Some Problems in Applying the Hassler Relative Permeability Method

Measurement of relative permeability in the laboratory and/or its prediction from models has become an increasingly important subject in reservoir modelling, as reservoir simulation has become the standard method for more accurate forecasting of production and ultimate recovery. However, as is often the case, when the history match in a reservoir simulator is unsatisfactory, the parameter which is most likely to be adjusted is the relative permeability. With developments in measurement and interpretation techniques it is now possible to obtain reliable relative permeability curves. This paper examines current developments in relative permeability measurements from laboratory coreflood experiments, including developments in data reduction and the interpretation of results. It is apparent that classical analytical techniques, such as the Johnson, Bossler and Nauman (JBN) method, cannot describe the majority of unsteady-state displacements, yet are often inconsiderately applied. Valid relative permeability data is only possible if the analytical or numerical model allows for capillarity, viscous instability, wettability and permeability heterogeneity, and can be tuned by carefully measured pressure, flow, and saturation data, and, if possible, non-invasive saturation monitoring. The use of upscaling techniques to bridge the gap between reservoir simulation models and geological scale models poses a great challenge on relative permeability measurements and predictions, at small scale (laminar) levels and for different rock facies. Introduction Relative permeability is used to describe multiphase flow in a porous medium. Such data are important input to many reservoir engineering calculations, providing a basic description of the way in which the phases will move in the reservoir. Definition of the flow process can have a significant effect on the predicted hydrocarbon production rate and duration, and is important in calculating the volume of recoverable hydrocarbon reserves. Although ways to determine relative permeability from measurements made in the field have been proposed, they are fraught with problems and have never been regularly used. The most common method for determining relative permeability has been laboratory special core analysis. Laboratory measurement of representative relative permeability data on a reservoir core-fluid system is a complex task. The experiments are costly and time consuming. Accuracy is limited to the specific core samples and is bounded by narrow saturation limits. A fundamental theoretical approach to modelling multiphase fluid flow in porous rocks is prevented by the complex nature of the problem. Major difficulties arise in mathematically describing flow through a porous system where the lengths, diameters and connectivity of channels are largely unquantifiable, and the complex nature of the hydrocarbon fluids which range from heavy oils to gas condensates with complex thermodynamic behaviours. The extension of Darcy's law to multiple phase flow is, in fact, a heuristic procedure suggested by the analogy with single phase flow. It does not provide an understanding of the physics of multiple phase flow. However, as the law has been substantiated by experiments, it must be assumed that it is at least partly correct and a physical understanding of it has to be sought.

