are added at the solid walls to account for the deviation from the Poiseuille’s flow (Klinkenberg 1941). Experimentally this will have an effect on the apparent permeability, and leads to a relation between the permeability at high pressure (classic case) and the real gas permeability K: PK →∞

1Pb

K KP→∞

⎛ ⎞⎟⎜ ⎟= +⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠ (41)

where <P> is the mean pressure between Pin and Pout.

The coefficient b depends on the gas and rock type. Tables and empirical relations exist to estimate b and to determine from gas permeability measurement. One popular method is to measure the gas permeability at different pressures and to extrapolate to the very high pressure case (Figure 24). The Klinkenberg effect has to be considered for rock of typically 0.01 Darcy at atmospheric pressure.

PK →∞

Steady state methods Let us define the variable y as for a liquid and for a gas. ( )in outy P P= − ( )2

in outy P P= − 2

The standard approach consists in plotting the experimental variable y as a function of flow rate and determine the permeability by using a linear or quadratic regression. However, this method is valid only if the parameters such as viscosity or atmospheric pressure (for gas) is the same for all the experiments. In addition, the standard approach cannot be used for a rigorous determination of the Klinkenberg effect.

Q

Consequently, we preferred to use a more general optimization technique using an inversion loop to determine the unknown parameters.



1/Pm 1/bar0.70.60.50.40.30.20.10.0

Perm

eabi

lity

K m

icro

darc

y70

60

50

40

30

20

10

0

Figure 24 - Klinkenberg correction.

Unsteady state : Pulse decay method See “ recommended practices for Core Analysis”, API

(to be developed)

Unsteady state : Luffel method No sleeve around the sample: all around the rocks sample (chip or cuttings), for very low permeability (below microdarcy)

Luffel and … SPE 26633 (1993)

(to be developed)

Darcylog method for cuttings (to be developed)



From NMR T2 (to be developed)

From Capillary Pressure Kozeny Carman

Purcell, Thomeer

(to be developed)



Two-phase flow

Introduction Two-phase displacements in a porous medium can be illustrated using a transparent medium made of crushed glass. In Figure- 25the medium is first saturated with water (white phase) and oil colored in red is injected on the left-hand side of the sample. The two lateral boundaries are impermeable and water displaced by the injected oil can flow on the right-hand side of the sample.

Figure- 25 - Injection of oil (red) in a transparent porous medium saturated with water (white). The profiles correspond to the average of the fraction of oil or water over a slice of the medium

In core analysis, we generally do not use the two-dimensional map of saturation but the saturation profile, either for oil or for water (Figure- 25). Saturations for water and

for oil in a volume V are defined as the ratio of the volume of water (respectively oil) to the total volume of liquid in the volume V. In this example, V is the volume of the slice of porous medium shown on the photograph.

wS oS



The following figures show sequences of various injections with the corresponding water saturation profiles (average water saturation inside transverse slices along the sample):

Figure 26 –Oil injection at high flow rate into a transparent porous medium saturated with water at three different steps and corresponding saturation profiles (flow rate = 0.75 cc/min).

Figure 27 - Oil injection at high flow rate into a transparent porous medium saturated with water at three different steps and corresponding saturation profiles (flow rate = 0.019 cc/min).

Figure 28 – water injection following oil injection in a transparent medium. A large amount of oil is left in the medium at the end of the displacement.



Oil injection at high flow rate showing a sharp front (Figure 26),

Oil injection at a lower flow rate showing a lore diffuse front (Figure 27),

Water injection following the oil injection showing a remaining saturation of oil being trapped in the medium (Figure 28)

Two-phase flow model In a porous medium fully saturated with one fluid, flow is described by the Darcy’s law (see permeability chapter):

(KU gradP )gρ

μ= − + (42)

where is the Darcy’s velocity, the permeability, P the pressure, U K μ the fluid viscosity, •

its density, and the gravity. In the multiphase case, fluids will interfere with each other: g

• Fluids and solid in contact involve the existence of an interface and then of capillary forces. This leads to a difference of pressure across the interface between the two fluids (e.g. in a bubble the gas pressure is higher) corresponding to the capillary pressure Pc.

• In monophasic flow the energy dissipation, characterized by the permeability, depends on the porous material characteristics (pore volume, topology, geometry, etc.). Adding another fluid will change all the local properties. In a Darcy’s approach (macroscopic, homogeneous, continuous medium) these changes are considered through coefficients depending on the volume of each phases called relative permeabilities Kr. For each fluid the Darcy’s velocity is then given by:

ρμ

ρμ

11 1

1

22 2

2

( )

( )

K Kr PQ A g

z

K Kr PQ A

z

⎧ ∂⎪⎪ =− −⎪⎪ ∂⎪⎨⎪ ∂⎪ =− −⎪⎪ ∂⎪⎩g

(43)

Capillary effects and two-phase flow theory are first introduced, with a description of the main characteristics of the capillary pressure and relative permeability curves. Types of boundary conditions are then presented. The second part is focused on the experiments with an overview on the experimental principles, their boundary conditions and the tools provided by CYDAR.

Capillary effects The non miscibility of two fluids in contact leads to the existence of an interface. The equilibrium is governed by the Laplace’s equation. It stipulates that the pressure



difference, (P2-P1), between two fluids across the interface is controlled by the interfacial tension γ and the radius of curvature R as follow:

γ

1 22

cP P PR

= − = (44)

The interfacial tension γ characterizes the energy spent in the creation of the interface. R will be such as the energy is minimized.

Typically a spherical liquid-gas meniscus with a radius μ1 m and a surface tension

gives a capillary pressure around 1 bar. That illustrates the importance of capillary effects in porous media where characteristic lengths are often micrometric. γ -10.05 mN.m∼

Now in a porous medium the solid phase has to be considered. The wettability illustrates the fluid’s affinity with the solid. It is characterized by the contact angle • which measures the contact line angle between a fluid and a solid. A fluid is wetting for θ π0 2≤ < and non-wetting for π θ2≤ (Figure 29). Mercury is typically a non-wetting fluid with a contact angle around 140°. Usually in a system gas-liquid-solid the gas is the non-wetting phase, but this depends on the physico-chemical properties of the solid.

Figure 29: Sketches of a liquid drop on a solid surface.

