Steady-state Relative Permeability Research Articles

Steady-state relative permeability measurements in rough-walled fractures: The effects of wettability and aperture

Tight subsurface geosystems provide significant prospects for the recovery and storage of mass and energy. Because of the tightness of these systems, natural and engineered fractures are the main conveyance conduits transporting the fluids from the tight rock matrix to the wellbore or vice versa. During the fracturing process, most of the newly induced fractures are considered water-wet, however during injection/production, the wettability of the fractures gradually changes. These changes are mainly due to the interactions between the resident and injected fluids, or the exposure of the virgin fracture surfaces to the resident fluids during production. This alteration in wettability can affect the trapping and flow dynamics, e.g., relative permeability, of the fluids flowing in the fracture. Furthermore, fluid injection and production into and from these systems impact the fractures pore pressure and hence the effective stress they are exposed to, resulting in variations in the fracture aperture. In this study, we present a comprehensive macro-scale experimental investigation focusing on the impact of wettability and aperture on steady-state two-phase relative permeability in rough-walled fractures. Using the stead-state measurement technique, we meticulously characterized oil-brine relative permeabilities and residual trapping in water-wet, oil-wet, and mixed-wet rough-walled fractures induced in Eagle Ford shale rock samples. Additionally, we assessed the influence of fracture aperture size on these properties during both drainage and imbibition flow processes. Our results indicate significant phase interference in oil-brine flow in non-water-wet rough-walled fractures, which renders the commonly used x-curve and Corey models inadequate to represent the steady-state oil-brine relative permeabilities measured in our study, leading to substantial overestimations. Mixed-wet fractures exhibited reduced relative permeabilities compared to oil-wet fractures, which in turn demonstrated lower relative permeabilities than water-wet counterparts. The observed trends stem from evolving fluid configuration and transport dynamics within the fractures as surface wettability transitions. Furthermore, an increase in fracture aperture corresponded with an increase in relative permeability. This was attributed to a broader flow area, diminished capillarity, attenuated impact of fracture wall roughness and tortuosity effects, and a reduction in bypassing and trapping events, all synergistically contributing to the mitigation of phase interference. Similarly, an increase in fracture aperture led to a marked decrease in post-waterflooding endpoint oil saturation and post-oil flooding endpoint brine saturation. Finally, we provided generalized correlation models based on experimental results from fractures with varying conductivities and wettability, yielding a more accurate representation of fracture relative permeabilities compared to traditional models. The data and correlation models produced in this study improve our understanding and prediction of multiphase flow in subsurface fractures and offer insights into rapid production decline in unconventional shale reservoirs.

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

Understanding multiphase flow in fractures filled with minerals and proppants is vital in various subsurface applications. Limited experimental data have led to reliance on correlations lacking physical basis. We conducted experiments to characterize relative permeability in rough-walled fractures packed with unconsolidated porous media. We tested fractures packed with water-wet 40/70 sand (silica) and surface-coated ceramic proppants. Brine and mineral oil were used as the wetting and non-wetting fluid phases, respectively. Steady-state (SS) drainage (non-wetting-phase displacing wetting-phase) and imbibition (wetting-phase displacing non-wetting-phase) tests were performed under a wide range of saturation histories (full-cycle and scanning-curves) to study relative permeability hysteresis of the propped fractures. Every SS drainage or imbibition test consisted of several discrete points at which fluid saturations and the corresponding relative permeability were measured by varying the fractional flow rates of fluids whilst maintaining a constant total flow rate. We analyzed residual non-wetting phase saturations and relative permeability trends to understand two-phase flow behavior in each proppant pack. High-resolution x-ray microtomography was used to understand the pore-scale topology, wettability, and to provide insights about the pore-scale displacement mechanisms involved in this study. The results showed that commonly used models to estimate relative permeabilities of fractures significantly overestimated the SS brine and oil relative permeabilities (denoted as krw and kro) measured in this study. Further analysis unveiled that the kro values during imbibition exceeded their drainage counterparts in both proppants, the ceramic proppant exhibited a lower initial water saturation and a higher end-point kro permeability at the end of the drainage displacement, as well as higher krw across all flooding processes. Updated fitting parameters for a Brooks-Corey-type relative permeability correlation are introduced. This study presents improved insights, extensive experimentally generated relative permeability data, and an updated relative permeability correlation, which can be collectively utilized to reduce uncertainties associated with continuum-scale forecasts of multiphase flow behavior in fractured subsurface formations.