Relative permeability and anisotropy of absolute permeability are commonly measured quantities in the oil industry. The anisotropy of relative permeability, however, has not been accurately documented previously. Recent measurements of relative permeability on Tensleep sandstone samples from Wyoming demonstrate the magnitude and character of anisotropy that can be expected. Core plugs were cut both parallel and perpendicular to the eolian sandstone bedding planes, then prepared for standard absolute gas permeability measurements, and run through an unsteady state relative permeability apparatus. When flooding perpendicular to bedding. the fractional flow curve is steeper than the curve obtained when flooding parallel to bedding. In other words, more piston like behavior is observed perpendicular to bedding. Such anisotropy characteristics can be used in a simple reservoir simulation to demonstrate the expected changes for a producing reservoir. A significant change is predicted when considering the recovery of oil from horizontal wells in a waterflood as opposed to vertical wells. Properly placed horizontal wells can enhance oil recovery in even the most mature waterflood projects. Introduction Relative permeability studies have been conducted through countless variations since the pioneering work of Buckley and Leverett. Their work, however, and most subsequent studies treat relative permeability as a scalar quantity that simply adjusts permeability k to account for a partial fluid saturation. Absolute permeability, on the other hand, is readily recognized as an anisotropic quantity being slightly different depending on the direction of flow parallel or across sedimentary bedding surfaces. Normally, when bedding surfaces are well defined, the permeability parallel to bedding is consistently higher than that measured perpendicular (flowing across) layers. This type of intrinsic permeability anisotropy exists regardless of fluid saturations. and is often measured in dried cores at 100% gas saturation. Relative permeability is then a function of water (or gas) saturation and can possibly exhibit a new dimension in permeability anisotropy. Previous work on relative permeability anisotropy is scarce. Data is abundant on relative permeability and measurements of absolute permeability anisotropy But the combination of relative permeability anisotropy is difficult to find. Corey and Rathjens show one core measurement where relative permeability measured in the vertical direction is slightly different than that measured parallel to bedding. That study. however, presented an insufficient number of measurements to conclusively determine the characteristics of such relative permeability anisotropy Kortekaas used a reservoir simulator to show the effects of relative permeability anisotropy through cross-bedding in sandstones. Orientation of bedding was shown to be important, but there were no actual laboratory measurements of relative permeability anisotropy. This current work provides fundamental laboratory data on relative permeability anisotropy, the interpreted characteristics of such anisotropy, and some suggestions as to how it affects actual oil recovery in waterflood projects. Sample Collection Dunn et al present most of the details regarding the geologic considerations of sample collection. The samples used for this study are Tensleep sandstones of Permian age from the Big Horn Basin in northwest Wyoming (Fig. 1). Tensleep sandstones and equivalents represent the largest producing formation in Wyoming. with over a billion barrels cumulative oil production. P. 317

Two-phase oil/water relative permeability measurements were conducted at ambient and high temperatures in two different rock-fluid systems; one using a clean Poly-Alpha-Olefin (PAO) oil and the other with Athabasca bitumen. The tests were performed in a clean sand-pack with the confining pressure of 800 psi, using deionized water as the aqueous phase. Both the JBN method and the history match approach were utilized to obtain the relative permeability from the results of isothermal oil displacement tests. The contact angle and IFT measurements were carried out to assess any possible wettability alteration and change in fluid/fluid interaction at higher temperatures. Results, Observations, Conclusions: The results of the clean system using the viscous PAO oil confirmed that the two-phase oil/water relative permeability in this ultra-clean system is practically insensitive to the temperature. The slight variation in oil endpoint relative permeability, especially at ambient condition, was attributed to variations in the packing of sand. It was found that the history matching derived two-phase relative permeability from the highest temperature test provides reasonably good history matches of the other displacements that were conducted at lower temperatures. In addition, it is shown that the JBN approach based relative permeability curves show larger variations, primarily due to insufficient volume of water injection at lower temperatures, which makes the practical residual oil saturation much higher than the true residual. In contrast with the ultra-clean system, the results obtained with bitumen showed much larger variations in relative permeability with temperature. Most of the reported studies involving history matching approach treat the low-temperature measurements as the base case and show that changes in relative permeability are needed to history-match the tests at higher temperatures. We have shown that the displacement done at the highest temperature provides a more reliable estimate of the relative permeability and, in some cases, this relative permeability can successfully history match tests done at lower temperatures. In view of the impracticality of injecting sufficient water to reach close to real residual oil saturation at low temperatures, it would be better to obtain relative permeability data at high temperatures for characterizing the two-phase flow behavior of viscous oil systems.

