DANMARKS OG GR0NLANDS GEOLOGISKE UNDERS0GELSE RAPPORT 1998/30

Determination of Saturation Functions andWettability for Chalk based on

Measured Fluid Saturations

Final Report for EFP-96 Project J.nr. 1313/96-0005

Dan Olsen, Niels Bech and Carsten Moller Nielsen

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GEOLOGICAL SURVEY OF DENMARK AND GREENLAND MINISTRY OF ENVIRONMENT AND ENERGY

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G E U S

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DANMARKS OG G RON LAN DS GEOLOGISKE UNDERS0GELSE RAPPORT 1998/30

Determination of Saturation Functions andWettability for Chalk based on

Measured Fluid Saturations

Final Report for EFP-96 Project J.nr. 1313/96-0005

Dan Olsen, Niels Bech and Carsten Mailer Nielsen

oG E U S

GEOLOGICAL SURVEY OF DENMARK AND GREENLAND MINISTRY OF ENVIRONMENT AND ENERGY

SUMMARY 5

INTRODUCTION 7

EXPERIMENTAL TECHNIQUE 9

Sample material 9

Fluid Data 9

Flooding experiments 9Step 1 Threshold pressure experiment 10Step 2 Transient flow drainage experiment 10Step 3 Stationary flow drainage experiment 11Step 4 Stationary flow internal imbibition experiment 12Step 5 Forced flow imbibition experiment 12

Experiment extensions 12NMR monitoring of transient flow drainage experiment 12Pump shutdown experiment 13

PARAMETER ESTIMATION TECHNIQUE 15

General Equations 15Drainage Case 17Imbibition Case I 19Imbibition case II 20Pump shutdown experiment 20

Test of the parameter estimation technique - model M15A 21Relative Permeabilities 21Capillary Pressure 22Drainage 25Imbibition 29

NMR MEASUREMENTS 31

Hardware 31

Pulse sequence 31

Data manipulation 32Step 1 Frequency dimension elimination. 33Step 2 Relaxation correction 34Step 3 Proton density correction 36Step 4 Calculation of saturation 36

GEUS 3

Signal processing 37

Limitations and accuracy of NMR method 37

RESULTS AND DISCUSSION 39

Model M15A 39

Application to core plug experiments 39

Plug M113 40Data 40Primary drainage 40Imbibition 41

Plug M16H 46Data 46Primary drainage 46

Plug M15M 50NMR monitoring of transient flow drainage experiment 50

CONCLUSIONS 53

NOMENCLATURE 55

REFERENCES 57

APPENDIX A 61

APPENDIX B 77

4 GEUS

Summary

A new procedure for obtaining saturation functions, i.e. capillary pressure and relative permeability, of tight core samples uses the pronounced end effect present in flooding experiments on such material. Commonly being a nuisance in core analysis, the end effect contains valuable information about the saturation functions. In core material with high capillary pressure, the end effect may allow determination of the saturation functions for a broad saturation interval.

A complex core flooding scheme provides the fluid distributions and production data necessary for the calculation. An NMR technique is used for fluid distribution determi­nation. A non-magnetic coreholder connected to a mobile flooding unit allows NMR measurement at selected times during the flooding scheme. Fluid distributions are usually measured in situations with stable flow, but the NMR technique allows meas­urement during transient flow.

The steady state situation at the end of an initial primary drainage experiment allows calculation of the drainage capillary pressure and the drainage relative oil permeability. The relative water permeability is calculated from the unsteady state data obtained during the transient part of this experiment. After a flow reversal, a new end effect de­velops in the opposite end of the core by an internal imbibition process, which at steady state allows calculation of the spontaneous imbibition capillary pressure and the imbibition relative oil permeability. Following a change from oil flooding to water flood­ing, the forced imbibition capillary pressure is calculated from transient pressure drop measurements. An undesirable interdependency of the saturation functions is avoided by their calculation from different data sets. Killough’s method is employed to account for the scanning effect in hysteresis situations for both capillary pressure and relative permeability. The procedure allows estimation of the scanning parameters. The satu­ration functions are determined by a least squares technique.

The procedure is demonstrated on chalk samples from the North Sea. The experimen­tal time is intermediate between the centrifuge and porous plate methods. The proce­dure is superior to the centrifuge and mercury injection methods respectively by elimi­nating errors from end effects and using reservoir relevant fluids.

Four capillary pressure determinations by the mercury injection method yields consis­tently lower capillary pressure relative to the new method. This is similar to previous findings, and is tentatively ascribed to problems with the mercury injection method. One determination of the forced imbibition capillary pressure by the centrifuge method agrees with the determination by the new method. The centrifuge method appears to be unable to measure the drainage and spontaneous imbibition capillary pressure on tight chalk samples.

GEUS 5

G EU S

Introduction

The project “Determination of saturation functions and wettability for chalk based on meas­ured fluid saturations in core samples” is financed by the EFP-96 research programme of the Danish Ministry of Environment and Energy, grant no. 1313/96-0005. The project is partly based on work done in the project “Rock parameters” (EFP-93 grant no. 1313/93- 0014).

The project is a collaboration between the Institute of Energetics (ET) at the Technical Uni­versity of Denmark (DTU) and the Geological Survey of Denmark and Greenland (GEUS). The duration of the project is from January 1st, 1996 to December 31st, 1997. During 1996 ET served as project coordinator. As of January 1st, 1997, the task as project coordinator was switched to GEUS due to changes in staff at ET. The following staff has participated in the project.

Jan Reffstrup ET until December 31st, 1996Jens Vinther Norgaard ET until July 31st, 1996Carsten Mo Her Nielsen ET from July 1st, 1996Jens Kjell Larsen ET from July 1st, 1996 to March 31st, 1997Dan Olsen GEUSNiels Bech GEUS from December 1 st, 1996

As of March 31st, 1998, the project has resulted in one international paper with peer review (Nielsen et a/., 1997), which was presented at the 1997 International Symposium of the Society of Core Analysts, Calgary, Canada, September 8-10, 1997. This paper presents results obtained during the first part of the project. An abstract for a paper, intended to de­scribe results for the full project period has been accepted by the organising committee for the 73rd SPE Annual Technical Conference and Exhibition, New Orleans, U.S.A., Septem­ber 27-30, 1998. The paper and the abstract are included in this report as Appendix A and B.

Petrophysical parameters are an integrated part of every simulation study concerning dis­placement processes in porous media. A prerequisite for meaningful modelling is reliable petrophysical data. Simulations and models rapidly evolve in the direction of more details and greater precision, and the underlying petrophysical data should evolve accordingly. Several workers have presented methods utilising core flooding techniques in combination with com­puter simulations to determine saturation functions of rock samples (Honarpour et at., 1995, Kohhedee, 1994, Chardaire-Riviere et ai, 1992). The aim of the present project is to de­velop an experimental method, which is particularly suitable for determination of the satu­ration functions capillary pressure and relative permeability for samples of chalk. Inherently chalk has high capillary pressure, and laboratory determinations of saturation functions on such material are complicated by strong scale effects, particularly by the end effect. This capillary retention of the wetting phase in displacement experiments results in a saturation gradient through the core sample. Although being a problem in many traditional core analyses, the end effect actually contains detailed information about the saturation func­tions of the sample, and can cover a large saturation interval. The idea of the work is to

GEUS 7

make use of this information to calculate relative permeability functions and capillary pres­sure curves.

Conventional methods for determination of saturation functions are mainly developed for rocks of lower capillary pressure and higher mechanical strength than chalk, and their ap­plication to chalk may be questionable. The mercury injection method for capillary pressure determination uses a fluid system with wetting characteristics, contact angle and surface tension much different from the oil-water fluid system of an oil reservoir. The centrifuge method for determination of capillary pressure is largely unsuitable due to the low me­chanical strength of chalk. In addition no general agreement exists on the flow equations for deriving capillary pressure from centrifuge data, cf. Hassler & Brunner (1945), Chris­tensen (1992) and Forbes etal. (1994). The porous plate method for determination of capil­lary pressure appears to be valid, but is extremely time consuming. The steady state and unsteady state methods for determination of relative permeability are experimentally viable, but must be coupled to computational procedures to account for the saturation gradients caused by capillary pressure. Classical Buckley-Leverett theory does not account for capil­lary pressure.

The method to be presented in this report is based on a parameter estimation technique, utilising saturation data obtained from a complex experimental core flooding procedure. It uses the prominent end effect, that is caused by the strong capillary forces of rocks with low permeability. It is based on the method of Norgaard et al. (1995), which uses the end effect for numerical simulation of capillary pressure functions for a primary drainage proc­ess, but is improved and extended to allow calculation of both drainage and imbibition saturation functions. The method assumes capillary continuity of the fluid phases, and ho­mogeneity of the sample with respect to the saturation functions.

The method depends on experimental data on the distribution of fluids during flooding ex­periments. A 1D NMR imaging technique is used for quantification of the fluid distribution (Olsen etal., 1996), and the technique is described in some detail. Nevertheless, the com­putational method is not dependent on a specific technique for quantifying the fluid distri­bution. Gamma ray attenuation (Nichols & Heaviside, 1985), X-ray CT scanning (Hicks, 1996, Gall, 1993, Chardaire-Riviere et al., 1992) or microwave scanning techniques (Honarpour etal., 1995) may be suitable. It should, however, be noted that the gamma ray attenuation and X-ray CT techniques require the core to be measured three times, while satu­rated with different fluid systems, resulting in problems with sample repositioning and instru­mental drift. Doping with reservoir incompatible chemicals may also be necessary.

8 G E U S

Experimental technique

Sample materialThe sample material used in the experimental work is chalk from the Danish sector of the North Sea. Results are presented for a total of 3 chalk samples. They are all cylindrical 1.5 inch plug samples, with lengths between 7.3 and 7.5 cm. To minimize effects of sample inhomogeneity on the flooding experiments, the samples were drilled with the cylinder axis parallel to any visible bedding or layering. The NMR method used for determination of fluid saturation is dependant on the spin-spin relaxation constant, T2, of the samples being above approximately 8 ms. Chalk of maastrichtian age from the Dan oil field is known to comply to this requirement (Olsen, 1997), and therefore the sample material for the project was taken from drill cores of maastrichtian age from the Dan field. The samples were cleaned by Soxhlet extraction with methanol and toluene prior to the flooding experiments. The samples are strongly water wet. Basic parameters for the samples are given in Ta­ble 1.

Fluid DataA synthetic formation brine with a salinity of 7% was used for all flooding experiments (Table 2). The brine composition is similar to formation water from the Dan field. Two types of oil were used during the project, n-decane and n-pentadecane. Basic parameters for the fluids are given in Table 3. Neither brine nor oil was doped with other chemicals. A prereq­uisite for the parameter estimation technique and the NMR measurements, both to be de­scribed later, is that the fluid system can be described as a two-phase system. In all ex­perimental work it is thus essential to avoid the presence of a free gas phase. This was in part accomplished by careful degassing of all experimental fluids.

Flooding experimentsInformation for calculation of both drainage and imbibition capillary pressure functions, as

Table 1. Basic parameters for samples.Sample id. M113 M16H M15MLength (cm) 7.5 7.4 7.3Diameter (cm) 3.8 3.8 3.8Kw (mD) 0.70 1.1 2.1*(%) 26.4 33.3 37.5T] (ms) @ 5^=100 500 450T2 (ms) @ 5^=100 Oil type n-decane

10.6n-decane n-pentadecane

GEUS 9

Table 3. Fluid parameters.Table 2. Brine com­position.

Cone, (g/l)NaCICaCb

66.866.99

Viscosity(CP)@ 21 °C

Density(g/ml)@ 21 °C

Proton density rel. to water@ 21 °C

Brine 1.12 1.048 0.9759n-decane 0.92 0.730 1.0185n-pentadecane 2.77 0.767 1.0454

well as drainage and imbibition relative permeability functions are obtained from a complex core flooding procedure. The procedure consists of a sequence of 5 flooding experiments as follows

Step 1 Threshold pressure experiment.Step 2 Transient flow drainage experiment.Step 3 Stationary flow drainage experiment.Step 4 Stationary flow internal imbibition experiment.Step 5 Forced flow imbibition experiment.

