Bài báo SPE (IPTC 18132 MS)

IPTC-18132-MS Oil -Water Relative Permeability Data for Reservoir Simulation Input, Part-I: Systematic Quality Assessment and Consistency Evaluation Ammar Agnia and Hossein Ali Algdamsi, Schlumberger; Mohamed Idrees Al Mossawy, Universiti Teknologi PETRONAS Copyright 2014, International Petroleum Technology Conference This paper was prepared for presentation at the International Petroleum Technology Conference held in Kuala Lumpur, Malaysia, 10 –12 December 2014 This paper was selected for presentation by an IPTC Programme Committee following review of information contained in an abstract submitted by the author(s) Contents of the paper, as presented, have not been reviewed by the International Petroleum Technology Conference and are subject to correction by the author(s) The material, as presented, does not necessarily reflect any position of the International Petroleum Technology Conference, its officers, or members Papers presented at IPTC are subject to publication review by Sponsor Society Committees of IPTC Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the International Petroleum Technology Conference is prohibited Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented Write Librarian, IPTC, P.O Box 833836, Richardson, TX 75083-3836, U.S.A., fax ϩ1-972-952-9435 Abstract The relative permeability concept has been used extensively in reservoir engineering As numerical reservoir simulation has become more popular as a tool for reservoir development, the role of relative permeability data became even more evident and important Its key use is to control the advancement and mobility of different fluids simultaneously coexisting in the porous media, and hence controlling the recovery of the fluids However, deriving a reliable relative permeability data set remained a major challenge In reservoir engineering, this challenge has been present for many decades and might be so in the foreseeable future Another challenge is to have a data set which is internally consistent and does not hinder the simulation performance Optimistically, with the significant literature accumulated over the years in deriving and using relative permeability, some techniques can be extracted for data quality check, control and assurance This paper covers the limitations of the conventional methods used for calculating relative permeability from displacement experiments It also compiles all contemporary techniques in a systematic workflow for quality assessment and consistency evaluation The workflow has been demonstrated with different synthetic and field examples This paper will provide a reference for reservoir engineers who have an interest in investigating, checking the quality, and preparing relative permeability data set usable for reservoir simulation process Introduction Special core analysis (SCAL) is the hub of the evaluation and management of hydrocarbon reservoirs Relative permeability is one the main constituent of the SCAL which importance is widely recognized for the prediction of oil recovery during displacement by water As any other piece of data, high quality and reliable relative permeability data set can reduce uncertainty in dynamic reservoir modelling and provide a sound foundation for reservoir engineering studies Conversely, ppoor quality data can result in lost time due to rework and additional studies, inadequate development plans, and inefficient investment Relative permeability curves can be generated from different sources such as mathematical models and experimental methods However, experimental methods are more desirable for two reasons First, they produce specific relative permeability relationships for specific reservoirs Second, it is best available IPTC-18132-MS approach to resemble the flooding process in the field provided that the experiments performed on representative core samples and fluids from the reservoir under study Therefore, our discussion will be restricted to relative permeability data are derived from laboratory experiments The main challenge in the derivation process is how to obtain a reliable relative permeability data set The term reliable will often be used in referring to relative permeability data set with is a good probability that the defined relationships are representative of the reservoir and inherently repeatable Judgment regarding reliability will be made according to analysis of results that are judged to have been obtained using valid laboratory procedures When the term valid is used in referring to laboratory measurements it will mean that none of the procedures used during the test are inconsistent with obtaining reliable results for the sample tested For instance, the use of an extracted core plug with altered wettability other than the reservoir one would generally mean that the results are invalid The source of unreliability may be attributed to the following reasons: The derived relative permeability data set are usually prone to excremental artefacts The state of the experiment does not fit the model’s assumptions used to derive the relative permeability data sets In another word, the methods of calculating relative permeabilities from data obtained from displacement experiments don’t describe all physical effects encountered in the experiment ● The dependence of relative permeability on wide range of variables and conditions (Fluid saturations, Saturation history, initial saturations values, wettability, pore geometry, overburden stress, clay and fines content, temperature, interfacial tension and viscosity, and displacement rate) (Dandekar 2013) ● The uncertainty involved in all stages of deriving the relative permeability (Grimstad et al 1997) ● ● The validation process can be portrayed as a systematic correction of the inherent errors of the experiments and the derivation techniques, in a way that the true characteristics of the relative permeability are preserved An example is the systematic approach in assessing the validity of extracted relative permeability profile from experimental result and assuring the reliable and representative portion of the data was utilized to produce the relative permeability characteristics and guarantee the consistency and quality of revealed end -point There are many techniques disseminated in the literature for checking the quality of relative permeability Those techniques are compiled to enable the reader of performing data quality assessment and consistency evaluation The quality check scheme followed within this paper starts with preliminary check for the quality of the experiment, relative permeability, and data sets provided to simulators Before starting it will be good to start the discussion with a brief review of the relative permeability concept and the methods used to measure it in the laboratory Overview Water Oil Relative Permeability The relative permeability is a macroscopic property that is defined through extensions of Darcy’s law to multiphase flow (Grimstad et al 1997) It is defined as the ratio of the effective permeability of a specific fluid to reference permeability The reference permeability can be any of the following (Qadeer, Brigham, and Castanier 2002): Absolute permeability to one of the phases Klinkenberg corrected gas permeability Non-wetting phase effective permeability at irreducible wetting phase saturation Wetting phase effective permeability at irreducible non-wetting phase saturation IPTC-18132-MS The choice of the reference permeability is not critical in itself However, it is very important to that it is stated and consistently applied (Qadeer, Brigham, and Castanier 2002, Glover 2002) Relative permeability data should be obtained by experiments that best model the type of displacement that is thought to dominate reservoir flow performance (Fanchi 2006) A number of laboratory measurement techniques for deriving the relative permeability are described in the literature Generally, three methods exist to measure relative permeability: steady-state (Osoba et al 1951), unsteady displacement (Johnson, Bossler, and Naumann 1959), and centrifuge (Hagoort 1980) In the steady-state method, relative permeability is measured directly; whereas it is indirect measurement in the unsteady-state and centrifuge methods (Mohanty and Miller 1991) The steady-state method has an advantage of simple calculations, and disadvantage of tedious long procedure The unsteady-state method takes less time but requires more complicated calculations (Al-Mossawy and Demiral 2011) The centrifuge method is fairly quick but gives the relative permeability of the displaced phase only (Mohanty and Miller 1991) Each of these methods will be discussed briefly in the following sections Steady-state Coreflood Steady-state experiment is run with simultaneous flow of two phases, both wetting and non-wetting, at different flow rates The flow rate of one phase is increased while the other is decreased gradually At each rate change, the pressure drop and the saturations are closely monitored Equilibrium, i.e steady-state condition, is established within the system once the pressure drop across the core and the saturations are not changing with time At that point, effective and relative permeabilities for individual phases are computed using the pressure drop, flow rates, length and cross sectional area of the core, and viscosities of fluids into Darcy’s law (Bennion and Thomas 1991) The saturations of the phases are varied by changing the ratio of the flow rates of the fluids Thus the relative permeability curves can be determined over a representative range of saturations (Qadeer, Brigham, and Castanier 2002) In general, between five and ten stages are usually needed to establish relative permeability curves The test does not necessarily depict reservoir fluid displacement mechanism where one fluid displaces the other, since the test is not truly a displacement test but rather an equilibrium flow test However, its virtue is that rate effects associated with viscous instabilities are eliminated Capillary pressure forces are usually ignored but the experiment can be designed in such way the end effects are eliminated (Bennion and Thomas 1991) The data obtained by steady-state method is at least as believable as the plausible model on which they are based (Darcy’s law), especially if convincing measures are taken to minimize the capillary end effects A major difficulty is the determination of saturation at each stage There are different methods that are used for in-situ determination of fluid saturation in cores such as measurement of electric capacitance, nuclear magnetic resonance, neutron scattering, X-ray absorption, gamma-ray absorption, volumetric balance, and weighing the sample techniques (Honarpour, Koederitz, and Herbert 1986) However, the problem is the identification of a single value to represent average saturation over the whole core plug at each equilibrium stage There is also uncertainty as to whether the fluid distributions are representative of the displacement process Saturations measurements such as weighing the sample will interrupt flow and can cause problems of capillary contact with the sample end pieces (Heaviside and Black 1983) An important advantage of the method is that it is possible to define relative permeability across a border saturation range, even for systems having favourable mobility ratios Although the technique allows flooding with hundreds of pore volumes of water, relative permeability still may not be measured at low oil saturations in high permeability and intermediate wettability rocks The steady-state method has also an advantage of simple calculations, but a disadvantage of tedious long procedure where days or weeks are often required to achieve equilibrium for each saturation point (Bennion and Thomas 1991) The main advantages of the steady-state technique are: Relative permeability data are calculated over the full saturation range Simple calculations IPTC-18132-MS The disadvantages are: There is also uncertainty as to whether the fluid distributions are representative of the displacement process (Heaviside and Black, 1983) It does not necessarily depict reservoir fluid displacement mechanism where one fluid displaces the other The experiments are time consuming and thus expensive Unsteady-state Coreflood In unsteady-state method, only one of the phases is injected into the core to displace the other The theory is based on Buckley-Leverett Equation (Buckley and Leverett 1942) as a process model The idea is to monitor the pressure differential across the core and production history of controlled multiphase displacement experiment, and then to back-calculate the relative permeability values with analytical methods Different approaches have been developed to determine relative permeability data from the unsteady-state displacement (Johnson, Bossler, and Naumann 1959, Jones and Roszelle 1978, Mitlin et al 1998, Toth et al 1998, 2001, Li, Shen, and Qing 1994) However, those approaches neglect the capillary pressure effects The cumulative production data also are processed to provide a basis for calculating average saturation levels to be associated with the relative permeability values The method is preferred to the steady-state for its rapidness and, to some extent, it can depict reservoir fluid displacement mechanism where one fluid displaces the other However, it is more susceptible to end effects, rate-dependent instability effects, and potential non-equilibrium between displacing and displaced fluids (Bennion and Thomas 1991) Nevertheless, the unsteady-state methods, comparatively speaking, supply the wanted data quickly with reduced expense Other models also have been presented based on the initial and final stages of the flow process (Corey 1954, Torcaso and Wyllie 1958, Pirson 1958, Wyllie and Gardner 1958, Honarpour, Koederitz, and Harvey 1982, Al-Mossawy and Demiral 2011, Ahmed 2001, Ibrahim and Koederitz 2000, Behrenbruch and Goda 2008) The main advantages of the unsteady-state method are: Appropriate “shock-front” development Ability to flow at appropriate reservoir flow rates Relatively fast (and therefore relatively inexpensive) Relatively low throughput (less prone to fines migration) The main disadvantages are: Inlet and outlet boundary effects, especially at low rates which invalidate Buckley-Leverett theory Relatively complex interpretation Centrifuge Oil Relative Permeability Tests The method is based on the concept of rotating a core at various angular velocities to initiate forced drainage or imbibition process For every rotational speed, the fluid production is collected in a pipette and measured at equilibrium The average saturation derived from the effluent production volume With the driving force remaining constant, the relative permeability of the displaced phase is directly proportional to the oil production rate The production is measured as function of the drainage time then relative permeability is produced by differentiation of the data according to the algorithm devised by (Hagoort 1980) The model interpret the centrifuge experiment as a gravity drainage process based on the following major assumptions: capillary effects are negligible, the invading fluid mobility is much larger than the displaced fluid mobility, and the acceleration along the sample is uniform There are several types of centrifuge experiments: single-speed, multi-speed, constantly-accelerating and constant flow rate centrifuge techniques (Bauget et al 2012) IPTC-18132-MS A major advantage of the test is that large pressure