An automatic history-matching algorithm has been developed to determine relative permeabilities and capillary pressure curves simultaneously from the production history of a displacement test of oil and water through a core. Relative permeability and capillary pressure curves are represented by power functions. Those representations contain some parameters which are found by minimizing an objective function. The objective function is formed by the sum of the square of the differences between experimentally measured and numerically simulated production data. The numerical simulator is an IMPES finite-difference program which models the one dimensional two-phase flow. The minimization is performed by a non-linear regression algorithm: the Quasi-Newton Approximation for the Least-Squares Problem (Broyden-Fletcher-Goldfarb-Shanno formulae). The automatic history Bitching algorithm is applied to analyze the influence of capillary pressure on the determination of relative permeability curves. The algorithm is tested with simulated data and with actual experimental data. Database consisting of simulated experiments with random errors are tested to demonstrate the feasibility of the method. Besides, they provide exact relative permeability curves for comparison purposes. In effect, results obtained by using simulated data show the convergence and uniqueness of the algorithm. But the main observation is that relative permeabilities estimates can be in error when capillary pressure terms are neglected. Several sets of actual experimental data at inlet constant pressure are also tested. Convergent solutions are always found. Three different objective functions are applied. They are formed by oil rates, total fluid rates or a combination of both. The solution is dependent on the chosen objective function. Best adjustments are found by applying a combination of oil and total fluid rates, and including capillary pressure.

This paper presents the results of a simulation study to evaluate the effects of gas-water fracture relative permeabilities on post-fracture performance for a well stimulated with a water-frac. More specifically, we investigate the effects of the fracture relative permeability curve shape on both short-term and long-term post-fracture production performance. We utilize a reservoir flow simulator coupled with a geomechanical model which allows us to account for fracture growth during the stimulation treatment and to model realistic fluid distributions in both the reservoir and fracture. We also incorporate new fracture relative (gas-water) permeability data measured in the laboratory for a range of proppant types. Unlike many previous studies that simply assume a linear shape, the new measured fracture relative permeability data are quite non-linear. We compare short-term fracture cleanup and long-term water frac production performance using both the measured non-linear as well as hypothetical linear gas-water fracture relative permeability curves. Generally, the non-linear relative permeability data—combined with low initial absolute fracture conductivities—create very low effective fracture conductivities to gas and cause ineffective fracture cleanup. Although fractures with high absolute conductivities clean up more effectively, we still observe significant residual water saturations—even after several hundred days of production. We also observe differences in the longer-term production performance caused by residual fracture water saturations that are much higher than originally thought. Finally, we assess the effects of relative permeability curve shape on post-fracture diagnostics using pressure transient testing. Evaluation of simulated pressure buildup tests suggests the computed fracture half-lengths are essentially equal to the model inputs, but the computed effective fracture conductivities are much lower.

Means for increasing tertiary oil recoveries from previously waterflooded viscous oil reservoirs are receiving added attention today as a result of industrywide efforts to improve U.S. oil producing rates and reserves. Injection of a bank of polymer solution that precedes injection of a miscible slug (e.g., a micellar fluid) can reduce reservoir permeability contrasts and result in improvement of the sweep efficiency of the process. To evaluate the potential magnitude of improved recovery and economics of prior polymer slug injection, there is a need for basic polymer/oil relative permeability data for use in performance evaluation calculations. Such relative permeability data were measured by steady-state procedures on a suite of 18 outcrop and formation core samples ranging in permeability from about 50 to 1,200 md. Six different Polyacrylamide polymers were tested, and resistance and residual resistance data were obtained on each. Data were obtained in both oil-wet and water-wet systems. The observation in these studies was that the presence of polymers in the water phase had a significant and consistent effect, lowering water relative permeability over the entire water saturation range. In many of the tests, the presence of flowing polymer or its residual effect during subsequent brine flow had no effect on oil relative permeability. In several tests, polymer contact actually improved oil mobility through increases in oil relative permeability at all levels of oil saturation. Permeability level and polymer type produced no clear-cut differences in flow behavior. The obvious differences in core wettability resulted in widely varying relative permeability characteristics, but again the effect of polymer contact was about the same, qualitatively, as obtained on the water-wet cores.

