Variation Of Relative Permeability Research Articles

Time-dependent model for two-phase flow in ultra-high water-cut reservoirs: Time-varying permeability and relative permeability

For the ultra-high water-cut reservoirs, after long-term water injection exploitation, the physical properties of the reservoir change and the heterogeneity of the reservoir becomes increasingly severe, which further aggravates the spatial difference of the flow field. In this study, the displacement experiments were employed to investigate the variations in core permeability, porosity, and relative permeability after a large amount of water injection. A relative permeability endpoint model was proposed by utilizing the alternating conditional expectation (ACE) transformation to describe the variation in relative permeability based on the experimental data. Based on the time dependent models for permeability and relative permeability, the traditional oil-water two-phase model was improved and discretized using the mimetic finite difference method (MFD). The two cases were launched to confirm the validation of the proposed model. The impact of time-varying physical features on reservoir production performance was studied in a real water flooding reservoir. The experimental results indicate that the overall relative permeability curve shifts to the right as water injection increases. This shift corresponds to a transition towards a more hydrophilic wettability and a decrease in residual oil saturation. The endpoint model demonstrates excellent accuracy and can be applied to time-varying simulations of reservoir physics. The impact of variations in permeability and relative permeability on the reservoir production performance yields two distinct outcomes. The time-varying permeability of the reservoir results in intensified water channeling and poor development effects. On the other hand, the time-varying relative permeability enhances the oil phase seepage capacity, facilitating oil displacement. The comprehensive time-varying behavior is the result of the combined influence of these two parameters, which closely resemble the actual conditions observed in oil field exploitation. The time-varying simulation technique of reservoir physical properties proposed in this paper can continuously and stably characterize the dynamic changes of reservoir physical properties during water drive development. This approach ensures the reliability of the simulation results regarding residual oil distribution.

Dynamic Optimization for Petrophysical Property Estimation in Unsteady-State Coreflooding Using Pyomo

Summary This paper deals with a mathematical modeling and optimization-based approach for estimating relative permeability and capillary pressure from average water saturation data collected during unsteady-state waterflooding experiments. Assuming the Lomeland-Ebeltoft-Thomas (LET) model for the variation of the relative permeability with saturation, the appropriate governing equations, boundary, and initial conditions were solved within the Pyomo framework. Using interior point optimization (IPOPT) with a least-squares objective function, the six parameters of the LET model that ensure the history matching between the measured and calculated average saturation are determined. Additionally, we inferred the capillary pressure function and performed a Sobol sensitivity analysis on the LET model parameters. The results showcase the reliability and robustness of our proposed approach, as it estimates the crucial parameters driving the variation of oil-water flow relative permeability across several cases and effectively predicts the capillary pressure trend. The proposed approach can be seen as an alternative to experimental and numerical simulation-based techniques for predicting relative permeability and capillary pressure curves.

The effect of H2S content on acid gas migration and storage in sandstone reservoir via numerical simulation

Acid gas injection, as one of the technologies for carbon dioxide capture, utilization and storage (CCUS) technologies, has been widely used in the development of sour oil and gas reservoirs. The current study numerically investigates the effect of hydrogen sulfide (H2S) content on the acid gas migration and storage in sandstone reservoir. The results suggest that the variations of acid gas density and viscosity induced by different H2S contents have little influence on the distribution of acid gas plume. However, the higher solubility of acid gas at higher H2S content results in a decrease of migration distance. The variation of relative permeability induced by different H2S contents has great influence on the acid gas migration. With the increase of H2S content, the horizontal migration distance decreases due to the decrease of gas relative permeability. Whereas, the water relative permeability increases and the gravity number increases with increasing H2S content. As a result, most of the horizontal flow occurs at the top of the reservoir and the lateral gas migration at the lower part of the reservoir is weakened. Furthermore, the variation of capillary pressure induced by different H2S contents has little influence on the distribution of acid gas plume during the injection period. However, capillary pressure hinders the migration of acid gas during post-injection period. Thus, for the sandstone reservoir with surface contamination, the horizontal and vertical migration distance decrease with increasing H2S content. For the sandstone reservoir with clean surface, the horizontal migration distance first increases and then remains unchanged with the increase of H2S content. After 50 years, the residual and mobile trapping of acid gas decrease with the increase of H2S content, and the solubility trapping of acid gas increases with the increase of H2S content.

