US20050216196A1 - Contamination estimation using fluid analysis models - Google Patents

Contamination estimation using fluid analysis models Download PDF

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US20050216196A1
US20050216196A1 US11/022,016 US2201604A US2005216196A1 US 20050216196 A1 US20050216196 A1 US 20050216196A1 US 2201604 A US2201604 A US 2201604A US 2005216196 A1 US2005216196 A1 US 2005216196A1
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contamination
fluid
fluids
viscosity
drawn
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Ridvan Akkurt
Mark Proett
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Halliburton Energy Services Inc
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    • E—FIXED CONSTRUCTIONS
    • E21—EARTH OR ROCK DRILLING; MINING
    • E21B—EARTH OR ROCK DRILLING; OBTAINING OIL, GAS, WATER, SOLUBLE OR MELTABLE MATERIALS OR A SLURRY OF MINERALS FROM WELLS
    • E21B49/00—Testing the nature of borehole walls; Formation testing; Methods or apparatus for obtaining samples of soil or well fluids, specially adapted to earth drilling or wells
    • E21B49/08—Obtaining fluid samples or testing fluids, in boreholes or wells
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N24/00—Investigating or analyzing materials by the use of nuclear magnetic resonance, electron paramagnetic resonance or other spin effects
    • G01N24/08—Investigating or analyzing materials by the use of nuclear magnetic resonance, electron paramagnetic resonance or other spin effects by using nuclear magnetic resonance
    • G01N24/081—Making measurements of geologic samples, e.g. measurements of moisture, pH, porosity, permeability, tortuosity or viscosity

Definitions

  • This invention relates to systems and methods for determining the level of mud filtrate contamination in formation fluids.
  • Drilling mud also known as drilling fluid
  • Drilling fluid is typically pumped down the center of the hollow drill stem to emerge again at the surface of the borehole. It lubricates the drill shaft, cools the borehole, carries away the drilling detritus, and serves as a wetting-phase, which facilitates the flow of hydrocarbons from the formation and into the borehole.
  • Various types of drilling muds are generally classified based on the type of filtrate used therein. The mud filtrate chosen dictates the mud's function and performance, as well as formation invasion effects.
  • Water-based mud (WBM) filtrates include, but are not limited to, freshwater, seawater, saltwater (brine) and others, or a combination of any of these fluids.
  • OBM oil-based mud
  • the filtrate is an oil product, such as diesel or mineral oil.
  • oil-based mud is characterized as any type of non-aqueous fluid.
  • oil-based mud also includes the recently developed variety of oil mud that is also referred to as synthetic-based muds.
  • synthetic-based muds include, without limitation, olefinic-, naphthenic-, and paraffinic-based compounds.
  • different factors affect the ability of the tool the accurately estimate the contamination levels at a given point during pumpout.
  • the OFA measures the optical density (OD) of the flowing fluid and uses empirical relationships to transform the OD into data on contamination by determining the composition of the measured absorbed light spectrum from the sample. Based on this absorption spectrum one can estimate the types of materials present in the fluid and the proportion of each material in the fluid. While the industry has learned how to interpret OFA data over the years, it still is not robust in certain applications where the color contrast is small, or is masked, as is frequently the case in light oils and condensates. One problem with this approach is that it assumes that the measured property is directly linked to the contamination, which may not necessarily be the case.
  • NMR nuclear magnetic resonance
  • the CMR tool is described, for example, in U.S. Pat. Nos. 5,055,787 and 5,055,788 to Kleinberg et al. and further in “Novel NMR Apparatus for Investigating an External Sample,” by Kleinberg, Sezginer and Griffin, J. Magn. Reson. 97, 466-485, 1992.
  • NMR devices, methods and pulse sequences for use in logging tools are also in U.S. Pat. Nos. 4,350,955 and 5,557,201.
  • the content of the above patents and publications is hereby expressly incorporated by reference for background. A brief discussion of the main NMR measurement parameters follows.
  • NMR measurements are based on exposing an assembly of magnetic moments, such as those of hydrogen nuclei, to a static magnetic field.
  • the assembly tends to align along the direction of the magnetic field, resulting in a bulk magnetization.
  • a magnetic field having direction perpendicular to the static magnetic field is applied to rotate the magnetic moments away from the direction of the bulk magnetization.
  • the rate at which the rotated moments return to the equilibrium bulk magnetization after application of the oscillating magnetic field is characterized by the parameter T 1 , known as the spin-lattice relaxation time.
  • T 1 values are in the range of milliseconds to several seconds.
  • T 2 spin-spin relaxation time constant
  • T 1 spin-spin relaxation time constant
  • Both relaxation times provide information about the properties of the formation fluid, such as the formation porosity and the composition and quantity of the formation fluid.
  • D diffusion coefficient
  • ⁇ viscosity
  • Viscosity and diffusivity are both related to the translational motion of molecules and therefore are interrelated.
  • T a molecule contains more energy and can move faster against a given “friction” ⁇ , therefore D is proportional to the temperature.
  • Diffusivity is a property that can be precisely determined by NMR techniques without disturbing or altering the fluid. The relationship D ⁇ T/ ⁇ has been verified over a wide range of viscosities at different temperatures and pressures by NMR spin-echo experiments.
  • diffusion In a uniform magnetic field, diffusion has little effect on the decay rate of the measured NMR echoes. In a gradient magnetic field, however, diffusion causes atoms to move from their original positions to new ones, which also causes these atoms to acquire different phase shifts compared to atoms that did not move. This contributes to a faster rate of relaxation.
  • FIG. 5 shows a simplified diagram of a downhole NMR fluid analysis apparatus, such as the MRILab®, that provides NMR measurements to which the contamination estimation methods of the present disclosure can be applied in an illustrative embodiment. Fluids enter the device at the top and pass through two sections, referred to as polarization and resonance sections, respectively. Measurements are performed as the fluid flow passes through the device.
  • U.S. application Ser. No. 10/109,072 which is hereby incorporated by reference, discloses details of this device, which are summarized for reference in Appendix A.
  • the sharpness parameter is derived from the T 1 distribution, which to a certain extent can be affected by the level signal-to-noise ratio (SNR) or distribution shape, which may lead to changes in the T 1 spectra that are unrelated to changes in contamination.
  • SNR signal-to-noise ratio
  • a contaminant could be any fluid originating from mud filtrate that invades the reservoir during the drilling of a well; contamination (c) is the volume fraction of the contaminant in a fluid sample, where 0 ⁇ c ⁇ 1.