Accurate estimation of relative permeability is vital for decision making in upstream applications from project appraisal to field development and evaluation of various field development options. As relative permeability is a function of both rock and fluid properties, it is harder to generalize it over a wide combination of rocks and fluids. In addition to this complexity, it is hard to gather coherent sets of data to develop a correlation covering the domain of interest for most projects. As a result, fast and reliable relative permeability prediction method is missing in literature. In this study, we identify Euler number (Arns et al., 2001) (a measure of fluid connectivity/distribution) and saturation as first-order predictors of relative permeability and develop a reliable correlation between them using machine learning of experimental special core analysis (SCAL) data and pore network simulation results. In order to achieve our objective, first, we developed a machine learning model based on random forest algorithm (Breiman, 2001) to analyse SCAL data that indicates a key missing feature in the traditional saturation-based relative permeability prediction. We identified this missing feature and proposed the Euler characteristic (a quantitative measure of fluid connectivity/distribution in porous medium) as a potential first-order predictor of relative permeability in combination with in-situ fluid saturations. For this, we first generated ‘artificial’ relative permeability data (supplementing what we already have) using pore network simulation (Valvatne and Blunt, 2004) by systematically varying a set of key parameters such as pore geometry, wettability and saturation history. Subsequently, we used machine learning to rank the importance of each parameter and identify possible correlative response to those selected variables. The investigation based on machine learning of pore network simulation results in combination with the available data suggests that phase saturation and Euler numbers are the two dominant parameters affecting the relative permeability. In particular, it shows that Variation in relative permeability with different rock-fluid parameters (that along with intial fluid distribution can cause the variation in Euler number) is significant even when saturation is fixed. In other words, the relative permeability is multivalued function of saturation, as hysteresis models also indicate. This suggests neither of saturation or Euler number alone is sufficient for relative permeability prediction.At a fixed saturation (zero-dimensional volumetric abundance) and Euler number coordinates, the relative permeability is very consistent and vary insignificantly across different cases, suggesting thesetwo parameters as first-order predictors. Euler number characterizes the fluid connectivity/distribution, while saturation represents the net volumetric fluid quantity. We believe that Euler number has been the missing first-order predictor in traditional saturation-based predictive relative permeability models.Most importantly, we identify and present the quantitative relationship between relative permeability and Euler characteristic, and present a reliable correlation to determine the relative permeability based on Euler number and saturation. To the best of our knowledge, this is the first successful attempt at directly investigating the quantitative relationship between Euler number and relative permeability based on machine learning of experimental SCAL data in combination with pore network simulation results. This work provides the necessary framework and lends itself for further research and development using additional data as time goes on along with more advanced numerical simulation and data analysis models.

Gas-water relative permeability of unconventional reservoir rocks: Hysteresis and influence on production after shut-in

Forced oil–water displacement is the crucial mechanisms of secondary oil recovery. The knowledge of relative permeability is required in the simulation of multiphase flow in porous media. Obvious dynamic effect of capillary pressure occurs in that the formation of ultra-low permeability reservoir (the permeability is <1 × 10−3 μm2) is tight and the pores and throats are very small. In addition, the significant capillary end effect causes serious errors when calculating relative permeabilities. For these reasons, the JBN method (neglecting capillary pressure) does not apply. Therefore, the dynamic capillary pressure and capillary end effects should be taken into account. This work focuses on calculating two-phase relative permeability of ultra-low permeability reservoir through considering the dynamic capillary pressure and eliminating the influence of capillary end effects. Firstly, laboratory core scale measurements of in situ water phase saturation history based on X-ray CT scanning technique were used to estimate relative permeability. Secondly, a mathematical model of two-phase relative permeability considering the dynamic capillary pressure was established. The basic problem formulations, as well as the more specific equations, were given, and the results of comparison using experimental data are presented and discussed. Results indicate that the dynamic capillary pressure measured at laboratory core scale in ultra-low permeability rocks has a significant influence on the estimation of unsteady-state relative permeability. The mathematical calculating method was compared with the history matching method and the results were close, suggesting reliability for ultra-low permeability reservoirs. Importantly, the proposed methods allow measurement of relative permeability from a single experiment. Potentially this represents a great time savings.

