Three-phase Oil Relative Permeability Research Articles

Identifiability of parameters of three-phase oil relative permeability models under simultaneous water and gas (SWAG) injection

We assess the relative performance of a suite of selected models to interpret three-phase oil relative permeability data and provide a procedure to determine identifiability of the model parameters. We ground our analysis on observations of Steady-State two- and three-phase relative permeabilities we collect on a water-wet Sand-Pack sample through series of core-flooding experiments. Three-phase experiments are characterized by simultaneous injection of water and gas into the core sample initiated at irreducible water saturation, a scenario which is relevant for modern enhanced oil recovery techniques. The selected oil relative permeability models include classical and recent formulations and we consider their performance when (i) solely two-phase data are employed and/or (ii) two- and three-phase data are jointly used to render predictions of three-phase oil relative permeability, kro. We assess identifiability of model parameters through the Profile Likelihood (PL) technique. We rely on formal model discrimination criteria for a quantitative evaluation of the interpretive skill of each of the candidate models tested. We also evaluate the relative degree of likelihood associated with the competing models through a posterior probability weight and use Maximum Likelihood Bayesian model averaging to provide model-averaged estimate of kro and the associated uncertainty bounds. Results show that assessing identifiability of uncertain model parameters on the basis of the available dataset can provide valuable information about the quality of the parameter estimates and can reduce computational costs by selecting solely identifiable models among available candidates.

Characterization of two- and three-phase relative permeability of water-wet porous media through X-Ray saturation measurements

We present experimental investigations of multi-phase (two-phase (oil/water, oil/gas) and three-phase (oil/water/gas)) relative permeabilities performed on laboratory scale rock cores. Two- and three-phase relative permeabilities data were obtained on two core samples (a Sand-pack and a Berea sandstone) by way of a Steady-State (SS) technique. Spatial and temporal dynamics of in-situ saturations along core samples were directly measured through an X-Ray absorption technology. The latter rendered detailed distributions of (section-averaged) fluid flow phases through the medium, which can then be employed for the characterization of relative permeabilities. The technique also enabled us to clearly identify the occurrence of end-effects during the experiments and to quantify the reliability of corrective strategies. For the oil/water settings we considered low and high viscosity oil, our findings supporting the observation that relative permeability to oil and water was sensitive to oil viscosity. Three-phase experiments were performed by following an IDI (Imbibition-Drainage-Imbibition) saturation path. The complete experimental data-base is here illustrated and juxtaposed to results obtained by the implementation of simple and commonly employed three-phase relative permeability models. In the three-phase setting, water and gas relative permeabilities display an approximately linear dependence on the logarithm of their own saturation. Consistent with the observation that oil behaves as an intermediate phase in our system, three-phase oil relative permeabilities lie in between those of their two-phase counterparts. Our data-set stands as a reliable reference for further model development and testing, as only a limited quantity of three-phase data are currently available.

Three-Phase Oil Relative Permeability in Water-Wet Media: A Comprehensive Study

We report experimental three-phase oil relative permeability in two water-wet media (a sandpack and a Berea sandstone core) along different saturation paths. Three oils with different viscosities, compositions, and spreading coefficients were used in the measurements. The data show that oil relative permeability can vary significantly along different saturation paths. Most importantly, we find that despite the significant (orders of magnitude) variation of oil relative permeability along different saturation paths, the oil relative permeability in each medium can be collapsed into a single relative permeability curve, once they are plotted as a function of mobile oil saturation. However, this collapsed curve varies depending on the porous media. We show that the same behavior occurs in the relative permeability data published over the past 50 years. These observations indicate that the key factor in differences between oil permeabilities in the same porous media is changes in the residual oil saturation. We examine the performance of most commonly used relative permeability models, i.e., Corey, Saturation-Weighted Interpolation (SWI), and Stone against our data. Given the importance of residual oil saturation, we fit the experimental data along different saturation paths by treating the residual oil saturation in these models as the fitting parameter while keeping the other parameters constant. We find that the Corey and SWI models fit the data very well while the Stone model performs poorly at low saturations. We find that residual oil saturation is a nonlinear function of gas/water saturation as opposed to linear relationship previously suggested by others.