The experiments are described below. Please refer to the chapter Parameter Estimation Technique for a description of the theoretical and computational basis. Flooding rates of 5 to 30 ml/h were used in the experiments.

Step 1 Threshold pressure experiment

The sample is saturated with brine, and is mounted in a Hassler type core holder. The ab­solute permeability to brine, Kw, is measured during a period of stable one-phase flow. A drainage experiment is set up with very slow linear upramping of the inlet oil pressure, while ApT and displaced fluid volume are recorded. A plot of Ap versus displaced fluid volume shows a distinct break at the point when the oil phase enters the sample. The inlet oil pressure at this point is the threshold pressure pth.

Step 2 Transient flow drainage experiment

After the threshold pressure experiment the sample is cleaned for oil and brine, and is re­saturated with brine. The sample is mounted in a core holder, and a transient flow drainage experiment is conducted at constant rate q0, with monitoring of Sw mecm and ApT as function

of time. The data is later used for calculation of . This experiment is similar to a conven­

tional unsteady state Buckley-Leverett experiment, but the use of the resulting data set is different. A non-magnetic core holder connected to a Mobile Flooding Unit (MFU, fig. 1) is used to allow NMR measurement during the following experimental steps. In the case of sample M15M the transient flow drainage experiment was monitored by NMR, cf. section Experiment extensions.

10 GEUS

pressure indicator^>ruc^ sample holder

(acrylic plastic)

sample:

injection pump (Pharmacia P-500)

confining pressure

storagetubing

manometer

reduction valve outlet: 0-7 bar(g)

Nitrogen

manometerreduction valve outlet: 15 bar(g)

Figure 1. Sketch of Mobile Flooding Unit (MFU).

Step 3 Stationary flow drainage experiment

During the transient flow drainage experiment the flow is sustained at controlled temperature and flow rate until a stable ApT is obtained, and until water production has ceased. This may take several weeks. At this time a stable fluid saturation gradient is established in the sample, and the stationary flow drainage experiment is conducted, with monitoring of Sw as function

of position in the profile, together with Apr. The data set is used for calculation of kdr° and

pf . This experiment is equivalent to the experiment developed by Norgaard et a!. (1995),

but by inclusion of the threshold pressure, pth, measured in experiment step 1, it is possible

to calculate k* together with pf. The fluid saturation profile is measured by NMR. The non­

magnetic core holder, and the MFU allows the measurement to be done while sustaining the fluid flow through the sample. A high degree of flow stability during the period of flow stabili­sation has proved to be crucial. It is found that even small and shortlived perturbations in flow

GEUS 11

rate may cause a significant change in ApT, probably reflecting a change in the fluid satura­

tion distribution close to the sample outlet.

Step 4 Stationary flow internal imbibition experiment

The oil flow direction is then reversed and the water contained in the end effect is dis­placed through the sample towards the former inlet, where a new end effect evolves due to the capillary retention of the water. The flow rate must be large enough to create a water production from the sample. If the flow rate is insufficient to cause water production, the water saturation measured in the downstream part of the sample may be controlled by the availability of water rather than by the capillary pressure of the sample, leading to errone­ous calculation of p^pt and dependent parameters. The stationary flow internal imbibition

data set is measured, when ApT is stable, and water production has ceased. Again, NMR is

used to monitor Sw as function of position in the profile, together with ApT. The data is used

for calculation of k™b and p'cmbpt, together with the capillary pressure scanning parameter e .

Step 5 Forced flow imbibition experiment

In the last experiment step a forced imbibition process is produced in the sample by flooding the sample with brine. During the transient flooding period ApT as function of time is

measured. The data set is used to calculate p™bforced.

Experiment extensions

NMR monitoring of transient flow drainage experiment

A transient flow drainage experiment on sample M15M was successfully monitored by NMR saturation profiling. NMR saturation profiles were acquired from the start of the ex­periment to some time after breakthrough. During 5 hours a total of 45 saturation profiles were acquired. Various setups were tested, and a setup yielding a saturation profile in 6 minutes proved useful. At an oil flooding rate of 5 ml/h the oil front moved through the sample at a rate of 0.3 mm/minute. During the acquisition of a saturation profile the oil front thus moved 1.8 mm, creating some noise in the profile. Nonetheless, the experiment pro­vided a quite detailed description of the flooding process, Fig. 20. Data sets of this type undoubtedly contain valuable information about the saturation functions of the sample, and they fit into the general calculation scheme described in the chapter Parameter estimation technique. However, it was outside the time limit of the project to include the transient NMR profiles in the parameter estimation technique.

12 GEUS

Pump shutdown experiment

In an experiment extension, named the pump shutdown experiment, which was conducted immediately after the stationary flow drainage experiment (experiment step 3), the dis­placement pump was simply stopped. After this the oil and water pressures become equal in the inlet and outlet chambers. In the core sample, however, the water pressure will be lower due to the capillary pressure. The water will remain inside the sample, and will redis­tribute until a stationary situation is reached. In this steady state situation neither the water nor the oil are flowing, and the pressures in both phases are constant throughout the sam­ple. The difference between the phase pressures equals the capillary pressure, which is also constant through the core sample. The pump shutdown experiment allows determina­tion of the scanning parameters , and the spontaneous part of the imbibition capillary pres­sure curve. However, the same information is gained from the stationary flow internal im­bibition experiment, and the pump shutdown experiment is thus redundant.

GEUS 13

G E US

Parameter estimation technique

The experimental data used for the determination of the saturation function, are the pro­duction data and saturation profiles obtained from the complex flooding procedure. The saturation functions are determined as analytical functions of the saturation.

General EquationsConsider the one-dimensional flow of oil and water in a horizontal core sample of length Land cross sectional area A. The following assumptions are made:

1. The cross sectional area A is constant.2. The absolute permeability K is constant.3. The oil is dead.4. There are no internal sources or sinks.5. The temperature is constant.

The mass conservation equations for oil and water are written

d{A4> PA} [ 3{p0q0} _(D

dr dx

d{A<t> | d{pwqw](2)

dr dx

The fluid flow is described by one-dimensional Darcy equations

AKkr0 dp0 \i0 dx

(3)

(4)

In a situation where water is the wetting phase and oil the non-wetting phase, the capillary pressure is defined by

(5)Pc =Po~ Pw

The saturation functions are determined as analytical function of the water saturation

K = f,{s,) (6)

G BUS 15

^nv =/w(Sw) (7)

Pc - /c {Sw ) (®)

The following constraint and relations apply

^+^,=1 (9)

Pj=Pj{Pj)’j = °’w (10)

§=§{pJ)>j = o,w (11)

The fluid densities and matrix porosity are functions of pressure.

Mathematically the flooding experiments are defined by initial and boundary conditions, listed below, which is divided into a drainage case and two imbibition cases. Inlet flow rates are constant within each case. The flow direction of imbibition cases I and II is reversed relative to the drainage case. The origin for the length scale is the sample inlet in the drainage case. In flooding situations with reversed flow direction the origin is the sample outlet. The length scale is thus fixed relative to the sample.

1) Drainage case:

(■*’()) “ 1 (12)

(13)

(14)

(15)

2) Imbibition case I. Spontaneous part of the imbibition process:

S„(x,0) = .St_(x) (16)

(17)

(18)

(19)

GEUS16

Here subscript, r°o, is the time where the drainage is finished and steady-state conditions prevail.

3) Imbibition case II. Forced part of the imbibition process:

(20)

p„(*.0)=p“w (21)

(22)

P„(0.*) = P" (23)

Again, subscript r°° is the time where the imbibition case I is finished and steady state conditions prevail.

As for the above initial and boundary conditions, the parameter estimation technique is divided into a drainage and imbibition part, as described below.

Drainage Case

The drainage case initial conditions Eqs. (12-13) correspond to a core plug saturated with water. The plug is drained by injecting oil at a rate of qf (Eq. (14)). The drainage is de­

scribed by the equation system Eqs. (1 - 15). Given enough time, flooding the sample at constant rate results in a steady state situation, where only the displacing fluid is flowing. Eq. (4) shows that the pressure gradient in the displaced water phase is zero. In other words, the water pressure is uniform throughout the sample. If the functional relationships of relative permeability to oil and capillary pressure with respect to water saturation (Eqs. (6) and (8)) are known, then the capillary pressure and the water saturation profile can be calculated from the following equations, assuming constant porosity and fluid densities:

dPc = Vo <lo

dx Kkro A(24)

= Vo q0 i dx Kkro A

(25)

The boundary conditions, at steady state are

Pc(x = 0) = Pd0rin{tca)~ Pt,Jtca) (26)

GEUS 17

where p*Jt°°) = p%0Ut.

pc(x = L) = 0 (27)

Sw(x = 0) = ^W! (28)

To determine the drainage capillary pressure and the drainage relative permeability to oil, data from the stationary flow drainage experiment is used, i.e. step 3 of the experimental procedure. In this experiment the steady state saturation profile of the sample and the oil phase pressure at the inlet is measured. The oil and water phase pressures at the outlet are considered equal to the atmospheric pressure. The functional relationships Eqs.(6) and (8) of the relative oil permeability and the capillary pressure are assumed to be uniquely described by two sets of parameters

fo 0*11 ^2 no ’ Sw ) (29)

Pc($W) = f c (Pi i ^2 ’1 - - ^rtc ’ Sw ) (30)

The unknown parameters ai,i = \,no and bi,i = l,nc are determined from Eqs. (24 -30)

by means of a parameter estimation technique. For a given set of coefficients a, and bi in

Eqs. (29) and (30) the capillary pressure, pc, and the water saturation, Sw, are calculated

from Eqs. (24) and (25) and the boundary conditions Eqs. (26 - 28). The coefficients at and

bl are determined so that they minimize the following least squares objective function

Jx (a, 5)M 2

+ (4f>:r-Ap;r)oo= + (P;,„i=i

+

(31)

The objective function is a sum of squared residuals, which are the differences in the calcu­lated and measured values for the water saturation along the sample, the differences in total pressure, the differences in the capillary threshold pressure and the differences be­tween calculated and measured outlet pressure. The squared residuals are scaled by means of weight factors to, to make their contribution to the objective function similar in

size.

The objective function Eq. (31) is minimized by use of a standard non-linear least square solver (MI0L2, Madsen etal. 1991).

In order to determine the relative permeability to water the transient flow drainage experi­ment is used, i.e. step 2 of the experimental procedure. The relative permeability to water is also represented by a functional relationship of the water saturation:

18 GEUS

(32)MM = /.(The coefficients c, are determined so that they minimize the following least squares objec­tive function, by use of the same non-linear least square solver as above

2N

(33)

The objective function consists of a sum of squared residuals that give the difference be­tween calculated and measured values of the total pressure drop in the oil phase across the sample at times f,, i = 1,N. For a given set of coefficients, ci, the relative permeability

to water is calculated. The relative permeability to oil and the capillary pressure are known from above.

The transient part of the drainage process is described by the equation system Eqs. (1 - 15). The system is solved by means of a commercial reservoir simulator (ECLIPSE 100, 1994). The extraction of results from the ECLIPSE output file is performed by means of software developed by Frandsen (1997). The objective function Eq. (33) is minimized by use of the non-linear least square solver referenced above.

Imbibition Case I

It is assumed in the present work that the water relative permeability exhibits no hysteresis (Brown & Holland, 1995, Killough, 1976). In other words, the relative permeability function determined for the drainage case also applies to the imbibition cases. In order to determine the imbibition part of the saturation functions the experiment is continued in two steps:

1. Experimental step 4. The end effect is shifted to the opposite end of the core (x = 0) by reversing the direction of the oil flow (Confer the initial and boundary conditions Eqs. (16 -19)).

2. Experimental step 5. The plug is flooded by water injected through the face x = L. (Confer the initial and boundary conditions Eqs. (19b - 19e)).

The development of an end effect at the new sample outlet follows an internal imbibition process, while the old end effect is broken down by an internal drainage process. The satu­ration profile at the new sample outlet can be used in the calculation of the saturation func­tions for the imbibition process. Recall that the bounding saturation curves are obtained under the initial condition of Sw= 100% and Sw= Swi for the drainage and imbibition proc­ess respectively. The reversed drainage process violates this condition and scanning curves between the bounding saturation functions will therefore control the process. The scanning process is described by a method developed by Killough (1976). The scanning curves are dependent on the local saturation history of the sample.

