Two-Stage Upscaling of Two-Phase Flow: From Core to Simulation Scale

In the coarse scale simulation of heterogeneous reservoirs, effective or upscaled flow functions, e.g., oil and water relative permeability and capillary pressure, can be used to represent heterogeneities at subgrid scales. The effective relative permeability is typically upscaled along with absolute permeability from a geostatistical model. However, the potentially important effects of smaller scale heterogeneities (on the centimeter to meter scale) in both capillarity and absolute permeability will not be captured by this approach. In this paper, we present a new two-stage upscaling procedure for two-phase flow. In the first stage, we upscale from the core (fine) scale to the geostatistical (intermediate) scale, while in the second stage we upscale from the geostatistical scale to the simulation (coarse) scale. The computational procedure includes numerical solution of the finite difference equations describing steady state flow over the local region to be upscaled, using either constant pressure or periodic boundary conditions. The two-stage method is applied to synthetic two-dimensional reservoir models with strong variation in capillarity on the fine scale. Results are presented in terms of both oil production rates and saturation fields. Accurate reproduction of the fine grid solutions (simulated on 500 × 500 grids) is achieved on coarse grids of 10 × 10 for different flow scenarios. It is shown that, although capillary forces are important on the fine scale, the assumption of capillary dominance in the first stage of upscaling is not always appropriate, and that the computation of rate dependent effective properties in the upscaling can significantly improve the accuracy of the coarse scale model. The assumption of viscous dominance in the second upscaling stage is found to be appropriate in all of the cases considered.

Functional relationships for unsaturated flow in soils, including those between capillary pressure, saturation, and relative permeabilities, are often described using analytical models based on the bundle‐of‐tubes concept. These models are often limited by, for example, inherent difficulties in prediction of absolute permeabilities, and in incorporation of a discontinuous nonwetting phase. To overcome these difficulties, an alternative approach may be formulated using pore‐scale network models. In this approach, the pore space of the network model is adjusted to match retention data, and absolute and relative permeabilities are then calculated. A new approach that allows more general assignments of pore sizes within the network model provides for greater flexibility to match measured data. This additional flexibility is especially important for simultaneous modeling of main imbibition and drainage branches. Through comparisons between the network model results, analytical model results, and measured data for a variety of both undisturbed and repacked soils, the network model is seen to match capillary pressure–saturation data nearly as well as the analytical model, to predict water phase relative permeabilities equally well, and to predict gas phase relative permeabilities significantly better than the analytical model. The network model also provides very good estimates for intrinsic permeability and thus for absolute permeabilities. Both the network model and the analytical model lost accuracy in predicting relative water permeabilities for soils characterized by a van Genuchten exponent n≲3. Overall, the computational results indicate that reliable predictions of both relative and absolute permeabilities are obtained with the network model when the model matches the capillary pressure–saturation data well. The results also indicate that measured imbibition data are crucial to good predictions of the complete hysteresis loop.

This paper presents graphical constructions that simplify the calculation of relative permeability from displacement data. These constructions convert raw data to relative permeability in a less tedious, more accurate manner than the usual computations. Fractional-flow saturation curves derived from waterflood displacements are always concave downward and never yield multiple-value saturations. Introduction To find oil and water relative permeabilities by the displacement or unsteady-state method, a small linear core usually is saturated with water, then oilflooded to irreducible water saturation. Subsequently, the core is waterflooded, and during the process, pressure drop (either constant or variable) across the entire core and water injection rate (constant or variable) are determined. Effluent fractions are collected and the amount of water and oil in each is measured. Augmented by the absolute permeability and pore volume of the core and by oil and permeability and pore volume of the core and by oil and water viscosities, these data are sufficient to develop relative permeability curves. The average saturation in the core at any time in the flood can be found from an over-all material balance. However, to calculate relative permeability, the saturation history at some point in the core must be determined, not the average saturation history. The Welge equation yields saturations at the effluent end of the core when the average saturation history is known. Similarly, to compute relative permeability, the point pressure gradient per unit injection rate is needed, not the pressure gradient per unit injection rate is needed, not the average. The equation developed by Johnson et al. converts average relative injectivity to a point value, accomplishing the required task. While the equations of Welge and Johnson et al. have been used successfully for years, they require tedious computation and are subject to error because of the evaluation of derivatives. The graphical techniques presented in this study are equivalent to these equations, but are easier to use and can give a more accurate evaluation of relative permeability. Lefebvre du Prey has presented graphical constructions based on curves of volume of oil produced vs time and pressure drop vs time to develop the required point functions. These constructions are limited to constant rate displacements. The constructions presented here are general and apply to constant rate, constant pressure, or variable rate-pressure displacements. Constant-rate and constant-pressure examples are given to help clarify the methods. The graphical techniques make it easy to see that double or triple saturation values, so extensively discussed in the past simply do not result from the fractional flow curve generated by a single displacement, such as a waterflood or an oilflood. Theory Ignoring gravity effects and capillary pressure, water and oil relative permeabilities (expressed as functions of saturation) are (1) (2) To use these equations, the fractional flow of water or oil and effective viscosity, lambda-1, must be determined as functions of saturation. JPT P. 807