  • contamination function is a temporal function that substantially matches the behavior of the contamination fluid fraction while pumping a sample from an invaded zone. More generally, a contamination function is applicable to multiple-component fluid systems including systems comprised of miscible or immiscible liquids, including liquids containing dissolved, suspended or dispersed solids.
  • fluid mixture includes in its meaning a mixture of liquids (either miscible or immiscible) or a liquid containing soluble, suspended or dispersed solids.
  • a contamination function is a mathematical model that may be derived, for example, through simulations or observation.
  • a mixing law (or mixing rule), as used in this disclosure, is a mathematical function that describes a property of a mixture in terms of the properties of its constituents.
  • a mixing law would thus allow for the property of the mixture to be predicted if the weight or volume functions for the constituents, and the properties of these constituents, are known.
  • mixing laws are provided in the illustrative embodiment in which the physical property of the fluid being monitored is its viscosity. Different mixing laws may apply for other physical properties. In general, all that is required for the mixing law is to provide a mathematical expression that relates a property of a fluid mixture in terms of the constituent components properties (i.e., the values of such property for 0% and 100% contamination, respectively).
  • the proposed contamination estimation approach is based on the use of a contamination function that describes the time behavior of a particular physical property of the mixture of fluids entering a tool, and a mixing-law or rule that is used to estimate the volume fractions of the constituent fluids given information about or measurements of the bulk physical property.
  • the physical property is viscosity, which is monitored indirectly using relaxation measurements (i.e., T 1 or T 2 ) obtained from an NMR fluid analyzer, such as the MRILab® fluid analyzer.
  • relaxation measurements i.e., T 1 or T 2
  • Other physical properties that can be used in alternate embodiments include resistivity, capacitance, density, hydrogen index, compressibility, speed of sound, pumping pressures, optical density, and others.
  • the determined values of one or more of the above fluid properties over time are fit to a parameterized contamination function.
  • the variables of this time function are determined, for example, through regression. Once the variables of the contamination function are determined, this function can predict how the fluid property changes over time. Then, the mixing law function that relates fluid fractions to the bulk fluid property can be used to estimate the actual contamination over past or future time periods.
  • the contamination function used to predict the fluid property is either independent or substantially independent from the fluid fractions in the mixing law. These embodiments are referred to as uncoupled or loosely coupled contamination models.
  • the mixing laws are taken into account in tracking the bulk fluid property over time, resulting in a coupled contamination modeling. Different examples of such models are illustrated in the specific embodiments described below.
  • the log-mean value of a NMR spectrum in particular the log-mean T 1 value (T 1Lm ) is shown to track the viscosity or other properties of the fluid and is used as an indirect property measurement. From the contamination function that best fits the empirical data, and a mixing law that includes estimates of the property values at two different contamination levels (typically at 0% and 100% contamination) one can compute the contamination index of the fluid (considered below) as a function of time.
  • the contamination function used in a particular experiment may be determined in advance through prior knowledge. Further, the accuracy of prediction can be improved if one has a priori knowledge about the monitored property value at 0% and 100% contamination.
  • several contamination functions corresponding to different physical properties can be used to monitor the time behavior of the fluid mixture entering the tool. These embodiments are based on the observation that in certain fluid mixtures one property may be more sensitive than others to the contaminant. Accordingly, an array of devices can be used in such embodiments to measure different fluid properties. Using this approach, for example, different contamination estimates can be combined into a single average contamination estimate. Individual contamination estimates may be weighted, possibly using nonlinear regression techniques. As is the case with MRILab® estimates of viscosity, the formation fluid properties can be more accurately predicted because the end points are used to determine the in-situ sample properties.
  • an object of this disclosure to provide methods for estimation of the level of contamination in the fluids flowing through an analyzer tool. Another object of the disclosure is to provide methods for estimating the pumping time needed to achieve a certain contamination threshold level. Yet another object of the disclosure is to provide methods for monitoring at least one physical property of the fluid entering the tool, such as its viscosity, and based on mixing laws that are known or can be derived use the monitored property to derive an estimate of the contamination level. Additional objective is to provide a computationally efficient algorithm for contamination estimation that can be implemented substantially in real time.
  • the invention is a method for estimating levels of contamination of formation fluids in a borehole, and a corresponding system implementing the steps of: (a) providing a first mathematical contamination function model which expresses a time behavior of one or more fluid properties of a mixture of formation fluids and contamination fluids drawn from a borehole, where said one or more fluid properties are sensitive to the fraction of contamination fluids in the mixture; (b) providing a second mathematical mixing law function model expressing at least one of said one or more fluid properties of a fluid mixture in terms of corresponding properties of formation fluids and contamination fluids in the mixture; (b) drawing fluids from the borehole into a fluid analyzer; (c) measuring at least one of said one or more properties of the drawn fluids by the analyzer; and (d) estimating the level of contamination in the fluids drawn from the borehole at one or more points in time based on the at least one measured fluid property and the provided mathematical models in step (a) and step (b).
  • the invention is a method for downhole formation testing, comprising the steps of: (a) providing measurement signals corresponding to a mixture of formation fluids and contamination fluids drawn from a borehole, the mixture entering a downhole fluid analyzer; (b) based on the provided measurement signals, determining parameters of a contamination function, which expresses the time behavior of one or more fluid properties of the fluid mixture drawn from the borehole; (c) computing from the determined contamination function of a value estimate of said one or more fluid properties for at least one low level of contamination and for at least one high level of contamination; and (d) computing a contamination index for the mixture of fluids drawn from the borehole at different time instants based on the computed value estimates and a fluid mixing law relating properties of the drawn fluids in terms of the corresponding properties of the formation fluids and the contamination fluids.