The wettability will affect the capillary pressure. For example in a circular channel of radius r (Figure 30) the curvature radius is θcosr R= , then equation (44) becomes:

γ θ2 cos

cPr

= (45)

We will see that in porous media the capillarity is usually interpreted through this law. In other word the pore space is usually modeled as circular tubes.

Figure 30: Sketch of a fluid at pressure P2 into a tube saturated with a fluid at pressure P1.

Remarks: • the physico-chemistry of the solid and the fluids will affect wettability. For

example, a petroleum reservoir can be oil-wet solid as well as water-wet, etc. • All the previous discussion is valid for smooth surfaces.



r

The characteristics of the capillary pressure curve will be discussed in the “Pc and Kr curves” part.

Nomenclatures and definitions Reference and non-reference fluids:

The wettability depends on the fluid, the rock interface, on many factors. For example it is well known that asphaltenes adsorption on pore wall changes wettability. Then a wetting fluid could become non-wetting. To avoid any confusion, in petrophysics one prefers to use reference and non-reference fluid. The nomenclature is as follow:

• The reference fluid is the aqueous phase in a liquid/liquid system; • In the case gas/liquid, the liquid is the reference fluid unless in the system

gas/mercury.

In the following the subscript r refers to the reference fluid and n to the non-reference one. The capillary pressure is then defines as follow:

(46) c nP P P= −

Remark: most of numerical simulators use the term of "wetting fluid", even if the wettability is unknown (wetting fluid = reference fluid).

Saturation:

Because there are two fluids there distribution in the porous medium will affect the flow. In porous media this distribution is measured by the saturation. In an elementary pore volume , the local saturation is the ratio between the fluid volume dV and : pdV pdV

,ii

p

dVS with i

dV= r n= (47)

Obviously, the sum of the fluid volumes will be equal to the total pore volume , then we

have . At the sample scale one can define a mean saturation by: pV

1r nS S+ =

1

,ii

p p V

VS S dV with

V V< >= = =∫ i r n (48)

Drainage-imbibition:

Commonly drainage is the decrease of water concentration, for example foam drainage. In petrophysics the drainage will be a flow process leading to the decrease of the reference fluid saturation. The imbibition is the opposite.



Flow equations The local Darcy’s law in monophasic flow is:

( )ρμK

U gradP= − + g (49)

with U the Darcy’s velocity, μ the fluid viscosity, K the absolute permeability, ρ the density and g the gravity. Let’s recall the homogeneous and isotropic assumptions here (see “Permeability” part). In the diphasic case this law is written for each phase:

( )ρμ

,ii i i

i

KKrU gradP g with i= − + = r n (50)

The coefficients Kr are the relative permeabilities and account for the permeability reduction due to the presence of the other fluid. Without considering physico-chemical effects, the presence of a fluid will change the flow resistance to the second one. The relative permeabilities depend on the saturations, but also depend on heterogeneities, physico-chemistry, pore space topology, etc.

In the Darcy approach fluids are assumed to be continuous and the system medium/fluids to be homogeneous at the considered volume (see “Permeability” part). For a unidirectional flow along the core axe, equation (50) gives:

ρ θμ

cos ,i ii i

i

KKr PU g wi

z

⎛ ⎞∂ ⎟⎜= − + =⎟⎜ ⎟⎜⎝ ⎠∂th i r n (51)

with θ the angle with the vertical. In monophasic case, the Darcy’s law is enough to describe the flow of an uncompressible fluid because the mass balance is obvious. The sample is fully saturated then the fluid volume inside the material is constant all over the experiment. But in two-phase flow, one has to consider the coupling of the previous equations with the mass balance written for each phase:

( )ρφ ρ 0S

div Ut

∂+ ⋅ =

∂ (52)

At one dimension and for an uncompressible fluid this leads to:

φ 0i iS Uwith i r n

t z

∂ ∂+ = =

∂ ∂, (53)

This approach is valid at macroscopic scale and assumes: • The existence of a Representative Elementary Volume (REV) which is the minimum

volume (macroscopic) where the fluids are continuous and homogeneous. • The instantaneous equilibrium.



]

Pc and Kr curves Capillary pressure

The capillary pressure is defined as the pressure difference between the non-reference and the reference phases (equation (46)). The curve is always represented as Pc versus the reference fluid saturation. Its characteristics are (Figure 31):

• The capillary pressure could be positive or negative, depending on the wettability. If the reference fluid is the wetting fluid (fluid 1 in Figure 30), then its pressure will be the smallest and Pc will be positive (Figure 31.a). In the opposite case Pc will be negative (Figure 31.b). And very often, porous materials have a mixed wettability and leads to a positive-negative capillary pressure (Figure 31.c).

• The capillary pressure is defined in a saturation interval [ , since a fluid

must have a volume minimum to be continuous at the macroscopic scale. Because then and so on.

min max,S S

r nS +S =1 r,min n,max S =1-S

• always decreases. Pc(Sr)• Because the capillary curve definition assumes the continuity of the fluids, at the

two bounds of saturation the curve has an infinite asymptote.

Figure 31: Typical shape of capillary pressure curves: a. reference wet; b. non-reference wet; c) mixed wettability.



Figure 32: Curves’ shape of drainage-imbibition cycles.

Between drainage and imbibition hysteresis is typically observed. It illustrates the fact that the energy provided to inject the mercury is not restituted when the liquid is extracted. For example compressing and decompressing a spring between hands is almost a reversible process. The same spring in a tube with contacts along the solid wall will not be reversible due to friction forces. Indeed by pushing the spring inside the tube the hands will furnish all the energy but a part of it will be dissipated due to friction. The spring released will not be able to provide the same amount of energy, and it won’t recover its full length. Then in a graph representing the work versus the spring length, the first compression will be above the first decompression. In our system, in a graph (Figure 32) representing the mercury pressure (work) versus the saturation (length) the first drainage curve (compression) will be above the first imbibition (decompression).

All this have several origins: • In injection “small pores” are invaded (Figure 33), in imbibition “big pores” are

emptied (Figure 34), then nV P∂ ∂ n won’t be the same.

• Contact angle hysteresis: In one case the non-reference fluid “pushes” the reference fluid, in the second it is the opposite. This leads to a change in the contact angle.

