Summary. Introduction

Transcription

1 A New NMR Method of Fluid Characterization in Reservoir Rocs: Exerimental Confirmation and Simulation Results R. Freedman, SPE, and S. Lo, Schlumberger; M. Flaum, SPE, and G.J. Hirasai, SPE, Rice U.; A. Matteson, SPE, Schlumberger; and A. Sezginer, Sensys Instruments Summary his aer introduces a new magnetic resonance fluid (MRF) characterization method. he MRF method is based on two ey ingredients a new microscoic constituent viscosity model (CVM) and a new multifluid relaxation model. he CVM rovides a lin between nuclear magnetic resonance (NMR) relaxation times and molecular diffusion coefficients in hydrocarbon mixtures such as crude oils. he multifluid relaxation model accounts for the 2 decay of sin-echo signals that arises from intrinsic sin-sin interactions, surface relaxation, and attenuation due to molecular diffusion of fluid molecules in a magnetic field gradient. he MRF method exloits the fact that the molecular diffusion coefficients of brine, oil, and gas molecules tyically have values that are well searated from one another. hus, the diffusion attenuation of a suite of measured NMR signals contains sufficient information to allow differentiation of brine, oil, and gas. he method involves the simultaneous inversion of a suite of sin-echo measurements with the new MRF multifluid relaxation model. he alication of the MRF method to magnetic resonance logging data can rovide a detailed formation evaluation. he information rovided includes total orosity, bul volume of irreducible water, brine and hydrocarbon saturation, hydrocarboncorrected ermeability, and oil viscosity. his aer discusses the theory underlying the CVM and validates the model by testing its redictions on hydrocarbon mixtures including live and dead crude oils. he robustness and accuracy of the multifluid inversion is demonstrated by a Monte Carlo simulation of a model carbonate roc that contains brine, oil, gas, and oil-base mud filtrate (OBMF). he MRF method is alied to suites of sin-echo measurements acquired in the laboratory on artially saturated rocs and shown to rovide accurate fluid saturation and oil viscosity estimates. Since the comletion of this wor, field test results have shown that the MRF method rovides a owerful and unique new formation evaluation tool. Introduction It is well nown in the industry that oil-bearing reservoirs can be misinterreted or even missed altogether by conventional resistivity-based interretation. One difficulty is the fact that many oil-bearing reservoirs exhibit anomalously low values of resistivity, which results in suriously high water saturation estimates. Other difficulties in the interretation of resistivity logs can be traced to fresh formation waters or waters with unnown or variable salinity. Problems also occur in formations with comlex lithologies for which use of default arameter values (e.g., m = n = 2) in Archie's equation can result in totally erroneous water saturation estimates. Coyright 2001 Society of Petroleum Engineers his aer (SPE 75325) was revised for ublication from aer SPE 63214, first resented at the 2000 SPE Annual echnical Conference and Exhibition, Dallas, 1 4 October. Original manuscrit received for review 22 October Revised manuscrit received 8 August Manuscrit eer aroved 24 August he urose of this aer is to introduce a new state-of-the-art MRF characterization method. he MRF method overcomes the aforementioned roblems inherent in resistivity interretation. It also rovides a wealth of formation evaluation information not obtainable by other well logging or laboratory methods. he MRF method is based on a new multifluid relaxation model. he method relies on the different sensitivities of sinecho measurements to the fluids resent in roc formations when a suite of measurements is acquired with different ulse arameters. In general, a measurement suite consists of sin-echo sequences acquired with different echo sacings, olarization times, alied magnetic field gradients, and numbers of echoes. Such measurements are sensitive to the viscosities and molecular diffusion coefficients of the fluids and therefore rovide the information needed for fluid characterization. he inversion of a measurement suite involves a nonlinear fitting of the full data suite to the MRF multifluid relaxation model. A ey ingredient in the multifluid relaxation model is a new microscoic henomenological model of relaxation and molecular diffusion in liquid hydrocarbon mixtures. his model is referred to as the CVM. It rovides an imortant lin between diffusion-free relaxation times and molecular self-diffusion coefficients in crude oils. his lin reduces the number of indeendent arameters needed to characterize the crude oil NMR resonse and imroves the accuracy and robustness of the inversion. he inversion of a suite of sin-echo measurements using the MRF multifluid relaxation model could be erformed without emloying the CVM by assuming that the relaxation time and diffusion coefficients are totally indeendent arameters. his aroach, however, involves solving for more unnowns and requires a quantity and quality of data that is not easily obtainable by a moving logging tool. Moreover, to obtain oil viscosity it would still be necessary, following the inversion, to invoe emirical correlations that relate the relaxation times and diffusion coefficients to viscosity. he CVM simlifies the rocess by using the emirical correlations at the outset to reduce the number of unnowns and directly obtain robust fluid roerties from the inversion. CVM theory redicts that there exist distributions of relaxation times and molecular diffusion coefficients in liquid hydrocarbon mixtures. CVM further redicts that the two distributions are not indeendent (i.e., if either is nown, the other one can be redicted). Moreover, CVM redicts that the mixture viscosity can be estimated from either the relaxation time distribution or the diffusivity distribution and that the two estimates are theoretically equivalent Scoe of the Paer. We resent the results of a research and develoment study that was conducted on three fronts to suort the develoment of the MRF method. he study included the following efforts: We conducted Monte Carlo numerical exeriments to test the robustness and accuracy of the inversion. Laboratory NMR exeriments conducted to test the redictions of the CVM consisted of sin-echo and ulse field gradient (PFG) measurements on the following hydrocarbon mixtures 452 December 2001 SPE Journal