differentials can be created across the core plug without creating an unstable displacement The higher pressure differentials imposed by centrifuge results in relative permeability data over an extended saturation range, i.e to lower oil saturations The design of experiments requires a balance between the Brownell-Katz (or Bond) number criteria and retention due to a capillary end effect (Hirasaki, Rohan, and Dudley 1995) Hagoort’s analytical calculation from a single-speed experiment underestimates the relative permeability maximum due to the time needed by the centrifuge to reach the selected speed (Bauget et al 2012) It also overestimates the final liquid saturation since capillary end effects are not taken into account (Bauget et al 2012) Notable limitations of the technique are; values of water relative permeability cannot be defined from a single-speed displacement of oil by water, disability of using live oil, and fines migration in cores with high content of illite For a multi-speed experiment, the relative permeability determination requires history matching with a numerical simulator (Bauget et al 2012) Which technique? The choice of measuring technique for relative permeability can be a challenging task For example, each of the above mentioned techniques is claimed to yield residual oil saturation and relative permeability Experimental artefacts and neglected physical phenomena could invalidate the classical interpretation theory and often make the interpretation of special core analysis experiment highly unreliable For instance, the capillary end effect is present during the laboratory water flood which cannot be squeezed away, except at extremely high rates or after excessively long flooding times, resulting in artificially high residual oil saturations Thus, the capillary pressure is interfering with a relative permeability measurement in the steady-state or unsteady-state apparatus Conversely, it’s possible to say that relative permeability effects interfere with a capillary pressure measurement in the centrifuge The centrifuge does, on the other hand, yield the true value of residual oil saturation The choice of test method should be made with due regard for reservoir saturation history, rock and fluid properties (Glover 2002) In addition, laboratory measurements of relative permeability should be representative of flow behavior in the reservoir For example, centrifuge experiments are often used in studies where gravity drainage is identified as the dominant recovery mechanism (Edwards et al 1998) The unsteady-state method is more often used because it is substantially quicker In some cases the steady-state method is needed to fully define the curves ranges However, this can lead to difficulties if data from the different methods is not in agreement (Heaviside and Black 1983) Integrating the results from different types of tests can be of great help in recognizing which parts of each data set are reliable and which are invalid The unsteady state test provides reliable results within the mid saturation range and the experiment artefacts start to show up at high saturation range Therefore the centrifuge results will be used as supplementary of the unsteady state results Numerical simulations of laboratory experiments should be an essential element to reconcile the experimental results Numerical simulations can be also used in the experimental design stage, and for quality control on contractor interpretation of experimental data In summary, thorough understanding of procedures and conditions used to obtain the results and the strengths and weaknesses of each test the experimental results can be reconciled Quality check of experiment Laboratory measurements are prone to experimental and physical erroneous effects that can propagate to the derivation of relative permeability, hence making it unreliable Therefore, flooding experiments should be designed in such way such that eliminate or minimize these effects on measurements For example, end effects are generally reduced by using high viscous pressure gradients and equilibration times are chosen sufficiently long to reduce “after-flow” effects due to low relative permeability (Kokkedee et al 1996, Masalmeh 2012) On the other hand, high rate flooding experiments may cause the displacement to become unstable and trigger fines migration A combination of proper experimental design such as IPTC-18132-MS Figure 1—Relation between oil recovery at breakthrough and scaling coefficient (Rapoport and Leas 1953) flooding at high rate, using longer cores (composite core), using capillary mixing sections numerical simulation of the experimental data is required The use of numerical simulation to aid proper interpretation of laboratory experiments proved to be an efficient tool to minimize the impact of the experimental artefacts on the results and conclusions of the study Reducing the experimental artefacts is not straightforward Therefore, the laboratory tests should be conducted ideally under conditions where the laboratory values for the key flow parameters match the field values (Mohanty and Miller 1991) A simple way of presenting the flow phenomena that depend on these parameters in a very complex manner is the scaling group concept in which key flow parameters interrelated and can be expressed by a different set of dimensionless numbers that defines the critical ranges of these parameters Relative permeability in turn depends on the pore structure, wettability and flooding conditions, which can be represented by a set of dimensionless groups (Mohanty 2002) Following this procedure, one could resolve inconsistencies observed between different experimental techniques and procedures The flooding condition can be represented by a set of dimensionless groups including capillary number, bond number and heterogeneity index(Mohanty and Miller 1991) Flow rate Rapoport and Leas scaling group (Rapoport and Leas 1953) has often been used to select the rate of water flood required for stabilized flow The term “stabilized flow” refers to flow where the shape of the front does not change with time The effect of capillary pressure in core floods is to spread the front, but at the same time there is a wave sharpening effect because of the convex-upward shape of the fractional flow curve These two effects tend to balance and make the wave approach an asymptotic limit or stabilized flow It is not obvious that a stabilized flow region exists in all the different wettability situations As an example, if the objective is to find rate-independent residual oil saturation, a stabilized flow region may be one of the rate selection criteria (Haugen 1990, Chen and Wood 2001, Skauge, Thorsen, and Sylte 2001, Rapoport 1955) The Rapoport and Leas scaling group is defined as: Eq.1 Where: L core length, v the interstitial velocity and ␮w water viscosity Experimentally, a critical value could be found beyond which a flood is stabilized This is done in practice by plotting oil recovery at breakthrough versus Lv␮w as in Fig.1 The value at which recovery becomes independent of the scaling coefficient gives the critical value (Kyte and Rapoport 1958) reported that Lv␮wϾ1 (cm2/min.cp) resulted in a minimal end effect However, questions can raises that whether: Conducting the flooding experiments at field rate is more representative, and Conducting the experiments at higher rate would lead to de-saturation of Sorw? IPTC-18132-MS It is important when designing displacement experiments for measurements of relative permeability to consider the following points: Table 1—Summary of critical ranges for capillary numbers (Mohanty and Miller 1991) Water Wet A high pressure gradient will be required to Mixed wet minimize capillary pressure end effects Field Range Laboratory Range Lv␮w Ͼ1 The