Relative permeability evolution with thermo-poromechanical process during N2 and scCO2 injection in brine saturated Deadwood sandstone from Aquistore

This study delves into the intrinsic and multiphase flow properties, specifically steady-state drainage relative permeability, of a subsurface Deadwood sandstone from the Aquistore CO2 storage site in Canada. Consecutive core-flooding experiments were conducted utilizing N2- and scCO2-brine pairs across a broad range of temperatures (20–70 °C) and isotropic effective stress (0–30 MPa). Moreover, we monitored crack initiation and propagation of the sandstone during uniaxial loading at an elevated temperature using an integrated approach that combines microCT scanning with an in-situ heating/loading test. Our findings reveal a 54 % decrease and a 3 % increase in the absolute permeability of the sandstone through isothermal compaction followed by thermal expansion processes, respectively. Elevating the temperature from 20 °C to 70 °C results in a systematic 24 % increase in irreducible brine saturation and nearly doubles the end-point N2 mobility, indicating an increased tendency of the rock surface towards the brine phase with temperature. Substituting N2 with scCO2 demonstrates a leftward shift in relative permeability and a decrease in irreducible brine saturation (from 0.36 to 0.31), consistent with low interfacial tension and the de-wetting effect during cyclic scCO2-brine injections. Micro-CT image analysis reveals micro-crack initiation at 10 MPa stress and 70 °C temperature, suggesting that a mixed impact of induced cracks, dynamic wettability, and thermo-mechanical deformation is responsible for the substantial increase in well injectivity over time in Aquistore. This novel experimental program provides indispensable insight into thermo-poromechanical and wettability controls on multiphase flow at the Aquistore injection site in Canada, with potential applicability to similar scenarios globally.

Integrated Physical and Digital Chalk Relative Permeability Evaluation: A Case Study

The Valhall chalk field has produced more than 1 billion barrels of oil equivalents over the last 40 years, primarily from the homogeneous Tor Formation. The underlying Hod Formation is more heterogeneous and is less maturely developed. The extent of heterogeneity poses a challenge in the evaluation of multiphase fluid flow properties. The objective of the work was to use digital core analysis to generate early relative permeability data to leverage and compare with conventional physical steady-state relative permeability data. Accurate digital and physical description of capillary pressure and relative permeability in the high-porosity chalk is complicated by both low permeabilities and heterogeneity. The main challenge with chalk is that flow occurs in a nano-environment. Physically, the nano-environment translates to low permeability, difficult rock preparation, and extensive experimental time, especiallyfor steady-state flow experiments. Representative three-dimensional (3D) digital rocks were generated using a combination of X-ray computed tomography (CT) and focused ion beam-scanning electron microscopy (FIB-SEM) methods. The digital rocks were used to simulate two-phase flow and generate relative permeabilities and sensitivity to wettability. Physical steady-state and digital relative permeabilities on several core plugs and subsets are compared in this study, which discusses the advantages of performing both as part of the formation evaluation process. The physical and digital results compare reasonably well. The physical results provide a relative permeability anchor, and the digital results provide the leverage of early results, parametric sensitivities, and quality assurance. Hence, integration between digital and physical core analysis yields a robust understanding and input for uncertainty modeling.

Comparative analysis of hydrogen, methane and nitrogen relative permeability: Implications for Underground Hydrogen Storage

Underground Hydrogen Storage (UHS) is an emerging technology which aims to store terawatt-scale energy in the subsurface to alleviate daily and seasonal fluctuations in the ever-increasing renewable energy market. Depleted gas reservoirs and aquifers will be the main targets of storage which requires an understanding of the interaction between formation or cushion gas with native brine. A comprehensive comparison of the petrophysical and two-phase flow properties of the hydrogen-water, methane-water and nitrogen-water systems at in-situ conditions is the focus of our work. We conduct steady-state relative permeability experiments at 2.13 MPa and 298 K using identical capillary numbers of Nc=2.92×10−7 for all three systems. The saturation and in-situ contact angles are determined with segmentation of 3D micro-CT images. The contact angle of each system is almost identical at the irreducible water saturation, Swi, 44°, 46°, 45°, respectively, as is the interfacial tension of 72.0, 71.5 and 71mN/m, respectively. The capillary pressure-saturation relationships for each system have a maximum difference of 3.8 % which leads to approximately equivalent relative permeability in the Bentheimer sandstone core. The end-point relative permeabilities are krg0=0.049, krg0=0.045 and krg0=0.062 at irreducible water saturations of Swi=0.24, Swi=0.25 and Swi=0.23, respectively. All three gases demonstrate equivalent flow properties which indicates that both nitrogen and methane are suitable analogue gases for laboratory multiphase flow experiments and cushion gas to improve safety and the economics of commercial UHS projects. The results enable history matched depleted gas reservoir performance to be used as a guide for UHS.