2 and crude oils: n-hexane-n-hexadecane, n-hexane-squalene, methane-n-hexadecane at seven gas/oil ratios (GORs), four dead crude oils, and a live crude oil at four measured GORs. NMR sin-echo measurement suites were acquired on artially and fully saturated Berea 100 and Indiana limestone roc samles. he fluid and roc roerties estimated from inversion of the data suites were comared with those estimated by indeendent laboratory measurements. his aer elucidates the foundations underlying the MRF method and demonstrates its viability using NMR laboratory measurements and simulation results. Previous Wor. o lace the MRF method in ersective, it is rudent to review the currently oular methods of NMR-only fluid characterization. Published aers on NMR-only fluid characterization alications have for the most art emloyed the differential methodology roosed by Aurt et al. 1 and Prammer et al. 2 his methodology involves maing two or more NMR sin-echo measurements with different wait times. Either the sin-echo measurements or the 2 distributions comuted from them are subtracted to yield a differential signal (either a differential 2 sectrum or sin-echo train) that is further rocessed to estimate hydrocarbon-filled orosity. In the literature, the differential methods are referred to either as the differential sectrum method (DSM) or time domain analysis (DA) method, deending on whether the subtraction is done in the 2 or time domain. he DA method is more robust than the DSM; however, subtracting sin-echo trains leads to a 40% increase in root mean square noise (rmsnoise) on the differential signal. For the differential methods to wor, the wait times must be selected so that the differential signal contains negligible contributions from the brine in the formation. o select roer wait times so that the brine contribution is canceled requires nowledge of the NMR roerties of the fluids in the formation. his is a limitation of the differential methods for evaluation of exloration wells. Moreover, interretation of the differential methods requires that the 1 distribution of the brine hase not overla with the 1 sectra of the hydrocarbon hases. his limits the alicability of the differential methods to shaly sands containing very low viscosity oils and gas. A recent aer by Aurt et al. 3 noted the limitations of the DSM and DA methods for oils with low to intermediate viscosities (e.g., 1 to 50 c). hese authors roosed a method called the Enhanced Diffusion Method (EDM) that attemts to exloit the fact that the brine hase is more diffusive than low to intermediate viscosity oils. By increasing the echo sacing so that diffusion dominates the 2 relaxation of the brine, an uer limit on the aarent brine 2 can be achieved. Oil-filled orosity is estimated by integrating the aarent 2 distribution for relaxation times greater than the uer limit. Although are the basic concet underlying the EDM is valid, there comlications in ractice that limit its reliability for detection of oil. hese include the following: aarent 2 distributions are broadened by regularization (smoothing) that is alied by the rocessing; intermediate viscosity crude oils have broad 2 distributions with short relaxation time tails that overla the brine 2 distributions; in exloration well logging it should not be assumed that the diffusivity of the crude oil is less than that of water; and, finally, in wells drilled with oil-base muds it is difficult using the EDM concet to searate the OBMF filtrate signal from that of the native oil. he MRF method overcomes the fundamental roblem associated with the differential methods. hat is, the differential methods assume that the brine and hydrocarbon signals can be cleanly searated by subtracting sin-echo measurements (or their associated 2 distributions) acquired with different ulse arameters. In ractice, crude oils exhibit broad relaxation time distributions; therefore, there will almost always exist some overla of the oil and brine 1 distributions, and in some cases there will be total overla that comletely violates the basic remise underlying the differential methods. he MRF method avoids this limitation by the use of a multifluid relaxation model that roerly accounts for crude oil and brine 1 and 2 distributions even when the two overla. he first ublication, to our nowledge, to aly a multifluid forward model for NMR fluid characterization is the wor of Looyestijn, 4 who used a diffusion rocessing method to comute oil saturations from NMR data acquired using different echo sacings. More recently a more general multifluid relaxation model and inversion method than the model used in Ref. 4 was discussed by Slijerman et al. 5 hese authors discuss a linear forward model and method for determining fluid volumes and relaxation times of different fluid tyes. he forward model and inversion assume a single relaxation time for the crude oil. his method does not comute the distributions of crude oil relaxation times and self-diffusion coefficients that exist in real crude oils. We show below that the distributions are required to accurately estimate oil viscosity and BVI. Multifluid Relaxation Model Consider a suite of N sin-echo measurements and let each measurement be characterized by a arameter set {W, E, G, NE } for = 1,N where for the -th measurement: W is the wait time (s), E = the echo sacing (s), G = the magnetic field gradient (Gauss/cm) and NE = the number of echoes acquired. In ractice, the number of different sin-echo measurements comosing a suite is a relatively small number (e.g., tyically N 6. ) he multifluid relaxation model can be written in the general form A j Nw j* E W = al ex 1 ex( ) (Brine) l = 1 2, l ( ) ξ * 2, l No j* E W + b ex 1 ex( ) (Oil) = 1 2, o( η, ) 1, o( η) (1) j* E W + A ex 1 ex( ) (Gas) g + A OBMF 2, g ( ) 1, g j* E W ex 1 ex( ), (OBMF) 2, OBMF ( ) 1, OBMF where A j = the amlitude of the j-th echo for the -th measurement. he first term in Eq. 1 contains the brine contribution to the signal on the j-th echo. he summation in the first term is over the set of N w amlitudes {a l } that comose the diffusion-free brine 2 distribution. he 2,l are a set of logarithmically saced relaxation times owing to bul and surface relaxation of hydrogen nuclei in the brine. he arameter ξ is an aarent 1 / 2 ratio for the brine. Note that the olarization functions (i.e., the factors containing W ) in Eq. 1 are valid for a stationary measurement. For a moving NMR logging tool, seed-deendent olarization functions are used to roerly account for seed and magnet reolarization effects. he aarent relaxation rate of the brine transverse magnetization including the effects of unrestricted molecular diffusion 6 in the magnetic field gradient is given by December 2001 SPE Journal 453