pressure gradient, however, should be small compared to the total operating pressure so that the incompressible fluid assumption is valid the core should be homogeneous; The driving force and fluid properties are held constant Nc Nc end 10-5 10-8 10-6 10-8 to 10-5 Ͼ0.1 Ͼ1.0 Ͻ0.01 0.01 to 10 Capillary number The key flow parameter that controls the final fluid saturation is the capillary number Nc which defined as a ratio of viscous to capillary (interfacial tension) forces (Fulcher, Ertekin, and Stahl 1985) and generally have the form (Masalmeh 2012) of: Eq.2 Where: u is Darcy’s velocity, ␮ is the viscosity of the displacing phase, ␴ is the interfacial tension, ␪ is the contact angle and ␸ is the porosity in fraction In general, the actual flow within the reservoir is driven by viscous or gravitational forces However, capillary forces are usually present and may dominate Within pore scale the flow paths are determined by the capillary forces (Chandler et al 1982) If viscous forces are increased, they may become comparable to the capillary forces at the pore scale which may alter the flow paths and thus the relative permeability (Maas 2011) This change in flow dynamics can be reflected as a capillary number dependence of the relative permeability This dependence can play an important role in flow processes which may not applicable to the field situation, and therefore should be avoided in laboratory experiments, if proper and reliable basic reservoir data are to be obtained (Boom et al 1995, Masalmeh 2012, Maas 2011) The capillary end effect can be represented by a dimensionless flow parameter, Nc end which is the ratio of a characteristic capillary pressure to the viscous pressure drop across the core, and can be approximated (Mohanty and Miller 1991) by: Eq.3 Where: ␴ is the interfacial tension, ␮o is oil viscosity, L core length, ␸ is the porosity in fraction, and K is the absolute permeability This parameter has a critical value range in which it affects the relative permeability derived by analytical techniques For example, in water-wet for Nc endϾ0.1, both oil and water permeability decrease as Nc end increases however, when Nc end Ͻ 0.1 it does not affect relative permeability (Mohanty and Miller 1991) Table-1 summarizes the critical ranges of these parameters for strongly water-wet and weakly mixed-wet media and their typical ranges in the field and in the laboratory Bond number The Bond number measures the relative strength of gravity to capillary forces as described below (Masalmeh 2012): Eq.4 IPTC-18132-MS Figure 2—In situ saturation profiles at rate ft/day (Chen and Wood 2001) Where: ⌬␳ is the density difference between the displacing and displaced phase, g is the Acceleration due to gravity and K is the absolute permeability If the driving force for displacement is gravity (as in centrifuge experiments) its dominance can be checked with the Bond number De-saturation effects can cause changes in capillary pressure and relative permeability at high flow rate and/or low interfacial tension that (usually) not occur under normal field conditions To avoid these effects, the critical Bond number (ratio between gravitational and capillary forces) should be checked (Skauge, Thorsen, and Aarra 1997) concluded with experiments that centrifuge experiments can be clearly influenced by Bond number variations Furthermore, oil relative permeability increases with centrifugal speed for all classes of permeability they tested They showed that the remaining oil saturation vary with Bond number Critical Bond number values at which a speed insensitive relative permeability can be obtained is in the order of 10-5 This Bond number requirement implies an upper limit for the centrifugal acceleration Instability number One of the conditions for the onset of instability during two phase immiscible displacement is that the mobility ratio is higher than (Mϭ krw ␮o/kro ␮wϾ1) An unstable displacement leads to premature breakthrough and a longer period of two-phase flow at the outlet (Mohanty and Miller 1991) At present, no techniques are available, which allow the interpretation of unstable data (Maas 2011) Therefore, in laboratory experiments, this situation is undesirable In one dimensional laboratory experiments, especially for light oil, there is less potential for fingers to grow In case that there is a potential for front instability, the experiment can be designed such that the rates are increased in steps where the high rate is only applied at the end if capillary end effect is still present The effects of fingering and capillarity cannot be suppressed simultaneously At low rates, fingering is small, but the capillary end effect is high On the other hand, at high rates, fingering is large, but the capillary end effect is low (McPhee and Arthur 1994) For the critical ranges of instability number the reader is referred to (Mohanty and Miller 1991) Saturation Profile In-situ saturation monitoring of fluids inside the core during the experiments provide an insight of the flow processes and which will greatly assist in the interpretation of the data Some systems are Equipped with an x-ray scanner, which can determine the average in-situ fluid saturation at any location along the core during an experiment revealing the presence of experimental artifacts, such as rate and end effects or capillary discontinuities at plug junctions as shown in Fig.2 and Fig.3 The technique can also identify instability in the flood front due to rock heterogeneities or viscous fingering Since water saturation measurement is made in-situ, it is inherently less prone to the errors suffered by traditional volumetric IPTC-18132-MS Figure 3—In situ saturation profiles at rate 39 ft/day (Chen and Wood 2001) techniques (Trewin and Morrison 1993) In-situ saturation profiles can also be used as observed data to validate the process of numerical simulation Experiment’s report The main entrance to any successful quality check is a coherent and transparent experiment’s report highlighting any experimental difficulties encountered and indicates the most reliable data Engineers should ensure that provision of this information is part of the contract Relative permeability result quality checks Identifying reliable special core analysis results The first challenge is to differentiate those laboratory results which are considered reliable representative of the reservoir from those which are clearly invalid or highly questionable SCAL flow parameters, like capillary pressure, relative permeability and residual oil saturation depend strongly on the wetting state of the rock material being investigated Therefore, for the determination of a set of representative SCAL data, the wettability state of the core sample in the laboratory should approach as close as possible the wettability state of the reservoir Water-oil results are more likely to be unreliable due to factors associated with wettability Wettability can be altered as the result of: The drilling mud used during the coring operation, When and how the core was preserved and The procedures and conditions used in making measurements in the laboratory Understanding the history of the core material used in the special core analysis program, including coring fluid, preservation techniques and laboratory procedures used SCAL data obtained on plug which is taken with such care from the reservoir, that ideally all the properties including wettability are conserved at the surface should be representative Key considerations for include the drilling mud used to obtain the core material, the timing and method of core preservation, the laboratory handling of the core, i.e cleaning, drying, etc., the type of test conducted, steady or unsteady state, waterflood, centrifuge, etc., The fluids used and fluid