GEUS 19

The imbibition case I experiment is described by the equation system Eqs. (1 -11) with the initial and boundary conditions Eqs. (16 -19).

Data from the stationary flow internal imbibition experiment, i.e. experiment step 4, is used to determine the imbibition relative permeability to oil, the spontaneous part of the imbibi­tion capillary pressure and the scanning parameter, e . In this experiment the total pres­sure drop and the saturation profile are measured in the steady state, which follows the reversal of the oil flow direction. The imbibition relative permeability to oil and the imbibition capillary pressure are represented by functional relationships similarly to Eqs. (29) and (30). The coefficients at, and £ are determined by the least squares solver so they minimize the following least square objective function

2M(34)+

Imbibition case II

Finally, in order to determine the forced part of the imbibition capillary pressure the core plug is flooded with water in the forced flow imbibition experiment, i.e. step 5 of the ex­perimental procedure. This second imbibition experiment is described by the equation system Eqs. (1 - 11) with the initial and boundary conditions Eqs. (20 - 23). The capillary pressure is represented by a functional relationship similarly to Eq. (30). The coefficients bl are determined so that they minimize the following least square objective function

2N

(35)

The objective function consists of the difference between calculated and measured values of the total pressure drop across the sample during the transient flooding period.

Pump shutdown experiment

This experiment provides an alternative way to calculate the scanning parameter £ , and the spontaneous part of the imbibition capillary pressure. It is, however, recommended that these be determined by the stationary flow internal imbibition experiment. As neither the oil nor the water is flowing in the pump shutdown experiment the steady state situation is in­dependent upon the relative permeabilities. It is therefore possible to determine the scan­ning parameter, £ , which governs the scanning between the drainage and imbibition capil­lary pressure curves, together with the spontaneous part of the imbibition capillary pressure curve. A saturation profile is measured when no further saturation change occurs. Note, that the pc value at Swi, is known from the drainage experiment, and that the saturation

where the capillary pressure is zero, Sw spt is determined from the steady state saturation

profile for the forced flow imbibition experiment. It is assumed that the spontaneous part of

20 GEUS

the imbibition capillary pressure curve can be described by a functional relationship similar to Eq. (30). The objective function to be minimized is

M

(36)i=l

As pump shutdown experiment is described by Eqs. (1-11) and Eqs. (16-19), and the sys­tem is solved by means of ECLIPSE 100 and the objective function Eq. (36) is minimized by the non-linear least square solver MI0L2.

Test of the parameter estimation technique - model M15AThe parameter estimation technique was tested on a model labelled M15A, defined by a set of synthetic saturation functions. The equation system Eqs. (1 - 11) was first solved with the initial and boundary conditions Eqs. (12-15) (exp. step 2 and 3), and then with theconditions Eqs. (16 - 19), first the pump shutdown case with q‘™b = 0, and then with re­

versed flow of oil (exp. step 4). Finally, the water flooding case (exp. step 5) was calculated using the conditions Eqs. (20 - 23). The four solutions represent the drainage and imbibi­tion experiments and they are considered the experimental results in this synthetic case. Except for the steady state following the primary drainage the solutions were obtained by means of ECLIPSE 100.

The M15A synthetic sample parameters, and some synthetic experimental conditions are given in Table. 4. The saturation functions are given as functional relationships of the water saturation. The relative permeabilities are represented by power laws. A negative flow rate value indicates a reversed flow direction.

Relative Permeabilities

The relative water permeability is given by the following expression, which is used both for drainage and imbibition, i.e. no hysteresis for the wetting phase permeability

(37)

where

(38)

The coefficient used for the “true” solution are given in Table 5.

GEUS 21

Table 4. Synthetic M15A parameters.L = 7.17 cm $or ~ 0.34

A = 11.22 cm-2 $wi = 0.15

<f> = 0.3352 ^w.spt 0.5

K = 0.919 mD Pc max ~ 3.493 atm

cr = 7.35*10"5 atm"1 Pcmin — -10.0 atm

Pw = 1.048 g/ml Po,oat ~ 1.0 atm

Pw = 1.095 cp Po.th = 1.328 atm

Cw = 0.0 atm II 7.997 cm3/h

Po = 0.729 g/ml = -10.7 cm3/h

p. = 0.902 cp = -10.0 cm3/h

Q = 1.0*10"4 atm"1

The relative oil permeability is given as

where hysteresis is taken in to account by

S*’dr =

(39)

(40)

S *,imb So-S',(41)

The coefficients for the “true” solution are given in Table 6.

The capillary pressure function is divided into a drainage and an imbibition case.

Capillary Pressure

In the drainage case the capillary pressure is represented by the following functions

Table 5. Coefficients in M15A relative Table 6. Coefficients in M15A relative water permeability function. oil permeability function.

Drainage Imbibitiona, 0.9 0.9

a2 1.5 1.25

D rainage/l mbibition

Ci 1.0

c2 1.99915

22 GEUS

(42)l’.’--+r If Sw<s(S„ + f>.)

Pc =A

sw + bA + A if (43)

where for the “true” solution the coefficients, bt, are determined to

=0.22213 atm

b2 = 0.123454

b3 —1.78753

bA = -1.04726

Si =0.693

(44 - 48)

and

*i

dP,

w ^ u 2

( >

v dSwj s so

(49)

(50)

A=^' A^:+*4

(51)

V J S„=S°

*1*3 (52)

The parameters fx - f3 are determined in such a way that the capillary pressure function

Eqs. (42-43) satisfy

Pc(Swi) = Pc,max

Pc(V = 0

(53)

(54)

(55)

dp c’Eq.f 42) dPc>Eq.( 43;

k dSw ,s =s2 V y

(56)

GEUS 23

The capillary pressure, p° at Sw = S° is a sixth unknown. For plug M15A (“true” solution)

we have

p° = 1.557 atm

We define the threshold pressure as

Pth = Pc +V dSwjss°

(i-4

(57)

(58)

The value obtained is

pth = 1.3278 atm

The simulation results in a maximum capillary pressure of

Pc,max. = Pc(Sm) = 3.493Stm

In the imbibition case the capillary pressure is given as:

Pc =4,+4

- + e,

fc = - 4/' + <4 "

where

d,=- 0.119897

d2 = 0.449663

and

Pc, max (SWi + 4 X SK,spt + dx )C _ C °w,spt °wi

*2 =■'w,spt + dl

e3 =

(59)

(60)

(61)

(62)

(63 - 64)

(65)

(66)

(67)

24 GEUS

=-f3OSw.jp, -4)^ (68)

2e6 + =0 (69)

The minimum capillary pressure for the synthetic M15A core plug is

Pc,min=Pc( l-SJ = -iOatm

(70)

(71)

The parameters e1-e6, Eqs. (65 - 70) are determined in such a way that the capillary

pressure function Eqs. (61 - 62) satisfy

Pc(SWi) = Pc,m

Pc(^,) = 0

r dpc,Eq.(61) dPc’Eq.( 62)

V y

(72)

(73)

(74)

Swspl is the saturation where the imbibition capillary pressure is zero. It is assumed here

that Sw<spt is known.

The scanning parameter, e is set equal to 0.07.

Drainage

Initial GuessThe initial values used for the unknown parameters in the saturation functions are given in Table 7.

The initial relative permeabilities are simply straight lines through the end points. The ini­tial guesses upon the parameters of the capillary pressure curve, Eqs. (42 - 43) are de­termined as follows:

1. S° is guessed.

GEUS 25

2. p°c=Pc(Sl) is guessed.

3. The coefficients bx - b4 are calculated to satisfy the conditions Eqs. (53 - 56)

Eqs. (49 - 52) then give:

fx =1.2381

f2 = 0.0877

/3 =1.8049

\dSw j s ,= -0.6988aim

(75)

(76)

ResultsThe calculated and “measured” capillary pressure as well as the oil and water relative per­meabilities for the drainage case are shown in Figs. 2 and 3. Good agreement between the calculated results and the input data is found. This demonstrates that the procedure for calculating drainage saturation functions from a saturation profile and production data is valid for ideal synthetic data. The values obtained of the various parameters in the analyti­cal saturation functions are compared in Table 7 to the initial guess and the true solution.

The resulting match of the steady state water saturation profile and the transient total pres­sure drop are also very good, cf. Figs. 4 and 5. Fig. 5 also shows that the pressure drop attains its maximum at oil breakthrough.

Table 7. Comparison of calculated and true drainage solutions for M15A.Parameter Initial guess Calculated solution True solution

a\ 1. 0.9076 0.9

a2 1. 1.51915 1.5

Ci 1. 1.00000 1.

C2 1. 1.99915 1.99915

h 0.7583 atm 0.24422 atm 0.22213 atm

h 0.1708 0.10056 0.123454

h 1.0 1.61741 1.78753

K -1.053 -1.03583 -1.04726

si 0.7 0.692 0.693

pI 2.0 atm 1.55653 1.557 atm

26 GEUS

Model M15A Capillary Pressure

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Drainage, calculated x Drainage, synthetic data

Imbibition, calculated o Imbibition, synthetic data

Sw (fraction)

Figure 2. Comparison of true and calculated capillary pressure functions for synthetic sample M15A.

Model M15A Relative Permeabilities

.2 0.8

58 0.5

S» 0.3

5 0.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

—krwdr, calculated x krwdr, synthetic data■ krodr, calculated o krodr, synthetic data= 'kroimb, calculated+ kroimb, synthetic data

0.8 0.9 1.0

Sw (fraction)

Figure 3. Comparison of true and calculated relative permeability functions for synthetic sample M15A.

GEUS 27

Model M15A Equilibrium Water Saturation Profiles

Figure 4. Comparison of true and calculated saturation profiles for synthetic sample M15A.

Model M15A Pressure Drop and Oil Production3.0

I1.5 8.

o O

+ 1.0

-- 0.5

- 0.00.0 1.0 2.0 3.0 4 0 Time (h)5 0

=Calc. pressure drop, normal flow •Calc, pressure drop, reverse flow ■Calc, oil production, normal flow

X Input pressure drop, normal flow O Input pressure drop, reverse flow

Figure 5. Pressure drop and oil production for synthetic sample M15A.

28 G BUS

Table 8. M15A, initial guesses, imbibition case.Parameter Initial guess Calculated solution True solution

a2 1.51915 1.25 1.25

<k 0. -0.119895 -0.119897

d2 0.483413 0.4591 0.449621

e 0.05 0.07 0.07

It was found that it is important to know the threshold pressure. If the threshold pressure is not available the simultaneous iterations on oil relative permeability and capillary pressure may converge towards an erroneous solution. This is because the water saturation gradi­ent, Eq.(25) is a function of the product of the relative oil permeability and the gradient of the capillary pressure with respect to water saturation. An additional fixpoint of the capillary pressure curve is needed other than the two end points. The relative water permeability could be determined from the transient total pressure drop alone. In other words, it was not necessary to include the water saturation profiles in the objective function Eq. (33).

Imbibition

Initial GuessThe initial guesses applied in the imbibition case are given in Table 8.

The drainage relative oil permeability is used as initial guess on the imbibition relative oil permeability. The unknown coefficients in the imbibition capillary pressure function , Eqs. (61 - 62) are calculated so that the conditions Eqs. (72 - 74) are satisfied. The coefficients e1-e6 given by Eqs. (65 - 70) are shown in Table 9. Note that e6 is obtained from the solution to the spontaneous part of the imbibition curve, which is computed first.

ResultsThe calculated and “measured” imbibition capillary pressure and relative permeability to oil are shown in Figs. 2 and 3. Again, good agreement between the calculated results and the input data is found. This demonstrates that the procedure for calculating imbibition satura-

Table 9. Comparison of calculated and true coefficients in the imbibition capillary pressure function for M15A.

Parameter Initial guess Calculated solution True solutionex (atm) 0.7485 0.114202 0.114193

e2 (atm) -1.497 -0.300448 -0.300428

e3 (atm) -2.0883 10* -1.969303 105 4.06347 105

<?4 (atm) 2.6222 10"3 5.969821 10"3 6.635036 10"3

e5 3.776 5.4156 6.

e6 (atm) 0.39522 0.39522 0.39519

GEUS 29

tion functions from steady state saturation profiles and transient pressure drop data is valid for ideal synthetic data. The values obtained of the various parameters in the analyti­cal saturation functions are compared in Tables 8 and 9 to the initial guess and the true solution.