3 2 1 1 ( H G E) = + γ Dw ( ),.. (2) ( ) 12 2, l 2, l D w () is the molecular diffusion coefficient of water molecules 7-1 at temerature and γ H = 2π 4258 Gauss -1 s is the roton gyromagnetic ratio. For restricted diffusion, the second term in Eq. 2 is modified. he second term in Eq. 1 accounts for the signal from the crude oil. he b are signal amlitudes from the -th molecular constituent in the crude oil mixture and 1,o (η ) are longitudinal relaxation times of the constituents. he aarent transverse magnetization relaxation rate of the -th molecular constituent in the crude oil is given by where 2,OBMF is the diffusion-free transverse relaxation time of the filtrate. It can usually be assumed that the longitudinal and transverse relaxation times are equal (i.e., 2,OBMF = 1,OBMF ). he OBMF diffusion coefficient D OBMF and relaxation times can be estimated from laboratory measurements of the OBMF viscosity and used as inuts in Eq. 1. If suitable laboratory measurements are not available, then the in-situ OBMF viscosity can be treated as an unnown arameter that is estimated in the inversion of Eq. 1. As noted above, if gas is dissolved in the OBMF, then the OBMF relaxation times have an exlicit deendence on GOR. o simlify the notation, the temerature deendence of 1,OBMF, 2,OBMF, and D OBMF is not shown in the equations above. Eq. 1 assumes that crude oil and OBMF in the flushed zone exist as two searate fluids (i.e., that there is a istonlie dislacement of the crude oil by the invading OBMF and that the mixing of the two fluids by diffusion is negligible). If this assumtion is not valid, then the two searate fluids can be treated as a mixture in Eq. 1. 2, o ( H G E) = + γ Do( η),...(3) ( η, ) ( η ) 12 2, o where 2,o (η ) is the diffusion-free (intrinsic) bul transverse relaxation time and D o (η ) is the molecular diffusion coefficient for the -th molecular constituent in the oil. o simlify the notation, the temerature deendence of 1,o (η ), 2,o (η ), and D o (η ) is not shown in Eqs. 1 and 3. Note that we have made the assumtion that reservoir roc surfaces are referentially waterwet so that the hydrocarbon relaxation times are not affected by surface relaxation. his assumtion is not essential (i.e., the MRF multifluid relaxation model can be modified to handle relaxation in mixed or water-wet rocs). he η are henomenological microscoic arameters referred to as constituent viscosities in the CVM. hese arameters rovide an imortant lin between diffusion-free relaxation times and molecular diffusion coefficients in liquid hydrocarbon mixtures. For live oils containing solution gas, the relaxation times 2,o (η ) and 1,o (η ) have an exlicit deendence on GOR. he deendence of 2,o (η ) and 1,o (η ) on η and on GOR for live oils is discussed in detail in the section on the CVM. he third and fourth terms in Eq. 1 reresent signals from gas and OBMF, resectively. he relaxation of gas and OBMF signals can be described by single exonential decays with amlitudes A g and A OBMF, resectively. he OBMF signal in Eq. 1 should be included in the evaluation of formations drilled with oil-base muds. he aarent gas transverse magnetization relaxation rate is ( H G E) = + γ Dg ( P, ),. (4) ( ) ( P, ) 12 2, g 2, g where 2,g (P,) = 1,g (P,) are gas relaxation times and D g (P,) is the gas molecular diffusion coefficient. Plots of these quantities as functions of temerature () and ressure (P) for methane gas are shown by Kleinberg and Vinegar. 7 he aarent OBMF transverse relaxation rate is given by 2, OBMF ( H G E) = + γ DOBMF,.....(5) ( ) 12 2, OBMF Inversion of the Multifluid Relaxation Model. An efficient, fast, accurate, and robust real-time algorithm has been develoed to erform the nonlinear inversion of the relaxation model. CPU times for inversion of a tyical data suite on a PC Windows machine with a 450-MHz Pentium rocessor were less than one second. Freedman 8 has ublished the mathematical details of the inversion algorithm used to obtain the results shown in this aer. o test the inversion, Monte Carlo simulations were erformed for model roc formations containing brine, crude oil, gas, and OBMF. he simulations were designed to exlore a wide range of fluid and roc roerties. he simulation results show that: he inversion is robust and accurate over a wide range of hydrocarbon and roc roerties. Gas can be easily identified because its diffusion coefficient is significantly greater than that of the flushedzone liquids. he inversion rovides accurate estimates of the gas amlitude in Eq. 1 even in the resence of crude oil (for examle, in a reservoir where the ressure is below the bubbleoint ressure). he largest source of uncertainty in the estimated gas volumes and saturations results from uncertainties in the gas hydrogen index. Crude oil saturations can be quantitatively estimated even in the resence of OBMF and gas, rovided that there exists sufficient viscosity contrast between the oil and OBMF. As exected, if the native oil and OBMF have similar relaxation times and diffusion coefficients, then it is not ossible to comute accurate saturations; the inversion will confuse the two fluids and cannot differentiate one from the other. In such cases the oil saturation will be overestimated and the OBMF saturation underestimated, or vice versa. he inversion is robust and accurate for a wide range of crude oil viscosity. his range is from very low (i.e., of the order of one c) viscosity to moderately high viscosity on the order of 100 c. he inversion is more difficult for higher-viscosity oils and is less accurate for oils with viscosity in excess of about 100 c. his results, in art, from loss of sensitivity to diffusion the diffusion-induced relaxation rate is negligible comared to the intrinsic (diffusion-free) relaxation rate for high-viscosity oils. Moreover, the relaxation times for high-viscosity oils are comarable to those of the BVI so that it is difficult to differentiate high-viscosity oil from BVI. 454 December 2001 SPE Journal