• Trapping fluids. • Change in the rock wettability which is not the case with mercury unless a very

specific physico-chemistry of the solid.

Figure 32 and Figure 33, representing a drainage and an imbibition in micromodel, illustrate perfectly the processes difference and the hysteresis.



Figure 33: Drainage micromodel experiment.

Figure 34: Imbibition micromodel experiment.

Relative permeability

The relative permeabilities (equation (50)) depend on the fluid saturations. Figure 35 shows their typical shapes.



Figure 35: Shape of relative permeability curves.

As the capillary pressure curve, these relative permeabilities have several characteristics in common whatever is the experiment:

• As Pc they are defined in a saturation interval [ ],min maxS S .

• Relative permeabilities are always positive. • For homogeneous media the curves tend towards zero following a power law. In

the heterogeneous case, non-monotonic curves have been observed. • The Kr maxima are usually less than the unity. However values higher than 1 have

been observed for very viscous non-wetting fluid, because the wetting fluid forming films facilitates the flow by lubrication.

The relative permeabilities are usually determinate from experimental results by history matching. This means Kr are inputs and adjusted until simulations give good results in comparison with measurements (pressures, production). Analytical models exist in form of power law based on the characteristics discussed above. They are known as Corey’s models, and have the following form:

( ) ( )α α* *min,max ,max

max min; ;

r nrr r n n

S SS Kr K S Kr K

S S

−= = =

−*1 S−

)

(54)

with and K the maxima of the relative permeabilities respectively for the

reference and non reference fluid; n and are power constant; and S S are the maximum and minimum saturations for each fluid (Figure 35).

n,maxK r,max

r nn min max

Local properties

It is important to point out that the properties, Pc(S and , discussed above are local. The main purpose is to be able to incorporate them in a local Darcy approach (simulator, modeling). Then it is not obvious that the measurements done at the sample exit, like mean saturation <S and pressure difference, are equal to the local ones. For instance, from centrifuge effluent volumes one may get the <S and but not the local curve

as it will be explained later.

r rKr(S )

>> Pc

Pc(S)



Wettability To be developed

USBM

Amott index

saturation

capi

llary

pres

sure

0 1

0 Sor

A

D

B

C

Swi

saturation

capi

llary

pres

sure

0 1

0

saturation

capi

llary

pres

sure

0 1

0

saturation

capi

llary

pres

sure

0 1

0 Sor

A

D

B

C

Swi

Figure 36 – Calculation of Amott wettability index

The Amott water index is defined as the ratio of volume (or saturation) of oil displaced during spontaneous imbibition to the volume of oil produced during the total imbibition. The Amott oil index is the ratio of water produced during spontaneous (negative) drainage to the total drainage. With the notations of Figure 36:

wI

oI

;w oA C

I IA B C

= =+ +D

I

(55)

And the Amott index is the difference between the two indices: AmottI

(56) Amott w oI I= −



The Amott index is obtained by a using spontaneous displacement followed by forced displacements, either using a centrifuge or injection with a pump.

Boundary conditions This is a very important part of the experimental interpretation and simulation. In fact this is the essential basis, erroneous conditions lead obviously to erroneous results. Here, it is all about how to express experimental boundary conditions to implement them in a simulator as mathematical conditions. Depending on the setup there are essentially two main types of boundary conditions:

• a given combination of flow rates and pressure (generally at the inlet), • a free surface at the inlet or the outlet.

Remarks: In porous media saturations can not be imposed but are always deduced from other measurements, capillary pressure for instance.

There are two fluids, and then boundary conditions can be different in each fluid. In the following the different cases are illustrated with oil (non reference fluid) and water (reference fluid), but can be generalized to any fluids.

Pressure and flow rate in a fluid

Flow rate:

A pump, to inject or to extract, will impose a given flow rate. Most of the experiments are done with constant flow rate by step using a piston pump with constant displacement. A zero flow rate may be imposed by mounting at an extremity a membrane impermeable to one of the fluids, like in a porous plate experiment (see below).

Pressure:

Several systems can be used to impose a pressure on a fluid either a pump controlled in pressure or a mechanical apparatus based on equilibrium with a spring (control valve, back pressure valve, pressure controller for gas cylinders, etc.). Another means is to have a large volume of gas at constant temperature. This pressure is generally imposed to be constant by steps, but it can be also a function of the flow rate when a membrane (or porous plate) is placed on the face of the sample.

Remark:

All the combinations of flow rate and pressure are not possible. For instance, there is no problem to inject 2 fluids with different flow rates, but injection with two different pressures is not possible, since the fluid at the higher pressure will fill the other fluid injection tubing (except if a porous plate is placed and a fluid injected behind the porous plate).

Free surface:

This kind of conditions is typically the case when a sample, or a part of it (outlet and inlet), is plunged into one of the fluids (e.g. centrifuge, gravity displacement, semi-dynamic method).



The free surface can correspond to two different boundary conditions. Let’s assume a sample saturated with oil and water and placed in water.

1. if oil is produced through the face as observed like in Figure 37, in large drops or films form. Capillary pressure is inversely proportional to the radius of curvature (equation (45)). Then the pressure difference between oil and water is small compared to the capillary pressure inside the medium where the characteristic pore size is smaller. Because the fluids are continuous, the capillary pressure inside the sample is also very small at the outlet (Figure 38. This leads to a boundary condition on the capillary pressure: at the outlet Pc<<1 (usually assumed to be zero). The condition is not only on the capillary pressure, because it depends on the fluids saturation (Figure 31) then it also imposes a saturation value at the outlet.

Figure 37 shows a water-wet sample saturated with oil (red liquid) placed in a container filled with water. On the free surface, water penetrates the medium under capillary forces (water-wet sample) and oil is pushed outside. At the surface, the difference of pressure between oil and water is very low since the radius of curvature is large compared to the pore radius inside the medium (Figure 38).

Figure 37 - Example of free surface boundary condition. A sample saturated with oil in water

(courtesy IFP).

R

Poil

Pw

R

Poil

Pw

Figure 38 – The two fluids are continuous on the free surface and pc at the boundary of the sample is equal to the pressure in the drop, close to 0 (Laplace’s law)

• Oil is not produced: there is no interface between the two fluids outside the sample, and the capillary pressure at the boundary is unknown. In the example of water filling the outlet, the first condition is the pressure of water being equal to the outlet pressure. The second condition is a zero oil flow rate. This case can be found: either at the end of an imbibition, when oil is no longer produced; or at the beginning of a drainage, before oil breakthrough.