The resulting matches of the steady state water saturation profile following the reversal of the oil flow direction and the transient total pressure drop are also very good, cf. Figs. 4 and 5.

It was found that the spontaneous part of the capillary pressure, the relative permeability to oil and the scanning parameter could be determined simultaneously from the reverse oil flow “experiment". Therefore, there was no need to perform the pump shutdown experi­ment.

30 GEUS

NMR measurements

HardwareA 4.7 T (200 MHz) SISCO experimental NMR scanner with a horizontal bore is used for the NMR work. The scanner belongs to Danish Research Centre of Magnetic Resonance, and is physically situated at Hvidovre Hospital in Copenhagen. It is equipped with a 13 cm diameter insert gradient set, capable of producing individual magnetic gradient up to 100 mT/m along three orthogonal directions. Gradient riserate is 2*10s mT/m*s.

An RF (Radio Frequency) coil of a slotted tube resonator design is used. It is optimized for homogeneity in the Z direction, i.e. parallel to the horizontal bore of the scanner, and

has a good signal homogeneity until a maximum length of approximately 90 mm. The physical inner diameter of the coil is 8.5 cm, and the maximum imageable diameter is approximately 6 cm.

The scanner is controlled by SISCO’s VNMR software (V.90.1.1) running on a Sun Workstation.

Pulse sequenceThe objective of the NMR measurements is to obtain fluid saturation information, that is spatially resolved along the direction of fluid flow through a sample. This direction is always equivalent to the Z direction of the scanner. The objective is accomplished by using a pulse sequence named TST (Olsen et a/., 1996), that was developed and improved by two earlier EFP projects (EFP J.no. 1313/93-0014 and 1313/95-0007). It is a one-dimensional (1D) pulse sequence. If the 1D axis of the pulse sequence is aligned parallel to the flow direction of the sample, each resulting fluid saturation value represents the average fluid saturation in a thin, disc-like slice of the sample, the flow direction being perpendicular to the disc. Assuming homogeneity of the sample, the flooding experiments of the present project are essentially 1D experiments, and the use of a 1D pulse sequence is appropriate. The 1D fluid saturation determinations are termed saturation profiles.

TST is a simple CSI (Chemical Shift Imaging) pulse sequence, containing one spatial and one chemical shift dimension (Fig. 6). Spatial information is obtained entirely by phase encoding. Slice selection gradients and read-out gradients are not used, making the pulse sequence unaffected by chemical shift induced spatial shifts. In the original version of the pulse sequence a full echo was sampled (Olsen et a/., 1996). In Olsen (1997) the sequence was improved to allow the user to specify any intermediate state between full echo acquisition and half echo acquisition. The improvement allows two important experimental modifications: First, shorter echo times, TE, may be used, resulting in higher signal intensity and more reliable relaxation modelling. Second, data acquisition may be extended until the

G EU S 31

signal has completely relaxed, resulting in a large reduction in the image artifacts previously being generated by the Fourier transformation of a truncated time signal. To minimize off-resonance effects, the carrier frequency of the pulse sequence is always chosen to be the mean of the water and oil resonances, and the shortest achievable hard pulses are used. On the SISCO system the shortest possible 90° pulse is approximately 100 gs. In practice data acquisition with a highly asymmetric echo, and a TE of 3 ms has proved to work very well. The line width of the oil and water resonances usually lies between 100 and 200 hz (FWHM), while the two resonances are separated by 700-800 hz. Therefore, mutual interference between the two resonances are slight, usually 5-10 % of the total signal intensity. The interference is significantly reduced by an interference correction procedure.

Quantitative treatment of the NMR data requires that the water and oil resonances are located at constant frequencies in the whole sample. Good homogeneity of the main magnetic field, B0, across the whole sample volume is therefore important. Homogeneity of

the B0 field may be disturbed by the presence of paramagnetic minerals like pyrite, or ferromagnetic minerals like magnetite. In the investigated samples the contents of paramagnetic and ferromagnetic minerals are too low to cause any significant inhomogeneity in the B0 field.

The saturation profiles of the project have from 64 to 256 pixels (data values). A typical profile is 9 cm long, which means that the spatial resolution ranges from 0.141 cm/pixel to 0.035 cm/pixel. The time requirement for a high quality 256 pixel saturation profile on a sample with Tx =600 ms is 2 hours, in an implementation with 5 different TE's and a TR (repetition time) of 3 s. Compliance on TR, resolution, number of TE's, phase cycling and spatial resolution may bring the acquisition time for a saturation profile down to 5 minutes, and still retain a fair signal to noise ratio.

Phaseencoding

Figure 6. Timing diagram of the TST pulse sequence-

Data manipulationFig. 7 shows a grey tone rendering of a data set from the TST pulse sequence. It represents a stack of spectra at consecutive spatial positions. Several manipulations are necessary to transform this two-dimensional (2D) data set to a quantitative one-dimensional (1D) saturation profile. The sequence of manipulations is:

32 G BUS

I------------------------------------ 1----------------------------------- 1-6000 0 6000

Frequency dimension (Hz)Figure 7. Example of a data set from the TST pulse sequence.

Step 1 Frequency dimension elimination. Step 2 Relaxation correction.Step 3 Proton density correction.Step 4 Calculation of saturation.

Step 1 Frequency dimension elimination.

For each spatial position, i.e. for each phase encoding step, the water and oil signal in­tensities are determined by integration of the modulus spectrum. An integration width of 500 hz has been selected, centered on the actual resonance. This reduces the 2D TST data set to two 1D intensity profiles.

G BUS 33

Step 2 Relaxation correction

In an infinite series of ideal RF pulses with flip angle 90°, repetition time TR, spin-lattice constant 7j, and magnetization in the Z -direction Mz, the magnetization just prior to each RF pulse is given by

Mz(t = TR) = Mz(t = oo)(l- exp(-TR/T1)) (77)

The effect of spin-lattice relaxation on signal intensity is eliminated if TR is sufficiently long to allow complete spin-lattice relaxation between RF pulses. To accomplish this, TR was selected to be at least 5 time as long as the longest 7J component of the sample. With this convention it was assured, that the error on signal intensity due to differential saturation of the NMR signals was less than 1 % of the detected signal. An exception to the rule was the transient flow drainage experiment on sample M15M, where a TR /7j of 3.6 was used, leading to a maximum error on signal intensity due to differential signal saturation of 2.7%.

Spin-spin relaxation (T2 relaxation) is a complex process, covered by a comprehensive literature, for a review see Halperin etal. (1989). Spin-spin relaxation in an inhomogeneous system in principle follows a multiexponential behaviour according to

M(t) = M(t — O)jP(a) exp —t

Tja) da (78)

where M(t) is the magnetization at time t, M(t-0) is the magnetization at time 0, P(a) is the volume probability density function for pore size a, and T2(a) is the spin-spin

relaxation constant for pores of size a (Halperin etal., 1989).

The spin-spin relaxation in a sample is compensated by a relaxation modelization on an array of data sets (Edelstein et at., 1988, Chen et al., 1993). The array of data sets is acquired, with identical acquisition parameters, except for different values of the echo time, TE. A spin-spin relaxation modelization is then performed for each pixel array, producing 1D data sets of the fitted parameters, which includes the magnetization at time zero, M(t = 0). The TE values of the data acquisition are selected for optimal definition of the signal relaxation. Downwards the setting of TE is restricted by system hardware constraints, i.e. RF power limitations, gradient strength limitations and minimum electronic switching times. Depending on the setup of the SISCO scanner, the smallest possible TE value, TE^, is presently 2.5 ms for a 1D saturation profile experiment. In arrayed TE

experiments the smallest TE value is usually selected to be close to TE^, in order to

trace the relaxation path as close to the M(t = 0) condition as possible. The largest TE value in a TE array is usually selected to be approximately 3 times the anticipated single­exponential T2 relaxation constant, at which time the magnetization M(t) has declined to

5 % of the M(t = 0) value.

34 G EUS

An important issue is the choice of spin-spin relaxation model. Single-exponential, bi­exponential and stretched exponential modellings were tested by Kim et al. (1992), while Kenyon et al. (1988) tested bi-exponential, tri­exponential and stretched exponential modelling. The conclusion of Kim et al. was that bi-exponential fitting is preferable, while Kenyon et al. found that stretched exponential fitting ispreferable. Working on chalk samples, Olsen (1997) found that M(t = 0) is most

confidently determined by single-exponential fitting. This apparent discrepancy mainly stems from the fact that M(t = 0) for the 1D saturation profiles isdetermined by a significant extrapolation outside the sampled data interval. In the

present project single-exponential modelling is used, i.e.

Pixel plots, water, 25FB97_M16H_14

♦ Pixel 100--------Fit pixel 100

SS= 3.668E-05

0.80 -

0 0.60 --

0.40 -

0.20 --

0.10 -

0.00 4-

Echo time (ms)

Figure 8. Example of single-exponential modelling of spin-spin relaxation for the water resonance in a ID saturation profile.

M(t) = M(t = 0)exp (79)

where E is the signal level (noise) at t = °°. An example of spin-spin relaxation modelization is presented in Fig. 8. The figure presents a single model for the relaxation of the water signal from a representative pixels in sample M16H. The figure illustrates the importance of obtaining a small TE^, in order to achieve a fairly short extrapolation along

the model function to obtain the M(t = 0) value. In the example of the figure the single­

exponential modelization works well, and the precision of the M(t = 0) determination is consequently good. This is the case for a large majority of pixels in 1D saturation profiles. Failure to produce good fit mainly occurs at the ends of a sample, where susceptibility contrasts destroy the homogeneity of the magnetic field. These poor fits cannot be improved by choosing another relaxation model. Lack of pixelwise precision may to some extent be alleviated by filtering, but at the cost of reduced spatial resolution.

GEUS 35

Step 3 Proton density correction

The fluid saturation, SF, of a phase, F, is defined as

(80)

where VF is the volume of phase F, and VT is the total available volume. The intensity of

the magnetization of the phase, MP, is proportional to the number of protons present in the

phase, NF, i.e.

MF = CNF = C PDF VF (81)or

(82)

The constant C is dependent on system specifications and setup. It is not necessary to determine C, since it cancels out in the further calculations. PDF is the proton density of

phase F. To calculate saturations from measured magnetizations we therefore need to know the proton densities of the involved fluids. In this project they are calculated from chemical analyses. Proton densities for the fluids of the project are given in Table 3.

Step 4 Calculation of saturation.

In a two-phase system of oil and water VT =V0+VW. From Eqs. (80) and (82) then follows

(83)

O

and

(84)

where SW(Z) and S (Z) are saturations of water and oil, at position Z in the profile.

MJZ) and M0(Z) are magnetization intensities of water and oil, at position Z in the pro­

file, corrected for relaxation and proton density. PDW and PD0 are the proton densities of water and oil. Eqs. (83) and (84) are valid if the pore space of the sample does not contain a free gas phase.

36 GEUS

Signal processingThe SISCO NMR scanner is controlled by SISCO's VNMR software running on a Sun workstation. This software is dedicated to control the NMR data acquisition and perform Fourier transformation of the data, but it is not well suited for signal processing. After a 2D Fourier transformation in the VNMR software the NMR data is therefore transferred to GEUS for further processing.

The signal processing of NMR data at GEUS is mainly done with the program SISP written in VAX Fortran. It is originally produced by GEUS for use in the JOULE project JOUF-0019, but since then it has been extended several times, e.g. for analysing X-ray CT image data. The key capabilities of SISP are listed below:

1. Input/output facilities in special file formats, e.g. SISCO's PHASEFILE format.2. Smoothing and filtering facilities.3. Statistical analysis of image data.4. Image arithmetic, i.e. pixelwise addition, subtraction, multiplication and division of

images.5. Zoom, translation, mirroring, cut & paste facilities.6. Image projection and integration facilities.7. T2 relaxation modelling facilities.