4 Fig. 1 Synthetic sin-echo measurement suite used for the Monte Carlo simulation of a model carbonate formation containing brine, crude oil, gas, and OBMF. he measurement suite consisted of six hase-corrected Carr-Purcel-Meiboom- Gill (CPMG) sin-echo sequences that have different wait times, echo sacings, magnetic field gradients, and echo numbers. he tic mars on the x-axis show the start and finish of echo acqusition for each CPMG. he time on the x- axis is the echo acquisition time and does not include the wait time that recedes each CPMG. his suite of measurements was used to invert the multifluid relaxation model in Eq. 1. In this simulation, the rmsnoise er echo was equal to 1.8.u. for the first two CPMGs and 0.6.u. for the other four CPMGs. Monte Carlo Inversion Results. It is instructive to discuss the results of an inversion of the multifluid relaxation model. he results shown for this examle are reresentative of the results from many other simulations that were erformed for a wide variety of fluid and roc roerties. Suites of noisy synthetic sinecho sequences were generated using Eq. 1 for a model formation containing brine, crude oil, gas, and OBMF. his examle reresents a challenging numerical test of the accuracy and robustness of the MRF inversion. Each data suite consisted of six different sin-echo measurements. It is worth noting that fewer measurements suffice in ractice. Freedman et al. 9 have used suites of data with four sin-echo measurements in recent field alications of the MRF method. A tyical data suite for one realization of the random noise that was used in the Monte Carlo simulation is shown in Fig. 1. he brine 2 distribution used for the simulation is shown in Fig. 2. It is similar to the 2 distributions found in many carbonate rocs. In this examle, the OBMF viscosity was assumed nown and equal to 2 c. Fig. 3 comares the results of the Monte Carlo simulation with the inut fluid saturations and oil viscosity. Also shown are the standard deviations in the fluid saturation and viscosity estimates. he results shown in Fig. 3 clearly demonstrate that Eq. 1 can be accurately inverted for fluid saturations and oil viscosity even when three hydrocarbon fluids are resent. For this examle, the number of unnown arameters estimated in the inversion of Eq. 1 is equal to 60 (41 brine amlitudes, 8 crude oil amlitudes, 8 crude oil constituent viscosities, OBMF amlitude, gas amlitude, and aarent brine 1 / 2 ratio). he MRF method can be also used to rovide robust estimates of diffusion-free brine and oil 2 distributions; however, this requires an alternative imlementation of the inversion algorithm that differs slightly from the one used in this aer. he alternative imlementation is discussed in Ref. 8 and is used to rocess field and laboratory data in Ref. 9. Monte Carlo simulations lie the one discussed in the revious aragrah serve to establish the numerical accuracy and robustness of the MRF inversion for synthetic data. o establish the viability of the MRF method for real fluid-saturated rocs, it is necessary to also show that the relaxation model is based on Fig. 2 he brine 2 distribution used in the carbonate Monte Carlo simulation referred to in Fig. 1. correct hysics. It is therefore necessary to confirm with laboratory exeriments that CVM rovides an accurate descrition of NMR relaxation and molecular diffusion in hydrocarbon mixtures and crude oils and that the MRF inversion can be used to accurately estimate fluid saturations and oil viscosities in controlled exeriments conducted on artially and fully saturated roc samles. his is discussed in the section entitled Exerimental Confirmation. Formation Evaluation Parameters Estimated From the Inversion of the Multifluid Relaxation Model. he following answer roducts rovided by the MRF method are comuted following the inversion of the multifluid relaxation model. Flushed-Zone Fluid Volumes. he brine-filled orosity, φ w,xo, is comuted from the equation φ wxo, 1 = HI w Nw al,... (6) l = 1 Fig. 3 Comarison of the inut fluid roerties with the results of a Monte Carlo simulation for a model carbonate formation containing brine, crude oil, gas, and OBMF. he solid bars show the inut fluid saturations, and the stried bars show the MRF Monte Carlo estimates. he inut and estimated oil viscosities are also shown. Note the accuracy and small standard deviations in the estimates. December 2001 SPE Journal 455

5 where it has been assumed that, excet for the hydrogen index correction, the sin-echo amlitudes used in the inversion are in calibrated orosity units. he brine hydrogen index can be estimated from the salinity of the formation water. 7 he a l are the amlitudes in the brine 2 distribution. BVI is comuted from the equation, 1 Ncut BVI = a,...(7) HI 1 l w l = where the summation is over the N cut brine amlitudes with relaxation times ( 2,l ) less than the bound-water 2 cut-off ( 2,cut ). he free-water volume, φ wf,xo, is comuted from the difference φ wf, xo φw, xo BVI, =...(8) 2,cut can be determined emirically from the maximum observable 2 after an initially water-saturated roc is centrifuged to remove the free water. 10 yical values of 2,cut are 33 and 100 ms for sandstone and carbonate rocs, resectively. he flushedzone oil volume, φ o,xo, is comuted from the equation φ oxo, No b 1 No = = HI HI = 1 o, o 1 and similarly for the gas and OBMF volumes: and g gxo, =, HI g b,.....(9) A φ...(10) A φ =..... (11) OBMF OBMF, xo, HIOBMF In Eq. 9 we have relaced the distribution of hydrogen indices of the molecular constituents in the crude oil by the macroscoic or measured hydrogen index (i.e., HI o, ΗΙ ο ). he hydrogen indices of dead crude oils can be estimated from the API gravity and are close to one for gravities greater than 25 o API. 7 he hydrogen index of OBMF can either be measured using NMR or comuted from the nown chemical formula, molecular weight, and number of hydrogen nuclei in the chemical formula. Formulas for the hydrogen index of live oils as a function of temerature, ressure, and solution GOR have been ublished by Zhang et al. 11 Flushed-Zone Fluid Saturations. he flushed-zone fluid saturations are comuted from the fluid volumes. For examle, the oil saturation S o,xo is comuted from the equation, φ φ S oxo, = φ + φ + φ + φ φ oxo, oxo, wxo, oxo, gxo, OBMF, xo,...