Effective Relative Permeabilities Based on Momentum Equations with Brinkman Terms and Viscous Coupling

Summary The relative permeability expresses the mobility reduction factor when a fluid flows through a porous medium in the presence of another fluid and appears in Darcy’s law for multiphase flow. In this work, we replace Darcy’s law with more general momentum equations accounting for fluid-rock interaction (flow resistance), fluid-fluid interaction (drag), and Brinkman terms responding to gradients in fluid interstitial velocities. By coupling the momentum equations with phase transport equations, we study two important flow processes—forced imbibition (coreflooding) and countercurrent spontaneous imbibition. In the former, a constant water injection rate is applied and capillary forces are neglected, while in the latter, capillary forces drive the process and the total flux is zero. Our aim is to understand what relative permeabilities result from these systems and flow configurations. From previous work, when using momentum equations without Brinkman terms, unique saturation-dependent relative permeabilities are obtained for the two flow modes that depend on the flow mode. Now, with Brinkman terms included, the relative permeabilities depend on local spatial derivatives of interstitial velocity and pressure. Local relative permeabilities are calculated for both phases utilizing the ratio of phase Darcy velocity and phase pressure gradient. In addition, we use the Johnson-Bossler-Naumann (JBN) method for forced imbibition (with data simulated under the assumption of negligible capillary end effects) to calculate interpreted relative permeabilities from pressure drop and average saturation. Both flow setups are parameterized with literature data, and sensitivity analysis is performed. During coreflooding, Brinkman terms give a flatter saturation profile and higher front saturation. The saturation profile shape changes with time. Local water relative permeabilities are reduced, while they are slightly raised for oil. The saturation range where relative permeabilities can be evaluated locally is raised and made narrower with increased Brinkman terms. JBN relative permeabilities deviate from the local values: The trends in curves and saturation range are the same but more pronounced as they incorporate average measurements, including the strong impact at the inlet. Brinkman effects vanish after sufficient distance traveled, resulting in the unique saturation functions as a limit. Unsteady state (USS) relative permeabilities (based on transient data from single-phase injection) differ from steady-state (SS) relative permeabilities (based on SS data from coinjection of two fluids) because the Brinkman terms are zero at SS. During spontaneous imbibition, the higher effect from the Brinkman terms caused oil relative permeabilities to decrease at low water saturations and slightly increase at high saturations, while water relative permeability was only slightly reduced. The net effect was a delay in the imbibition profile. Local relative permeabilities approached the unique saturation functions without Brinkman terms deeper in the system because phase velocities (involved in the Brinkman terms) decreased with distance. In both systems, scaling and simulations demonstrate that the relative change in relative permeabilities due to Brinkman terms increases with the Brinkman coefficient, permeability, and inverse squared distance from the inlet.

A Method to Correct Steady-State Relative Permeability Measurements for Inhomogeneous Saturation Profiles in One-Dimensional Flow

AbstractTraditionally, steady-state relative permeability is calculated from measurements on small rock samples using Darcy’s law and assuming a homogenous saturation profile and constant capillary pressure. However, these assumptions are rarely correct as local inhomogeneities exist; furthermore, the wetting phase tends to be retained at the outlet–the so-called capillary end effect. We have introduced a new method that corrects the relative permeabilities, analytically, for an inhomogeneous saturation profile along the flow direction. The only data required are the measured pressure drops for different fractional flow values, an estimate of capillary pressure, and the saturation profiles. An optimization routine is applied to find the range of relative permeability values consistent with the uncertainty in the measured pressure. Assuming a homogenous saturation profile systematically underestimates the relative permeability and this effect is most marked for media where one of the phases is strongly wetting with a noticeable capillary end effect. Relative permeabilities from seven two-phase flow experiments in centimetre-scale samples with different wettability were corrected while reconciling some hitherto apparently contradictory results. We reproduce relative permeabilities of water-wet Bentheimer sandstone that are closer to other measurements in the literature on larger samples than the original analysis. Furthermore, we find that the water relative permeability during waterflooding a carbonate sample with a wide range of pore sizes can be high, due to good connectivity through the microporosity. For mixed-wet media with lower capillary pressure and less variable saturation profiles, the corrections are less significant.