Joint estimation of absolute and relative permeabilities using ensemble-based Kalman filter

An ensemble-based, sequential assimilation procedure is developed and successfully applied to estimate absolute and relative permeabilities jointly under multi-phase (oil, gas and water) flow condition in the porous media. Two-phase oil–water and gas–oil relative permeabilities are represented by power-law models, and Stone's Model II is used to calculate three-phase oil relative permeability. Prior absolute permeability field is also generated with isotropic Gaussian covariance. The proposed method is validated by a twin numerical setup with three different case studies, in which a synthetic 2D reservoir under multi-phase flow condition is considered. In the first case, there is simultaneous flow of oil, water, and gas phases within the reservoir. In the second, only oil and water phases are present; and, only oil and gas phases are mobile for the third case. By assimilating measured production rates of the producers and bottomhole pressure of the injector, relative permeability curves and absolute permeability field are accurately estimated. Therefore, simultaneous estimation of absolute and relative permeabilities from measured production data is feasible if there exists a prior model for the absolute permeability field and relative permeability curves.

Prediction of three-phase oil relative permeability through a sigmoid-based model

Abstract We present a novel methodology for the prediction of three-phase oil relative permeability, k ro , through a sigmoid-based model. This modeling choice allows us to incorporate key effects of the pore-scale phase distributions into the proposed effective empirical model for k ro . Our sigmoid-based model is able to reproduce: (i) oil remobilization induced by gas injection in water-wet media; (ii) a smooth transition towards the layer-drainage regime for low oil saturations; and (iii) the consequent reduction of residual oil saturation in a three-phase system. The ability of our model to embed the effects of these processes into the estimation of oil relative permeabilities is discussed through comparison with previous experimental measurements. Two different procedures, respectively aimed at describing primary gas injection and secondary waterflooding, are proposed for data analysis and interpretation. A remarkably good agreement between model predictions and direct three-phase relative permeability measurements from the literature is observed when model parameters are estimated solely on the basis of two-phase information. The robustness of the proposed methodology is also assessed by comparing estimates of model parameters based on two-phase data against their counterparts estimated within a Maximum Likelihood framework on the basis of three-phase measurements.

The effect of saturation history on three-phase relative permeability: An experimental study

We investigate the effect of different saturation histories relevant to various oil displacement processes (including secondary and tertiary gas injections) on three-phase gas/oil/brine relative permeabilities of water-wet sandstone. It is found that three-phase water (wetting phase) relative permeability is primarily a function of water saturation and shows no dependency upon saturation history. Three-phase gas (nonwetting phase) relative permeability is also a function of gas saturation as well as the direction of gas saturation change. Three-phase relative permeability to oil (intermediate-wetting phase) appears to depend on all phase saturations, and saturation history have no significant impact on it. Three-phase oil relative permeability shows weak sensitivity to initial oil saturation prior to gas injection. The functional forms of oil relative permeability with saturation, particularly at low oil saturations, are also examined. It is observed that, at high oil saturations where networks of oil-filled elements govern oil flow, oil relative permeability exhibits a quartic form with oil saturation whereas, at low oil saturations where flow is believed to be controlled by layer drainage, it shows a quadratic form . The quadratic form of three-phase oil relative permeability is consistent with the theoretical interpretation of layer drainage at the pore scale.

Flow coupling during three-phase gravity drainage

We measure the three-phase oil relative permeability k(ro) by conducting unsteady-state drainage experiments in a 0.8 m water-wet sand pack. We find that when starting from capillary-trapped oil, k(ro) shows a strong dependence on both the flow of water and the water saturation and a weak dependence on oil saturation, contrary to most models. The observed flow coupling between water and oil is stronger in three-phase flow than two-phase flow, and cannot be observed in steady-state measurements. The results suggest that the oil is transported through moving gas-oil-water interfaces (form drag) or momentum transport across stationary interfaces (friction drag). We present a simple model of friction drag which compares favorably to the experimental data.

Impact of Three-Phase Relative Permeability Model on Recovery in Mixed Media: Miscibility, IFT, and Hysteresis Issues

This paper explores the importance of proper choice of three-phase relative permeability model for mixed-wet media, in contrast with commonly available water-wet rock models, that is, Stone1 and Baker. The mixed-wet condition is likely the most frequently encountered wetting condition worldwide.(1, 2) Two formulations are available under these conditions. Blunt(3) developed a model that is somewhat complex to employ in simulators. In contrast, Jerauld(4-6) formulated a model that is relatively easier to implement. Jerauld’s model incorporates the effect of interfacial tension (IFT) between phase pairs, as well as the ability to model mixed wettability. In this paper, to evaluate the effect of three-phase relative formulation, Stone 1 is used as a paradigm of strong wetting and serves as a base comparison for a mixed-wet formulation, namely, Jerauld’s model. We illustrate the differences in performance prediction in three-phase oil relative permeability (kro) for the same two-phase relative permeability da...