Interval pressure transient tests (IPTT) are commonly used in the industry to obtain permeability distribution along the wellbore. Despite the remarkable progress achieved in IPTT analysis in last two decades, interpretation of IPTT is still unclear if test interval consists of oil zone with vertically connected transition zone and water aquifer. One easy and common approach for interpreting such data is to assume that oil zone, transition zone, and water aquifer as a distinct zone and then perform non-linear regression to measured IPTT pressure data set using single phase layer cake model. In this approach, due to the single-phase nature of the model, capillary pressure and relative permeability effects are totally ignored, however, it is assumed that water saturation data is available from log analysis and saturation weighted viscosity value represents the transition zone. As expected, it is determined only effective horizontal and vertical permeability values from this analysis rather than absolute permeability. It is also important to highlight that reliable effective permeability data is estimated solely in the zone where a test is performed. In fact, accurate prediction well and reservoir performance require values of absolute permeability information. As a second approach to analyze IPTT data, it is assumed that water saturation, relative permeability, and capillary pressure data are known parameters and optimization technique is used based on numerical reservoir simulation to estimate absolute permeability values. Like the previous case, horizontal and vertical permeability values are reliably obtained only in the zone where a test is performed. The only difference with the previous approach is that estimated permeability values are absolute permeability values rather than effective permeability. In practice in IPTT jobs, water cut is generally measured with pressure data in various depths in the transition zone to fine tune free water level. It is also important to note that relative permeability and capillary pressure data are not always available before IPTT interpretation. As a last approach in this study, both pressure and water cut data are simultaneously used in the optimization to find each zone’s horizontal and vertical permeability values, relative permeability and capillary pressure curves. Fairly satisfactory estimations are obtained from simultaneous regression of water cut and pressure data obtained from IPTT.

In routine core analysis, porosity and permeability, both relative and absolute, are measured on rock samples which are not under net overburden (confining) pressure. Using these data to predict reservoir performance or estimate reserves can lead to serious errors since all reservoirs are under net overburden pressure. Data collected from constant rate, dynamic displacement experiments were utilized to study the effect of net overburden pressure on porosity and absolute and relative permeabilities. These experiments were conducted on small, consolidated rock samples under net overburden pressures up to 41.37 MPa (6000 psi) and room temperature. The pore pressure was maintained atmospheric. Examination of the data shows a decrease in porosity and permeability with increase in overburden pressure. A correlation between porosity and overburden pressure and also between permeability and overburden pressure has been developed using linear regression analysis. Both correlations are found to be logarithmic. The irreducible water saturation and residual oil saturation increase with increased overburden pressure levels. While the relative permeability to oil decreases with increasing overburden pressure, a corresponding negligible decrease in water relative permeability is observed.