A note on dynamic rock typing and TEM-function for grouping, averaging and assigning relative permeability data to reservoir simulation models

The Salado Two-Phase Flow Laboratory Program was established to address concerns regarding two-phase flow properties and to provide WIPP-specific, geologically consistent experimental data to develop more appropriate correlations for Salado rock to replace those currently used in Performance Assessment models. Researchers in Sandia`s Fluid Flow and Transport Department originally identified and emphasized the need for laboratory measurements of Salado threshold pressure and relative permeability. The program expanded to include the measurement of capillary pressure, rock compressibility, porosity, and intrinsic permeability and the assessment of core damage. Sensitivity analyses identified the anhydrite interbed layers as the most likely path for the dissipation of waste-generated gas from waste-storage rooms because of their relatively high permeability. Due to this the program will initially focus on the anhydrite interbed material. The program may expand to include similar rock and flow measurements on other WIPP materials including impure halite, pure halite, and backfill and seal materials. This conceptual plan presents the scope, objectives, and historical documentation of the development of the Salado Two-Phase Flow Program through January 1993. Potential laboratory techniques for assessing core damage and measuring porosity, rock compressibility, capillary and threshold pressure, permeability as a function of stress, and relative permeability are discussed. Details of actual test designs, test procedures, and data analysis are not included in this report, but will be included in the Salado Two-Phase Flow Laboratory Program Test Plan pending the results of experimental and other scoping activities in FY93.

SummaryWe present results from simulations of two-phase flow directly on digitized rock-microstructure images of porous media using a lattice Boltzmann (LB) method. The implemented method is performed on a D3Q19 lattice with fluid/fluid and fluid/solid interaction rules to handle interfacial tension and wetting properties. We demonstrate that the model accurately reproduces capillary and wetting effects in pores with a noncircular shape. The model is applied to study viscous coupling effects for two-phase concurrent annular flow in circular tubes. Simulated relative permeabilities for this case agree with analytical predictions and show that the nonwetting-phase relative permeability might greatly exceed unity when the wetting phase is less viscous than the nonwetting phase.Two-phase LB simulations are performed on microstructure images derived from X-ray microtomography and process-based reconstructions of Bentheimer sandstone. By imposing a flow regulator to control the capillary number of the flow, the LB model can closely mimic typical experimental setups, such as centrifuge capillary pressure and unsteady- and steady-state relative permeability measurements. Computed drainage capillary pressure curves are found to be in excellent agreement with experimental data. Simulated steady-state relative permeabilities at typical capillary numbers in the vicinity of 10−5 are in fair agreement with measured data. The simulations accurately reproduce the wetting-phase relative permeability but tend to underpredict the nonwetting-phase relative permeability at high wetting-phase saturations. We explain this by pointing to percolation threshold effects of the samples. For higher capillary numbers, we correctly observe increased relative permeability for the nonwetting phase caused by mobilization and flow of trapped fluid. It is concluded that the LB model is a powerful and promising tool for deriving physically meaningful constitutive relations directly from rock-microstructure images.