Application of Machine Learning to Interpret Steady-State Drainage Relative Permeability Experiments

Summary A meticulous interpretation of steady-state or unsteady-state relative permeability (Kr) experimental data is required to determine a complete set of Kr curves. In this work, different machine learning (ML) models were developed to assist in a faster estimation of these curves from steady-state drainage coreflooding experimental runs. These ML algorithms include gradient boosting (GB), random forest (RF), extreme gradient boosting (XGB), and deep neural network (DNN) with a main focus on and comparison of the two latter algorithms (XGB and DNN). Based on existing mathematical models, a leading-edge framework was developed where a large database of Kr and capillary pressure (Pc) curves were generated. This database was used to perform thousands of coreflood simulation runs representing oil-water drainage steady-state experiments. The results obtained from these simulation runs, mainly pressure drop along with other conventional core analysis data, were used to estimate analytical Kr curves based on Darcy’s law. These analytically estimated Kr curves along with the previously generated Pc curves were fed as features into the ML model. The entire data set was split into 80% for training and 20% for testing. The k-fold cross-validation technique was applied to increase the model’s accuracy by splitting 80% of the training data into 10 folds. In this manner, for each of the 10 experiments, nine folds were used for training and the remaining fold was used for model validation. Once the model was trained and validated, it was subjected to blind testing on the remaining 20% of the data set. The ML model learns to capture fluid flow behavior inside the core from the training data set. In terms of applicability of these ML models, two sets of experimental data were needed as input; the first was the analytically estimated Kr curves from the steady-state drainage coreflooding experiments, while the other was the Pc curves estimated from centrifuge or mercury injection capillary pressure (MICP) measurements. The trained/tested model was then able to estimate Kr curves based on the experimental results fed as input. Furthermore, to test the performance of the ML model when only one set of experimental data is available to an end user, a recurrent neural network (RNN) algorithm was trained/tested to predict Kr curves in the absence of Pc curves as an input. The performance of the three developed models (XGB, DNN, and RNN) was assessed using the values of the coefficient of determination (R2) along with the loss calculated during training/validation of the model. The respective crossplots along with comparisons of ground truth vs. artificial intelligence (AI)-predicted curves indicated that the model is capable of making accurate predictions with an error percentage between 0.2% and 0.6% on history-matching experimental data for all three tested ML techniques. This implies that the AI-based model exhibits better efficiency and reliability in determining Kr curves when compared to conventional methods. The developed ML models by no means replace the need to conduct drainage coreflooding or centrifuge experiments but act as an alternative to existing commercial platforms that are used to interpret experimental data to predict Kr curves. The two main advantages of the developed ML models are their capability of predicting Kr curves within a matter of a few minutes as well as with limited intervention from the end user. The results also include a comparison between classical ML approaches, shallow neural networks, and DNNs in terms of accuracy in predicting the final Kr curves. The research presented here is an extension of the state-of-the-art framework proposed by Mathew et al. (2021). However, the two main aspects of the current study are the application of deep learning for the prediction of Kr curves and the application of feature engineering. The latter not only reduces the training/testing time for the ML models but also enables the end user to obtain the final predictions with the least set of experimental data. The various models discussed in this research work currently focus on the prediction of Kr curves for drainage steady-state experiments; however, the work can be extended to capture the imbibition cycle as well.

An accelerated process-based method for the accurate computation of relative permeability from direct simulations of two-phase flow on micro-computed tomography images of porous media

Pore-scale forces have significant effects on the macroscopic behavior of multi-phase flow through porous media. We develop a robust and accurate accelerated process-based method for the computation of relative permeability from direct simulations of pore-scale two-phase flow on micro-computed tomography images. In the pressure drop calculation, we take advantage of an existing analysis that establishes a relationship between pore-scale forces and Darcy-scale pressure drops using an energy-conservation approach. We establish a thermodynamically consistent approximation of Darcy-scale viscous pressure drops as the rate of energy dissipation per unit flow rate of each flowing phase for the first time within the context of a free-energy lattice Boltzmann method (LBM). In addition, we propose and test a new computationally efficient partial-mirror periodic boundary condition for a fully coupled visco-capillary pore-scale flow simulator based on a free-energy LBM. The new boundary condition is imposed only in the main flow direction and significantly reduces the computational cost of the process-based relative-permeability computation protocol at a small compromise on accuracy.We first compute primary-drainage and subsequent imbibition relative permeability curves for a reservoir sandstone rock sample. We use this real-reservoir dataset to validate the pressure drop computation method and the partial-mirror periodic boundary condition. We then simulate the entire drainage and imbibition cycle using an extensively studied Berea sandstone dataset. We quantitatively demonstrate that pore-scale direct numerical simulation-based relative permeability curves computed with our novel process-based method agree well with experimental steady-state relative permeability measurements. We also quantitatively demonstrate that the new partial-mirror periodic boundary condition accelerates the relative permeability computations 4 to 13 times.