Experiments in Core Analysis Different experimental set-ups are used to determine the properties used to described two-phase flow, both in laboratory and reservoirs: capillary Pressure curves , wettability index and relative permeabilities. In this chapter, we will describe the main experiments and the specific boundary conditions.

Spontaneous displacement

Spontaneous displacements, either imbibition or drainage, are mainly used to determine the Amott wettability index.

It is a simple setup in one dimension, with or without gravity: a sample, partially saturated with a fluid, is placed in a container with a second fluid.

Figure 39 – Spontaneous imbibition of a sample saturated with oil (red) immerged in water.

The fluids are displaced by capillary forces and flows take place in all the directions if all the faces of the sample are open (Figure 39). However, spontaneous displacements are usually modeled as one-dimensional displacements. The sample has only one face (outlet) in contact with the second liquid, the rest is closed. Due to the capillary forces the fluid surrounding the sample may penetrate it. If it is the reference fluid we have a spontaneous imbibition ( increases), otherwise it is a spontaneous drainage ( increases). rS nS

Boundary conditions:



zz

Figure 40 – Model for one-dimensional spontaneous displacement without gravity

effects.

The sample is placed in a cover allowing only one face (outlet) to be in contact with the surrounding fluid (Figure 40):

• Inlet: no flow for both fluids : and

rQ =0

nQ =0• Outlet : free surface with the two

possible conditions:

after breakthrough

0c n rP P = P = P= ⇒ out

before breakthrough:

Imbibition: and 0

Drainage: and 0r out n

n out r

P P Q

P P Q

= =

= =

Porous Plate

Porous plate (Figure 41) is a direct measurement of the capillary pressure curve. At the inlet and the outlet a semi-permeable membrane to one of the fluids is placed. Then for each face only one of the two fluids can flow trough, with their pressure fixed.

At the equilibrium, pressures and saturations are uniformly distributed. This leads to the equality between the macroscopic properties measured at the two faces and the local ones. Then, the mean saturation is determined from the effluent volumes and the capillary pressure directly from the difference between the pressures of the fluids.

A B

water

oil

Semi-permeable

plates

water

oilA B

P

Figure 41: Principle of the determination of capillary pressure by the porous plate method.

Due to the equilibrium, the local capillary pressure curve is directly obtained by varying the fluid pressures: at each equilibrium step a Pc corresponds to a volume of reference fluid, i.e. to a given saturation.

Boundary conditions



n

t

n

t

Let’s take as example a drainage. A non-reference fluid is injected by pressure steps at the inlet through a membrane permeable to this fluid. The reference fluid is produced at the outlet through a membrane permeable to it. Pressures are imposed at the outside extremities (after the membranes). By assuming no pressure drop trough the membrane, we have for a drainage:

• inlet: no flow of reference fluid ,rQ = 0 n iP = P .

• outlet: no flow of non-reference fluid , nQ = 0 r ouP = P

In imbibition the fluids are interchanged: • inlet: no flow of non-reference fluid , . nQ = 0 r iP = P

• outlet: no flow of reference fluid , . rQ = 0 n ouP = P

Pressure drop through the membrane can also be accounted by introducing the resistance of the membrane such as R

P R QμΔ = (57)

By considering the membrane as a porous medium of permeability , section k A and thickness ε .(equation (57) combined with Darcy’s law):

RA k

ε= (58)

Injection of one fluid

This is the standard experiment to determine relative permeabilities using transient flow. The typical setup is shown Figure 42. A fluid is injected into the sample initially saturated with the other one. The effluent volumes and pressure drop are measured, and in some cases the saturation profiles along the sample using X-ray or gamma ray.

Figure 42: Unsteady-state experiment.

Boundary condition:

Only one fluid is injected, either at constant rate or constant pressure. The injected fluid is noted 1 and the displaced one is noted 2. The boundary conditions are:

• inlet : either constant rate or constant pressure for the injected fluid. The flow of the displaced fluid is zero:

1 1

2

Injected fluid: or

Displaced fluid: 0in inP P Q Q

Q

= =

=



• outlet: free surface with the displaced fluid. With the injected fluid two cases have

to be considered: before breakthrough (BT) the flow rate of the injected fluid is zero, and after breakthrough both fluids are produced and Pc=0. In both cases the pressure in the displaced fluid, always continuous, is equal to the outlet pressure:

2

1

1 2

Displaced fluid:

Injected fluid before BT: 0

after BT: 0

out

out

P P

Q

Pc P P P

=

== ⇒ = =

Simultaneous injection of two fluids

The two fluids are injected simultaneously using a typical setup shown Figure 43.


Let’s consider an imbibition: a sample is saturated with the non-reference fluid. Then both non-reference and reference fluids are injected. Until the breakthrough only the non-reference fluid is produced. The boundary conditions are then:

• inlet : constant flow rates for both fluids

, ,r r in n n inQ Q Q Q= =

• outlet: free surface with both fluids. Before breakthrough and after breakthrough . In both cases the pressure in non-reference fluid, always continuous, is equal to the outlet pressure.

rQ = 0Pc=0

before BT: 0

after BT: 0

n out

i

r n

P P

Q

Pc P P

=

== ⇒ =

Figure 43 - Steady-state experiment.



Gravity displacement

The sample, saturated with two fluids at given saturations, is placed vertically in one of the fluids. Only outlet and inlet are not closed. Due to gravity forces (difference of densities) the surrounding fluid penetrates the porous medium. In a case of an imbibition the sample is placed into the densest fluid, and into the less dense for a drainage, see Figure 44.

Figure 44 - Gravity displacement, fluid 1 is denser than fluid 2.


In the example of a gravity drainage with oil and water, the sample is placed in oil. At the bottom, there can be either a surface in contact with oil or water. The mechanism is simple: water, heavier than oil, will drain at the bottom of the sample. The general set of boundary conditions is two free surfaces. As specified above, free surface conditions can lead to spontaneous displacements. Then in addition to the flow driven by gravity, spontaneous processes may happen, even "counter-current" flow. The basics conditions without any spontaneous displacements are:

• Inlet: imposed pressure in the surrounding fluid at the entrance, and Q=0 for the other fluid (for gravity drainage, pressure in oil, and ). wQ =0

• Outlet: free surface at the exit with the two possibilities as described above (for gravity drainage, the exit face is assumed to be filled with water).