properties, test conditions, i.e pressure, temperature, pressure differential, flooding rate, etc 10 IPTC-18132-MS Extraction of valid water-oil relative permeability data Conventionally, SCAL laboratory data are interpreted analytically In the interpretation, the effect of capillary pressure in a relative permeability experiment might be fully ignored In reality, both capillary pressure and relative permeability affect flow behaviour in any laboratory experiment accordingly laboratory data invariably needs to be refined prior to use in a reservoir simulation model or in hand calculations due to the following: First, it is sometimes necessary to disregard one portion of curves due to a particular weakness of the measurement technique An example of this is the unreliability of the high water saturation portion of unsteady state relative permeability results conducted at low rates on some samples Second, it is often necessary to extend the laboratory results beyond the saturation range covered in the test The review should summarize the amount of valid data of the various types for the field in question consequently it is often appropriate to classify the results of the review and divided into groups such as: ● ● ● Data considered to be valid for further use Data of questionable validity, but which may be required if there is inadequate clearly valid data, Data which is clearly invalid or highly questionable Integrating the results from different types of tests and recognizing which parts of each data set are reliable and which are invalid Thorough understanding of procedures and conditions used to obtain the results and the strengths and weaknesses of each special core analysis data set As it’s known that the unsteady state test provides reliable results within the mid saturation range and the experiment artefacts start to show up at high saturation range Therefore the centrifuge results will be used as supplementary of the unsteady state results Residual oil saturation Sor is very important characteristic in defining the relative permeability curves Centrifuge experiment may not achieve the true value of Sor, in this case Sor from post imbibition capillary pressure curve may be utilized which is not the subject of this paper Numerical simulations of laboratory experiments should be key element in Especially the use of different experimental techniques in combination with numerical simulations, aimed to history-match the different experimental data sets with a single set of capillary pressure and relative permeability curves, allows us to effectively combine the strengths of the various techniques, whilst at the same time consistently taking account of all artefacts Moreover, numerical simulations can be used in the experimental design stage, and for quality control on contractor interpretation of experimental data Water Oil relative permeability consistency check Below are the consistency checks that can be applied to the crude relative permeability (Stiles 2013, McPhee 2007, Glover 2002): There should be agreement between kro values at intermediate water saturations from valid waterflood tests and centrifuge oil relative permeability tests Results from the waterflood should generally be given greater weighting over intermediate water saturations where they not agree with results from the centrifuge kro test Values of residual oil saturation at the end of the various tests (on the same rock type) should be ordered as follows, with the residual oil saturation decreasing from top to bottom: ● ● ● Reservoir conditions waterflood, Centrifuge kro test, Centrifuge Pc test The krw end-point value at the end of an unsteady-state waterflood should be less than the krw end-point IPTC-18132-MS 19 Figure 13—Refined kro and krw, compared to reported laboratory data value for Sor using trial and error procedures with the data from relative permeability tests Inconsistent data should be studied for possible cause and may be discarded The basic Equation used follows: Eq.13 Eq.14 Eq.15 Eq.16 Eq.17 Eq.18 Eq.19 Eq.20 Eq.21 Where: ϭ value at kro ϭ does not Equal laboratory Sor The above parameters shoulf satisfy the limits given as per Table-3 The method is used as per the following steps: 20 IPTC-18132-MS Plot the laboratory values raised to an exponent of 0.25 versus laboratory Sw as shown in Fig.14 An analysis of the resulting plot will normally indicate either of the following two possibilities: Table 3—Wayne Beeks’ technique parameters limits Parameter Kro at Swi 1.0 A 0 at 0 B C Data plot in a good straight line This Krw indicates that the exponent n in equation is equal to 4.0 Extrapolation of line to gives the theoretical end point value ● Data plot exhibits a curved relationship This is expected result since an exponent n value of was derived under ideal and theoretical conditions Extrapolate the curved line to a value of equal to The estimated theoretical end point Sor value equal to (1-Sw) when Equals ● Calculate value of using values determined in Step and laboratory Sw values Plot values versus on log-log scale as shown in Fig.15 Analysis of this plot has the following two possibilities: Data plot in a good straight line This indicates that the value used to determine is correct Determine the slope of the line which Equals to n ● Data plot exhibits a curved relationship; if the curve is concave upward, the value used to determine must be reduced ● Estimate a new value of and recalculate Return to step and repeat as many times as necessary to obtain a good straight line on the log-log plot of versus Intermeadiate data points should be given the most weight since values at low (Sw - Swi) and high Sw values might be questionable Change the value of 0.25 if necessary With the theoretical end-point value obtained determine the slope n and plot ( )1/n versus Sw as shown in Fig.16 This step is an essentially check and data should be in a good straight line Plot Kro versus on log-log scale as shown in Fig.17 and determine the intercept A Plot value versus the final determined in step on log-log scale as shown in Fig.18 Analysis of the plot has two possibilities: Data plot on straight line This is the expected result Intermediate data points should again be given the most weight ● Data not plot in a good straight line; if satisfied that there is no problem with the data and good result was obtained, force a straight line through the data point This failure to obtain a good straight line can be considered to be caused by laboratory errors in most cases Determine the slope m and extrapolate the line to a value of ϭ 1.0, the intercept value is Equal to the following: Eq.22 ● Plot Krw value versus the final on log-log and determine the intercept C as shown in Fig.19 Final check is the C coefficient should equal A ϫ B Calculate values for kro and krw over the full saturation range, i.e from Soϭ1-Swi to SoϭSor, using the characteristics defined from the previous steps IPTC-18132-MS 21 Figure 14 —kro raised to an exponent of 0.25 versus laboratory Sw Figure 15—Estimation of the slope n 22 IPTC-18132-MS Figure 16 —Check of the slope n Figure 17—Estimation of the intercept A History matching of displacement experiment Classical analytical techniques, such as JBN, cannot adequately describe fluids displacement occurring during the unsteady-state experiment Most of those techniques are based on (Buckley and Leverett 1942) model and its modification by (Welge 1952) The model neglects both capillary pressure and gravitational effects and assumes perfectly dispersed flow with no core heterogeneities in its basic derivation This means that the classical analytical techniques, cannot account for end effect phenomena and the dispersing effect of capillary pressure on saturation shock fronts within porous media (Bennion and