  • FIGS. 1A and 1B show T 1 data and data analysis from several experiments performed under downhole conditions
  • FIG. 2 shows the results of sample quality vs. time (seconds) simulations for an oil-based mud (OBM) systems, where sample quality is given in percentages;
  • OBM oil-based mud
  • FIG. 3 shows the results of sample quality vs. time (seconds) simulations for a water-based mud (WBM) system at 10 cc/sec pumping rate, where sample quality is given in percentages;
  • WBM water-based mud
  • FIG. 4 illustrates the use of NMR fluid analyzer measurements for contamination estimation in one embodiment
  • FIG. 5A is a schematic diagram of a downhole NMR fluid analyzer
  • FIG. 5B is a simplified version
  • FIGS. 6 A-D illustrate horizontal cross sections of the analyzer shown in FIGS. 5A and B;
  • FIG. 7 illustrates a schematic block diagram of the electronics of the analyzer in one embodiment
  • FIG. 8 illustrates a saturation recovery pulse sequence diagram used for T 1 measurements
  • FIG. 9 shows T 1 saturation-recovery data for three different fluids as seen in an NMR fluid analyzer
  • FIG. 10 shows a re-plot in the T 1 domain of the data shown in FIG. 9 ;
  • FIG. 11 illustrates a pulse sequence employed for diffusivity measurements
  • FIG. 12 shows a diffusivity measurement
  • FIG. 13 shows the results of contamination estimation for the data shown in FIG. 1A using a Coupled Log-mean T 1 model
  • FIG. 14 shows the results of contamination estimation for the data shown in FIG. 1A using an immiscible fluid model.
  • the principal objective of contamination estimation is to secure fluid samples with low levels of contamination for proper Pressure Volume Temperature (PVT) analysis.
  • Another objective is to predict when to stop pumping with a high level of confidence as to not waste rig time on unnecessary clean up pumping.
  • Improved contamination estimates increase the accuracy of the log analysts' data interpretation, and help to provide better estimates of permeability and anisotropy from the pumpout data.
  • the proposed contamination estimation approach described in different embodiments below is based on the use of one or more contamination functions describing the time behavior of a particular physical property of a mixture of fluids entering a tool; and one or more mixing-law models that can be used to estimate the volume fractions of the constituent fluids given information about or measurements of the physical property.
  • the physical property is viscosity, which is monitored using indirect measurements, i.e., T 1 measurements obtained from an NMR fluid analyzer.
  • Other physical properties that can be used in different embodiments include resistivity, capacitance, density, hydrogen index, compressibility, speed of sound, pumping pressures, optical density, and others.
  • Contamination estimation in accordance with this approach is done for both miscible fluids (i.e., OBM applications) and immiscible fluids (i.e., WBM applications). Accordingly, different contamination models are based on different set(s) of specific assumptions or requirements, and each model may have its particular strengths over the others.
  • CI ( t ) fc ( ⁇ 1 , ⁇ 2, ⁇ m, , t ) (4) where ⁇ 1/2 are the end points in the property values (0% and 100% contamination), and ⁇ m is the measured property value.
  • a bulk fluid property may be assumed to depend linearly on the corresponding properties of the constituent fluid volume fractions (i.e., on a linear combination of the contamination and virgin reservoir fluids). For example, light absorption is proportional to the volume fraction of the constituent fluids.
  • the approaches in this application are based on the observation that the log-mean value of an NMR parameter varies with contamination over time.
  • Previous research has shown that the echo amplitudes in NMR measurements of a fluid entering a fluid analyzer change with time, and as a result have developed time dependent models where an empirical relationship was used to curve fit the echo amplitude changes and to relate the combined time decays to contamination.
  • the embodiment(s) discussed below are based on the observation that log-mean T 1 (T 1Lm ) shifts with contamination over time, rather than on the tracking of individual T 1 amplitudes. This observation is also applicable to the variation in the log-mean value of other NMR parameters with contamination over time.
  • the log-mean T 1 shift over time can be attributed to changes in viscosity as the fluid being pumped changes to different proportions of filtrate oil or water.
  • T 1Lm is inversely related to viscosity and can be used to estimate contamination when combined with a mixing law that is based on viscosities of two miscible fluids. Because the log-mean T 1 is used in the illustrative embodiments discussed below, some explanation and definitions are in order.
  • the contamination estimation models are applied to a log-mean value derived from one or more NMR measurements.
  • the derivation of the log-mean value of a T 1 spectrum is presented below in a preferred embodiment.
  • similar relations apply for deriving the log-mean value of other spectra obtained by NMR measurements, such as a T 2 spectrum or diffusivity spectra. Modifications and variations of the approaches, disclosed herein, based on the use of a log-mean value of an NMR-based measurement will be apparent to one of ordinary skill in the art without undue experimentation.
  • the contamination estimation models are applied to log-mean T 1 values.
  • the typical range for the T 1 relaxation parameter is from about 1 ms to 20,000 ms.
  • all of the amplitudes that cover the measured range are used in the computation of T 1Lm .
  • T 1Lm may be computed for a subset of T 1 spectrum values, either to reject noisy components in the spectrum, or to focus around a certain T 1 range, where the change in energy contained in that particular band has a large dynamic range as a function of time.
  • the T 1 spectrum can be subdivided into several bands, and the corresponding T 1Lm for each band is then computed. This subdivision can be based on computational simplicity or observations concerning the temporal characteristics of different T 1 sub-bands.
  • contamination estimation is based on the use of one or more of the bands of a subdivided T 1 spectrum.
  • a band of the T 1 spectrum can be selected on the basis of one or more desired criteria, such as maximum dynamic range, longest center T 1 value, and others.
  • contamination estimation is based on a combination of the bands of the subdivided T 1 spectrum.
  • the contamination estimates resulting from the application of the one or more contamination estimation models to different bands of the subdivided T 1 spectrum can be combined in a number of different ways, either linearly or non-linearly, to form an averaged contamination estimate.
  • FIGS. 1A , B give an example of the computation of T 1Lm from the T 1 distributions in OBM logging experiments.
  • Panel 10 shows MRILab® T 1 measurements (in milliseconds) from ninety-one successive OBM logging experiments (uncorrected for flow).
  • Vertical scale 12 shows the experiment number in sequential order, from earlier times (top) to later times (bottom).
  • Panel 14 shows the apparent hydrogen index for each experiment, which is derived from the area under the T 1 distribution (discussed below). The hydrogen index is about 0.7 for a majority of the experiments. The tool has stopped pumping at the very bottom of the chart, therefore at the very end the hydrogen index is about 1.
  • the fit error for each wait time group is shown in panel 16 .
  • Panel 10 also shows the corresponding T 1Lm (connected diamond markers), calculated from the MRILab® T 1 distributions for each experiment (solid curves). Variations in the T 1 distributions and T 1Lm include contributions due to changes in contamination and flow rate. The center of the T 1 distributions is around 3.5 seconds for a majority of the experiments.
  • the relationship between the viscosity of crude oil and the log-mean value of other NMR parameters can be empirically estimated or approximated using Eq.
  • the T 1Lm can be used to estimate contamination when combined with a mixing law that is based on viscosities of two miscible fluids, as discussed in further detail below.