Laboratory data developed from flooding gas-filled sandstone cores containing an irreducible wetting phase can be correlated with absolute permeability and used to approximate gas and water relative permeability permeability and used to approximate gas and water relative permeability at selected saturations. The technique furnishes a set of data allowing relative permeability to be approximated for any permeability range used to characterize the reservoir. Introduction Gas-water relative permeability data are required in mathematical models to predict the advance of water into a gas zone, the residual gas saturation in the water-encroached zone, and in some instances, the gas-water ratios for a given water saturation. Laboratory measurement of these data for a range of increasing water saturations requires steady-state testing at elevated mean pressure in the core and X-ray scanning capabilities. The required steady-state and X-ray equipment, and, hence, data, are currently not available to most engineers needing gas-water relative permeability information. Because of the favorable water-gas viscosity ratio of about 50 that exists in the laboratory tests, displacement of gas by water injection results in pistonlike movement of the fluids. Hence, there is pistonlike movement of the fluids. Hence, there is no subordinate gas and water production over a wide water-saturation range from which unsteady-state relative permeability can be calculated. However, the waterflood tests furnish end-point values on the gas and water relative permeability curves, as well as the residual (trapped) gas saturation following water displacement. Samples selected for laboratory testing should cover permeability and porosity ranges and rock types found in permeability and porosity ranges and rock types found in the reservoir. Before using the relative permeability information in engineering calculations, the end-point data must be correlated. Values then can be assigned from the correlations to each rock type, or to statistically determined permeability ranges representative of the reservoir. In certain cases, these correlations furnish all the information required, since many models of water displacing gas assume piston-like displacement in the reservoir, with only gas flowing above the rising water-gas level and water flowing below this level. In other cases, it is necessary to develop gas and water relative permeability data at intermediate saturations between terminal values. Since relative permeability curves reflect pore geometry, fluid-rock permeability curves reflect pore geometry, fluid-rock wetting characteristics, and the direction of saturation change, any technique selected to furnish these data must simulate the actual reservoir conditions. In some situations, water-oil relative permeability data can be normalized, related to the measured gas and water end-point values, and then used to approximate the gas and water relative permeability at intermediate saturations. The normalization permeability at intermediate saturations. The normalization technique and variations of it have evolved as engineers have been faced with resolving difficulties in combining water-oil relative permeability data to yield average curves, or data compatible with water saturations determined from capillary pressure tests. Combining the normalized water-oil data with pressure tests. Combining the normalized water-oil data with measured gas-water end-point values is a new variation of this old theme. Water-oil relative permeability data may not be available for normalization, or may not be suitable if they represent less than a strongly water-wet condition that is usually assumed representative when water displaces gas. JPT P. 199

A damped iterative EnKF method to estimate relative permeability and capillary pressure for tight formations from displacement experiments

This paper deals with the generation of dynamical pseudo functions: pseudo relative permeabilities, pseudo capillary pressures and, for compositional simulations, -factors. The generation of pseudo functions is complex due to the fact that they are flow dependent. It is therefore necessary to reproduce the flow conditions, for which they will be used, to compute these parameters. For a homogeneous areal reservoir, the oil and water pseudo relative permeabilities of a cell are shown to be dependent on the angle of the flood with the downstream interface. Dual Scale Simulations are presented. They are simulations of models containing a limited number of cells, that enable to reproduce 1) the macroscopical flow in the reservoir and 2) the fine scale flow in a coarse cell. The results of these simulations can be post-processed into dynamical pseudo functions. This method is applied to a 3D model of a lean gas injection scheme in a stratified reservoir containing near critical fluids. Its validation is obtained by comparison of the results with fine grid simulation results in terms of local reproduction of fine scale saturation and pressure patterns; computation of pseudo functions; accurate correction of the coarse model with the pseudo function. Introduction Nowadays, flow simulation grid cells are too large to take into account small scale heterogeneities (such as laminations or impermeable barriers) and to reproduce some small scale physical phenomena (as gravity tongues, fingering, fronts, high velocity zones close to the perforations…). Pseudo functions are a way to transfer this otherwise lost small scale information into the flow simulation model (or 'coarse' model). The generation and use of dynamical pseudo functions (such as pseudo relative permeabilities, pseudo capillary pressures) is a global problem, containing various steps:–the grouping of the cells into 'rock types', that will be characterized by specific pseudo functions;–the reproduction of the fine scale flow patterns in a cell to compute the local pseudo functions;–the post processing of this information into pseudo functions thanks to an 'upscaling method';–the knowledge of the dependence of the pseudo functions with flow conditions, in order not to have to recompute these functions each time the flow changes, during a sensibility study of any parameter of the model (well description, permeability array…). This paper concentrates on the second item. Indeed, several authors have pointed out the fact that the pseudo functions are flow dependent (or 'boundary conditions' dependent), which means that the pseudo functions of a specific cell must be generated under the flow conditions proper to that cell. To solve this dependence problem, we present here the Dual Scale Simulations. The principle of these simulations is to simulate simultaneously the coarse grid model and the finely gridded cell where the pseudo functions are to be computed. This way, the macroscopical flow is reproduced, and hence, the flow patterns in the coarse cell are obtained under correct flow conditions. The flow data can then be post-processed into accurate pseudo functions. Why are the pseudo functions flow dependent? The dependency of the pseudo functions can be understood comparing the upscaling of absolute and relative permeabilities. Upscaling the absolute permeability A common way to compute 'effective' absolute permeability is to solve, with appropriate boundary conditions, the Laplace equation, standing for a one phase, incompressible fluid conservation equation: P. 475