Comparison of Intercept Methods for Correction of Steady-State Relative Permeability Experiments for Capillary End Effects

Summary Steady-state relative permeability experiments are performed by coinjection of two fluids through core plug samples. The relative permeabilities can be calculated using Darcy’s law from the stabilized pressure drop and saturation of the core if capillary end effects and transient effects are negligible. In most cases, such conditions are difficult to obtain. Recent works have presented ways to extrapolate steady-state pressure drop and average saturation measurements affected by capillary end effects collected at different rates to obtain correct relative permeabilities at correct saturations. Both the considered methods are based on linear extrapolations to determine intercepts. Gupta and Maloney (2016) derived their method intuitively and validated it with numerical and experimental data. Andersen (2021a) derived a method from fundamental assumptions and presented an intercept method in a different form where the saturation and relative permeabilities are found directly and uniquely from straightline intercepts. All system parameters, including saturation functions and injection conditions, appear in the model. In this work, the two methods are compared. It is proven theoretically that Gupta and Maloney’s method is correct in that it produces the correct saturation and pressure drops corrected for capillary end effects. Especially, a constant pressure drop was assumed and here proved to exist, as a result of capillary end effects in addition to the Darcy law pressure drop with no end effects. Their method assumes a well-defined end effect region with length xCEE, but this length can be defined almost arbitrarily. This choice has little impact on average saturation and pressure drop, however. They also assumed that for a defined end effect region, the average saturation was constant and equal to the slope in their saturation plot. It is shown that if the region is defined, the average saturation is indeed constant, but not given by the slope. The correct slope is predicted by the Andersen model. We also comment on theoretical misinterpretations of the Gupta and Maloney method. A few works have correctly calculated that the pressure drop over the end effect region is independent of rate, but not accounted for that its length is rate dependent. We show that the combined pressure drop is equal to a constant plus the Darcy pressure drop over the full core. Examples are presented to illustrate the model behaviors. Literature datasets are investigated showing that (a) apparently rate-dependent CO2-brine relative permeability endpoints can be explained by capillary end effects and (b) the intercept methods can be applied to correct shale relative permeabilities.

Lattice Boltzmann Study of the Steady-State Relative Permeabilities in Porous Media

Lattice Boltzmann Study of the Steady-State Relative Permeabilities in Porous Media

Role of Viscous Forces in Foam Flow in Porous Media at the Pore Level

A pore network model is proposed to simulate the complex process of in situ foam generation, destruction, and propagation as a drainage process. Motivated by the need to account for viscous flow effects, which arise from the viscous drag of moving bubbles, the statistical physical method of the modified invasion percolation with memory algorithm is extended. The model is capable of capturing the flow characteristics of (weak) continuous gas and (strong) discontinuous gas foams based on a minimum number of input parameters, that is, pore throat size distribution, regeneration probability, network dimensionality, and size. The dependence of the flowing foam fraction on the applied pressure gradient is predicted. The steady-state relative permeabilities during the simultaneous flow of a liquid and a gas, with lamella generation, destruction, and mobilization by flow displacement, are computed. The proposed model adequately portrays literature results with the decrease in the gas relative permeability upon introduction of foams in agreement with the reported results for similar porous media. The results find application in optimizing the current population balance models and guide foam-based enhanced oil recovery projects.

Analytical modeling and correction of steady state relative permeability experiments with capillary end effects – An improved intercept method, scaling and general capillary numbers