  • a mixing law or rule is a mathematical expression that describes a property of a mixture in terms of the properties of its constituents. This allows for the property of the mixture to be predicted if the weight or volume fractions or functions for the constituents, and the properties of the constituents are known.
  • ⁇ m is the measured viscosity of the mixture
  • ⁇ 1 and ⁇ 2 are the end point viscosities (i.e., the viscosity of each component of the mixture)
  • s 1 and s 2 are their respective volume fractions (or saturations)
  • n typically 4 in the viscosity mixing rule in Eq. (8), developed by Todd, M. R., et al. “The Development, Testing and Application of a Numerical Simulator for Predicting Miscible Flood Performance,” Journal of Petroleum Technology (1972).
  • s 1 and s 2 may represent a mass fraction, a mole fraction, or a volumetric fraction.
  • n in the Power-Law mixing rule in Eq. (9) is an adjustable value that depends upon the components and the proportions in the mixture.
  • the Arrhenius rule in Eq. (10) is also known as the log mixing rule.
  • the fourth mixing rule in Eq. (11), is a modified version of the classic Arrhenius expression, which was originally proposed by Lederer, E. L., Proc. World Pet. Cong., vol. 2, pp. 526-28, London (1933).
  • the constant ⁇ is found empirically and has values between 0 and 1.
  • the volume fractions s 1 and s 2 are associated with the contaminant and the oil, respectively.
  • Rhames et al. examined this equation for mixtures with low viscosity ratios ⁇ 1 / ⁇ 2 , and found that this function expression provided an excellent fit to their data. See Rhames, M. H., et al. “Viscosity Blending Relationships of Heavy Petroleum Oils”, Analytical Chemistry, vol. 20, pp.
  • one or more of the viscosity mixing rules can be applied to the log-mean value of an NMR spectrum, or the log-mean value of one or more subdivided bands of the NMR spectrum, for estimating the viscosity of a formation fluid contaminant and/or a hydrocarbon phase in a formation fluid.
  • the contamination estimation methods in accordance with the approach proposed herein will vary according to the chosen viscosity mixing rule. Several variations of the contamination estimation models are described below.
  • the contamination estimation approach in this application is based on the use of a contamination function, which is a temporal function that substantially matches the behavior of the contamination fluid fraction while pumping a sample from an invaded zone.
  • the idea in accordance with the proposed approach is to fit the determined values of one or more fluid properties, such as viscosity, over time to a parameterized contamination function as shown in the table.
  • the variables of this time function are determined, for example, through regression. Matching could be done over the individual functions listed.
  • a detailed numerical simulation study was performed to determine the pumpout contamination versus time.
  • FIG. 2 shows the results of a large number of sample quality simulations with OBM systems using a sample quality function developed from the Landmark VIP reservoir simulator. Invasion was simulated first and used as the initial condition for the sampling pumpout sequence. The contamination curves were developed by tracking the volume fraction of the fluids entering the sampling tool.
  • FIG. 3 shows how these simulations can be closely mimicked using the Error Function contamination function in Eq. (17).
  • the parameter a 1 controls the shape of the curve (inflection point at 50% contamination) and has units of time.
  • the dimensionless parameter a 2 scales the independent variable t. Typical ranges for a 1 and a 2 , for OBM systems are
  • the other contamination functions in the preceding table can be matched to the simulated curves in a similar manner.
  • the objective is to adjust the vector of unknown parameters such that the time function c(t) matches the measured fluid property data.
  • the application of the contamination estimation methods of this disclosure vary according to the contamination model chosen to be applied, for example, i.e., to an NMR spectrum, or to one or more subdivisions of the NMR spectrum.
  • the methods may also change dependent on whether the contamination model is applied in conjunction with one or more mixing rules.
  • MFM Miscible Fluid Models
  • This section provides four variations of the contamination estimation methods used in different embodiments, based on four different contamination estimation models. It will be appreciated that other contamination estimation models can be formulated from variations of the approaches discussed herein. While the contamination estimation models are described relative to T 1 spectra and the log-mean T 1 value (T 1Lm ), the variations of the contamination estimation methods, based on contamination estimation models that are expressed in terms of other NMR measurement parameters, including, but not limited to, T 2 and diffusivity measurements, are also envisioned and will be appreciated by those of skill in the art.
  • T 1Lm is correlated to viscosity and that the viscosity mixing laws predict two very clearly defined end points, one can reason that T 1Lm is a bounded function, where the asymptotes at time zero and infinity correspond to the T 1Lm of the contaminant and the formation fluid (e.g., the crude), respectively.
  • the parameter c(t) is assumed to conform to the shape of one of the contamination function models discussed above.
  • the problem to be solved is an over-determined non-linear least squares problem, where the parameters in the contamination function are solved using well-known nonlinear least-squares (NLLS) solvers.
  • NLLS nonlinear least-squares
  • the level of contamination of a formation fluid or an estimation of the pumping time needed for achieving a given contamination level can be provided based on application of the ILMT1 contamination estimation model.
  • the model is solved (using NLLS techniques) to compute the parameters a 1 , a 2 , a 3 , and p.
  • the limits of the function are then calculated using the computed parameters to obtain the T 1Lm for the endpoints (see Eqs. 19 and 20).
  • Eq. (21) is used to compute the contamination at time t k .
  • the contamination estimate at time t k can then be used to estimate the pumping time necessary for achieving a given contamination level. This estimation can be based on examining the time-function of the contamination parameter.
  • the inverse of the log-mean value i.e., 1/T 1Lm
  • the log-mean value T 1Lm
  • the main advantage of this approach is that the inverse of the log-mean value of the NMR parameter could provide information about other materials properties, e.g., 1/T 1Lm is directly proportional to viscosity.
  • ELMT1 Explicit Log-mean T 1
  • the T 1Lm values are fit using a contamination model, as in the ILMT1 model, in order to compute the two limits given in Eqs. (19) and (20).
  • a viscosity mixing rule e.g., Eqs. (8) through (11) is applied to compute c(t k ) for a given mixture.
  • contamination estimates at different times can be used to compute the time necessary to achieve a given contamination level at a given pump time.
  • the contamination model and the mixing rule are applied in two discrete steps.
  • the contamination model is coupled with the viscosity mixing rule.
  • the resulting nonlinear system is then solved for a greater number of unknowns.
  • IFM Immiscible Fluid Model
  • the IFM can be applied for NMR measurements either in the time domain or in the T 1 domain.