Injection of carbon dioxide (CO2) into subsurface reservoirs is a pivotal component of carbon capture, utilization, and storage (CCUS) technologies, aimed at mitigating the adverse impact of anthropogenic greenhouse gas emissions on climate change. A critical aspect influencing the efficiency of CCUS is the interaction between CO2 and brine within sandstone reservoirs, particularly how CO2 storage affects the wettability of the rock and, consequently, dynamics of fluid flow characterized by the relative permeability and capillary pressure. This study aims to quantitatively assess the relative permeability between CO2 and brine in Berea sandstone, a common analog for reservoir rocks, and the implications of it in CO2 storage. Prior to performing the quantitative assessment, a detailed analysis of static reactivity of the rock-fluid system has been completed.This study integrates experimental and machine learning (ML) approaches to assess the effects of CO2 storage on CO2-brine relative permeability in Berea sandstone. We conducted a 30-day CO2 storage experiment under reservoir conditions (1500 psi and 150°F), analyzing rock and fluid properties before and after storage. These experimental data provided valuable insights into the physical and geochemical changes occurring due to CO2 exposure.Before applying ML approaches, data for this study were collected from a combination of available experimental measurements and literature sources, which were curated to include key input parameters known to influence relative permeability behavior. The dataset consisted of the following features: porosity (%), permeability (mD), interfacial tension (IFT, mN/m), salinity (mg/L), pressure (MPa), threshold capillary pressure (kPa), temperature (°C), and gas saturation (Sg). These features were used as input variables, while the target variables were the gas relative permeability (krg) and water relative permeability (krw).To predict relative permeability curves, an XGBoost machine learning model was trained on the dataset. The input data were processed into feature matrices along with corresponding target variables for both gas and water relative permeability. The model demonstrated high accuracy, with prediction errors for both water and CO2 relative permeability being less than 10% absolute relative error (ARE) when compared to experimental values. After developing the model, we applied the experimental data collected before and after CO2 storage, allowing us to quantify the effect of CO2 storage on changes in relative permeability. This machine learning approach offers a fast and cost-effective alternative to the time-consuming and expensive core flooding experiments. The insights from this study are expected to improve CCUS predictive models and optimize CO2 injection strategies and storage capacity in sandstone reservoirs, thereby contributing to climate change mitigation efforts.

Fracture permeability and relative permeability of coal and their dependence on stress conditions

Significant advances have been made in methods for accurate measurements of saturations and fluid distributions. Further research is needed to reduce (or properly account for) capillary end effects, to control hysteresis, and to minimize wettability changes involved in flow experiments. Studies are needed on modeling complex displacements in reservoirs with flow tests performed at idealized laboratory conditions. Similarly, improvements in interpretation of laboratory data and in scaling up for field use are still required. Until additional advances in technology are made, the best course of action is to generate both steady- and unsteady-state laboratory data, under simulated reservoir conditions, on carefully selected and preserved cores.