8. Extensible by user delivered subprograms.

Limitations and accuracy of the NMR methodSpin-spin relaxation and resolution of the NMR resonances restrict the choice of samples suitable for NMR saturation profiling. The samples in this work have T2 values above 10 ms and good resolution of the NMR resonances. Comparison of the mean saturations of satura­tion profiles with conventional bulk saturation determinations indicates an accuracy for the method of 2 p.u. on the mean saturations (Olsen, 1997). The accuracy of the pixel satura­tions is inferior, perhaps around 5 p.u., but probably dependant on the setup. The reproduci­bility of the pixel saturations is 2 p.u. (1 standard deviation), as estimated from replicate analyses and saturation profile smoothness. The mean NMR saturations for the samples of the project was regularly verified by gravimetric analysis.

GEUS 37

38 GEUS

Results and discussion

Model M15AThe parameter estimation technique was tested on a model named M15A. This test model is thoroughly described in the chapter Parameter Estimation Technique. A set of synthetic saturation functions was used as input to a simulated drainage experiment using a reser­voir simulator (ECLIPSE 100, 1994). The resulting saturation profile and pressure drop data were then used to determine the saturation functions for the drainage case. It was found important to include the threshold pressure in the calculations to obtain a unique determination of the saturation functions. Next, a reversal of the oil flow direction was simulated, where the outset is the steady state drainage situation. The resulting steady state saturation profile was then used to calculate the imbibition relative permeability to oil and the spontaneous part of the capillary pressure, assuming no hysteresis for the relative permeability to water. Finally, a reverse waterflooding experiment was simulated. The re­sulting transient pressure drop was used to determine the forced part of the imbibition capillary pressure. The synthetic saturation profiles are shown in Fig. 4, and the pressure drop in Fig. 5. The results for the saturation functions are shown in Figs. 2 and 3. Good agreement exists between the calculated results and the input data. This demonstrates that the procedure for calculating saturation functions from saturation profiles and produc­tion data is valid for ideal synthetic data.

Application to core plug experimentsThe primary drainage and reverse flow imbibition experiments have been conducted on a number of chalk core plugs. Results from three plugs are presented herein. For the two plugs labelled M113 and M16H, saturation functions are presented, as determined by use of the five-step flooding procedure described in the section Flooding experiments and the parameter estimation technique. For the third plug, M15M, measured saturation profiles from the transient flow drainage experiment, described in the section Experiment exten­sions, are presented. In all cases the analytical representations of the saturation functions were similar to those applied for the synthetic plug M15A, cf. Eqs. (37), (39), (42 - 43) and (61 - 62).

Table 10. Plug M113 parameters.L = 7.467 cm A = 11.2398 cm2 <J) = 0.2636 K = 0.70 mD

cr = 1.76 10'3 atm"1 Pw = 1.049 g/cm3 Pw = 1.12 cp cw = 4.58 10"5 atm"1

Po = 0.73 g/cm3 Po = 0.92 cp C0= 1.16 10"4 atm"1 Swi = 0.0636c

w,spt = 0.343 Sor = 0.28 Pc.™ =5.527 atm Pth = 1.836 atm

= 10.0 cm3/h sr = - 23.0 cm3/h q‘”b = - 5.0 cm3/h P o.out = 1.0 atm

GEUS 39

M113 saturation profiles

Normal flow direction

Position in profile (mm)Primary drainage 10 ml/h — —Internal imb.10 ml/h reversed

Internal imb. 23 ml/h reversed ----------- Forced imb. 5 ml/h reversed

Figure 9. Measured saturation profiles on sample Ml 13.

Plug M113

Data

The M113 plug parameters and some experimental conditions are given in Table 10, and the measured saturation profiles in Fig. 9.

Primary Drainage

Initial GuessThe initial values used for the unknown parameters in the saturation functions are given in Table 11.

40 GEUS

Table 11. Plug M113, initial guesses, drainage case.

ax =1.0 h = 0.64906 atm Cj = 1.0

a2 = 1.0 h = 0.088869 c2 = 3.0

h = 1.0

h = -1.0179

si = 0.8

vl = 2.0 atm

The initial relative permeabilities are simply straight lines through the end points and the initial guesses upon the parameters of the capillary pressure curve, Eqs. (42 - 43) are de­termined by1. guessing S°.

2. Guessing p\ =pc(S°).

3. Calculating the coefficients bx - bA to satisfy the conditions Eqs. (53 - 56)

ResultsThe calculated capillary pressure as well as the relative permeabilities to oil and water for the drainage case are shown in Figs. 10 and 11. The values obtained for the various pa­rameters in the analytical saturation functions are shown in Table 12.

The resulting match of the steady state water saturation profile and the transient total pres­sure drop are shown in Figs. 12 and 13. It is seen that some deviations exist but the gen­eral curve shapes are similar.

Imbibition

Initial GuessThe initial guesses applied in the imbibition case are given in Table 13.

The drainage relative permeability to oil is used as initial guess on the imbibition relative permeability to oil. The unknown coefficients in the imbibition capillary pressure function,

Table 12. Plug M113, solution, drainage case.

CL, =1.0 = 2.7504 atm

a, =2.1223 = -0.0100 = 4.6827

= 0.2908

= -1.0006

GEUS 41

Plug M113 Capillary Pressure

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0■ 1 Drainage Pc --------- Imbibition Pc-------- Hg 1 drainage ---------Hg 1 imbibition-------- Hg 2 drainage ---------Hg 2 imbibition- - - Centrifuge forced imb.

Sw (fraction)

Figure 10. Calculated and measured capillary pressure determinations on sample Ml 13.

Plug M113 Relative Permeabilities

—.. Drainage kroDrainage krw

“■■“Imbibition kroSw (fraction)

Figure 11. Calculated relative permeability functions on sample Ml 13.

42 GEUS

Plug Ml 13 Equilibrium Water Saturation Profiles

Drainage, measured Drainage, calculated

Internal imb., measured---------Internal imb., calculated

Position (m)

Figure 12. Calculated and measured saturation profiles for sample Ml 13.

Plug M113 Pressure Drop, drainage case

——Pressure drop, measured

Pressure drop, calculated

Time (h)

Figure 13. Calculated and measured pressure drop for sample M113, drain­age case.

GEUS 43

Plug M113 Pressure Drop, forced imbibition

Time (h)Measured pressure drop

Calculated pressure drop

Figure 14. Calculated and measured pressure drop for sample Ml 13, imbibi­tion case. At time 0.5 h the flowrate was reduced from 10 ml/h to 5 ml/h.

Eqs. (61 - 62) are calculated so that the conditions Eqs. (72) - (74) are satisfied. The rela­tive permeability to oil and the spontaneous part of the capillary pressure is determined from the steady state total pressure drop and water saturation distribution following a re­versal of the oil flow. The forced part of the imbibition capillary pressure curve is calculated from the transient total pressure drop measured during the reverse water flooding of the core plug.

ResultsThe calculated imbibition capillary pressure and relative permeability to oil are shown in Figs. 10 and 11. The values obtained of the various parameters in the analytical saturation functions are compared in Table 13 to the initial guess. The scanning parameter was esti­mated to 0.023845. The resulting match of the steady state water saturation profile follow­ing the reversal of the oil flow direction is shown in Fig. 12. Again, some deviations exist

Table 13. Plug Ml 13, initial guesses and calculated solution, im­bibition case.

Parameter Initial guess Calculated solution2.1223 1.67423

dx 0. -0.016434

d2 0.6491 0.679998

E 0.05 0.023845

44 GEUS

between the calculated and measured results but the general curve shape is predicted well. This is, however, not the case with the transient pressure drop during the reverse water flooding, Fig. 14. The calculated pressure drop is smaller than the measured one. The reason for this behaviour is not clear. It may be that the relative permeability to water is overestimated for the upper part of the water saturation interval. This in turn could indi­cate that the functional form, Eq. (37), is not satisfactory. It might be better to use a com­bination of two different exponential functions or an exponential function and a straight line. This behaviour is not observed in the primary drainage experiment, Fig. 13. It may also be that the assumption of no hysteresis in the relative water permeability does not hold.

The determined capillary pressure curves are compared with results obtained from a stan­dard mercury injection technique and centrifuge method on the same plug, Fig. 10. The mercury injection curve is scaled by Leverett’s J-function using a standard value of 480 mN/m for the interfacial tension for the mercury-gas system and a measured value of 38 mN/m for the oil-water system. No correction for contact angle is applied as the sample was strongly water wet (Anderson, 1987). The drainage part of the capillary pressure curve is similar in shape to the mercury drainage curves, but the mercury injection data has a wider and more flat plateau. A similar situation has been reported previously (Norgaard et a!., 1995, Christoffersen, 1992). The spontaneous part of the capillary pressure curve shows a significant deviation with respect to the mercury curves. The mercury injection technique gives a much lower capillary pressure at wetting phase saturations below 0.2, together with a lower irreducible water saturation. The lower irresidual saturation for the mercury injection technique may be caused by the high applied mercury drainage pres­sure, up to 2000 bar. This can cause failure of the matrix, i.e. mercury can penetrate into porespace, which previously was isolated for the non-wetting phase (Christoffersen, 1992). This may also explain the lower wetting phase saturation obtained at pc = 0 for the mer­cury curves compared to the presented spontaneous capillary pressure curve. The forced imbibition capillary pressure curve compares well with a curve determined by the centrifuge technique, Fig. 10. The shape indicates strong wetting preference to water. This is also verified by the measured saturation profile for the waterflood, Fig. 9, which shows an al­most uniform water saturation. The centrifuge data do show a lower residual oil saturation, which may be explained from the performance of the centrifuge experiment. The primary drainage of the centrifuge experiment was not completed, i.e. there was water production at the last pressure step applied in the centrifuge. Therefore, the residual oil saturation, for the centrifuge data, is obtained by scanning curves from an incomplete primary drainage process, which will give a saturation between Sw = 1.0 and Sw = 1 - Sor (Killough (1976), Aziz & Settari (1979)). A primary drainage capillary pressure curve could not be calculated from the centrifuge experiment, because the Hassler and Brunner calculation model (Hassler & Brunner, 1945) do not apply, when water production has not ceased.

G EUS 45

Table 14. Plug M16H parameters.

L = 7.339 cm A =11.1273 cm2 <t> = 0.3333 K = 1.1 mD

cr = 1.76 10"3 atm"1 p„ =1.049 g/cm3 = 1.12 cp cw = 4.58 10 3

9o = 0.73 g/cm3 \i0 = 0.92 cp c<>= 1.16 10'4 atm"1 swi = 0.05cu wspt = 0.75 Sor =0.23 Pc,max = 4.7076 atm Pm = 1.6778 atm

it = 10.0 cm3/h qimb _ . 30 o cm3/h =-10.0 cm3/h P o, out = 1.0 atm

Plug M16H

Data

The M16H plug parameters and some experimental conditions are given in Table 14. Only drainage saturation functions have been determined for plug M16H, but a full set of satura­tion profiles has been measured, Fig. 15.

Primary Drainage

Initial GuessThe initial values used for the unknown parameters in the saturation functions are given in Table 15.

Again, the initial relative permeabilities are simply straight lines through the end points and the initial guesses upon the parameters of the capillary pressure curve, Eqs. (42 - 43) are determined by

1. guessing S°.

2. Guessing p° =pc(S°).

3. Calculating the coefficients bx - bA to satisfy the conditions Eqs. (53 - 56)

ResultsThe calculated capillary pressure as well as the relative permeabilities to oil and water for the drainage case are shown in Figs. 16 and 17. The values obtained of the various pa­rameters in the analytical saturation functions are shown in Table 16.

The resulting match of the steady state water saturation profile and the transient total pres­sure drop are shown in Figs. 18 and 19.

46 GEUS

M16H saturation profiles

Direction of normal flow

30 40 50Position in profile (mm)

Primary drainage 10 ml/h ----------Internal imb. 10 ml/h reversedInternal imb. 25 ml/h reversed ——Internal imb. 30 ml/h reversed

— —Forced imb. 10 ml/h reversed ---------Secondary drainage 10 ml/h---------Primary drainage 10 ml/h repeat

Figure 15. Measured saturation profiles on sample M16H.

Table 15. Plug M16H initial guesses, drainage case.

=1.0 bx =0.64906 atm ii o

ollsf b2 =0.088869

b3 =1.0

b4 =-1.0179

3: =0.8

p°c = 2.0 atm

c3 = 3.0

GEUS 47

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0— Drainage Pc----- Hg 1 drainage Pc ------ Hg 2 drainage Pc (fraction)

Figure 16. Calculated and measured capillary pressure functions for sample M16H.