  • the NMR data derives from MRILab® T 1 measurements.
  • An assumption of the IFM model is that the measured response, whether in the time or in the T 1 domain, is a linear combination of the signatures of the two end members, where the weighting is governed by the contamination level.
  • the IFM method used in a preferred embodiment includes two steps. First, the two end members are determined from the available data. Given the end-member signatures, the contamination levels are then determined in the second step.
  • the first step of the IFM model is to determine the end-point signatures x and y.
  • Equations (26) and (27) are given as discrete functions only for ease of notation, and the IFM is not so limited.
  • the c k of Eq. (27) are not ordered in time, in that the unknown vectors x and y would be the same if the columns of B are reshuffled.
  • an advantage of the IFM model is that the end point vectors can be estimated independent of the dynamics of fluid flow.
  • the contamination function is generally a monotonically decreasing function of time. However, if the flow rate changes during the measurements, e.g., if pumping is stopped, the mud filtrate can flow back in around the probe in the analyzer. This may result in a higher level of contamination when the pump starts again. As a result, if contamination is modeled as a function of time while determining the two end points, then some otherwise valid data points would be interpreted as misfits.
  • the parameters can be solved using well-known NLLS solvers. For example, a separable non-linear least squares approach can be used where a bi-linear problem is solved to get x, y first, followed by c k .
  • the second step of the IFM contamination estimation model includes fitting the c k obtained from the first step using a contamination model.
  • the error function is also a solution for well defined cases of WBM invasion, having a strong resemblance to the form of the sample quality function previously discussed for the case of OBM.
  • FIG. 3 shows the results of a large number of sample quality simulations with WBM systems using the sample quality function discussed above based on the Error Function contamination function in Eq. (17). Typical ranges for a 1 and a 2 , for WBM systems are:
  • the input data can be weighted based on flow rate, hydrogen index, fit error, noise level, and others.
  • contamination can be estimated in different embodiments either in the time-domain, or T 1 domain, where preferably the user can specify either the time or T 1 range.
  • Eq. (27) solves for two end-points, the IFM model could be extended in straightforward manner to the case of three or more fluids by weighting.
  • fluid 1 being clean crude from a previous logging station
  • fluid 2 being the OBM filtrate that enters when flow starts
  • fluid 3 being crude at the current depth, which may have a different viscosity from fluid 1 .
  • the input curve for the algorithm is T 1Lm .
  • T 1Lm The input curve for the algorithm.
  • the T 1 distribution is computed by standard inversion algorithms, and then converted to T 1Lm using Eq. (5).
  • the sample being analyzed is a mixture of two fluids—the contaminant and the formation fluid.
  • the T 1Lm curve one can obtain viscosity indices for these fluids, i.e., the values of ⁇ 1 —the viscosity of the contaminant, and ⁇ 2 as the viscosity of the native fluid.
  • the volumetric fraction of the contaminant and the formation fluid at each experiment is computed by applying a fluid mixing-model.
  • the algorithm can be summarized in the following steps:
  • time-function and mixing-model are uncoupled. It will be appreciated that both uncoupled and coupled estimates can be provided based on principles discussed above. In a particular embodiment, several tests may be performed to determine the estimation model that optimally fits the data.
  • this disclosure also provides methods for applying the contamination estimation models to the non-ideal conditions encountered while drilling.
  • the equation of the contamination estimation models are developed based on an assumption of ideal behavior of the formation fluids.
  • One assumption of the contamination estimation models is that there are two end point fluids, e.g., the contaminant and the hydrocarbon.
  • the fluid that is measured at the beginning of each experiment may not contain either of the end-point materials.
  • the fluid measured in the experiments at earlier times may be the fluid left in the flow line from a previous station, possibly measured at a different depth where the reservoir fluid could be completely different.
  • the fluid in the flowline may be water left in the tool during calibration in the shop.
  • contamination is usually modeled as a monotonic phenomenon, in that contamination is taken to decrease as a function of time as the pump-out time increases. For example, T 1Lm (or its reciprocal) may actually decrease until the fluid left in the flowline is pumped out, then reverse direction during the relevant portion of the clean-up process.
  • top graph of FIG. 4 An example of such behavior can be seen in the top graph of FIG. 4 , where the diamonds and circles represent actual data points.
  • the top graph shows a log-log plot of contamination estimation models in the specific embodiments to a viscosity vs. time measurement, where the diamonds ( ⁇ ) and the circles ( ⁇ ) are data points, while the curve is the fit of a model to the data points.
  • the middle graph shows spin-echo data from a number of experiments.
  • an inflection-point detection algorithm may be applied to identify the relevant time window that contains data from the fluids of interest. Once the relevant time window is identified, data outside this window can be excluded from the inversion.
  • the application of an inflection-point detection algorithm to a data set is demonstrated in the top graph of FIG. 4 , where the data points represented by diamonds are within the desired range, while the data points represented by circles are determined to be outside this range. Data points in the window that fall within the desired range are weighted more in application of the contamination estimation models as compared to the data points outside the window.
  • an arctan-like function is used in the weighting.
  • Factors that can influence this residual contamination include overbalance, permeability, and (to a lesser degree) anisotropy.
  • the permeability influences how quickly the mudcake forms and the flow rate at which the formation tester can pump a sample. As a result, the residual contamination increases with reduced permeability.
  • the residual contamination can be estimated, and in most cases is less than 1%.
  • the present disclosure also provides methods of adjusting the contamination models to take into account the residual contamination. Using additional simulations, it is possible to develop a correlation function, where the residual contamination (related to the overbalance and permeability) is estimated before pumping starts.
  • the limits of the contamination estimation functions e.g., Eqs. 12-18
  • Eqs. 12-18 used to estimate the contamination while pumping, i.e., the two end points, are then rescaled, so that the projected sample contamination is asymptotic to the residual contamination.
  • Implicit LogMean T 1 approach the choice of the contamination function model is the ArcTan model in Eq. (8).
  • the task is then to find the vector x that minimizes the above function, like in any other least squares problem.
  • the illustrative embodiments using viscosity d k is the log mean of T 1
  • other parameters such as log-mean T 2 , log-mean D 0 , as well as the hydrogen index, pressure values, or others can be used instead. It will thus be appreciated that the approach proposed herein is not limited to a particular fluid property, or a particular modeling function but rather can be extended without due experimentation to different properties, different mathematical model functions or function combinations.