New Developments in Steamflood Modeling

This article, written by Assistant Technology Editor Karen Bybee, contains highlights of paper SPE 97719, "New Developments in Steamflood Modeling," by M. Kumar, SPE, C. Satik, SPE, and V. Hoang, SPE, Chevron Energy Technology Co., prepared for the 2005 SPE International Thermal Operations and Heavy Oil Symposium, Calgary, 1–3 November. Using results from fine-scale, multipattern, geostatistical models, the full-length paper reviews key issues related to steamflood modeling. Pattern-element and single-sand models used in many previous studies are not sufficient to explain observed field performance, and larger heterogeneous models give more-realistic recovery predictions. Current computing improvements make larger-scale steamflood modeling viable compared with what was possible earlier, and a realistic steamflood performance is attained when necessary details are included in the model. Introduction Steam injection is the most widely used enhanced-oil-recovery method. Current oil production by steam injection is estimated to be more than 1.1 million BOPD. Most conventional heavy-oil steamflooding projects in California, Canada, Indonesia, and Venezuela employ vertical wells, although use of horizontal producers is growing. On the other hand, extra-heavy oils may require both horizontal injectors and producers, such as in steam-assisted gravity drainage. Oil recovery can exceed 20% of the original oil in place (OOIP) for cyclic steaming and more than 50% OOIP by continuous steam injection. Geologic Models The geologic model used in this study is from portions of the Kern River field, one of the largest oil fields in the U.S. on the basis of OOIP and reserves. The Kern River field is a shallow heavy-oil field 5 miles northeast of Bakersfield, California. The field has been on steam injection since the mid-1960s. Production is from from several distinct sand zones with high permeabilities and porosities deposited in a braided river environment. The sandstones are typically medium- to very coarse-grained, poorly to very poorly sorted, and have little to no detrital clay. The high-quality reservoir sandstones are interbedded with poor-quality sandstones, siltstones, and mudstones, that may be barriers to fluid flow. The individual sand bodies, typically 50 to 100 ft thick, are separated by competent and correlatable shale layers and are steamflooded one at a time. However, some of the shales may not be continuous over the entire project area. In addition, shale continuity varies both areally and vertically. As a result, significant fluid migration can occur between sands, making reservoir management and analysis more challenging. The Kern River field properties of low reservoir pressure, high permeability, and high oil saturations are all favorable for steamflooding. Flow Simulation Model Input Parameters. An average sand porosity of 32% was used in the study. Average permeability was approximately 2,500 md. Vertical permeability was considered to be one-half the horizontal permeability. Initial reservoir temperature and pressure were 90°F and 260 psia, respectively. Initial oil saturation was approximately 50%, and initial gas saturation was 0% in the oil zone. Crude gravity was 14°API and its molecular weight was 400. Dead-oil viscosities range from 4,749 cp at 90°F to 3.6 cp at 400°F. Steam quality at the sandface was 70%. Measured relative permeability values were used. Endpoint saturations and relative permeabilities were considered to be independent of temperature. Three-phase oil relative permeabilities were calculated by use of linear interpolation.

Simultaneous Estimation of Absolute and Relative Permeability by Automatic History Matching of Three-Phase Flow Production Data