Plug M16H Relative Permeabilities

® 0.4

Drainage krw Drainage kro

Sw (fraction)

Figure 17. Calculated relative permeability functions for sample M16H.

48 GEUS

Plug M16H Drainage Equilibrium Water Saturation Profile

Drainage, measured

Drainage, calculated

Position (m)

Figure 18. Calculated and measured saturation profiles for sample M16H.

Plug M16H Pressure Drop, drainage case

Pressure drop, measured

Pressure drop, calculatedTime (h)

Figure 19. Calculated and measured pressure drop for sample M16H.

G BUS 49

Table 16. Plug M16H solution, drainage case.

a, =1.0 = 4.8244 atm

a, = 3.2645 = 0.03115 = 3.3959

= 0.1917

= - 1.0003

The drainage capillary pressure curve for plug M16H is also compared to mercury data obtained on the same plug, Fig. 16. The results are very similar to plug M113.

Plug M15M

NMR monitoring of transient flow drainage experiment

During the transient flow drainage experiment on sample M15M the displacement process was monitored by NMR saturation profiling as described in the chapter Experimental tech­nique. The oil flooding rate was 5 ml/h, and the oil front moved through the sample at a rate of approximately 0.3 mm/minute. During the first 23 hours of the experiment a total of 58 fluid saturation profiles were measured. Of these 40 saturation profiles were measured from the onset of oil flooding until oil breakthrough. Oil breakthrough occurred 4 hours 10 minutes after the onset of oil flooding. A total of 4 different set-ups for the TST pulse se­quence were tested during the period of rapid saturation change. A set-up with 32 phase encoding steps and 6 minutes acquisition time, called Type 1, proved to be the best all­round set-up. This set-up is fairly insensitive to artifacts caused by fluid movement during acquisition, but only has a spatial resolution of 2.5 mm. An alternate set-up, called Type 3, with 64 phase encoding steps during an acquisition time of 6 minutes proved valuable, because it resulted in useful saturation profiles with a spatial resolution of 1.25 mm for the first hour of the experiment, and from approximately 3 hours and onwards. During the pe­riod from one to three hours within the experiment the Type 3 saturation profiles were useless due to artifacts caused by fluid movement. The useful Type 3 saturation profiles were very similar to the Type 1 saturation profiles, indicating that a spatial resolution of 2.5 mm was sufficient to define the form of the flooding front. The other set-up types that were tested were less useful due to artifacts.

A selection of saturation profiles is presented in Fig. 20. The figure presents a mix of dif­ferent set-up types. The main part of the flooding is traced by Type 1 profiles. At the start (time=22 min) of the experiment, and just after breakthrough (time=243 min) two Type 3 profiles are included. At time=258 minutes and later set-ups with resolutions of 1.25 or 0.63 millimeters are presented.

50 GEUS

M15M transient flow drainage saturation profiles

0 10 20 30 40 50 60 70 80Position in profile (mm)

■..... .. Time=0 min —■”Time=12 min “ I ime=22 min —-------Time=37 min

““■“Time=86 min “ ..........Time=111 min —-------Time=135 min

^”Time=184 min = “Time=209 min —------ Time=233 min

—" ■ Time=243 min """■—Time=258 min D=====3ITime=295 min —-------Time=522 min

■™*Time=1369 min - End fitting

Figure 20. Measured saturation profiles for sample M15M.

At the start of the experiment the saturation gradient in the sample is very steep, up to 0.17 mm"1. As the flooding proceeds, the maximum saturation gradient decreases until a value of approximately 0.04 mm"1 at the time when the oil front is situated in the centre of the sample (time=135 minutes). From this time the maximum saturation gradient increases until a value of 0.16 mm'1 just around breakthrough. Later in the experiment a very steep saturation gradient is retained, but only in the volume adjacent to the outlet end.

GEUS 51

52 G E U S

Conclusions

* A procedure to determine saturation functions for low permeable structures by a parame­ter estimation technique has been developed. The procedure utilises the strong capillary retention of the wetting phase in such structures. Capillary pressure functions and relative permeability functions, that are unbiased by capillary end effects, are produced for both drainage and imbibition situations.

* A complex five-step flooding procedure is presented from which information for calcula­tion of both the drainage and imbibition part of the saturation functions is gained. The pro­cedure utilises a non-magnetic core holder connected to a mobile flooding unit to allow saturation profile measurement in an NMR scanner, while the flooding is in progress.

* Saturation profiles are obtained from a 4.7 T NMR scanner, with a one-dimensional chemical shift imaging pulse sequence. The mean accuracy of the pixel fluid saturations is better than 5 p.u., and reproducibility of pixel fluid saturations is better than 2 p.u. The ac­curacy of the mean fluid saturation of a sample is 2 p.u. Transient flooding processes may be monitored with a time resolution of 6 minutes.

* The drainage saturation functions are calculated from a measured saturation profile, the pressure drop across the sample, and the threshold pressure obtained during a primary drainage process. The parameter estimation technique is verified on a synthetic test case, and good agreement is found between the calculated and the true solution for both capil­lary pressure and relative permeabilities. The calculated drainage capillary pressure curves for two chalk samples are compared with scaled mercury injection data obtained from the same samples. The plateaus of the mercury data are consistently situated at significantly lower capillary pressures. It is tentatively suggested that the mercury capillary data are invalidated by the highly aberrant surface tension and contact angle of the mercury- vacuum system, compared to an oil-brine system, and by destruction of the chalk pore structure by the high pressure injection of mercury. Drainage relative permeabilities for the same two samples are calculated, but no independent verification is available.

* The imbibition saturation functions are calculated from measured saturation profiles and the pressure drop across the sample during two imbibition processes. The parameter esti­mation technique is again verified on a synthetic test case, and good agreement is found between the calculated and the true solution. The calculated spontaneous imbibition capil­lary pressure curve for a chalk sample is compared with scaled mercury injection data ob­tained from the same sample. Again it is found that the plateau of the mercury data is situ­ated at significantly lower capillary pressures, and in addition the cross-over point Pc=0 is

situated at a much lower wetting phase saturation. The suggestion of invalid mercury re­sults is advanced as for the drainage case. Contrary to the mercury data, the forced imbibi­tion capillary pressure curve for a centrifuge experiment is found to be in good agreement with the results of the project.

GEUS 53

* The parameter estimation technique allows estimation of the capillary pressure scanning parameter e.

* The parameter estimation technique requires that the used functional forms of the satura­tion functions are able to capture the appearances of the true functions. In the case of the relative permeability to water, this may not be true.

* In the present procedure is assumed that the relative permeability to water does not ex­hibit hysteresis. There is some indication that hysteresis is present. It may be an improve­ment to include estimation of the imbibition relative permeability to water in the parameter estimation technique.

* The wettability of a sample can be obtained directly from the set of capillary pressure curves.

54 GEUS

Nomenclature

a coefficient, or pore size V volumea parameter vector X direction of flow downstreamA cross sectional area Z axial directionb coefficient £ scanning parameter

b parameter vector viscosity

Bo main magnetic field P density

B; RF field $ porosity

c coefficient Cl) weight factorS parameter vectorC compressibility Subscriptsd parameter c capillary pressuree parameter forced forced imbibitionE noise level F arbitrary fluid phasef parameter, arbitrary function i coefficient no. i

F arbitrary fluid phase in inlet of core sampleJ objective function mean meanKo relative oil permeability nc number of coefficients

K relative water permeability no number of coefficients

K absolute permeability nw number of coefficients

L length of core sample 0 oilM number of measured saturation points, or residual oil saturation

magnetization (NMR) out outlet of core sampleN number of time steps, r rock

number of protons (NMR) spt spontaneous

P pressure T total

P probability density function th threshold pressure

Pc defined by Eq. (54) w water

PD proton density wi irreducible water saturation

q flow rate z axial direction

s saturationsi defined by Eqs. (42 - 43) Superscripts

s* scaled saturationt time c calculated% spin-lattice relaxation time dr drainage

T2 spin-spin relaxation time imb imbibitionTE echo time m measuredTR repetition time

GEUS 55

G EUS

References

Anderson, W.G., “Wettability Literature Survey - Part 4: Effects of Wettability on Capillary Pressure”, Journal of Petroleum Technology, Oct. 1987,1283-1300,1987.

Aziz, K.; Settari, A., “Petroleum Reservoir Simulation”, Elsevier Applied Science Publishers LTD, GB, 1979.

Braun, E.M.; Holland, R.F.; “Relative Permeability Hysteresis: Laboratory Measurements and a Conceptual Model”, SPE Reservoir Engineering, August 1995,222-228,1995

Chardaire-Riviere, C.; Chavent, G.; Jaffre, J.; Liu, J.; Bourbiaux, B.J., “Simultaneous Esti­mation of Relative Permeabilities and Capillary Pressure,” SPE Formation Evaluation, Dec. 1992, 283-289, 1992.

Chen, S.; Qin, F.; Kim, K.; Watson, A.T., “NMR Imaging of Multiphase Flow in Porous Media”, AlChE Journal 39, 925-934,1993.

Christensen, R.L.; “Geometric Concerns for Accurate Measurement of Capillary Pressure Relationships With Centrifuge Methods”, SPE Formation Evaluation Dec. 1992 311-314,1992.

Christoffersen, K.R., “High-Pressure Experiments with Application to Naturally Fractured Chalk Reservoirs, 1. Constant Volume Diffusion, 2. Gas-Oil Capillary Pressure”, Doctor Dissertation, University of Trondheim, 214 pp, 1992.

ECLIPSE 100,95A Release, Intera Information Technologies Ltd., Highlands Farm, Hen­ley-on-Thames, Oxfordshire, U.K., 1994.

Edelstein, W.A.; Vinegar, H.J.; Tutunjian, P.N.; Roemer, P.B.; Mueller, O.M.; “NMR Imaging for Core Analysis”, SPE paper 18272,101-112,1988.

Forbes, P.L.; Chen, Z.A.; Ruth, D.W.; “Quantitative Analysis of Radial Effects on Centrifuge Capillary Pressure Curves”, SPE paper28182,15 pp, 1994.

Frandsen, P.E.; “SIPE - Small Interface Module for Parameter Estimation”, Personal communication, 1997.

Gall, B.L., “CT Imaging of Surfactant-Enhanced Oil Recovery Experiments”, Proceedings of the 205th National Meeting of the American Chemical Society, Denver, U.S.A., 120-125,1993.

G E U S 57

Halperin, W.P.; d’Orazio, F.; Bhattacharja, S.; Tarczon, J.C.; “Magnetic Resonance Re­laxation Analysis of Porous Media”, In: Klafter, J. & Drake, J.M. (eds.): Molecular Dynamics in Restricted Geometries, 311 -350, John Wiley & Sons, New York, 1989.

Hassler, G.L.; Brunner, E., “Measurement of Capillary Pressures in Small Core Samples”, Trans. AIME160,114-123,1945.

Hicks, P.J., “X-Ray Computer-Assisted Tomography for Laboratory Core Studies”, Journal of Petroleum Technology Dec. 1996,1120-1122,1996.

Honarpour, M.M.; Huang, D.D.; Dogru, A.H., ’’Simultaneous Measurements of Relative Permeability, Capillary Pressure, and Electrical Resistivity with Microwave System for Saturation Monitoring,” SPE paper 30540,1995.

Kenyon, W.E.; Day, P.I.; Straley, C.; Willemsen, J.F.; “A Three Part Study of NMR Longi­tudinal Relaxation Properties of Water-Saturated Sandstones”, SPE Formation Evaluation, September 1988, 622-636,1988.

Killough, J.E., “Reservoir Simulation with History-Dependent Saturation Functions,” SPEJ, Feb. 1976, 37-48,1976.

Kim, K.; Chen, S.; Qin, F.; Watson, A.T., “Use of NMR Imaging for Determining Fluid Saturation Distributions During Multiphase Displacement in Porous Media”, SCA paper 9219, 18 pp, 1992.

Kohhedee, J.A., “Simultaneous Determination of Capillary Pressure and Relative Perme­ability of a Displaced Phase”, SPE paper 28827,1994.