Centrifuge

A centrifuge experiment is very similar to a gravity experiment except that the volume

forces ρg (the density by the gravity constant) are replaced by ρω2r , where r is the distance from the centrifuge axis and ω is the rotational speed. In the standard experiment, the sample is completely immerged in one fluid in the core holder (non-reference fluid for drainage, reference fluid for imbibition). Like in gravity experiments, the general two conditions are two free surfaces. Then spontaneous displacement could be considered. However, spontaneous displacements are generally performed (and measured) outside the centrifuge: sample preparation. The option "centrifuge" of CYDAR verifies that the spontaneous displacement has been performed by calculating the initial capillary pressure along the sample.



Figure 45: Imbibition, the fluid 1 is the non-reference phase.


The boundary conditions are the followings, in the case of oil-water imbibition: • inlet: pressure imposed in oil Po (calculated using the centrifuge forces in the

surrounding oil assumed at equilibrium), no flow rate in water (no spontaneous imbibition since the core is at equilibrium saturation, see above)

• outlet: free boundary in water ( ), with additional condition of no counter current flow.

o wP =P

There are still some discussion in the profession about the exit boundary condition. In some special experiments, the outlet can be immerged in oil. In this case, spontaneous drainage is possible since an oil wet sample is placed in oil, and a counter flow can be observed. To our knowledge, there has been no experimental evidence of this counter flow in the literature.

Semi-dynamique Method

In this experiment, designed to measure capillary pressure and relative permeability at reservoir conditions (Lenormand, Eisenzimmer et al. 1995), one fluid is injected and the other one "washes" the outlet face. Let’s consider a drainage (Figure 51):

• the outlet is washed by the reference fluid at a given pressure Pr; • the non-reference fluid is injected at a flow rate ; nQ

• the pressure of the non-reference fluid is measured at the entrance nP ;

• the local saturation is measured at the entrance rS .



Figure 46: Principle of the Semi-dynamic method.


Pc>0

The positive Pc curves (both drainage and imbibition), are obtained by injecting the non-reference fluid at increasing then decreasing flow rates, and the reference fluid is injected to wash the outlet face. The boundary conditions are then:

• Inlet: two flow rates at entrance, imposed by the pump and nQ rQ = 0

• Outlet:

free surface in reference fluid after breakthrough,

zero flow rate of the injected fluid before breakthrough.

Pc<0

The negative Pc curves (both drainage and imbibition), are obtained by injecting the reference fluid at increasing then decreasing flow rates, and the non-reference fluid is injected to wash the outlet face. The boundary conditions are then:

• Inlet: two flow rates at entrance, imposed by the pump and rQ nQ = 0• Outlet: free surface in non-reference fluid.

Summary of the boundary conditions

Tableau 2: Types of boundary conditions depending on the experiments: Q on a flow rate, P on a pressure.



Experiments Inlet Outlet

Spontaneous 2 Q Free surface

Porous plate 1 P + 1 Q 1 P + 1 Q

Kr Unsteady State 2 Q or 1 Q + 1 P Free surface

Kr Steady State 2 Q Free surface

Gravity Free surface Free surface

Centrifuge Free surface Free surface

Semi-Dynamic 2 Q Free surface

The different experiments will be described in the following chapters devoted to Kr and Pc measurements.

Kr and Pc Measurements

Pc measurements Porous Plate

Porous plate (Figure 41) is a direct measurement of the capillary pressure curve. At the inlet and the outlet a semi-permeable membrane to one of the fluids is placed. Then for each face only one of the two fluids can flow trough, with their pressure fixed.

At the equilibrium, pressures and saturations are uniformly distributed. This leads to the equality between the macroscopic properties measured at the two faces and the local ones. Then, the mean saturation is determined from the effluent volumes and the capillary pressure directly from the difference between the pressures of the fluids.



Figure 47 – Porous plate apparatus using a thin sample and semi permeable membranes instead of porous plates (courtesy IFP)

Centrifugation

The method consists in rotating a core at various angular velocities, ω , and so to perform forced drainage and imbibition. Fluid production is then measured at equilibrium for every rotation speed. The average saturation <S>, derived from the effluent production volume, is linked to the capillary pressure, Pc(r), at equilibrium as follow:

π 2

1( )

core

S SL R

= ∫ Pc dv (59)

with L the core length, R its radius and S(Pc) the local capillary curve. All the difficulty is to inverse equation (59), i.e. to derive S(Pc) when <S(Pc)> is known. There is no exact solution to this problem.

A large amount of experimental and theoretical work has been devoted to this problem and can be found in the publications of the Society of Core Analysis. The conclusion of the



study was that only two methods were valid: the Forbes' method and the inversion based on splines (Chardaire-Riviere, Forbes et al. 1992).

• the Hassler and Brunner method based on an assumption of uniform centrifuge forces, leading to an analytical calculation of the local Pc curve. This solution is not accurate but has a historic interest. In addition, this solution is used as an intermediate for the Forbes' method. In the Hassler Brunner method, the local Pc is given by:

HBd S

S S PdP

< >=< >+ (60)

• the Forbes' method, based on an approximation that minimizes the errors. In drainage, the Forbes' method is a combination of two functions:

β α

2

(1 ) ; 12 2

inForbes

out

b b rS S S b

r

⎛ ⎞⎟⎜ ⎟= + − = −⎜ ⎟⎜ ⎟⎜⎝ ⎠ (61)

α αα

1 1;

1 1 2 1

d S bS S P

dP b

< > − −=< >+ =

+ + −1

(62)

β

ββ β

β ββ

1

1(1 ) 1 2 1

( ) ; 21 1 1theshold

Pthresholdthreshold HBP

P bS S y S y dy

bP

+

+

⎡ ⎤+ +⎢ ⎥= + =⎢ ⎥+ − −⎢ ⎥

−

⎣ ⎦∫ (63)

For imbibition, only the alpha function is considered. • The splines method (Nordtvedt and Kolitvelt 1991). The calculation is based on

analytical integration using the decomposition of the spline function into polynomial.