This paper was prepared for the 48th Annual Fall Meeting of the Society of Petroleum Engineers of AIME, to be held in Las Vegas, Nev., Sept. 30-Oct. 3, 1973. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgement of where and by whom the paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal provided agreement to give proper credit is made. Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers Office. Such discussion may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines. Abstract An apparatus has been constructed for laboratory measurement of water-oil relative permeabilities by the steady-state technique at elevated temperatures. Experiments were conducted on consolidated porous teflon and Berea sandstone cores with three oils: Kaydol, Protol and tetradecane having viscosities at room temperature of 130, 68 and 2 cp, respectively. For the oil-water systems, teflon was strongly oil wet and sandstone cores were water wet. Measurements were made under isothermal conditions at room temperature and were repeated at 200 degrees F for the teflon and 300 degrees F for the sandstone cores. Results of these measurements indicated that the effects of temperature on relative permeability curves were similar in both oil-wet and water-wet systems. At higher temperature, relative permeability to oil was found to be higher and residual oil saturation was lower. The rate of increase in relative permeability to water with increasing water saturation appeared to be higher at higher temperature. These effects seemed to be larger in systems with Kaydol and Protol than with tetradecane. Interpretations of results were made in light of fluids, fluid-fluid and fluids-solid properties. The changes in viscosity ratio with temperature appear to be a factor that influences relative permeability curves. Introduction The oil recovery mechanism involved in hot water flooding is generally believed to be the results of viscosity reduction and thermal expansion. The effect of lowering of the viscosity leads to a more efficient displacement. Test results have indicated that relative permeabilities to oil and to water, and the equilibrium saturations of residual oil and irreducible water also change due to an increase in temperature. Presently, the mechanism leading to these changes is not known. In the work on the effect of temperature on relative permeability of consolidated rocks, Weinbrandt and Ramey found that an increase in temperature from room temperature to 175 degrees F in Boise sandstone has caused the increase in irreducible water saturation, decrease in residual oil saturation and increase in the relative permeability to oil. Decrease in the relative permeability to water at saturations of below room-temperature residual oil but increase at flood-out were observed. By comparing their results to those of Owens and Archie, the latter dealing with the effect of wettability on relative permeability, Weinbrandt and Ramey suggested that the changes in relative permeability resulting from an increase in temperature may have been due to wettability changes.

Steady-state two-phase relative permeability measurements in proppant-packed rough-walled fractures

The existing method to infer relative permeability from resistivity data was modified by including more parameters such as residual oil saturation. Both oil-water relative permeability and resistivity were measured simultaneously in the same core sample at a room temperature in order to verify the modified model. Altogether 16 core samples in 2 wells from Daqing oil field, China have been tested. The permeability ranged from about 10 to 800 md. The oil-water relative permeability data were measured using a dynamic displacement technique. Oil-water relative permeability data were inferred from the resistivity data measured in the laboratory and logged from the well using the modified model. The model data were then compared to the experimental data. We demonstrated that the relative permeability of both oil and water calculated from the resistivity data measured in the same core samples and logged from the same wells were close to the experimental data measured using a dynamic displacement approach. The modified model had a greater accuracy compared with the existing models. Using the modified model, it would be possible to obtain the different distribution of relative permeability characteristics in different kinds of formations in a reservoir. It may also be feasible to infer relative permeability data while drilling if resistivity well logging is being taken. Introduction Relative permeability is one of the important parameters controlling multiphase fluid flow in porous media. These data are traditionally obtained with experimental measurements. However, relative permeability is expensive, difficult, and time-consuming to measure in the laboratory, especially for the rocks from unconventional oil and gas reservoirs such as shale plays, tight sands, and extremely low permeability reservoirs. It is also difficult to maintain exact reservoir conditions in taking a core or a fluid sample from the reservoir and bringing it to surface and it is almost impossible to conduct the measurements in real time. Consequently, there has been a decades-long research effort to develop methods and procedures to infer relative permeability using network modeling. Recently, the industry has been researching new methods to extract relative permeability in-situ including the utilization of specially designed permanent downhole electric resistivity array, pressure, and flow rate measurements. Relative permeability can also be derived from other parameters such as capillary pressure data. Mahmoud et al. (2013) predicted the capillary pressure from well logging data in carbonate reservoir and sandstone reservoir. Purcell (1949) reported a mathematical model to calculate the relative permeability from capillary pressure data. From then on, many researchers worked on this area. Li (2005, 2007 and 2010), Li and Horne (2006) and Li and Williams (2006) have made a lot of contribution for estimating the relative permeability using resistivity well logging data. Based on the reaserch on the interrelation between capillary pressure, resistivity and relative permeability reported by Li (2010), Alex et al. (2012) considered to modify the model in double porosity systems. They developed a method to calculate relative permeability and caplillary pressure from resistivity well logging data in naturally fractured reservoirs.