Madsen, K.; Hegelund, P.; Hansen, P.C., “Non-gradient Subroutines for Non-Linear Opti­mization”, Report Ni-91-05, Institute for Numerical Analysis, Technical University of Den­mark, June 1991.

Nicholls, C.I.; Heaviside, J., “Gamma Ray Absorption Techniques Improve Analysis of Core Displacement Tests”, SPE paper 14421,1985.

Nielsen, C.M.; Larsen, J.K.; Bech, N.; Reffstrup, J.; Olsen, D.; “Determination of saturation Functions of Tight Core Samples Based on Measured saturation Profiles”, SCA paper 9721,11 pp, 1997.

Norgaard, J.V.; Olsen, D.; Springer, N.; Reffstrup, J., “Capillary Pressure Curves for Low Permeability Chalk Obtained by NMR Imaging of Core Saturation Profiles,” SPE paper 30605,1995.

Olsen, D., “Quantitative NMR Measurements on Core Samples”, Danmarks og Gronlands Geo/. Unders. Rapp. 35, 31 pp, 1997.

58 G EUS

Olsen, D.; Topp, S.; Stensgaard, A.; Norgaard, J.V.; Reffstrup, J., “Quantitative 1D Satura­tion Profiles on Chalk by NMR,” Magnetic Resonance Imaging, 14,7/8, 847-851,1996.

GEUS 59

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APPENDIX A

GEUS

G E US

Proceedings

1997 International Symposium of the Society of Core Analysts

Calgary, Alberta, Canada 8-10 September 1997

The papers included in this volume are the proceedings from the 1997 SCA international Symposium in Calgary, Alberta, Canada, 8-10 September, 1997

The papers were subjected to a review and the authors encouraged to make the suggested corrections. The papers are the responsibility of the authors. The Society and the Board make no warranty of the quality or accuracy of the material included.

SCA - 9721

DETERMINATION OF SATURATION FUNCTIONS OF TIGHT CORE SAMPLES BASED ON MEASURED

SATURATION PROFILES

C. M. Nielsen, Technical University of Denmark; J. K. Larsen, Technical University of Denmark; N. Bech. Geological Survey of Denmark and Greenland; J. Reffstrup, Dansk

Olie & Naturgas A/S; D. Olsen Geological Survey of Denmark and Greenland.

ABSTRACTThe end effect of displacement experiments on low permeable porous media is used for determination of relative permeability functions and capillary pressure functions. Satura­tion functions for a drainage process are determined from a primary drainage experiment. A reversal of the flooding direction creates an intrinsic imbibition process in the sample, which enables determination of imbibition saturation functions. The saturation functions are determined by a parameter estimation technique. Scanning effects are modelled by the method of Killough. Saturation profiles are determined by NMR.

INTRODUCTIONLaboratory determinations of saturation functions in low permeable structures are compli­cated by strong scale effects, particularly by the end effect. This capillary retention of the wetting phase in displacement experiments results in a saturation gradient through the core. Although being a problem in many traditional core analyses the end effect actually con­tains detailed information about the saturation functions of the structure, and can cover a large saturation interval. The objective of the present work is to make use of this informa­tion to calculate relative permeability functions and capillary pressure curves.

The method of Nergaard et al. (1995), which uses the end effect for numerical simulation of capillary pressure functions for a primary drainage process, is improved to allow calcu­lation of both drainage and imbibition saturation functions.

A ID NMR imaging technique is used for quantification of the end effect (Olsen et al., 1996). However, the presented method is not dependent on a specific technique for quanti­fying the end effect. Gamma ray attenuation. CT scanning (Chardaire-Riviere et al., 1992) or microwave scanning techniques (Honarpour et al., 1995) may be suitable.

THEORYIn a core sample the relation between a gradient in pressure and the flow rate is described by the law of Darcy. Assuming the flow to be one dimensional and horizontal this relation is written for the two fluid phases

l

SCA - 9721

dp„ q„(1)dx kkm A

dp„ qv(2)

dx kkrw A

In a situation where oil is the non-wetting phase and water is the wetting phase, the capil­lary pressure is given by

Pc = P,,-P» (3)

Given enough time, flooding the sample at constant rate results in an equilibrium situation, where only the displacing fluid is flowing. Eqs. (1-2) show that the pressure gradient in the displaced phase is zero and the phase pressure of this fluid is uniform throughout the sam­ple. Then the capillary pressure can be calculated from Eq. (3) by determining the phase pressure of the displacing fluid along the sample. If the saturation is known as a function of position, the capillary pressure curve can be obtained for the saturation interval defined by the end effect. In the present study the saturation profile of the sample and the phase pres­sures at the inlet and outlet are measured. The pressure profiles are calculated by integra­tion of the Darcy equations, Eqs. (1) and (2). To perform the integration, the relative per­meabilities as functions of saturation must be determined. This is done by use of transient and steady state data from the displacement experiments, by a parameter estimation tech­nique. described below. The description of the procedure is divided into a drainage and an imbibition part.

DrainageThe procedure will determine the relative permeabilities and the capillary pressure for a primary drainage process. The gradient in capillary pressure at steady state (dpjdx - 0) may be written, by use of Eqs. (1-3)

= _Ji-.ii. (4)dx kkrn A

Assuming pc to be a function of saturation, fc(SJ, the gradient in water saturation is given by

dSw _ q„ 1dx kkm A dfc

Eqs. (4) and (5) can be solved with the boundary conditions

Pc (X = 0) = Pm - Pom (6)

y: 1-? II r- 11 0 (7)

y ii o II (8)

and the relationships

SCA - 9721

kr»{Sj) = f„(a\’a2i-,anS„) (9)

Pc(S„) = fe(b,,b2,...,bf,Sw) (10)

The saturation functions are represented by functional relationships of the water saturation. The relative permeability to oil and the capillary pressure are determined by using the measured saturation profile together with the total pressure drop over the sample, (pin-pout). For a given set of coefficients a, and b, in Eqs. (9) and (10) the capillary pressure, pc, and the water saturation. 5,,, are calculated from Eqs. (4) and (5) and the boundary conditions in Eqs. (6-8). The coefficients, a, and b, are determined so that they minimise the following least square objective function, by use of a standard non-linear least square solver

j„{a.b) . £($:(*,)-$;(*,)) + (Ap; -Aptf + (p;„-P:f (ii)/=!

The objective function is a sum of squared residuals, which are the differences in the calcu­lated and measured values for the water saturation along the sample, the difference in total pressure and the difference in the capillary threshold pressure. The squared residuals are weighted to make their contribution to the objective function similar in size. By Eqs. (4-11) the relative permeability to oil and the capillary pressure are determined by iteration.

In order to determine the relative permeability to water the transient part of the drainage experiment is used. The relative water permeability is again represented by a functional relationship of the water saturation

= f»(c\>c2 (12)

The coefficients c, are determined so that they minimise the following least square objec­tive function, by use of the same non-linear least square solver as above

•v 2jM = 2(APr(',)-Ap;«,)) (i3)

;=l

The squared residuals are the total pressure difference as function of time during the tran­sient part of the drainage. For a given set of coefficients, c, , the relative permeability to water is calculated. The relative permeability to oil and the capillary pressure are known from above. These three parameters are then given as input to a reservoir simulator (ECLIPSE 100. 1994), which is used to compute the total pressure drop across the sample during the drainage process.

ImbibitionIn order to determine the imbibition part of the saturation functions the end effect is shifted from one end of the core to the opposite by reversing the direction of the oil flow. The de­

3

SCA - 9721

velopment of an end effect at the new outlet follows an intrinsic imbibition process, while the old end effect is broken down by an intrinsic drainage process. The saturation profile at the new outlet can be used in the calculation of the saturation functions for the imbibition process. Recall that the bounding saturation curves are obtained under the initial condition of Sv. = 100% and Sn. = Sm for the drainage and imbibition process respectively. The re­versed drainage process violates this condition and scanning curves between the bounding saturation functions will therefore control the process. The scanning process is described by a method developed by Killough (1976). The scanning curves are dependent on the lo­cal saturation history of the sample.

Only the scanning of the capillary pressure curves is described here. The scanning of the relative permeability functions is described by similar equations. The scanning capillary pressure curves originating from the primary drainage capillary pressure curve are gener­ated by the expressions below. Similar expressions are used for the scanning from the bounding imbibition curve and for the intermediate case of reversal on a scanning curve.

P, = Pcj +F{Pc -Pcj) (14)

pcd and pC! are the bounding capillary curves for the drainage and imbibition. The local saturation history is accounted for by the factor F (Aziz and Settari, 1979)

F1 1

~+ 8 e

_______ 1_________ 2

(1 - S„r ) - S»hy + 8 8(15)

Fig. 1 shows the concept of scanning. Swhy is the point of reversal on the bounding curve, S’ir is the trapped oil saturation and s is a shape factor. S,‘, is equal to Sor when S„hy=Sm (bounding curve for imbibition). The trapped oil saturation is calculated as (Land, 1968)

= (1 (1 - Swhv ) (16)

Calculation of pc is performed in a way similar to the drainage case, recalling that kro re­fers to the imbibition process. For every measured saturation along the sample the F factor can be calculated assuming a value for s . Given pc from the reversed imbibition profile and pcd from the drainage, then pci can be determined from Eq. (14).

EXPERIMENTALNMR fluid saturation profile determinationThe flow regime considered is essentially one dimensional, along the length of a core sample, when care is taken to align the length of the sample parallel to any layering or lamination present. A one dimensional fluid saturation determination method is therefore appropriate, and an NMR technique was developed for the purpose (Olsen et aL 1996. Olsen. 1997). The technique is based on an NMR pulse sequence that is remarkably

4

SCA - 9721

simple, consisting of a spin echo sequence with phase-encoding, but without slice selection or read-out gradients. Signals are acquired as asymmetric echoes in order to keep echo time (TE) as short as possible. The cost of acquiring an asymmetric echo instead of a symmetric echo is a decrease in frequency resolution, which in the present case is subordinate to the improvement in minimum echo time. TEmin. At present, the shortest possible TEmin is 2.4 ms. The pulse sequence is unaffected by chemical shift artifacts in the spatial dimension, and is tolerant to flow. By using a non-magnetic core holder and a Mobile Flooding Unit, described elsewhere (Norgaard et al., 1995), the NMR technique may be applied while core flooding is in progress.

A 4.7 T SISCO scanner was used for the NMR measurements. ID fluid saturation profiles were obtained with a pixel accuracy better than 5 %-points, and a pixel reproducibility better than 2 %-points (la). Spatial resolution is 0.4 mm for a 100 mm profile.

Material and Fluid DataThe material used in the experimental work was chalk of Maastrichtian age from the Dan­ish sector of the North Sea. Results are presented for a 1.5” plug sample labelled Ml 13. It was cleaned by Soxhlet extraction with methanol and toluene. The porosity and absolute permeability were 26.36% and 0.70 mD respectively. The chalk was water wet and a syn­thetic formation brine with a salinity of 7%. density of 1048 g/1 and viscosity of 1.12 cp at 20 ° C was used. The non-wetting phase was n-decane with a density and viscosity of 730 g/1 and 0.92 cp at 20 ° C.

Flooding ProcedureTo establish both the drainage and imbibition part of the saturation functions a complex flooding procedure is used. The 100% brine saturated sample is first flooded with oil until steady state, production and displacement pressure being recorded. At equilibrium the satu­ration profile is measured. This stage is represented by the primary oilflood in Fig. 2. The decrease in the brine saturation towards the inlet of the core has been created by a primary drainage process. The oil flow direction is then reversed and the water contained in the end effect is displaced through the sample towards the new outlet, where a new end effect will evolve due to the capillary retention of the water. This build up of an end effect has fol­lowed an intrinsic imbibition process. It is done in two steps. First the oil rate is kept at the same rate as the primary drainage. At this condition the new end effect is not fully devel­oped at steady state. Fig. 2. The second step is to increase the flow rate until the pressure gradient matches the pressure gradient at steady state in the primary drainage. The end ef­fect now becomes developed and S„. at the new inlet is forced to the value of 5,,., obtained at the primary drainage. Fig. 2. The new end effect is used to calculate the spontaneous imbibition part of the saturation functions. The forced imbibition part is obtained by Hooding the core with brine in the same direction as the reversed oil flow, resulting in an almost constant saturation through the sample. Fig. 2.