Figure 48 – Centrifuge with automatic recording of fluid production (courtesy IFP)



•

water saturation1.000.800.600.400.200.00

capi

llary

pre

ssur

e ba

r2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

Local Pc

Experimental points

calculated average curve

water saturation1.000.800.600.400.200.00

capi

llary

pre

ssur

e ba

r2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

Local Pc

Experimental points

calculated average curve

Figure 49 – Example of capillary curve (local) determined from effluent production (experimental points) by the Forbes's method. The comparison between the calculated average

curve and the data points is a verification of the accuracy of the method.

Kr Measurement Detailed information can be found in (Christiansen 2001)

Analytical steady-state method

The steady-state method is based on a simultaneous injection of the two fluids at different flow rates.

For an imbibition (water/oil), the fractional flow of water ( /(w w w of Q Q Q= + ) is increased, step by step, by changing the two flow rates. Generally the total flow rate is kept constant. The first displacement consist in the injection of only oil at , then the flow rate of water is increased when the flow rate of oil is decreased. The final step consists in injecting only water.

Swi



When the effluent production is stabilized (steady-state), the pressure drop is measured. The relative permeability is calculated as if the saturation was uniform along the sample, an assumption valid only if capillary pressure is negligible. With this assumption, the relative permeabilities are also the same everywhere along the sample and can be derived from the difference of pressure between inlet and outlet using diphasic Darcy’s law:

; o ow w

w o

Q K KrQ K Kr P P

A L Aμ μ L

Δ Δ= = (64)

The corresponding saturation is derived from the effluent balance or using in-situ saturation measurements.

This simple analytical method is not rigorous since capillary pressure is not taken into account. A more accurate determination needs a numerical interpretation accounting for capillary effects (see Kr by history matching below).

Kr JBN (or Jones and Roszelle)

This method allows the determination of approximated relative permeability (Kr) from a displacement using analytical equations. The method is know as JBN (Johnson, Bossler et al. 1959), Welge or Jones and Roszelle (Jones and Roszelle 1978). The calculation is based on the very crude assumption that the capillary pressure is negligible.

It is now recognized that this approximation is always wrong, since the range of saturation obtained during a displacement is always the result of capillary and viscous forces. It has been demonstrated that this analytical approach can cause a over-estimation of the trapped residual oil ( ) of more than 10% (Joss Mass from Shell, SCA). The only acceptable method for Kr determination is to use numerical simulations taking into account a measured or estimated capillary pressure.

Sor

Experimental setup

The typical setup is shown Figure 50. A fluid 1 is injected into a sample initially saturated with a fluid 2. The pressure gradient is measured as well as the effluents volumes V1 and V2.

Figure 50: Unsteady-state experiment.

Initial conditions: • We will see that to be able to determinate the relative permeabilities analytically

we must assume initial saturations uniformly distributed.



Assumptions:

• Immiscible and uncompressible fluids • No capillary pressure: P1=P2=Pout. • No gravity • Unidirectional flow along the core axe.

Boundary conditions: • Inlet: the injection flow rate is Q1 at a pressure P1. • Outlet: the total flow rate is Q=Q1+Q2 and P1=P2.

Basic equations

Darcy’s law gives the flow rates as follow:

μ

μ

11

1

22

2

K Kr PQ A

z

K Kr PQ A

z

⎧ ∂⎪⎪ =−⎪⎪ ∂⎪⎨⎪ ∂⎪ =−⎪⎪ ∂⎪⎩

(65)

with A the cross section area of the sample, Kr the relative permeabilities and μ the viscosities. The mass balance equations allow to take into account the saturation variations:

∂ ∂φ∂ ∂∂ ∂φ∂ ∂

1 1

2 2

0

0

S QA

t zS Q

At z

⎧⎪⎪ + =⎪⎪⎪⎨⎪⎪ + =⎪⎪⎪⎩

(66)

φ is the porosity. These two systems of equations allow to calculate the saturation functions and . From equations (65), one may write the total production flow rate:

1S (z,t) 2S (z,t)

∂

μ μ ∂1 2

1 2( )

Kr Kr PQ AK

z=− + (67)

The fractional flow defining by i if Q Q= , , are then only function of the relative permeabilities:

1,2i =

( )μ

μ μ1 2

1 2

ii

i

Kr

fKr Kr

=+

(68)

JBN analytical calculation

Self-similar solution

The principle is to be able to work with a differential equation with one variable instead of a set of equations with z and t. This is what is called the self-similar solution. Then, depending on boundary and initial conditions, we want to work with one variable y which



will be z/V(t), where V(t) is the total injected volume. In the specific case of a constant flow rate y=z/t. By definition V(t):

τ τ0

( ) ( )t

V t Q d=∫ (69)

In the following for simplicity only equations for one phase (fluid 1 for instance) are given, because they are exactly the same for the two fluids. First of all, mass balance equation is rewriting in terms of V instead of t:

∂ ∂ ∂ ∂∂ ∂ ∂ ∂

1 1S S VQ

t V t= = 1S

V (70)

∂ ∂φ∂ ∂

1 110

S QA

V Q z+ = (71)

We can now write it with the fractional flow:

∂ ∂φ∂ ∂

1 1 0S f

AV z

+ = (72)

Let’s do the variable change y=z/V. This is possible because the fractional flow is function of the Kr and then depends only on the saturation. This is true because of the null capillary pressure assumption. We have:

∂ ∂∂ ∂

1 1

1

f df S1z dS z= (73)

and for the saturation:

∂∂∂ ∂∂ ∂

1 1

1 1

1S dS

z V dy

S dS y y dS

V dy V V dy

⎧⎪⎪ =⎪⎪⎪⎨⎪⎪ = =−⎪⎪⎪⎩1

(74)

and then:

φ1 1

1

1(

dS dfA y

V dy dS− + =) 0 (75)

The first term leads to a trivial solution dS1/dy = 0. The self-similar solution is given by:

φ11

1( )

dfS A

dS= y (76)

We obtain the same equation for the fluid 2:

φ22

2( )

dfS A

dS= y (77)



To be relevant the solution must not only verify equations (76) and (77) but boundary conditions must only depend on y. In x=0 (y=0), the fractional flow of the injected fluid must be independent of t, which is verified because only one fluid is injected.