  • one of these properties may be more sensitive than others to the contaminant.
  • an array of instruments can be used in a preferred embodiment to measure individual properties and the approaches disclosed below applied to each measurement.
  • different contamination estimates can be combined into a single average contamination estimate.
  • Individual contamination estimates may be weighted, preferably using nonlinear regression techniques. As is the case with MRILab® estimates of viscosity, the formation fluid properties can be more accurately predicted because the end points are used to determine the in-situ sample properties.
  • the mixing rules described in the above illustrative embodiments are applicable to two fluids—generally a contaminant and the native formation fluid.
  • two fluids generally a contaminant and the native formation fluid.
  • the viscosity mixing rules given for 2 fluids (Eqs. 8 thru 11), have counterparts for 3 or more fluids.
  • FIG. 1A illustrates that the flow, dependent on the pumpout rate affects the T 1 distributions.
  • the magnitude of the flow effect can be appreciated by comparing the T 1Lm values, which are in the order of 3 seconds while flowing at 30 cc/min, to stationary values in the order of almost 10 seconds (near the bottom of the log). Faster flow rates result in less polarization in the case of long T 1 s.
  • the flow rate does not affect the T 1 distributions substantially, and the flow effect can be ignored.
  • the fluid has low viscosity (long T 1 s), but the flow rate is low, then once again, the T 1 distributions are not affected for practical purposes. In either case, the effects of flow are minimal and no corrections are needed.
  • the contamination algorithms presented herein may function well without flow corrections to the data even in the worst case of long T 1 s and high flow rates, as long as the transition from stationary to non-zero flow is made quickly and the flow rate is not varied for the remainder of the time.
  • the most significant effect of flow is to change the apparent viscosity.
  • the artifacts caused by high flow rates are analogous to that of viscosity in the case of dead vs. live oil.
  • the apparent viscosity may not be close to the true viscosity, the volume fractions of the two end points, i.e., contamination is still accurate. It will be appreciated, however, that if the flow effects reduce the dynamic range in T 1Lm , the sensitivity of the contamination estimates may be reduced. These issues can be taken into account with proper adjustment of the pumpout rate of the device in operation.
  • the second section contains a plot of the estimated contamination vs. time.
  • the estimated contamination is about 2 percent towards the end of the measurement.
  • the third section is a plot of the input data (solid curve) vs. the fit (connected diamonds). The flow rate is shown in the bottom plot, for reference. In the third section, one can see the T 1 distributions for the two end members.
  • the top graph of FIG. 4 shows viscosity vs. time data points (diamonds and circles) from an MRILab® Fluid analyzer, obtained from a well drilled with OBM.
  • An inflection-point detection algorithm was applied to identify the relevant time window (diamonds) that contains data from the two fluids of interest.
  • the data in the relevant window (diamonds) is weighted by an arc-tan function as compared to the data outside the window (circles).
  • the top graph of FIG. 4 also shows the results of the application of the ELMT 1 model to the data from the MRILab® (shown by the curve fit).
  • the estimated contamination value of 4.7% is obtained from application of the ELMT 1 model, using a combination of the Modified Ahrrenius mixing rule in Eq. (11) and the ArcTan-Shifted 2 contamination model in Eq. (15).
  • Independent laboratory results were obtained from the analysis of the actual fluid samples secured with the RDT.
  • the laboratory measured contamination value is 4.0%.
  • the lower plot of FIG. 14 shows the T 1 distribution for the two end-members. It is interesting to note that the algorithm used in the analysis chooses the most commonly occurring T 1 spectrum for the crude (see FIG. 1A , experiment range 30 to 80), and has considered most everything else as a contaminant, including the very long T 1 values encountered only at the end of the measurement.
  • the upper plot shows the contamination values as a function of experiment number.
  • the monotonically decaying smooth curve is the contamination curve.
  • contamination values are on the order of 2 percent. The results shown here are obtained by constraining the contamination values between 0 and 1, which explains the abundance of points occurring at 0 and 1.
  • FIGS. 5A and 5B show simplified diagrams of a downhole NMR fluid analysis apparatus, such as the MRILab®, that provides NMR measurements to which the contamination estimation methods of the present disclosure can be applied in an illustrative embodiment.
  • Fluids enter the device at the top and pass through two sections, referred to as polarization and resonance sections, respectively. Measurements are performed as the fluid flow passes through the device.
  • the fluid entering the system is initially subjected to a strong magnetic field to achieve rapid polarization of the hydrogen nuclei. NMR measurements take place in the lower section, where the field strength is lower.
  • two separate radio frequency coils are used for pulse transmission and for reception. This split scheme allows for a transmitter coil 30 that is longer than the receiver 35 . The practical effect of this split is that relaxation times estimates are less sensitive to the actual flow rate(s) because signals are received from only a portion of a larger volume of fluid that is being pulsed.
  • T 1 relaxation times largely independent of flow velocities, up to a certain practical limit.
  • Transmission and reception both operate at 4.258 MHZ at room temperature, consistent with 1,000 Gauss field strength. At higher temperatures, the operating frequency is reduced to track the reversible reduction in magnetic field strength.
  • the magnetic field in the measurement volume of the device shown in FIGS. 5A and 5B in general is not entirely uniform.
  • This volume can be split conceptually into an interior region, where the field gradient is negligible, and a fringe region, where the field changes with an approximately uniform gradient.
  • the fringe region may comprise about one third of the total sensitive volume.
  • T 1 measurements and at short pulse-to-pulse spacing such as 0.25 ms
  • the effect of the gradient is not noticeable.
  • the main flow is diverted and a sample is stagnated within the NMR chamber.
  • the pulse-to-pulse spacing (Te) is increased to induce diffusion-dependent signal dephasing.
  • the uniform and the fringe regions are large compared to the largest possible diffusion length; therefore, these regions are essentially isolated from each other for the duration of a single pulse-echo train.
  • FIG. 6A is a detailed cross-sectional view of the polarization section of the apparatus that the flowing fluid encounters after it enters the measurement chamber.
  • the purpose of this pre-polarization section is to polarize hydrogen nuclei in the fluid(s) as rapidly as possible, so that they will exhibit full polarization under the operating field.
  • the flow tube 60 is made from ceramic, glass or PEEK material, and is surrounded by a Faraday shield 45 .
  • the magnet consists of segments 25 that are magnetized as indicated by the arrows and are made of material with very low temperature coefficient.