Steady state relative permeability experiments are performed by co-injection of two fluids through core plug samples. Effective relative permeabilities can be calculated from the stabilized pressure drop using Darcy’s law and linked to the corresponding average saturation of the core. These estimated relative permeability points will be accurate only if capillary end effects and transient effects are negligible. This work presents general analytical solutions for calculation of spatial saturation and pressure gradient profiles, average saturation, pressure drop and relative permeabilities for a core at steady state when capillary end effects are significant. We derive an intuitive and general “intercept” method for correcting steady state relative permeability measurements for capillary end effects: plotting average saturation and inverse effective relative permeability (of each phase) against inverse total rate will give linear trends at high total rates and result in corrected relative permeability points when extrapolated to zero inverse total rate (infinite rate). We derive a formal proof and generalization of the method proposed by Gupta and Maloney (2016) [SPE Reserv. Eval. Eng. 19, 02, 316–330], also extending the information obtained from the analysis, especially allowing to calculate capillary pressure. It is shown how the slopes of the lines are related to the saturation functions allowing to scale all test data for all conditions to the same straight lines. Two dimensionless numbers are obtained that directly express how much the average saturation is changed and the effective relative permeabilities are reduced compared to values unaffected by end effects. The numbers thus quantitatively and intuitively express the influence of end effects. A third dimensionless number is derived providing a universal criterion for when the intercept method is valid, directly stating that the end effect profile has reached the inlet. All the dimensionless numbers contain a part depending only on saturation functions, injected flow fraction and viscosity ratio and a second part containing constant known fluid, rock and system parameters such as core length, porosity, interfacial tension, total rate, etc. The former parameters determine the saturation range and shape of the saturation profile, while the latter number determines how much the profile is compressed towards the outlet. End effects cause the saturation profile and average saturation to shift towards the saturation where capillary pressure is zero and the effective relative permeabilities to be reduced compared to the true relative permeabilities. This shift is greater at low total rate and gives a false impression of rate-dependent relative permeabilities. The method is demonstrated with multiple examples. Methodologies for deriving relative permeability and capillary pressure systematically and consistently, even based on combining data from tests with different fluid and core properties, are presented and demonstrated on two datasets from the literature. The findings of this work are relevant to accurately estimate relative permeabilities in steady state experiments, relative permeability end points and critical saturations during flooding or the impact of injection chemicals on mobilizing residual phase.

Effect of Temperature on Bitumen/Water Relative Permeability in Oil Sands

The steam-assisted gravity drainage (SAGD) process is among the most applicable thermal enhanced oil recovery (TEOR) methods in North American heavy oil reservoirs. In addition to improving oil mobility through viscosity reduction, the high-temperature condition that a TEOR method causes may significantly alter the fluid flow performance and behavior, as manifested by the relative permeability to fluids in the reservoir. Therefore, in a SAGD and cyclic steam stimulation (CSS), the relative permeabilities can change within the temperature-gradient region developed around the SAGD chamber or between two cycles of CSS processes. However, there is no unique two-phase relative permeability model, especially for bitumen systems to cover a wide range of temperatures. Therefore, the primary objective of this study is to history-match the results of the recent study done by Esmaeili et al. using a new methodology and to conduct limited steady-state relative permeability measurements at different temperatures to confirm the validity of displacement-based relative permeability curves and then to propose a reliable general relative permeability model as a function of temperature. Twelve history-matched bitumen/water relative permeability sets are obtained in a wide range of temperatures from 70 to 220 °C under a confining pressure higher than 1000 psi and the test pressure in the range of 350–400 psi. An in-house developed reservoir simulator was used to obtain relative permeability curves from isothermal displacement data using the history-matching approach. The results demonstrated that both production and pressure-drop data can be matched using the optimized values of parameters of the generalized Corey relative permeability model. Furthermore, a more reliable value for actual residual oil saturation was determined by applying our new methodology in history matching. A comparative analysis with the steady-state relative permeability data allowed us to determine the extent to which the unsteady-state relative permeability is affected by viscous fingering. A considerable shift in both oil and water relative permeability at higher temperatures is noticed. Furthermore, it is also noted that the endpoint relative permeability to oil and water is also altered significantly. The implicit relative permeability data obtained from this study are consistent with the results of the JBN method that were reported in our previous study. By making the parameters of the Corey relative permeability model dependent on temperature, a new model of temperature-dependent bitumen/water relative permeability in sand was developed. It is anticipated that this model can facilitate inclusion of temperature-dependent relative permeability in reservoir simulation studies.

Experimental Study of the Effect of Solvent Addition and Temperature on Two-Phase Bitumen/Water Relative Permeability