(12) where the total NMR orosity (φ ) has been defined. he other fluid saturations are comuted using analogous equations. Oil Viscosity. he macroscoic (i.e., measured) crude oil viscosity (η o ) can be comuted from the logarithmic mean of the crude oil constituent viscosity distribution using the CVM; i.e., No log( ηo) = f log( η ),...(13) = 1 where f and η are the roton fraction and the constituent viscosity, resectively, associated with the -th crude oil molecular constituent. he f are defined by f b = No b n = 1 n,....(14) Note that the sum of the f terms is equal to one. A derivation of Eq. 13 is given in the later section on the CVM. Hydrocarbon-Corrected Permeability. he MRF method rovides a more accurate estimate of BVI (and therefore ermeability) in hydrocarbon-bearing zones than was reviously ossible. BVI is by definition equal to the total bound-water volume. It contains contributions from both caillary and claybound waters. he imur-coates (-C) ermeability equation requires BVI as an inut; i.e., K C 4 4 FFV 2 = 10 φ ( ),...(15) BVI where φ Τ is the total orosity in units of.u./100 and K C is the ermeability in millidarcy (md) units. FFV is the free-fluid volume and includes movable brine and hydrocarbons. As already noted, the existing NMR fluid-characterization methods cannot accurately searate brine and oil 2 distributions. Instead, the revious methods comute aarent 2 distributions that include contributions from both the brine and hydrocarbon signals. Summation of the amlitudes in the aarent 2 distribution over all relaxation times less than 2,cut roduces an estimate of the bound-fluid volume (BFV). In wet zones, BFV = BVI; however, in oil-bearing zones, if the oil 2 distribution has amlitudes with 2 values less than 2,cut, then BFV > BVI. his same effect can also occur in gas-bearing zones if the gas signal has a relaxation time less than 2,cut. he solid shaded area in Fig. 4 shows the difference between BFV and BVI in an oil zone. In hydrocarbon-bearing zones, use of BFV in lace of BVI in the -C equation can result in underestimation of ermeability. Comutations have shown, in some cases of ractical interest, that using BFV in Eq. 15 can result in ermeability estimates in oil zones that are essimistic by more than an order of magnitude. Fig. 4 A schematic showing the difference between BFV and BVI in an oil zone where the brine and crude oil 2 distributions overla. BFV and BVI are the areas under the aarent and brine 2 distributions, resectively, that are to the left of 2,cut. he solid shaded area shows the difference between BFV and BVI. he MRF method can searate overlaing crude oil and brine 2 distributions and therefore can accurately comute the BVI required for ermeability estimation. Accurate BVI estimates are also needed to rovide reliable estimates of the otential for water roduction from oil- and gas-bearing zones. 456 December 2001 SPE Journal

6 Constituent Viscosity Model (CVM) Published exerimental NMR and viscosity data for a wide variety of crude oils and other hydrocarbon liquid mixtures 12,13 have shown that their measured viscosities (η o ) are inversely roortional to the logarithmic means ( 2,LM ) of their diffusionfree 2 distributions. hese data can be fit by a macroscoic emirical correlation of the form a 2, LM =,...(16) η ( ) o where a is an emirically determined constitutive constant and is the samle temerature in K. he above correlation is for dead hydrocarbon liquids that do not contain solution gas. For live hydrocarbon mixtures and crude oils, the 2,LM have an exlicit deendence on GOR, 13 which is discussed below. Since crude oils are mixtures consisting of many different tyes of hydrocarbon molecules, 14 rotons in crude oils do not relax with a single relaxation time. Protons situated on different molecules exerience different local fluctuating diolar fields and therefore relax at different rates, giving rise to a distribution of relaxation times. yical crude-oil relaxation time distributions consist of a single ea accomanied by a long tail that extends toward shorter relaxation times. he longer relaxation times in the ea are associated with the lighter, smaller molecules in the crude oil, whereas the shorter relaxation times in the tail are associated with the heavier, larger molecules. One of the missing lins in our revious ability to reliably estimate crude oil viscosity from NMR well-log data was the lac of a microscoic hysical model for diffusion-free NMR relaxation and molecular diffusion in hydrocarbon mixtures. In secial cases, in which the brine and crude oil transverse relaxation time distributions are well searated, accurate estimates of 2,LM from NMR log data are ossible, and the viscosity of the crude oil can be reliably redicted using Eq. 16. However, in most situations encountered in ractice, the brine and crude-oil distributions overla, and it has not been reviously ossible to reliably estimate the 2,LM of the oil distributions from NMR welllog data. Constituent Viscosities. he CVM maes the hyothesis that the diffusion-free relaxation time 2, of the -th molecular constituent in a liquid hydrocarbon mixture is of the same form as the macroscoic emirical viscosity correlation for 2,LM ; i.e., a =...(17) η 2,, ( ) where η is the constituent viscosity in centioise (c) units. An identical equation can be written for the sin-lattice relaxation time ( 1, ). For most crude oils, 1,LM = 2,LM ; however, it has been found that 1,LM > 2,LM for crude oils with high ashaltene content. he η are henomenological molecular variables that have a hysical basis similar to the friction coefficients used in the Langevin equation treatment of a article diffusing in a viscous medium. 15 Recalling the definition of the logarithmic mean, we can write the equation for an n-comonent mixture f1 f 2 f n 2, LM 2,1 2,2 2, n,...(18) where f is the roton fraction of the -th molecular constituent. Combining Eqs. 16, 17, and 18 leads to the result that the macroscoic mixture viscosity is the logarithmic mean of the η distribution; i.e., one finds that f f f 1 2 n o = n LM η η1 η2 η ( η ),...(19) which is equivalent to Eq. 13 shown reviously. In arriving at the equation above we have used the fact that n f = 1,...