SCA - 9721

RESULTS AND DISCUSSIONThe parameter estimation technique was tested on a model labelled M15A. A set of syn­thetic saturation functions was used as input to a simulated drainage experiment using a reservoir simulator (ECLIPSE 100, 1994). The resulting saturation profile and production data were then used to determine the saturation functions. The results are shown in Fig. 3 and 4. Good agreement between the calculated results and the input data is found. This demonstrates that the procedure for calculating drainage saturation functions from a satu­ration profile and production data is valid for ideal synthetic data.

Experimental data was obtained from a series of flooding experiments performed on the sample Ml 13. The production and pressure data were recorded and at steady state the fluid distribution was measured as shown in Fig. 2. For the drainage case the saturation profile together with the measured total pressure difference were used to determine the relative oil permeability and the capillar}-’ pressure curve for the primary drainage, Fig. 5 and 6. Drainage relative permeability data for chalk, that are comparable to Fig. 5, are unknown to the authors. The lack of data is probably due to the fact that it is difficult to generate mea­ningful results for material with pronounced end effects by means of the standard methods. The verification of the synthetic data set M15A. demonstrates that the calculation procedu­re can reproduce realistic relative permeability functions. The determined capillary pressu­re was compared with results obtained from a standard mercury injection technique also presented in Fig. 6. The mercury injection curve was scaled by Leveretts J-function using a. standard value of 480 mN/m for the interfacial tension for the mercury-air system and a measured value of 38 mN/m for the oil-water system. No correction for contact angle was applied as the sample was strongly water wet (Anderson, 1987). The shapes of the curves are similar, but the mercury injection data has a wider and more flat plateau. A situation similar to Fig. 6. has been reported previously (Norgaard et ah, 1995). In general data from the air-mecury and brine-oil systems cannot be considered directly comparable (Christoffersen, 1992). The threshold pressure for the sample was measured independently by an injection experiment and incorporated into the objective function Eq. (11) for precise location of the plateau of the calculated capillary pressure curve.

The measured steady state saturation profiles from the complex flooding procedure show how the old end effect is displaced through the sample by reversing the flow direction, Fig. 2. From Fig. I and 2 follows that the imbibition is controlled by scanning curves starting from the primary drainage curve when the saturation of reversal Swfiy is above S,vi. In fact only the first pixel in the new outlet will follow the bounding imbibition curve during the increase in saturation. The scanning curves provide the information to calculate the imbi­bition bounding curve by use of the scanning method of Killough (1976) as outlined in the theory. The saturation profile from the reversed oilflood with the fully developed end effect will give part of the spontaneous imbibition curve. Information for the forced imbibition curve (pc < 0) is contained in the profile from the reversed waterflood. Fig. 2 shows that the reversed waterflood gives a narrow saturation interval indicating that the sample is

6

SCA - 9721

strongly water wet. The parameter estimation technique remains to be demonstrated for the imbibition process.

The reversed flooding of Ml 13 produced a hump on the saturation profile, Fig. 2. The hump has a negative saturation gradient on the side towards the outlet of the sample. The hump was examined by the test model M15A. A reversed oilflood was simulated with and without the use of scanning curves, Fig. 7. The simulations without the scanning option failed to produce the hump. The simulations also revealed that the hump is a result of a complex scanning process, where part of the sample follows more than one scanning cur­ve. This is a consequence of the scanning being a function of the local saturation history. When shifting the end effect obtained from the primary drainage, part of the sample will experience a transient higher water saturation than the final steady state saturation.

CONCLUSIONSA procedure to determine saturation functions for low permeable structures by a parameter estimation technique has been developed. The procedure utilises the strong capillary reten­tion of the wetting phase in such structures.

* The drainage saturation functions are calculated from a measured saturation profile, the total pressure drop and the threshold pressure from a primary drainage process. The technique was verified on a synthetic test case and good agreement was found between the calculated and the true solution. The calculated capillary pressure curve for a chalk sample was compared with scaled mercury injection data obtained from the same sample. The plateau of the mercury data is situated at a significantly lower capillary pressure.

* A complex flooding procedure is outlined by which the imbibition part of the saturation functions can be calculated. The flooding procedure shifts the end effect to create a satura­tion profile, where the increase in saturation has followed a spontaneous imbibition pro­cess. An additional flooding creates a profile, which has followed a forced imbibition pro­cess.

* In order to obtain the imbibition bounding curves for the saturation functions the local saturation history must be described by the use of scanning curves.

* The saturation functions can be obtained from standard flooding experiments with mode­rate pressure gradients across the sample.

* The technique can be used on native samples and with reservoir like fluids.

7

SCA - 9721

AcknowledgementThe Danish Ministry of Environment and Energy is acknowledged for funding the work through the EFP-96 programme. The Danish Research Centre of Magnetic Resonance is acknowledged for providing NMR facilities.

NOMENCLATURE

a coefficient Subscriptsa parameter vectorA flow area c capillary pressureb coefficient cd capillary pressure (drainage)b parameter vector ci capillary pressure (imbibition)c coefficient i coefficient no. ic parameter vector in inlet of core samplef arbitrary function l number of coefficientsF F factor (scanning) 0 oilJ objective function or residual oil saturationk absolute permeability out outlet of core sampleK relative permeability T totalKn relative oil permeability th threshold pressure^nr relative water permeability w waterL length of core sample why point of reversal (scanning)M number of measured saturation wi irreducible water saturation

points along the sampleN number of time stepsP pressure Superscripts<7 flow rate c calculated5 saturation m measuredt time * trapped oilX direction of flow downstreamz scanning parametrep viscosity

REFERENCESAnderson. W.G.. “Wettability Literature Survey - Part 4: Effects of Wettability on Capilla­ry Pressur e," Journal of Petroleum Technology (Oct. 1987), 1283-1300.

Azizi. K.: Settari. A.. Petroleum Reservoir Simulation, Elsevier Applied Science Publishers LTD. GB.( 1979), 395-401.

Chardaire-Riviere. C.; Chavent. G.: Jaffre. J.; Liu. J.; Bourbiaux. B.J.. “Simultaneous Estimation of Relative Permeabilities and Capillary Pressure." SPE Formation Evaluation,(Dec. 19921283-289.

8

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Christoffersen. K.R.. High-Pressure Experiments with Application to Naturally Fractured Chalk Reservoirs, 1. Constant Volume Diffusion, 2. Gas-Oil Capillary Pressure, Doctor Dissertation. University of Trondheim. (1992), 214 pages.

ECLIPSE 100 , 95A Release. Intera Information Technologies Ltd., Highlands Farm. Henley-on-Thames. Oxfordshire, U.K. (1994).

Honarpour. M.M.; Huang, D.D.; Dogru, A.H., ’’Simultaneous Measurements of Relative Permeability, Capillary Pressure, and Electrical Resistivity with Microwave System for Saturation Monitoring,” SPEpaper 30540, (1995).

Killough. J.E., “Reservoir Simulation with History-Dependent Saturation Functions,” SPEJ, (Feb. 1976).

Land, C.S., “Calculation of Imbibition Relative Permeability for Two- and Three-Phase Flow from Rock Properties,” SPEJ, (June 1968) 243,149-156.

Norgaard, J.V.; Olsen, D.; Springer, N.; Reffstrup, J., “Capillary Pressure Curves for Low Permeability Chalk Obtained by NMR Imaging of Core Saturation Profiles,” SPE paper 30605. (1995).

Olsen, D.. Quantitative NMR Measurements on Core Samples, Danmarks og Gronlands Geol. Unders. Rapp. (1997) 35. 31 pages.

Olsen. D.: Topp, S.: Stensgaard. A.; Norgaard, J.V.; Reffstrup, J., “Quantitative ID Satu­ration Profiles on Chalk by NMR,” Magnetic Resonance Imaging, (1996) 14, No. 7/8, 847- 851.

9

kr (f

ract

ion)

pc

(l>n

r)

SCA - 9721

Scanning curves, Killough’s method

0.4Sw (fraction)

Primary drainage ImbibitionScanning. Swhy 0.15 Scanning. Swhy 0.25 Scanning. Swhy 0.35 Scanning. Swhv 0.45

| 1 "— Scanning. Swhy 0.55

ig. 1. Example of scanning from theprimary drainage capillary pressure.

Ml 13 saturation profiles

: Primary flow direction

Position in profile (mm)

i —— Primary oilflood 10 ml/h I

——Reverse oilflood 10 ml/h j Reverse oilflood 23 ml/h j

|-------------Reverse waterflood 5 ml/h)

Fig. 2. Measured saturation profiles on sample Ml 13.

kr, synthetic data M15A, drainagei.o -

0.9 -

0.8 -

0.7 -

0.6 —

0.5 -

0.4 -

0.3

0.2

0.1

0.00.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Sw i fraction)

j "|,A'~ krw. synthetic input data ! “""^knv. calculated output data

—O— kro. <> mhetie input data ■

kro. calculated output data

Fig. 3. Verification of synthetic relperm data. kro is approximated by a polynomial of order three, k™ is approximated by an exponential function.

pc, synthetic data M15A, drainage400000.0 -

350000.0

300000.0 4

250000.0 4

200000.0

150000.0

100000.0

50000.0

0.0

Sw (fraction)

-A- Synthetic input data

"■^"Calculated output data

Fig. 4. Verification of synthetic capillary pressure data.

10

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kro, calculated Ml 13, drainage

1.0--------------------------------------------0.9 -

0.8 -

0.7 -

0.6 -

0.3 -

0.2 -

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 i.O

Sw (fraction)

I ■■■kro, calculated

pc, calculated Ml 13, drainage

600000 t

550000 -

500000 -

450000 -

400000 --

350000 --

300000 --

250000 --

200000 -

150000 -

100000 -

50000 --

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Sw (fraction)

—pc, calculated

-------- pc. Hg injection

Fig. 5. Measured data. kr0 is approximated by a polynomial of order three.

Fig. 6. Measured data. Calculated pc com­pared to pc from Hg injection.

Effect of scanning

0.4 - -

l 0.3

Position in profile (mm)

^■^Synthetic data M15A. corrected for scanning

Synthetic data M15A. not corrected for scanning

---------M113 measured profile 23 mi/h

Fig. 7. Qualitative illustration of scanning. The synthetic data M15A were generated with saturation functions differing from sample Ml 13. Therefore the saturation levels differ.

11

APPENDIX B

G EU S 77

78 GEUS

Abstract accepted for the 73rd SPE Annual Technical Conference and Exhibition 27-30 September 1998, New Orleans.

Paper title:Determination of oil-water saturation functions of chalk core plugs from two-phase flow experiments.

Authors:Bech, Niels Olsen, DanNielsen, Carsten Moller

Abstract:A new procedure for obtaining saturation functions, i.e. capillary pressure and relative permeability, of tight core samples uses the pronounced end effect present in flooding experiments on such material. Commonly being a nuisance in core analysis, the end effect contains valuable information about the saturation functions. In core material with high capillary pressure, the end effect may allow determination of the saturation functions for a broad saturation interval.

A complex core flooding scheme provides the fluid distributions and production data necessary for the calculation. The steady state situation at the end of an initial primary drainage experiment allows calculation of the drainage capillary pressure and the drainage relative oil permeability. The relative water permeability is calculated from the unsteady state data obtained during the transient part of this experiment. After a flow reversal, a new end effect develops in the opposite end of the core by an internal im­bibition process, which at steady state allows calculation of the spontaneous imbibition capillary pressure and the imbibition relative oil permeability. Following a change from oil flooding to water flooding, the forced imbibition capillary pressure is calculated from transient pressure drop measurements. An undesirable interdependency of the satura­tion functions is avoided by their calculation from different data sets. Killough’s method is employed to account for the scanningeffect in hysteresis situations for both capillary pressure and relative permeability. The procedure allows estimation of the scanning parameters. The saturation functions are determined by a least squares technique.

The procedure is demonstrated on chalk samples from the North Sea. The experimen­tal time is intermediate between the centrifuge and porous plate methods. The proce­dure is superior to the centrifuge and mercury injection methods respectively by elimi­nating errors from end effects and using reservoir relevant fluids. An NMR technique is used for saturation distribution determination.

GEUS 79