Summary The expanding solvent-steam-assisted gravity drainage (ES-SAGD) is a newly proposed thermal recovery technique showing promising efficiency in terms of a smaller steam-to-oil ratio and greater production rate to recover heavy oils and bitumen from oil-bearing formations, where a solvent is coinjected with the steam in the SAGD process. Numerical simulation of the ES-SAGD process requires reliable relative permeability data. The number of reported measurements of relative permeability involving bitumen systems is limited in the literature, mostly because of the experimental difficulties involved in such measurements. The relative permeability data sets for Canadian bitumen, in the presence of solvents, are simply not available in the open literature. The fluid-flow behavior of bitumen/water systems in the presence of solvent is an important matter that must be assessed before the implementation of any ES-SAGD process; therefore, the objective of the current study is to evaluate the impact of a light hydrocarbon solvent (n-hexane) on bitumen/water relative permeability under SAGD conditions. For this purpose, two-phase bitumen/water relative-permeability measurements were conducted in sandpacks over a wide range of temperatures from 70°C to 220°C using Athabasca bitumen, deionized water, and a light hydrocarbon solvent. A well-instrumented experimental setup was developed to perform the relative permeability measurements with the capability of applying confining pressure on the sand and measuring the pore-pressure profile with several intermediate pressure taps. Isothermal oil-displacement tests were carried out with solvent premixed with bitumen. The history-matching approach and Johnson-Bossler-Naumann (JBN) method were used to translate the oil displacement data into the relative-permeability curves. The results obtained with a solvent from this study and without any solvent reported in our previous study are compared to assess the solvent's impact on relative permeability. In addition, the steady-state relative permeability was measured to assess the reliability of unsteady-state relative permeability. The interfacial tension (IFT) and contact-angle measurements using the same fluids were carried out to determine the fluid/fluid interaction and wettability state of the system under high-pressure/high-temperature (HP/HT) conditions. The results of the present study confirmed that the two-phase diluted bitumen/water relative permeability is sensitive to temperature, especially in terms of the endpoint relative permeability to bitumen and water. Furthermore, adding normal hexane (below the asphaltene precipitation threshold) not only improves the displacement efficiency of water flooding because of the significant decrease in oil viscosity but also modifies the wettability and IFT of this system. At the same temperature, the two-phase oil/water relative permeability for bitumen/water systems is significantly different when the oil is diluted with the solvent. Also, the impact of solvent is more pronounced at lower temperatures. Furthermore, the consistency between the steady-state and unsteady-state relative permeability data proved that the effect of viscous fingering was small enough.

Steady state relative permeability experiments with capillary end effects: Analytical solutions including derivation of the intercept method

Steady state relative permeability experiments are performed by coinjection of two (or more) immiscible fluids. The relative permeabilities can be calculated directly from the stabilized pressure drop and saturation of the core if capillary end effects and transient effects are negligible. In most cases such conditions are difficult to obtain.This work presents an analytical solution in form of explicit expressions for the spatial profiles of pressure gradients and saturation, average saturation and pressure drop for a core being injected simultaneously with two phases at steady state when capillary end effects are significant. When arbitrary saturation functions are applied, such parameters and distributions can only by obtained by numerical integration.By assumption of simplified saturation function correlations the differential equation describing steady state can be integrated. A new dimensionless capillary number N is obtained which contains the fluid and rock parameters, but also the saturation function parameters (relative permeability and capillary pressure), fluid viscosities, injected flow fraction, total flow rate and more. It is shown that when this number is of magnitude 1, end effects cover parts of the core, but parts of the core are also unaffected. For N>10 the end effects are negligible, while for N<10 end effects are dominant.This paper gives the first formal proof of the intercept method from basic assumptions. It is shown that when the inlet saturation is sufficiently close to that of a no capillary pressure situation; the average saturation changes linearly with the inverse of total rate towards the saturation corresponding to no capillary forces; also, the pressure drop divided by the pressure drop of a no end effect situation goes linearly towards 1 with the inverse of total rate.

Pore-Scale Simulations of Simultaneous Steady-State Two-Phase Flow Dynamics Using a Lattice Boltzmann Model: Interfacial Area, Capillary Pressure and Relative Permeability

The dynamics of simultaneous flow of immiscible two-phase fluids at the steady state in the capillary force-dominated regime were investigated. It was described by the key state variables, including interfacial area between fluids, capillary pressure and relative permeability, as functions of fluid saturation. Based on the two-dimensional pore-scale simulations using the Shan-Chen multi-component lattice Boltzmann model (SCMC-LBM), the steady-state interfacial area, capillary pressure and relative permeability versus saturation relationships subjected to flow conditions, such as initial fluid distribution, saturation history and magnitude of hydraulic gradient, were examined. Also, the intrinsic differences in the interfacial area and capillary pressure versus saturation curves between at dynamic equilibrium in the steady-state infiltration and at quasi-static equilibrium in the transient displacement were explored. As the SCMC-LBM simulations revealed, the relative permeabilities of both fluids were insensitive to initial fluid distribution or saturation history because of the simultaneous steady-state flow dynamics, but dependent on the magnitude of hydraulic gradient due to the existence of a threshold hydraulic gradient. The hysteresis behaviours of interfacial area-saturation and capillary pressure–saturation curves were captured in the transient displacement but absent in the steady-state infiltration, and the unique interfacial area-capillary pressure–saturation surface at dynamic equilibrium did not tend to overlap with the one at quasi-static equilibrium. The hysteretic capillary pressure behaviour for these two flow patterns was of great theoretical and practical significances.