Thomas 1991) IPTC-18132-MS 23 Figure 18 —Estimation of the slope m and the intercept B Figure 19 —Estimation of the intercept C The crude assumption of neglecting capillary pressure is always wrong, since the range of saturation obtained during a displacement is always the result of capillary and viscous forces (Lenormand 2006) (Kokkedee et al 1996) state “Recent state-of-the-art SCAL measurements at representative wettability conditions and taking account of both relative permeability and capillary pressure artefacts in the interpretation, systematically point to lower residual oil saturations for a growing number of reservoirs all over the world The reduction in residual oil saturation typically amounts to 10-15 saturation percent 24 IPTC-18132-MS as compared to old estimates Crucial is the use of a combination of experimental techniques and the use of a numerical simulator for history matching the obtained laboratory data” Furthermore, the analysis theory is based on fractional flow data which can only predict relative permeability data after water breakthrough In strongly water wet rocks, a water displacement results in an almost piston like flow of water through the core resulting in a very steep and localized region of fractional flow This, therefore, results in only a very small cluster of relative permeability data points being obtained at saturations near the maximum level (Bennion and Thomas 1991) Thus the relative permeability data at intermediate saturation levels is not defined unless significant extrapolation effort is considered (Sigmund and McCaffery 1979) pointed out that those techniques can produce some anomalies and abnormal shape in relative permeabilities curves due to microscopic heterogeneities within core samples (Bennion and Thomas 1991) explained this issue as a result of non-monotonicity in the resultant fractional flow data, which can often occur in heterogeneous core samples, and since these techniques are based on the evaluation of derivatives of the fractional flow curves severe deviations in the computed relative permeability data can occur Valid relative permeability data can only be acquired if the interpretational model allows for capillarity, viscous instability, wettability dependence, and permeability heterogeneity to be considered Displacements yet are often inconsiderately applied The only way to account for the above mentioned phenomena is to use numerical simulation Numerical simulation of experimental fluid displacement is always referred to as history matching It was first proposed by (Archer and Wong 1973) The procedure requires the relative permeability data to be fitted with representative function and controlled by end points and curvature parameters The results of the analytical calculation are used as a first guess The simulated effluent production and pressure difference are compared to the experimental values and the difference is quantified by calculating an objective function The controlling parameters of relative permeability function are adjusted in order to minimize the objective function The adjustment can be made manually or using an automatic optimization program The performed history could either use: Relaxed Approach: Simultaneous tuning of parameters that define both functions of relative permeability input and capillary pressure to match laboratory experiments measurements Restricted Approach: Tuning of relative permeability function parameters with fixed input capillary pressure Based on the quality of the match the best fit curves are obtained for both relative permeability and capillary pressure The heterogeneity can be accounted for by adding the absolute permeability distribution to the history matching variables The absolute permeability can be populated geostatically and distribution can be either stochastically or guided by a CT scan for the core Below, from Fig.20 to Fig.24, are the results for the history matching process of the unsteady-state experiment which its relative permeability used previously Optimum history matching process requires the following tools: Flexible functional representation of relative permeability Flexible functional representation of capillary pressure 1-D flow simulator Rich objective function estimator for the best fit Optimization tool with its auxiliaries Geostatical tool for stochastic property distribution or, more preferably, guided by a CT scan images Although this process looks appealing to account for most of the physical phenomena that can appear during the displacement, yet it is not without limitations As any other inverse problem this method will IPTC-18132-MS 25 Figure 20 —Relative permeability curves from Lab, Refinements, and Simulation Figure 21—History matching of displacement experiment raw output suffer from the non-uniqueness Another drawback is the choice of the functional representation of relative permeability which limits the possibility of novel shapes for the relative permeability curves(Hussain, Cinar, and Bedrikovetsky 2010) QC for simulation use Those who are involved in numerical reservoir simulation can realize the impact of good relative permeability data on the simulator performance In many cases, poor simulation convergence may be 26 IPTC-18132-MS Figure 22—Range of relative permeability curves used for history matching process Figure 23—Simulation of displacement experiment caused by relative permeability data issues In reservoir simulation relative permeabilities are used to calculate fluid mobility to solve the flow Equations between cells and from cell to well Eq.23 The primary solution variables in reservoir simulation are pressure and saturations Eq.24 The solution scheme is formulated in terms of errors associated with each phase in each cell of the reservoir model which is commonly referred to as residual Eq.23 and Eq.25 The nonlinear system of the residual Equations Eq.24 is solved iteratively with Newton-Raphson method The solver nonlinear iterations utilize derivatives (Jacobian Eq.26) to solve for the solution variables (pressure and saturations) Therefore, KINKS, drastic and/or erratic changes in relative permeability data can greatly affect the derivative of relative permeabilities with saturation and disturbed the solution IPTC-18132-MS 27 Figure 24 —Saturation profile revealed by simulation of displacement experiment Figure 25—Water relative permeability Eq.23 Eq.24 Eq.25 Eq.26 Eq.27 28 IPTC-18132-MS Figure 26 —Water relative permeability derivative Figure 27—Residual of water and oil phases o worst converging cell Figure 28 —Number of non-linear iterations needed for every timestep Although such issues may not stop the simulation completely, however, they can severely slow down the simulation performance by shortening the timesteps required for the solution which results in many solver iterations In some cases, the solution can be directed to saturation where the iteration will circle within a residual cycle In another word, the first iteration has a positive residual value, then negative residual value, and so on In this case the simulator may try to alter the saturation changes and chop the timestep to suppress possible oscillations Any saturation change alteration made will consequently increase material balance errors for the subsequent non-linear iteration and therefore reduces the chances of convergence In Appendix-A shows a simple mathematical example illustrates the performance of Newton-Raphson method within a steep slope region A synthetic example of oil with water is demonstrated below (Fig.25 and Fig.26) that shows the effect of relative permeability data on the simulator performance Although the plot of the water relative permeability curve may appear reasonably fine, the derivative one is severely disturbed at saturation values (0.285) and (0.565) respectively IPTC-18132-MS 29 Figure 29 —Number of linear iterations needed for every timestep Figure 30 —Cumulative CPU time consumed throughout the full simulation When this relative permeability data was in a simulation model, the simulation performance has been drastically hindered by and the timesteps was chopped more frequently to adapt with those drastic saturation changes Fig.27 shows the water and oil residuals for one of the worst converging cells are shown in Both residuals are oscillating within the same a residual cycles Adjusting relative permeability curves slightly can solve the problem in many cases The adjustment should be minimal and in such a way that ensure the slope of the data monotonically increases with increasing saturations and without sudden sharp changes Once the relative permeability data was adjusted to have smooth and monotonic derivative the simulation performance increased 79% in the CPU time and total number of non-linear iteration dropped from 1268 to 106 iterations Fig.28, Fig.29 and Fig.30 show the performance of the simulation model in terms of non-linear iterations, linear iterations, and cumulative CPU time There is another process in reservoir simulation which can insidiously produce steep and a drastic derivative changes in relative permeability data is the End-Points-Scaling In some cases, especially when 30 IPTC-18132-MS it is applied without limits, End-Points-Scaling can squeeze the area between the residual oil saturation and the maximum water saturation in the table resulting in a very steep and almost near vertical line which can cause convergence problems Accordingly, it is recommended to check the output of the End-PointsScaling process and confine it with limits if needed The intention from this section is to draw the attention of the reservoir engineer to the impact of the relative permeability data quality on the simulation performance In summary, KINKS, drastic and/or erratic changes, and discontinuities should be avoided as much as it is possible in relative permeability data and their derivatives and should have smooth and monotonic derivatives Those checks should also be done for End-Point-Scaling output especially if the process is applied without limits Conclusions Compilation of all contemporary techniques for relative permeability quality check and validation was presented to provide a reference for reservoir engineers who have an interest in investigating, checking the quality, and preparing relative permeability data set usable for reservoir simulation process This part was made to facilitate the road for the subsequent parts of this topic References Ahmed, Tarek H 2001 Reservoir engineering handbook Boston, Gulf Professional Pub (Reprint) http://www.knovel.com/knovel2/Toc.jsp?BookIDϭ797 Al-Mossawy, Mohammed Idrees, Birol Demiral 2011 An Improved Relative Permeability Model to Match Displacement Experiments (in International Journal of Applied 1(2) Archer, John S., S.W Wong 1973 Use of a Reservoir Simulator To Interpret Laboratory Waterflood Data (in English) Society of Petroleum Engineers Journal 13(6): 343–347 http://www.onepetro.org/mslib/app/Preview.do?paperNumberϭ00003551&societyCodeϭSPE Bauget, F, S Gautier, R Lenormand et alet al 2012 GAS-LIQUID RELATIVE PERMEABILITIES FROM ONE-STEP AND MULTI-STEP CENTRIFUGE EXPERIMENTS (in Behrenbruch, Peter, Hussam Mohammed Goda 2008 Two-Phase Relative Permeability Prediction: A Comparison of the Modified Brooks-Corey Methodology with a New Carman-Kozeny Based Flow Formulation Proc Bennion, D B., F B Thomas 1991 Recent Improvements in Experimental and Analytical Techniques for the Determination of Relative Permeability Data from Unsteady State Flow Experiments Proc., Annual Technical Meeting, Port of Spain, Trinidad Boom, W., K Wit, A M Schulte et alet al 1995 Experimental Evidence for Improved Condensate Mobility at Near-wellbore Flow Conditions Proc Buckley, S E., M C Leverett 1942 Mechanism of Fluid Displacement in Sands (in English) http://www.onepetro.org/mslib/app/Preview.do?paperNumberϭSPE-942107-G&societyCodeϭ SPE Chandler, Richard, Joel Koplik, Kenneth Lerman et alet al 1982 Capillary displacement and percolation in porous media (in Journal of Fluid Mechanics 119: 249 –267 http://dx.doi.org/ 10.1017/S0022112082001335 Chen, AL, AC Wood Rate effects on water-oil relative permeability Vol 19, 1–12 Corey, Arthur T 1954 The interrelation between gas and oil relative permeabilities (in Producers monthly 19(1): 38 –41 Dandekar, Abhijit Y 2013 Petroleum reservoir rock and fluid properties, CRC press (Reprint) Edwards, JT, MM Honarpour, RD Hazlett et alet al Validation of gravity-dominated relative permeability and residual oil saturation in a giant oil reservoir Fanchi, John R 2006 Principles of applied reservoir simulation Amsterdam, Gulf Professional Pub (Reprint) http://www.knovel.com/knovel2/Toc.jsp?BookIDϭ2078 IPTC-18132-MS 31 Fulcher, R A., Jr., Turgay Ertekin, C D Stahl 1985 Effect of Capillary Number and Its Constituents on Two-Phase Relative Permeability Curves (in Glover, Paul WJ 2002 Formation evaluation, Université Laval: Quebec City, Quebec, Canada (Reprint) Grimstad, AA, K Kolltveit, JE Nordtvedt et alet al The uniqueness and accuracy of porous media multiphase properties estimated from displacement experiments Vol 9709 Hagoort, J 1980 Oil Recovery by Gravity Drainage (in Haugen, Jack Scaling criterion for relative permeability experiments on samples with intermediate wettability Heaviside, John, C.J.J Black 1983 Fundamentals of Relative Permeability: Experimental and Theoretical Considerations Proc., SPE Annual Technical Conference and Exhibition, San Francisco, California Hirasaki, GJ, JA Rohan, JW Dudley 1995 Interpretation of oil/water relative permeabilities from centrifuge displacement (in SPE Advanced Technology Series 3(01): 66 –75 Honarpour, Mehdi, L F Koederitz, A Herbert Harvey 1982 Empirical Equations for Estimating Two-Phase Relative Permeability in Consolidated Rock (in Honarpour, MM, F Koederitz, A Herbert 1986 Relative permeability of petroleum reservoirs (in Hussain, Furqan, Yildiray Cinar, Pavel G Bedrikovetsky 2010 Comparison of Methods for Drainage Relative Permeability Estimation From Displacement Tests Proc Ibrahim, M N Mohamad, L F Koederitz 2000 Two-Phase Relative Permeability Prediction Using a Linear Regression Model Johnson, E.F., D.P Bossler, V.O Naumann 1959 Calculation of Relative Permeability from Displacement Experiments Proc Jones, S.C., W.O Roszelle 1978 Graphical Techniques for Determining Relative Permeability From Displacement Experiments (in English) Journal of Petroleum Technology 30(5): 807–817 http:// www.onepetro.org/mslib/app/Preview.do?paperNumberϭ00006045&societyCodeϭSPE Kokkedee, JA, W Boom, AM Frens et alet al Improved special core analysis: scope for a reduced residual oil saturation Vol 9601, 1–13 Kyte, J R., L A Rapoport 1958 Linear Waterflood Behavior and End Effects in Water-Wet Porous Media (in Lenormand, R 2006 Conventional and Special Core Analysis, Application to Reservoir Modeling (in IFP Institute Français du Pétrole Li, K, P Shen, T Qing 1994 A New Method for Calculating Oil-Water Relative Permeabilities with Consideration of Capillary Pressure (in Maas, Jos Maas Special Core Analysis Simulation http://www.jgmaas.com/scores/ Masalmeh, SK Impact of capillary forces on residual oil saturation and flooding experiments for mixed to oil-wet carbonate reservoirs 27–30 McPhee, C A., K G Arthur 1994 Relative Permeability Measurements: An Inter-Laboratory Comparison Proc McPhee, Colin 2007 Core Analysis Training Course, Senergy Ltd (Reprint) Mitlin, V S., B D Lawton, J D McLennan et alet al 1998 Improved Estimation of Relative Permeability from Displacement Experiments Proc Mohanty, K.K., A.E Miller 