Abstract This paper focuses on the simultaneous estimation of the absolute permeability field and relative permeability curves from three-phase flow production data. Irreducible water saturation, critical gas saturation, and residual oil saturations are assumed to be known. The two-phase relative permeability curves for an oil-gas system and the two-phase relative permeability curves for an oil-water system are represented by power law models. The three-phase oil relative permeability curve is calculated from the two sets of two-phase curves using Stone's Model II. The adjoint method is applied to three-phase flow problems to calculate the sensitivity of production data to the absolute permeability field and the parameters defining the relative permeability functions. Using the calculated sensitivity coefficients, absolute permeability, and relative permeability fields are estimated by automatic history matching of production data. Introduction The main objective of this paper is to consider the feasibility of estimating absolute permeability fields and parameters that define relative permeability functions by automatic history matching of production data obtained under multiphase flow conditions. While the topic is not new, to the best of our knowledge, no paper in the petroleum engineering literature has considered this problemunder three-phase flow conditions. It appears that Archer and Wong(1) were the first authors to consider the estimation of relative permeability curves by applying a reservoir simulator to history match laboratory core flood data. They estimated only parameters that define the shape of relative permeability curves for simple empirical relative permeability models and adjusted relative permeabilities by a trial and error method during the history matching. Sigmund and McCaffery(2) were the first to apply nonlinear regression to the problem of history matching laboratory core flood data. They used power law expressions to model relative permeability curves and estimated only the two exponential parameters in these formulas. Kerig and Watson(3) considered a similar problem. They calculated predicted data from the Buckley- Leverett model, used cubic splines to parameterize relative permeability curves and compared relative permeability estimates obtained with such a representation to those obtained using a power law functional form. They showed that, in general, power law models do not contain enough degrees of freedom to represent the truth well, whereas cubic splines with a small number of knots appear to be sufficiently flexible to yield more accurate estimates of true relative permeability curves. In their results, they assume absolute permeability is known. Lee and Seinfeld(4) considered the simultaneous estimation of the absolute permeability field and relative permeabilities for a two-dimensional, two-phase flow oil-water system. They assumed power law relative permeability curves and assumed that the end point values of relative permeabilities were known. Thus, only the two exponents in the power law relative permeability functions were estimated. They modelled the two-dimensional isotropic heterogeneousermeability field using bi-cubic B-splines. In the specific xamples considered, they matched pressure and water cut data at wells producing from an oil reservoir under waterflood. Tikhonov(5) regularization was used to stabilize the nonlinear least squares problem.

Key characteristics of three-phase oil relative permeability formulations for improved oil recovery predictions

Significant three-phase regions can occur in a range of reservoir development strategies and oil relative permeability may then critically affect ultimate oil recovery. Unfortunately, three-phase oil relative permeabilities are not generally well characterized. In this paper we focus on theoretical methods of estimating three-phase oil relative permeabilities, as typically applied in reservoir simulation. In the absence of good physical models, we propose applying a mathematical filter to the many existing methods before fitting to measured data. First, the key characteristics of methods for predicting three-phase oil relative permeabilities are discussed, including choice of variables, behaviour at low oil saturations and three-phase residual oil saturations. Second, a numerical comparison of both predicted oil relative permeabilities and predicted incremental oil recoveries for immiscible WAG over waterflood is presented. None of the four most commonly used formulations assessed passed the mathematical filter successfully. Shortcomings were found in both of Stone's commonly used formulations for estimating expected recoveries. A wide range of incremental oil recoveries for immiscible WAG was found from choosing different formulations or different three-phase residual oil saturations. Some recommendations for best practice have been made.

Comparison of the three-phase oil relative permeability models

A comparative study of seven different methods for predicting three-phase oil relative permeabilities in the presence of gas and water phases is presented. Predicted oil relative permeabilities from these correlations have been compared with published three-phase experimental data obtained in Berea sandstone core samples. Some of the correlations under study have been recently developed and have never been tested against the laboratory data. The comparison shows that the commonly used models such as Stones' often do not give accurate predictions of the experimental data. It is concluded that the recently developed models fit the experimental data as well as or better than the previously developed and widely used three-phase oil relative permeability models.

Determination of Three-Phase Relative Permeabilities under Reservoir Conditions by Hot Water and Steamflood Experiments

In order to help the physical and numerical interpretation of Emeraude's steam pilot, two-phase waterfloods at four temperatures (between 30 and 240°C) and a steamflood were performed in the laboratory using the same porous medium (compacted silt) and under reservoir conditions. Dynamic isothermal displacements were interpreted with a thermal simulator taking into account capillary end effects. The corresponding oil-water relative permeability curves were obtained by matching observed pressure drop and oil production. Results show that temperature influences the end-point saturations but not the shape of the curves. The steamflood experiment was carried out in an adiabatic core holder. Oil stripping and production of a large amount of CO2 caused by dissolution of carbonates were pointed out. The numerical interpretation of this experiment, by making use of the oil-water relative permeabilities, provided the three-phase oil relative permeability which is an essential datum for numerical interpretation of a steam drive pilot. Then a parameter study was used to quantify the influence of the different mechanisms involved in hot water and steam floods.