Pore occupancy, relative permeability and flow intermittency measurements using X-ray micro-tomography in a complex carbonate

We imaged the steady-state flow of brine and decane (oil) at different fractional flows during dual injection in a micro-porous limestone, Estaillades, using X-ray micro-tomography. We applied differential imaging to: (a) distinguish micro-porous regions from macro-pores, and (b) determine fluid pore occupancy in both regions, and relative permeability at a capillary number, Ca = 7.3 × 10−6. The sample porosity was approximately 28%, with 7% in macro-pores and 21% in pores that could not be directly resolved (micro-porosity). Fluid occupancy in micro-porosity was classified into three sub-phases: micro-pore space with oil, micro-pore space with brine, and micro-pores partially filled with oil and brine. Our method indicated an initially higher oil recovery from micro-porosity, consistent with waterflooding in a water-wet rock. The fractional flow and relative permeabilities of the two fluids were obtained from measurements of the pressure differential across the sample and the saturation calculated from the images. The brine saturation and relative permeabilities are impacted by the presence of water-wet micro-porosity which provides additional connectivity to the phases. Furthermore, we find that in addition to brine and decane, a fraction of the macroscopic pore space contains an intermittent phase, which is occupied either by brine or decane during the hour-long scan time. Pore and throat occupancy of oil, brine and intermittent phase were obtained from images at different fractional flows using the generalized pore network extracted from the image of macro-pores. The intermittent phase, where the occupancy fluctuated between oil-filled and brine-filled, was predominantly located in the small and intermediate size pores and throats. Overall, we establish a new experimental methodology to: (i) quantify initial and recovered oil in micro-pores, (ii) characterise intermittent flow, and (iii) measure steady-state relative permeability in carbonates, which is shown to be greatly influenced by micro-porosity.

Steady-State Relative Permeability Measurements of Tight and Shale Rocks Considering Capillary End Effect

Relative permeability (kr) data are the key factors for describing the behaviour of the multi-phase flow in porous media. During the kr measurements of low-permeability rocks, high capillary pressure can cause a significant liquid hold-up at the core outlet. This liquid hold-up, which is known as capillary end effect (CEE), is the main difficulty for laboratory measurements of relative permeability (kr) for tight and shale rocks. In this paper, a novel method is proposed to correct the CEE during the steady-state relative permeability (SS-kr) measurements. The integrity of the proposed method is evaluated by a set of artificially generated data and the experimental SS-kr data of an Eagle Ford shale sample. It is shown that accurate kr data can be obtained using the proposed technique. This technique can be used to estimate reliable kr data without any saturation profile measurement equipment, such as CT scan or MRI.

Validation of Permeability and Relative Permeability Data Using Mercury Injection Capillary Pressure Data

This paper reports on a study with the objective to validate a set of core analysis data using a combination of mercury injection capillary pressure (MICP) data and statistical correlation techniques. The data set is from an off-shore reservoir in Atlantic Canada. Analysis of this reservoir was complicated by the fact that the permeabilities of the samples were high, greater than 2400 mD. The analysis was done using an existing data set, not a data set specifically tailored for the techniques used in the analysis. The data analyzed included samples that represented seven zones in a single well. Porosities and permeabilities were available for the MICP samples. Electrical properties, along with porosities and permeabilities, were available on samples from each zone, but not from the same depths as the MICP samples. Steady-state relative permeabilities (SSRP) were available for stacked samples in each zone; one of the samples in the stack was a companion sample for one of the MICP samples from that zone. The MICP results were used to validate the permeability measurements using both the Swanson method (SM) and the Ruth-Lindsay-Allen (RLAM) method. The SM, using published correlation parameters, significantly under-predicted the permeabilities; the RLAM, which uses no correlation parameters, gave predictions within a maximum error of just over 33% and a mean error of -12%. The MICP data was used to validate the shapes of the SSRP curves using the Gates and Tempelaar-Lietz method (GT-LM), the Burdine method (BM), and a modified Burdine method (MBM). The GT-LM, which uses no correlation parameters, provided good predictions of the wetting phase SSRP curves but very poor predictions of the non-wetting phase SSRP curves. The BM, using published correlation parameters, provided poor predictions of the wetting phase SSRP curves but improved predictions of the non-wetting phase SSRP curves. The MBM provided good predictions of the wetting phase SSRP curves and acceptable predictions of the non-wetting phase SSRP curves. The MBM method does use a correlation parameter but a single value was used for all seven zones. This work provides a protocol for validating core analysis data that can be implemented in a straightforward manner to determine the “quality” of the data. The results emphasize the importance of MICP as an experimental technique. A proposed modified workflow is presented that would optimize the validation protocol.