Principle of the calculation

The saturation S(t) and relative permeabilities Kr(t) are calculated at the outlet and the curve Kr(S) is deduced. This implies the production of two fluids, and then the approach is only valid after the breakthrough. The steps are the following:

• Mean saturations calculation in the sample from the production and injection volumes <S1> and <S2>.

• Saturations at the outlet are deduced from <S1> and <S2>. • At the outlet the volumes of effluents give the fractional flow and then the Kr

ratio. • The pressure gradient •P allows calculating the relative permeabilities.

Mean saturation

By definition we have:

φ

1 10 0

1L L

p

AS S dz

V L< >= =∫ ∫ 1S dz (78)

Because fluids are incompressible the mean saturation is simply the initial value plus the injected quantity minus the production. For the fluid 1 a volume V is injected, i.e. a saturation ( the pore volume =A.L ), but nothing for the fluid 2. and are produced, then:

Si

V/Vp Vp 1V 2V

1 1 1

2 2 2

/ /

/P

P

S Si V V V

S Si V V

< >= + −

< >= −PV

(79)

Calculation of the local saturations

By part integration of equation (78) one obtains:

[ ] 1

1

( )1 1 0 (0)

1 1 S LL

SS S z z

L L< >= − 1dS∫ (80)

By using the self-similar solution given by (76), and because V does not depend on S, we have:

[ ] 1

1

( ) 11 1 0 (0) 1

1 S LL

SP

V dfS S z

L V dS< >= − ∫ 1dS (81)

which leads to:

1 1 1( ) ( ) 1P

VS S L f L

V

⎛ ⎞⎟⎜ ⎟< >= − +⎜ ⎟⎜ ⎟⎜⎝ ⎠ (82)

At the inlet, only one fluid is injected, then the fractional flow is one. Equations (79) and (82) with (where ) give: 1 1 V f (L)=V '/V'V V'=dV/dt=Q



'1

1 1 1

'2

2 2 2

( ) ( )/'

( ) ( )/'

P

P

VVS L Si V V

V

VVS L Si V V

V

= + −

= + −

(83)

Kr1 and Kr2 Calculation

The ratio of the fractional flows (equation (68)) gives the relative permeabilities ratio:

μ μμ μ

'1 1 1 1

'2 2 2 2 2

Kr f V

Kr f V= = 1 (84)

Now we need another equation to be able to determinate the Kr. To do so we will calculate the mobility ( )μ μ1 1 2 2M Kr Kr= + from the pressure difference:

∂∂

Δ0

L PP dz

z=∫ (85)

By using equation (67) to express the pressure gradient and the variable change φY=A (x/V) , we obtain:

μ μφ

Δ( )

2 (0) 1 1 2 2

1 '/ /

Y L

Y

V VP

Kr KrA K=

+∫ dY

0

(86)

with and ( )0Y x = = ( ) φ /Y x L A L V= = . The total flow rate V’ and the injected volume V

do not depend on Y, then:

φ

μ μφΔ

/

2 0 1 1 2 2

' 1/ /

A L VV VP

Kr KrA K=

+∫ dY (87)

We can derive it with respect to t, the integration bounds depending on t we have:

φ

μ μφφ

μ μφ

Δ/

2 0 1 1 2 2

21 1 2 2

( ' )/'

/ /

' 1 (/ /

A L Vd V V dt dYP

Kr KrA K

V V d A L V

Kr Kr dtA K

= ++

+

∫ …

… / ) (88)

In equation (88) the integral may be express in terms of PΔ (equation (87)) and by deriving /A L Vφ we have:

μ μ

Δ Δ2 2

1 1 2 2

' '' ''

' ( /V VV V L

P PVV V AK Kr Kr

+= −

+ / ) (89)

and then

μ μΔ Δ

3

1 1 2 2 2'

( / / )( ' '') ' '

L VKr Kr

AK P V VV P VV+ =

+ − (90)



By using equation (68), one may able to calculate Kr1 and Kr2:

μ

μ

Δ

Δ

' 21 1

1 2

' 22 2

2 2

'

( ' '') ' '

'

( ' '') ' '

L V VKr

AK P V VV P VV

L V VKr

AK P V VV P VV

=+ −

=+ −

(91)

To summarize

We obtain two systems of equations one for the saturations and the second for the relative permeabilities: equations (92) and (93). The curves Kr(S) may be deducted by eliminating the time variable.

'1

1 1 1

'2

2 2 2

( ) ( )/'

( ) ( )/'

P

P

VVS t Si V V

V

VVS t Si V V

V

⎧⎪⎪⎪ = + −⎪⎪⎨⎪⎪⎪ = + −⎪⎪⎩

(92)

μ

μ

' 21 1

1 2

' 22 2

2 2

'( )

( ' '') ' '

'( )

( ' '') ' '

L V VKr t

AK P V VV P VV

L V VKr t

AK P V VV P VV

⎧⎪⎪⎪ =⎪⎪ + −⎪⎨⎪⎪⎪ =⎪⎪ + −⎪⎩

(93)

Here is the pore volume, are the effluent volumes and the total volume of

injected fluid, P is the pressure difference between the two extremities of the sample. The derivatives with respect to t are noted ′ and ′′. It is possible to simplify these equations in two cases: a constant flow rate and a constant pressure.

pV 1,2V V

Constant flow rate

Here the first derivative of the volume is constant, and , then we may rewrite equations (92)and (93) as follow:

V'=Q V=Qt

(94) '

1 1 1 1'

2 2 2 2

( )/

( )/

P

P

S Si tV V V

S Si tV V V

⎧⎪ = + −⎪⎪⎨⎪ = + −⎪⎪⎩

μ

μ

Δ Δ

Δ Δ

'1 1

1

'2 2

2

'

'

L VKr

AK P t P

L VKr

AK P t P

⎧⎪⎪⎪ =⎪ −⎪⎪⎨⎪⎪⎪ =⎪⎪ −⎪⎩

(95)

Constant PΔ

The relative permeabilities may be simplified because : P'=0



'1

1 1 1

'2

2 2 2

( ) ( )/'

( ) ( )/'

P

P

VVS L Si V V

V

VVS L Si V V

V

⎧⎪⎪⎪ = + −⎪⎪⎨⎪⎪⎪ = + −⎪⎪⎩

(96)

μ

μ

' 21 1

1 2

' 22 2

2 2

'

( ' '')

'

( ' '')

L V VKr

AK P V VV

L V VKr

AK P V VV

⎧⎪⎪⎪ =⎪⎪ +⎪⎨⎪⎪⎪ =⎪⎪ +⎪⎩

(97)

In order to obtain smooth relative permeability curves, it is recommended to fit first the experimental data for pressure drop and fluid production. However, since the calculation derives both Kr and saturation as functions of time, it is possible that the relative permeabilities are not single value functions of saturation. To avoid this problem, it is recommended to fit and smooth also the intermediate function of time Kr(t) and Sat(t), using monotonic functions, a method equivalent to the well known "Jones and Roszelle" calculation.