  • FIG. 6B is a detailed cross-sectional view of the second section of the apparatus, which is located between the polarization section and the measurement section(s) (labeled “Resonance Section” in FIG. 5B ).
  • the purpose of this section is to allow the hydrogen spins to settle to an equilibrium polarization that is close to a non-flowing magnetization corresponding to an external field of 1,000 Gauss.
  • FIG. 6C is a detailed cross-sectional view of the transmit portion of the measurement section of the apparatus, which follows the stabilization section.
  • FIG. 6D is a detailed cross-sectional view of the last section of the apparatus, which is the transmit/receive section.
  • the NMR time constants T 1 of the fluid(s) under investigation are determined by varying the delay time between a broadband saturation pulse and a read-out sequence. If the flow velocity does not exceed 10 cm/s, the measurement is generally independent of the flow speed and of the flow profile.
  • the electronics used in the NMR fluid analyzer which is illustrated in a block diagram in FIG. 7 , is similar to that of an NMR spectrometer.
  • the comparatively low frequency of 4.2 MHz used in for some of the measurements allows many traditionally analog functions to be realized readily as digital signal processing (DSP) algorithms.
  • a frequency source 34 controlled by a pulse programmer, sends its signal to a power amplifier 37 , which in turn drives the transmitter antenna 30 . All timing functions, like pulse widths and acquisition windows, are fully programmable.
  • the signal from the receiver antenna 35 is amplified, synchronously demodulated and integrated.
  • the system also performs its own calibration. All pertinent calibration factors are stored in the tool itself and after calibration echo amplitudes are reported on a scale of 0-2.
  • the two coils of the device 30 , 35 are connected to resonating capacitors 31 .
  • These capacitors are of the NPO (no temperature coefficient) and PTC type (positive temperature coefficient), shunted in parallel, as shown.
  • the resultant temperature characteristic is such that with increasing temperature, when the static magnetic field weakens (typically 1% per 100°C.), the capacitance increases at twice the rate (typically 2% per 100°C.).
  • the resultant LC circuit resonant frequency drops at half the capacitor rate (1% per 100° C.) and therefore follows the NMR resonance, making re-tuning of the circuit unnecessary.
  • the controller 33 In a transmit mode, the controller 33 gates the signal generator 34 of the apparatus and the power amplifier 37 to produce a radio frequency pulse in both coils.
  • the high voltage applied causes all crossed diodes 39 to conduct, thereby connecting the two coils.
  • receive mode the crossed diodes stop conducting and signal is only received from the lower coil 35 .
  • the signal is amplified, digitized and fed into the digital signal processor 33 for demodulation and further processing.
  • the MRILab® described above determines hydrogen density, self-diffusion rates and NMR relaxation rates of fluids during the pump-out phase, from which one can compute sample viscosity and GOR.
  • fluid samples are analyzed under true reservoir conditions and results are available substantially in real time.
  • the MRILab® measures the hydrogen index and the NMR polarization time constant (T 1 ) of flowing fluids, which pass through the device.
  • T 1 the NMR polarization time constant
  • the hydrogen in the fluid is first polarized by a set of magnets and then pulsed via an antenna coil to excite the magnetic resonance response.
  • the excitation and refocusing pulses are fed to a long transmitter coil that traverses the magnet section.
  • a smaller receiver coil located at the bottom of the flow-tube, picks up the NMR echo.
  • the separated coil arrangement permits NMR measurements while flowing.
  • the timing sequences for the excitation pulses are field-programmable for additional flexibility.
  • the device records the NMR amplitude corresponding to a number of distinct wait times ranging from 1 millisecond to 16 seconds.
  • the amplitudes are calibrated in hydrogen index units, where 1 unit equals the hydrogen density in water under atmospheric conditions.
  • the MRILab® can also be switched to T 2 mode of operation.
  • the measurement of the signal decay time T 2 is flow-sensitive and is generally valid when a sample is stagnant within the MRILab®, at which time the NMR signal decay induced by self-diffusion can be observed. Diffusivity is inversely proportional to viscosity, a relationship that holds true in dead oils as well as in gas-oil mixtures.
  • FIG. 8 illustrates a pulse sequence employed with the fluid analyzer in a specific embodiment. It will be appreciated that it is a standard saturation recovery sequence, where an initial saturation pulse is followed by a variable delay. In a preferred embodiment, the delay is programmable and is typically stepped through the values 1, 2, 4, 8, 16, 16384 ms in cyclical fashion. The recovered magnetization at the end of the delay is determined by a short read out sequence, consisting of two pulses and one spin echo. The height of the echo, if plotted as function of delay time, traces out a recovery curve that is converted into a T 1 distribution by standard inversion methods.
  • the inversion algorithm is a variant of the method employed to calculate T 2 distributions from wireline data, as disclosed in U.S. Pat. No. 5,517,115. With the above sequence it takes about 33 seconds to complete a measurement cycle. The signal-to-noise ratio of the system is so high that additional averaging may be unnecessary.
  • the sequence described above is insensitive to fluid flow and can be used to continuously monitor the T 1 profile of pumped fluids. Other measurement sequences may be used in alternative embodiments.
  • the T 1 measurement sequence is initiated by a frequency swept saturation pulse.
  • the frequency is selected such that the entire range of frequencies in the resonance sections of the apparatus is affected. In a specific embodiment, this range is typically the NMR center frequency +/1% (4.2 MHz+/40 kHz). Pulse amplitude, frequency sweep rate and pulse length are adjusted to effect saturation.
  • a variable delay is inserted.
  • consecutive measurements with delay values of 1 ms, 2 ms, 4 ms, . . . , up to 16384 ms are used.
  • the nuclear magnetization builds up again to its equilibrium value.
  • fluid volume moves into the receiver coil volume, while unprepared fluid enters the resonance volume.
  • the measurement is independent of the actual flow rate.
  • a short pulse sequence consisting of a ⁇ /2 pulse, followed by a ⁇ pulse.
  • the RF phase of these pulses is shifted by 90° against each other to cancel the effects of B 0 and RF field imperfections.
  • This is equivalent to the start of a Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence.
  • the time between these pulses is typically 0.125 ms; a spin-echo forms 0.125 ms after the ⁇ pulse.
  • This echo is digitized, quantified and its amplitude is taken as a measure of the recovered magnetization as function of the saturation recovery delay.
  • the ⁇ /2 and ⁇ pulses can be narrow band and need not be frequency swept. The reason is that they are only relevant for the receiver coil section which has a very tightly controlled field and resonance frequency distribution.