1991 Factors Influencing Unsteady Relative Permeability of a MixedWet Reservoir Rock (in English) SPE Formation Evaluation 6(3): 349 –358 http://www onepetro.org/mslib/app/Preview.do?paperNumberϭ00018292&societyCodeϭSPE Mohanty, Kishore K 2002 IMPACT OF CAPILLARY AND BOND NUMBERS ON RELATIVE PERMEABILITY Other Information: PBD: 30 Sep 2002 32 IPTC-18132-MS Odeh, A.S., B.J Dotson 1985 A Method for Reducing the Rate Effect on Oil and Water Relative Permeabilities Calculated From Dynamic Displacement Data (in English) Journal of Petroleum Technology 37(11): 2051–2058 http://www.onepetro.org/mslib/app/Preview.do?paperNumberϭ 00014417&societyCodeϭSPE Osoba, J S., J G Richardson, J K Kerver et alet al 1951 Laboratory Measurements of Relative Permeability (in Pirson, SJ 1958 Petrophysics: Oil Reservoir Engineering (2nd ed) New York, USA: McGraw-Hill Book Co., Inc (Reprint) Qadeer, Suhail, William E Brigham, Louis M Castanier 2002 Techniques to Handle Limitations in Dynamic Relative Permeability Measurements SUPRI TR, Stanford University Dept of Petroleum Engineering Rapoport, L A 1955 Scaling Laws for Use in Design and Operation of Water-Oil Flow Models, Society of Petroleum Engineers (Reprint) Rapoport, L.A., W.J Leas 1953 Properties of Linear Waterfloods (in English) Journal of Petroleum Technology 5: 139 –148 Sigmund, P.M., F.G McCaffery 1979 An Improved Unsteady-State Procedure for Determining the Relative-Permeability Characteristics of Heterogeneous Porous Media (includes associated papers 8028 and 8777) (in English) Society of Petroleum Engineers Journal 19(1): 15–28 http:// www.onepetro.org/mslib/app/Preview.do?paperNumberϭ00006720&societyCodeϭSPE Skauge, A, T Thorsen, MG Aarra 1997 Accuracy of gas-oil relative permeability from two-phase flow experiments (in Skauge, Arne, Trond Thorsen, Andre Sylte Rate selection for waterflooding of intermediate wet cores 17–19 Stiles, Jess 2013 Using Special Core Analysis in Reservoir Engineering Imperial College London, Imperial College London (Reprint) Timmerman, EH 1982 Practical reservoir engineering, PennWell Corporation (Reprint) Torcaso, Michael A., M R J Wyllie 1958 A Comparison of Calculated krg/kro Ratios With a Correlation of Field Data (in Toth, Janos, Tibor Bodi, Peter Szucs et alet al 1998 Practical Method for Analysis of Immiscible Displacement in Laboratory Core Tests (in English) Transport in Porous Media 31(3): 347–363 http://dx.doi.org/10.1023/A%3A1006570117639 Toth, Janos, Tibor Bodi, Peter Szucs et alet al 2001 Direct Determination of Relative Permeability from Nonsteady-State Constant Pressure and Rate Displacements Proc Trewin, B, S Morrison 1993 Reconciliation of core and log residual oil saturation through application of in situ saturation monitoring (in Welge, H J 1952 A Simplified Method for Computing Oil Recovery by Gas or Water Drive (in English) http://www.onepetro.org/mslib/app/Preview.do?paperNumberϭSPE-000124G&societyCodeϭSPE Wyllie, M.R.J., G.H.F Gardner 1958 The Generalized Kozeny-Carmen Equation-Its Application to Problems of Multi-Phase Flow in Porous Media (in World Oil 146 (121) IPTC-18132-MS 33 Appendix-A A simple mathematical example which clearly demonstrates the performance of Newton-Raphson method is the Error Function If the error function is solved with Newton-Raphson method to find the solution that makes (yϭ0) starting with initial condition of (xoϭ 0.996) and convergence tolerance of (0.001) the run will fail after five iterations Fig.31 In the first four iterations will be cycling around the solution (yϭ0) but cannot get it due to the sharp slope at that region Figure 31—Newton-Raphson method performance with the Error Function [...]... pressure and gravitational effects and assumes perfectly dispersed flow with no core heterogeneities in its basic derivation This means that the classical analytical techniques, cannot account for end effect phenomena and the dispersing effect of capillary pressure on saturation shock fronts within porous media (Bennion and Thomas 1991) IPTC -18132- MS 23 Figure 18 —Estimation of the slope m and the intercept... www.onepetro.org/mslib/app/Preview.do?paperNumberϭ00006045&societyCode SPE Kokkedee, JA, W Boom, AM Frens et alet al Improved special core analysis: scope for a reduced residual oil saturation Vol 9601, 1–13 Kyte, J R., L A Rapoport 1958 Linear Waterflood Behavior and End Effects in Water-Wet Porous Media (in Lenormand, R 2006 Conventional and Special Core Analysis, Application to Reservoir Modeling (in IFP... Influencing Unsteady Relative Permeability of a MixedWet Reservoir Rock (in English) SPE Formation Evaluation 6(3): 349 –358 http://www onepetro.org/mslib/app/Preview.do?paperNumberϭ00018292&societyCode SPE Mohanty, Kishore K 2002 IMPACT OF CAPILLARY AND BOND NUMBERS ON RELATIVE PERMEABILITY Other Information: PBD: 30 Sep 2002 32 IPTC -18132- MS Odeh, A.S., B.J Dotson 1985 A Method for Reducing the Rate Effect... Oil Recovery by Gas or Water Drive (in English) http://www.onepetro.org/mslib/app/Preview.do?paperNumber SPE- 000124G&societyCode SPE Wyllie, M.R.J., G.H.F Gardner 1958 The Generalized Kozeny-Carmen Equation-Its Application to Problems of Multi-Phase Flow in Porous Media (in World Oil 146 (121) IPTC -18132- MS 33 Appendix-A A simple mathematical example which clearly demonstrates the performance of Newton-Raphson... simulator performance Although the plot of the water relative permeability curve may appear reasonably fine, the derivative one is severely disturbed at saturation values (0.285) and (0.565) respectively IPTC -18132- MS 29 Figure 29 —Number of linear iterations needed for every timestep Figure 30 —Cumulative CPU time consumed throughout the full simulation When this relative permeability data was in... There is another process in reservoir simulation which can insidiously produce steep and a drastic derivative changes in relative permeability data is the End-Points-Scaling In some cases, especially when 30 IPTC -18132- MS it is applied without limits, End-Points-Scaling can squeeze the area between the residual oil saturation and the maximum water saturation in the table resulting in a very steep and... Mobility at Near-wellbore Flow Conditions Proc Buckley, S E., M C Leverett 1942 Mechanism of Fluid Displacement in Sands (in English) http://www.onepetro.org/mslib/app/Preview.do?paperNumber SPE- 942107-G&societyCodeϭ SPE Chandler, Richard, Joel Koplik, Kenneth Lerman et alet al 1982 Capillary displacement and percolation in porous media (in Journal of Fluid Mechanics 119: 249 –267 http://dx.doi.org/ 10.1017/S0022112082001335... Black 1983 Fundamentals of Relative Permeability: Experimental and Theoretical Considerations Proc., SPE Annual Technical Conference and Exhibition, San Francisco, California Hirasaki, GJ, JA Rohan, JW Dudley 1995 Interpretation of oil/water relative permeabilities from centrifuge displacement (in SPE Advanced Technology Series 3(01): 66 –75 Honarpour, Mehdi, L F Koederitz, A Herbert Harvey 1982 Empirical... for kro and krw over the full saturation range, i.e from Soϭ1-Swi to SoϭSor, using the characteristics defined from the previous steps IPTC -18132- MS 21 Figure 14 —kro raised to an exponent of 0.25 versus laboratory Sw Figure 15—Estimation of the slope n 22 IPTC -18132- MS Figure 16 —Check of the slope n Figure 17—Estimation of the intercept A History matching of displacement experiment Classical analytical... ● ● ● Were special efforts made to preserve natural wettability? Was the test conducted on individual or composite cores? Was reservoir oil or refined oil used? Was the test conducted at room or reservoir conditions? What test procedure was used: steady-state, unsteady-state or centrifuge? At what rate was the core flooded? What was the pressure differential at the end of the flood? Was special attention