Steam Assisted Gravity Drainage (SAGD) can cause dilatant shear failure in unconsolidated heavy oil reservoirs. Our experimental work documents changes induced by such shear failure in absolute permeability, relative permeability to oil and water, and residual saturations. Tests were performed on 2 inch diameter synthetic specimens made from lower fine to medium grain size Ottawa sand formed with wet vibration to an initial porosity of about 32%. Loading paths included triaxial compression and radial extension. Axial and volumetric strains were measured directly during deformation, and single and two phase permeability tests were executed at axial strains of 1 to 10%. Triaxial compression tests followed a path of increasing mean stress (under constant confining stress) while radial extension tests provided a decreasing mean stress loading path (reducing confining stress under constant axial stress). The relative permeability was determined by the unsteady state method for relative permeability endpoint assessment. Results show similar absolute permeability trends to those published by others, but our multiphase measurements appear unique. Relative permeabilities are more strongly influenced by shearing, with water relative permeabilities being increase by as much as a factor of 2. The effect seems to be slightly greater for finer grain sand than for the medium grain size. Absolute permeability changes were greatest for the samples with the lowest initial porosity, which would include those that were poorer sorted. Poorer sorting of the sand pack seems to reduce the impact on relative permeability changes, however. Residual oil saturations were reduced from 0.28 to 0.13 in the lower fine Ottawa sample, suggesting the possibility for significant improvement in oil recovery when the sand is sheared. During triaxial testing, it was interesting to note that at 50 psi effective confining stress, absolute permeability increased with shearing, but at 200 psi effective confining stress, absolute permeability was reduced, even though the volumetric strain was still dilatant. In addition to providing new data on the magnitude of absolute permeability and multiphase flow properties for unconsolidated sands, this paper also demonstrates that dilatant shear failure does not guarantee increased flow capacity for such rocks.

We use a three-dimensional mixed-wet random network model representing Berea sandstone to compute displacement paths and relative permeabilities for water alternating gas (WAG) flooding. First we reproduce cycles of water and gas injection observed in previously published experimental studies. We predict the measured oil, water and gas relative permeabilities accurately. We discuss the hysteresis trends in the water and gas relative permeabilities and compare the behavior of water-wet and oil-wet media. We interpret the results in terms of pore-scale displacements. In water-wet media the water relative permeability is lower in the presence of gas during waterflooding due to an increase in oil/water capillary pressure that causes a decrease in wetting layer conductance. The gas relative permeability is higher for displacement cycles after first gas injection at high gas saturation due to cooperative pore filling, but lower at low saturation due to trapping. In oil-wet media, the water relative permeability remains low until water-filled elements span the system at which point the relative permeability increases rapidly. The gas relative permeability is lower in the presence of water than oil because it is no longer the most non-wetting phase. We show how to use network modeling to develop a physically-based empirical model for three-phase relative permeability. We demonstrate that the relative permeabilities are approximately independent of saturation path when plotted as a function of flowing saturation. The flowing saturation is the saturation minus the amount that is trapped. The amount of oil and gas that is trapped shows a surprising trend with wettability - weakly water-wet media show more trapping of oil and gas than a water-wet system due to the complex competition of different three-phase displacement processes. Further work is needed to explore the full range of behavior as a function of wettability and displacement path.

A method for the simultaneous determination of capillary pressure and relative permeability curves of oil/water systems from coreflood experiments is presented. The Levenberg-Marquardt optimization technique is used to match results from a numerical simulator to laboratory coreflood data by applying a nonlinear least-squares procedure. The simulator is an IMPES finite-difference program which models the simultaneous one-dimensional flow of oil and water through a core. Exponentional functions are used to describe relative permeabilities and capillary pressure curves. Results show that relative permeabilities can be in error when capillary pressure terms are neglected and that the adjustment of a variable which combines oil and total flow rate output data leads to better matches than the adjustment of cumulative oil recovery.

We use a three-dimensional mixed-wet random network model representing Berea sandstone to compute displacement paths and relative permeabilities for water alternating gas (WAG) flooding. First we reproduce cycles of water and gas injection observed in previously published experimental studies. We predict the measured oil, water and gas relative permeabilities accurately. We discuss the hysteresis trends in the water and gas relative permeabilities and compare the behavior of water-wet and oil-wet media. We interpret the results in terms of pore- scale displacements. In water-wet media the water relative permeability is lower in the presence of gas due to an increase in oil/water capillary pressure that causes a decrease in wetting layer conductance. The gas relative permeability is higher for displacement cycles after first gas injection at high gas saturation due to cooperative pore filling, but lower at low saturation due to trapping. In oil- wet media, the water relative permeability remains low until water-filled elements span the system at which point the relative permeability increases rapidly. The gas relative permeability is lower in the presence of water than oil because it is no longer the most non-wetting phase.