  • T 1 distributions for some example fluids are shown in FIGS. 10 and 11 .
  • the data points were acquired according to the sequence in FIG. 8 and inverted from the time domain to the T 1 domain.
  • the horizontal axis, “time,” in FIG. 9 denotes the time elapsed between the saturation pulse and the readout sequence, the vertical axis is signal amplitude in arbitrary units.
  • the results are easier to interpret after inversion from time domain to T 1 domain, as shown in FIG. 10 .
  • 53 points are specified for the inversion result.
  • the single, sharp peak at 2-3 seconds is characteristic of water
  • the rounded peak in the “oil window” 0.5-1 second indicates oil and the broad response from the crude oil in the bottom panel is characteristic for complex hydrocarbons.
  • the figure shows examples of T 1 saturation recovery data for three different fluids: brine, Diesel oil and a crude oil.
  • the data points illustrated have been acquired by circulating different fluids through the analyzer. Shown from top to bottom are: water (mild brine) with a single relaxation peak in the “water window” at 2 seconds; next a simple hydrocarbon (diesel) with a single relaxation peak in the “oil window” at 0.5-1 second; and a complex hydrocarbon (crude), which shows a characteristic asymmetric distribution that starts in the few tens of milliseconds and extends to the “oil window.” These samples were under atmospheric conditions at ambient temperature. At elevated temperatures, Eq. (2) predicts an increase in T 1 proportional to the absolute temperature in addition to increases due to reduction in viscosity.
  • the determination of long relaxation times no longer depends on how long an echo train persists.
  • small perturbations in the applied field have relatively limited effect.
  • the saturation pulse prepares a much larger sample volume than what is actually used for the readout portion. Therefore, as long as the flow rate is low enough, and the readout is based on a fluid sample that was present anywhere within the resonance regions during the saturation pulse, the measurement is valid.
  • the T 2 parameter In contrast to T 1 , the T 2 parameter generally cannot be determined on a flowing sample. Distributions of T 2 times are determined by standard CPMG sequences on samples that have been stagnated momentarily. Stagnation can be achieved by closing a valve below the analyzer apparatus and diverting the flow stream around the sample chamber. The time required for a T 2 measurement is almost entirely determined by the polarization time (“wait time”) and is on the order of several seconds.
  • the hydrogen density or the total number of hydrogen atoms within the measurement volume is a by-product of any T 1 or T 2 measurement. It can be represented as the area under any T 1 distribution and is typically normalized to the hydrogen density of reference oil at measurement temperatures. At room temperature, the reference oil and water have the same hydrogen density. The reported number is the relative hydrogen index (HI) in the range 0-2, with accuracy around 1%.
  • HI relative hydrogen index
  • Hydrogen density can be automatically converted to hydrogen index (HI), which is the hydrogen density of a material relative to that of water at ambient conditions.
  • HI hydrogen index
  • the spin density of the fluid is proportional to the hydrogen index.
  • the mass density ⁇ m , the hydrogen index, and the hydrogen-to-carbon ratio R are related as follows: HI ⁇ m 9R/(12+R) (29)
  • HI x (9/4 ⁇ )+(1 ⁇ x )1, (30) where x is the volumetric gas fraction (m 3 /m 3 ) and ⁇ , in g/cm 3 , is the density of methane.
  • x is the volumetric gas fraction (m 3 /m 3 )
  • ⁇ in g/cm 3
  • the density of methane follows from its temperature and pressure, and Eq. (30) can be used to derive a first-order approximation for the gas fraction x.
  • Diffusion measurements can be performed using the NMR fluid analyzer using steady-gradient spin-echo (SGSE) experiments.
  • the experiments require that the fluid flow is temporarily stopped.
  • the concept of using the fringes of a uniform field volume for diffusometry derives from so called SSF-SGSE methods.
  • SSF-SGSE methods SSF-SGSE methods.
  • the main advantage of the SGSE method over pulsed-field gradient spin-echo (PFGSE) diffusometry is instrumental simplicity and superior stability.
  • the main drawback is a limit on sensitivity, which, for the downhole implementation, is approximately 10 ⁇ 6 cm 2 /s.
  • the sensitive volume of the apparatus can be divided into an interior, homogeneous region and an exterior gradient region.
  • the field in the fringe volume which makes up about 1 ⁇ 3 of the total volume, can be approximated by a single field gradient value G 0 .
  • G 0 At short echo spacing (0.25 ms), the effect of the field gradient is too small to be relevant.
  • the pulse sequence used both for diffusion measurements and for diffusivity calibration is shown in FIG. 11 .
  • two CPMG sequences with a short echo spacing (typically 0.25 ms) and a long spacing (T e ) are alternated.
  • the long echo spacing is selected as an integer multiple of the short spacing.
  • echoes line up in time, i.e., occur at the same elapsed time, since the excitation pulse and the ratio of their amplitudes can be formed.
  • the system parameter K 0 is the gradient volume divided by the total volume.
  • the hydrogen gyromagnetic ratio ⁇ is equal to 26,754 rad/s/gauss. Both K 0 and G 0 are temperature-dependent and are determined during calibration.
  • This curve is fit to a uni-exponential model plus an offset.
  • the two curves are the A 1 and A 2 signals for water at room temperature.
  • the lower graph of FIG. 12 is the ratio curve and the best fit uni-exponential model. Since D for water is known as 2.5 ⁇ 10 ⁇ 5 cm 2 /s, these curves determine the calibration parameters G 0 and K 0 .
  • the two curves in the upper graph of FIG. 12 are spin echo amplitudes at different echo spacings.
  • the accelerated decay for the longer spacing is a manifestation of diffusion in the gradient region of the magnetic field.
  • the ratio curve (lower graph) is the sum of an exponential and a constant term, corresponding to the gradient field region and the uniform field region, respectively.
  • the best fit model curve is also plotted; however, it is indistinguishable from the data.
  • the viscosity ⁇ is measured in cp, the temperature T in Kelvin and the diffusivity D in cm 2 /s.
  • the temperature may be obtained from the RDT fluid temperature sensor.
  • the proportionality factor is determined by fitting Eq. (1) to data from pure alkanes and methane alkane mixtures.

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US20060250130A1 (en) 2006-11-09
WO2005065277A3 (en) 2006-07-13
BRPI0418081A (pt) 2007-04-17
WO2005065277A2 (en) 2005-07-21
US7372264B2 (en) 2008-05-13

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