(20) = 1 Eq. 19 is one of the ey results of the CVM and is a viscosity mixing law for hydrocarbon mixtures. An imortant oint to emhasize is that the η are not the viscosities of the ure mixture constituents they differ from the latter because of intermolecular interactions in the mixture. Diffusion Coefficient Distributions. he CVM also maes the hyothesis that in hydrocarbon mixtures and crude oils there exists a distribution of molecular diffusion coefficients of the form b D =,...(21) η ( ) where b is an emirical constitutive constant that has been determined from Stejsal and anner 13,16 PFG diffusion measurements. D is the molecular diffusion coefficient of the -th constitutent in the mixture. he D are not the diffusion coefficients of the ure mixture comonents because, lie 2,, they are modified by intermolecular interactions in the mixture. An equation analogous to Eq. 18 can be written for D LM. Note that the mixture diffusion coefficients in Eq. 21 have the same deendence on /η as that of the diffusion coefficients for ure liquids derived by Einstein in his ioneering theoretical study of Brownian motion. 15 In the CVM, each hydrocarbon molecule in the mixture is assumed to relax and diffuse as it would in its ure-state liquid, excet that the microscoic constituent viscosity relaces the macroscoic ure-state viscosity. he microscoic mixture details such as molecular comosition, molecular interactions, and molecular sizes are contained solely in the constituent viscosities that can be determined from NMR measurements. he constitutive constant a in Eqs. 16 and 17 has been found for a wide variety of crude oils measured at 2 MHz to be well aroximated by the value a = s c K -1, which is used in this aer for all crude oil comutations. For ure n-alanes and methane-n-alane mixtures, Lo 13 found a larger value, a = s c K -1. We believe that the larger value of a found for the ure alanes and the methane-alane mixtures is caused by the absence of the short relaxation time tail that is resent in crude oil relaxation-time distributions. he constitutive constant b in Eq. 21 has been found to be well aroximated by the value b = cm 2 s -1 c K -1 for dead and live hydrocarbon mixtures and crude oils. his value is used for the diffusion comutations in this aer. Predictions of the CVM. In addition to Eq. 19, CVM theory leads to several new theoretical redictions that have been confirmed by laboratory exeriments. On substitution of Eqs. 17 and 21 into 19, we find that a b η = = ( η ),...(22) o LM 2, LM DLM where D LM is the logarithmic mean of the diffusion coefficient distribution. Eq. 22 redicts that the viscosity of a liquid hase hydrocarbon mixture or crude oil can be comuted from either the logarithmic mean of the 2 or the D distribution. he first equality in Eq. 22 is consistent with resent-day nowledge (e.g., Eq. 16), whereas the second equality is new. Moreover, the two methods December 2001 SPE Journal 457

7 of comuting η o are theoretically equivalent, and both methods are also equivalent to comuting η o from Eq. 19. CVM theory also redicts that the ratios of mixture constituent diffusion coefficients to constituent relaxation times are constant. Division of Eq. 21 by 17 leads to D 2, b DLM = =,...(23) a 2, LM where the second equality in Eq. 23 follows from Eq. 22. A consequence of Eq. 23 is that 2 and D distributions for hydrocarbon mixtures and crude oils are not indeendent (i.e., if one is nown, the other can be comuted). Extension of CVM to Live Hydrocarbon Mixtures and Crude Oils. he CVM can be extended to live hydrocarbon mixtures with existing macroscoic emirical correlations that relate 1,LM and D LM to η ο /Τ and GOR. In this section we first review these correlations and then discuss the modified CVM equations. Macroscoic Correlations for Live Hydrocarbon Mixtures. he existing macroscoic emirical correlations for live hydrocarbon mixtures were established using NMR inversion recovery ( 1,LM ) and PFG (D LM ) measurements acquired on binary mixtures of methane-n-hexane, methane-n-decane, and methanen-hexadecane. 13,17 he mixture data were acquired for a wide range of temeratures and ressures with the liquid and vaor hases in thermodynamic equilibrium. his range of temeratures and ressures corresonds to a wide range of GOR. Diffusion-free measurements of 2,LM [e.g., using Carr-Purcell-Meiboom-Gill (CPMG) sin-echo sequences] were not ossible in these mixtures because of large diffusion effects on the methane signal in the inhomogeneous field of the NMR sectrometer. Analysis of the liquid hase methane-n-alane mixtures data led to the following emirical macroscoic live oil correlations for 1,LM and D LM, a = =...(24) η 1, LM 2, LM, o f (GOR) where GOR is defined here as cubic meters of solution gas er cubic meters of stoc tan liquid (m 3 /m 3 ) at standard conditions. Standard conditions of temerature and ressure are 60 F and one atmoshere, resectively. GOR can be converted from units of m 3 /m 3 to units of cubic feet of gas er barrel of stoc tan liquid (ft 3 /bbl) at standard conditions by multilying by a conversion factor equal to he emirically determined function f(gor) in Eq. 24 is given by where f 10 α (GOR) 10, =...(25) α = 0.127[ log ] 2 10 (GOR) log 10(GOR) 2.80,..... (26) he macroscoic emirical correlation for live hydrocarbon mixtures that relates D LM to the mixture viscosity has the same form as the one for dead mixtures; i.e., D LM b =, (27) η o In Eq. 24 we have used the fact that 1,LM = 2,LM, because it has been shown for methane-n-alane mixtures 13 that the fast motion limit ω ο τ c <<1 is satisfied for the 90-MHz Larmor frequency used in the exeriments. In the inequality, τ c is a tyical rotational diffusion correlation time in the mixture that for low viscosity fluids is on the order of to seconds, and ω ο is the Larmor angular frequency. In a crude oil, there is a broad distribution of correlation times. We have exerimental data that suggest that the correlation times associated with the larger molecules in a crude oil are so long that the fast diffusion condition breas down at 90 MHz and the equality 1,LM = 2,LM is no longer valid. his is discussed in more detail in the section on live crude-oil measurements. Examination of Eq. 24 shows that the NMR relaxation times have an exlicit deendence on GOR. Of course, both the relaxation times and the diffusion coefficients have an imlicit deendence on GOR because the effect of solution gas is to reduce the mixture viscosity. he latter effect tends to increase the relaxation times and diffusion coefficients for live hydrocarbon mixtures and crude oils, whereas the effect of f(gor) is to reduce the relaxation times because f ( GOR) 1. Modified CVM Equations for Live Mixtures and Crude Oils. Emloying the emirical macroscoic correlation in Eq. 24 that was established for methane-n-alane mixtures, the CVM ostulates that a =...