This simple analytical method is not rigorous since capillary pressure is not taken into account. A more accurate determination needs a numerical interpretation accounting for capillary effects (see Kr by history matching below).

Kr by history matching

Both analytical steady-state and JBN use analytical calculations but assume that capillary pressure is negligible. There is no analytical ways to account for Pc and accurate calculations require numerical simulations. The principle is the following

• Relative permeabilities are fitted with an analytical law (Corey, Splines, …) controlled by a few numbers of parameters

• A first guess of Kr is chosen to start the simulation. The first guess can be the results of the analytical calculation (Analytical steady-state or JBN).

• The simulated effluent production and pressure difference are compared to the experimental values and the difference is quantified by calculating an objective function (generally least square differences)

• The parameters of the Kr functions are adjusted in order to minimize the objective function. The adjustment can be made manually by try and error or using an automatic optimization program.

It has been demonstrated, that this history matching can lead to final residual oil saturations ( ) smaller than the analytical results, especially for oil wet samples (paper by Joss Mass, SCA).

Sor



r

Pc and Kr by SDM In this experiment, designed to measure capillary pressure and relative permeability at reservoir conditions (Lenormand, Eisenzimmer and Ph 1995), one fluid is injected and the other one "washes" the outlet face. Let’s consider a drainage (Figure 51):

• the outlet is washed by the reference fluid at a given pressure ; rP

• the non-reference fluid is injected at a flow rate ; nQ

• the pressure of the non-reference fluid is measured at the entrance ; nP

• the local saturation is measured at the entrance . rS

Figure 51: Principle of the Semi-dynamic method.

The flow rate of the non reference fluid is increased by steps. At equilibrium the reference fluid will be at the uniform pressure Pr (since it is no flowing). Then the capillary pressure is directly given by . These give at the entrance. Ideally the is measured at the inlet (by X ray for example), otherwise it can be derived from the average saturation as follow:

c in out nP P P P P= − = − c rP (S ) rS

r<S >

( ) nr n n n

n

d SS P S Q

dQ

< >=< >+ (98)

The relative permeability is derived using the Darcy’s law:

μn n

nn

L dQKr

KA dP= (99)

The above example is at positive Pc because the reference fluid’s pressure is fixed at the outlet and will be lower than the pressure at the inlet, other there would be no flow. The drainage is obtained by increasing the injection flow rate, the imbibition by decreasing it.

Now if the reference fluid is injected with the same setup, the method is exactly the same but the capillary pressure will be negative because . The imbibition is done by increasing the injection flow rate, and the drainage by decreasing it (Lombard, Egermann et al. 2004).

out n r inP P P P= < =



Figure 52 – Semi-dynamic method for Pc and Kr determination at reservoir condition (courtesy IFP)



References

1 Darcy, H. (1856). Les fontaines publiques de la ville de Dijon, exposition et application des principes à suivre et des formules à employer dans les questions de distribution d'eau. Paris, Victor Dalmont.

2 Forchheimer, P. (1901). "Wasserbewegung durch boden." Zeits. V. Deutsh. Ing. 45: 1782-88.

3 Klinkenberg, L. (1941). "The permeability of porous media to liquids and gases." Drilling and Production Practice: 200-213.

4 Purcell, W. R. (1949). "Capillary pressures, their measurement using mercury and the calculation of permeability therefrom." Trans AIME(February): 39-48.

5 Johnson, E. F., D. P. Bossler and V.O.Naumann (1959). "Calculation of relative permeability from displacement experiments." Journal of Petroleum Technology(January): 61-63.

6 Jones, S. C. and W. O. Roszelle (1978). "Graphical techniques for determining relative permeability from displement experiments." Journal of petroleum technology(may ): 807-817.

7 Nordtvedt, J. E. and K. Kolitvelt (1991). "Capillary pressure curves from centrifuge data by use of splie functions." SPE Reservoir Engineering(November): 497-501.

8 Chardaire-Riviere, C., P. Forbes, J. F. Zhang, G. Chavent and R. Lenormand (1992). Improving Centrifuge Technique by Measuring Local Saturations. SPE 24882, Annual Technical Conference and Exhibition of the Society of Petroleum Engineers, Washington, USA.

9 Dullien, F. A. L. (1992). Porous media. Porous structure and fluid transport. San Diego, Academic Press Limited, inc.

10 Lenormand, R., A. Eisenzimmer and D. Ph (1995). Improvements of the Semi-Dynamic Method for Capillary Pressure Measurements. paper 9531, Society of Core Analysis, San Francisco, USA.

11 Christiansen, R. L. (2001). Two-phase flow through porous media. Golden, USA, Colarado School of Mines.

12 Lenormand, R. (2003). Interpretation of mercury injection curves to derive pore size distribution. SCA 2003-52, Society of Core Analysts, Pau, France.

13 Lombard, J. M., P. Egermann, R. Lenormand, S. Bekri, M. Hajizadeh, K. H. Hafez, A. Modavi and M. Z. Kalam (2004). Heterogeneity study through representative capillary pressure measurements. Impact on reservoir simulation and field predictions. SCA 2004-32, Society of Core Analysts, Abu Dhabi, U.E.A.


cydar theory overview exp 5

Documents

mercury pore size distribution

pore size distribution

darcys law historic

local darcys law

lenormand conventional

mercury imbibition

experimental principles

experimental data