Hot-Water and Steamflood Laboratory Experiments Under Reservoir Conditions

Summary Hot waterfloods at four temperatures (between 30 and 240°C [86 and 464°F]) and a steamflood were performed on the same porous medium (compacted silt) under reservoir conditions. Dynamic isothermal displacements were interpreted with a thermal simulator, taking into account capillary end effects. The corresponding oil/water relative permeability curves were obtained by matching observed pressure drop and oil production. Results show that temperature influences the endpoint saturations but not the shape of the curves. The steamflood experiment was carried out with an adiabatic core holder. The oil recovery was 70% higher than the recovery by the hot waterflood at the same temperature (240°C [464°F]). Oil stripping and production of a large amount of CO2 caused by dissolution of carbonates were pointed out. In the numerical interpretation of this experiment, we used oil/water relative permeabilities to provide the three-phase oil relative permeability. These experiments allow a better understanding of steam displacement mechanisms and supply essential data for numerical interpretation of a steamflood pilot.

Relative Permeability During Cyclic Steam Stimulation of Heavy-Oil Reservoirs

Introduction Reliable relative permeability data are important to the overall planning for prospective thermal recovery projects. Empirically derived relative permeability functions presented here have characteristics quite different from those determined experimentally. The empirical functions emerged from simulations of numerous ongoing thermal projects in unconsolidated heavy-oil reservoirs from Trinidad, West Germany, Canada, and the U.S.The cyclic steam process involves strongly changing saturation and pressure conditions during alternating imbibition and drainage cycles. Three-phase flow effects are to be expected in this process, where both steam and mobile water are found in the presence of oil due to steam condensation. Experimental procedures for determining relative permeability on a microscopic scale often ignore dependency on saturation history (hysteresis), changing pressure levels, and three-phase effects. Hence, relative permeability in this process (and in other thermal processes) often is considered a weakly determined parameter. The thermal simulator, which has been available since 1973, provides insight into the nature of relative permeability on a reservoir scale. Thermal Reservoir Simulator A volatile-oil steamflood simulator developed by Todd, Dietrich & Chase Inc. was used during analysis of the field thermal projects. This model is a fully implicit, finite-difference-based program that can treat up to four mass conservation equations and an energy balance. There are three phases: aqueous, liquid hydrocarbon, and vapor. The four mass species are water, a nonvolatile heavy oil, and two volatile components. Although only water and a dead-oil component were used in the majority of field projects, all four mass species were needed in some projects. The volatile components have been used to model (1) noncondensable-gas and CO2 additives to steam, (2) solution-gas and free-gas effects, and (3) distillation of hydrocarbon pseudocomponents.The model was employed in two-dimensional areal and vertical (cross section) simulations and in three dimensions, using Cartesian (x, y, z) and cylindrical (r, o, z) coordinates. Relative permeability functions were changed between the injection and production stages of the cyclic steam process to model the previously noted imbibition and drainage hysteresis effects. Empirical Relative Permeability Functions Relative permeability relationships shown in Fig. 1 are typical of those developed empirically (through simulation) to reproduce observed producing water/oil ratios in cyclic steam projects worldwide. In this work, two-phase data similar to those shown in Figs. 1a and 1b were used with a modified form of Stone's expression to generate three-phase oil relative permeability functions. These functions were used with adjustments for temperature dependency (Table 1) to match drainage production behavior. In each field application, very low water relative permeability curves (0.0025 is lesser than krwro is lesser than 0.25) were required to reproduce field performance. These empirically developed curves are much lower than those routinely measured (0.05 is lesser than krwo is lesser than 0.25) during imbibition water/oil relative permeability tests using unconsolidated core material.Experimentally determined imbibition water relative permeability curves often are used successfully to match observed steam injection pressures and rates. Physical Basis Several complementary conjectures are offered regarding a physical basis for the empirically derived relative permeability functions. Hysteresis of drainage and imbibition nonwetting-phase relative permeability curves is an established phenomenon. Only recently are experimental results appearing in the literature that indicate wetting-phase hysteresis. Jones and Roszelle showed that water relative permeability exhibits pronounced hysteretic behavior in water-wet sandstone, and Schneider and Owens demonstrated oil relative permeability hysteresis in oil-wet carbonate. These results support the speculation that water relative permeability (in water-wet sands) is much lower on the production cycle (drainage) than on the injection (imbibition) cycle during cyclic steam stimulation.A second complementary conjecture is related to mechanical stress. JPT P. 1987^