(28) η 2,, f (GOR) which is the generalization to live hydrocarbon mixtures of Eq. 17 that is valid for dead hydrocarbon mixtures and crude oils. It is easy to show using Eqs. 18, 24, and 28 that Eq. 19 for the mixture viscosity as a function of the constituent viscosities is also valid for live mixtures. It further follows from Eqs. 24 and 27 that a b η = = ( η ),...(29) o LM 2, LM f(gor) DLM he correlation in Eq. 27 says that the CVM constitutive equation (i.e., Eq. 21) for dead mixtures is also valid for live mixtures. herefore, using Eqs. 21 and 28 shows that the ratio of D to 2, for live hydrocarbon mixtures is given by D b f(gor) DLM = =, (30) a 2, 2, LM Exerimental Confirmation he first art of this section comares the CVM with exerimental results for dead and live hydrocarbon mixtures and crude oils. hese results confirm the validity of the CVM for hydrocarbon mixtures. he second art of this section comares water saturations estimated from MRF inversions of NMR data suites acquired on artially and fully saturated reservoir rocs with indeendent estimates of the water saturations. he oil viscosities estimated from the inversions are comared with the nown viscosity of the oil in the roc samles. Comarison of CVM Predictions with NMR and Viscosity Measurements for wo-comonent Mixtures. Pulsed field gradient (PFG) and CPMG sin-echo measurements were conducted at 2 MHz on deoxygenated binary hydrocarbon mixtures. hese included mixtures with two different molecular comositions: n-hexane-n-hexadecane (C6-C16) and n-hexane-squalene (C6-C30). he room temerature viscosities of ure C6, C16, and C30 are aroximately 0.3, 3, and 11 c, resectively. For each mixture, measurements were made at three different concentrations (i.e., weight fractions) of the constituents. Samle temeratures were held constant at 30 C for all measurements. 458 December 2001 SPE Journal

8 Fig. 5 A lot of NMR measurements of D vs. 2, for n- hexane-squalene mixtures at three concentrations. Also shown are the measured ure comonent endoints. he straight line is the relationshi redicted by Eq. 23 of the CVM for dead hydrocarbon mixtures. Diffusion-free relaxation times, 2,, and roton fractions, f, of the two mixture constituents were estimated for each mixture concentration by fitting the observed CPMG decays to a fourarameter, two-exonential model. As exected, the roton fractions estimated from the fits agreed very well with the nown roton fractions in each samle. hese estimates were used in Eq. 18 to comute 2,LM for each mixture. he sin-echo data were acquired using a 2.4-ms echo sacing, and 16,000 echoes were collected so that the long relaxation times of the mixture constituents could be resolved by inversion of the data. Inversion recovery measurements were also conducted on ure C6 to determine its 1 relaxation time. he good agreement between 1 from the inversion recovery and 2 estimated from a CPMG roved that diffusion effects were not resent in the CPMG decays, thus demonstrating that the 2, were free of diffusion effects. Diffusion coefficients, D, and roton fractions, f, of the two mixture constituents were estimated for each mixture concentration by fitting a sequence of Stejsal-anner 16 PFG sinechoes to a two-exonential relaxation model. hese estimates were used to comute D LM for each mixture. Fig. 5 shows a log-log lot of the measured D vs. 2, for the C6-C30 mixtures at the three concentrations shown in the figure, lus the ure comonent endoints. Also shown on the lot is the CVM-redicted (see Eq. 23) straight line with sloe one and intercet b/a. Note that the measured data oints fall along the line redicted by the CVM. Fig. 6 shows good agreement between the measured C6-C30 mixture viscosities for the three different mixture concentrations and those redicted by the CVM using Eqs. 22. heoretically, according to the CVM, the viscosity estimates obtained from 2,LM and D LM should be equal. he minor differences seen in Fig. 6 result from model and measurement errors. Results similar to those shown in Figs. 5 and 6 were also found for the C6-C16 mixtures and therefore, to save sace, will not be shown here. Fig. 6 A lot of measured vs. CVM-estimated viscosities for n-hexane-squalene mixtures at three concentrations. Also shown are the measured and estimated ure comonent viscosities. he average absolute ercent deviations of the CVM-estimated viscosities from the measured viscosities are 33.1 and 5.2% as comuted from D LM and 2,LM, resectively. Comarison of CVM Predictions with NMR and Viscosity Measurements for Dead Crude Oils. Crude oils are comlex mixtures that in ractice have unnown and highly variable molecular comositions. 14 One of the attractive features of the CVM is that the molecular comosition and other details such as molecular sizes and molecular interactions are contained in the η distributions that are estimated by inversion of NMR laboratory or well-logging measurements. Both CPMG and PFG measurements were erformed at 2 MHz on four crude oils from the Rice U. database to test and comare the viscosity estimators in Eq. 22. We also alied the CVM estimator of crude oil viscosity based on 2,LM to ublished Belridge field crude-oil data acquired at 2 MHz. 12 PFG data were not available for the Belridge crudes, so the D LM viscosity estimator could not be tested for these oils. Fig. 7 shows a comarison of CVM-estimated viscosity vs. measured viscosity for 31 Belridge crude oils and the four crude oils from the Rice U. database. he dead oil measurements were made at 25 and 30 C for the Belridge and Rice oils, resectively. CVM redicts that 2 and D distributions for hydrocarbon mixtures are not indeendent if either is nown, the other one can be estimated using Eq. 23 for dead mixtures or Eq. 30 for live mixtures. Measured 2 and D distributions were determined from CPMG and PFG measurements. o test the CVM rediction, D distributions were comuted from the measured 2 distributions and comared with the measured D distributions. Fig. 8 comares the measured and comuted D distributions for the four Rice U. crude oils. Clearly, the ea ositions and general shae of the measured and comuted distributions are similar as redicted. However, the measured D distributions from the PFG exeriments are missing the very small diffusion coefficients that are seen in the D distributions comuted from the CPMG measurements. Because the echo sacings used in the PFG Exeriments were in the range from 60 to 80 ms, the signals from very small diffusion coefficients (that corresond to very short relaxation times) were not recoverable. December 2001 SPE Journal 459

9 From Eq. 30 the ratio of b/a determines the y-intercet of the line redicted by CVM on a log-log lot of D vs. 2, f(gor). Fig. 9 comares the theoretical CVM line with exerimental measurements at seven different GORs. he GORs used in these comutations were estimated from measured Rice U. GOR data for methane-n-hexadecane mixtures. he live mixture data fall along the line redicted by CVM. Fig. 10 shows the comarison of mixture viscosities estimated from CVM with those comuted using Suerra, an interactive comuter database for the rediction of transort and equilibrium roerties of ure hydrocarbons and hydrocarbon mixtures. 18 he reason that there are fewer data oints for 2,LM than for D LM is that it was not ossible to resolve two distinct comonents in the C1-C16 relaxation time distributions for all of the GORs used in the exeriments. Fig. 7 A lot of measured vs. CVM-estimated viscosities for 31 Belridge field dead crude oils and 4 Rice U. dead crude oils. he average absolute ercent deviations of the CVM-estimated viscosities from the measured viscosities are 11.9 and 26.8% as comuted from D LM and 2,LM, resectively. Comarison of CVM With Live Crude-Oil Measurements. Measurements of viscosity and GOR were made on a live North Sea crude oil in equilibrium with methane gas at 35 C for four saturating bubbleoint ressures, e.g., 1,800, 2,800, 3,500, and 4,000 sia. he measured viscosity and API gravity of the dead oil at room temerature are 9.2 c and 29.7 API, resectively. he live oil viscosity and GOR measurements were made by a Schlumberger comany secializing in laboratory ressure/volume/temerature (PV) and flow-assurance measurements and are shown in able 1. Note that although the GORs are relatively low, the live oil viscosities are significantly reduced comared to those of the dead oil. NMR relaxation time and diffusion measurements at 35 C were also made on the live crude oil at the four bubbleoint ressures. he live crude-oil NMR exeriments and results are discussed below. Fig. 8 A comarison, for four Rice U. crude oils, of D distributions measured by PFG measurements with D distributions comuted from the measured 2 distributions using CVM Eq December 2001 SPE Journal

10 Live Crude-Oil PFG Measurements. Stejsal-anner PFG measurements at 30 C of diffusion coefficient distributions were first made at 2 and 90 MHz on the dead crude oil. As we exected, the D distributions were very similar and did not show a frequency deendence. he logarithmic mean diffusion coefficients for the dead oil were almost identical; i.e., D LM = cm 2 /s at 2 MHz and cm 2 /s at 90 MHz. After it was shown that the crude oil D distributions were indeendent of frequency, the live oil PFG were erformed at 90 MHz and 35 C. he D LM data measured on the live crude oil are shown in able 2. Fig. 11 shows excellent agreement between the CVM viscosities comuted using Eq. 27 and the measured viscosities. he data oint at 0.3 c reresents a live oil samle with a GOR of 2,200 SCF/bbl (391 m 3 /m 3 ) at reservoir conditions. Both viscosity and NMR relaxation time data were measured on this live oil by Ael et al. 19 Note the excellent agreement between the viscosity estimated using CVM Eq. 24 and the measured viscosity. Live Crude-Oil Relaxation ime Measurements. Inversion recovery measurements of 1 distributions at 30 C were first made made on the dead crude oil at 2 and 90 MHz. he dead oil 1 measurements showed a strong frequency deendence (i.e., 1,LM = 176 ms at 2 MHz and 369 ms at 90 MHz). hus, the 1 relaxation rate is slower at 90 MHz than at 2 MHz. he 1 distribution measured at 90 MHz showed an obvious shift toward longer relaxation times comared to the one at 2 MHz. he frequency deendence of 1 for an intermediate viscosity oil is not surrising because, as noted reviously, the fast motion condition, ω ο τ c <<1, is not necessarily satisfied for some of the larger molecules in the crude oil. 1 measurements were also made at 35 C on the dead and live crude oil at 90 MHz. CPMG measurements of 2 distributions at 30 C were first made on the dead crude oil at 2 MHz and 90 MHz. he 2,LM of the dead crude oil at 2 and 90 MHz were nearly identical (i.e., 2,LM = 184 ms at 2 MHz and 185 ms at 90 MHz); however, the shortest relaxation times in the 90- MHz 2 distribution were shifted to the right, and the longest times were shifted to the left comared to those at 2 MHz. Aside from this squeezing effect on the 90-MHz 2 distribution, the frequency deendence of 2 is much weaer than that of 1. he measured 2,LM of the dead and live crude oil at 90 MHz and 35 C are shown in able 2. Fig. 11 shows good agreement between the CVM viscosities comuted with CVM Eqs. 24 through 26 and the measured viscosities. he 90-MHz live oil 2 measurements were made with two echo sacings: 3 and 4 ms. hese measurements showed no deendence on the echo sacing, thus roving that the measured 2 distributions were free of diffusion effects. Estimation of Water Saturation and Oil Viscosity in Rocs by Inversion of Laboratory NMR Data. he MRF inversion was tested with suites of CPMG sin-echo data acquired at 2 MHz. he data suites were acquired on Berea 100 sandstone (SS) and Indiana limestone (LS) roc samles at 27 C. he samles were cylindrical cores (3.75-cm long and 2.0-cm diameter) that were either fully brine saturated or artially saturated with 0.2 ohm-m brine and S6 oil. S6 oil is a Cannon viscosity standard with a Fig. 9 A lot of NMR measurements of D vs. 2, for methanen-hexadecane mixtures at seven GORs. he straight line is the relationshi redicted by Eq. 30 of the CVM for live hydrocarbon mixtures. Fig. 10 A lot of CVM-estimated viscosities vs. those comuted using Suerra for methane-n-hexadecane mixtures at different GORs. he average absolute ercent deviations of the CVM-estimated viscosities from the Suerra viscosities are 22.2 and 18.4% as comuted from D LM and 2,LM, resectively. December 2001 SPE Journal 461