Bearing Fault Numerical Model for the Closed-slot Rotor Submersible Motor

This study proposes a numerical model of the bearing fault of a motor with a closed-slot rotor using the finite element method (FEM). The rotor's radial motion can be regarded as static eccentric at the defect time points and healthy at other time points. The frequency of the harmonic component is analyzed corresponding to bearing fault in stator current according to the radial movement of the motor shaft. Moreover, the relative permeability variation region is established to achieve the radial motion of the rotor with bearing fault. Firstly, the relative permeability variation region is established in the health and static eccentric models. Then, the defect time points are estimated and the static eccentricity model by transient field is analyzed. Finally, the relative permeability of the variable region in the static eccentric model is imported into the variable region of the health model at the defect time points. The simulation results show that the air gap flux density of the bearing fault model is different from that of the health model and static eccentric models. In addition, the stator current contains harmonic components of the bearing fault. The analysis results prove the applicability of the proposed model.

Finite Element Model to Simulate Two-Phase Fluid Flow in Naturally Fractured Oil Reservoirs: Part I.

Naturally fractured reservoirs host more than 20% of the world’s total oil and gas reserves. To produce from such reservoirs efficiently, a good understanding of the reservoir behavior at various conditions is essential. This allows us to predict the reservoir performance in advance and assess its economic feasibility. However, the production from such reservoirs is challenging due to (a) uncertainty associated with the fracture map, (b) complex physics phenomena of fluid and rock interaction, and (c) lack of comprehensive knowledge of the extent, orientation, and permeability sensitivity of the fracture network. This paper addresses the abovementioned challenges by presenting a three-dimenisonal (3-D) two-phase fluid flow model in a poroelastic environment. The model is based on a hybrid methodology by combining single continuum and discrete fracture network approaches. Also, the capillary pressure effect, saturation, and relative permeability variations are considered. A mathematical formulation for three-dimensional, two immiscible fluid flows including rock deformation for the fracture network and the rock matrix is presented. A standard Galerkin-based finite element method is applied to discretize the poroelastic governing equations in space and time. The characteristic Galerkin discretization method is used to stabilize the solution of the convection equation in a finite element approach. The 3-D model is validated against IMEX commercial software and finite element package to test its robustness. The results show that the developed model has the ability to predict the two-phase flow behavior precisely, which can be used to assess the production performance of naturally fractured reservoirs. Numerical results of fluid flow profiles for single and multiple fractures with different orientations that match experimental oil drainage tests and field case studies are presented that proves the reliability of the developed multiphase flow model. Results of simulated well production data show an excellent match with the field production data.

Relative Permeability Variation Depending on Viscosity Ratio and Capillary Number

AbstractThe relative roles of parameters governing relative permeability, a crucial property for two‐phase fluid flows, are incompletely known. To characterize the influence of viscosity ratio (M) and capillary number (Ca), we calculated relative permeabilities of nonwetting fluids (knw) and wetting fluids (kw) in a 3D model of Berea sandstone under steady state condition using the lattice‐Boltzmann method. We show that knw increases and kw decreases as M increases due to the lubricating effect, locally occurred pore‐filling behavior, and instability at fluid interfaces. We also show that knw decreases markedly at low Ca (log Ca < −1.25), whereas kw undergoes negligible change with changing Ca. An M‐Ca‐knw correlation diagram, displaying the simultaneous effects of M and Ca, shows that they cause knw to vary by an order of magnitude. The color map produced is useful to provide accurate estimates of knw in reservoir‐scale simulations and to help identify the optimum properties of the immiscible fluids to be used in a geologic reservoir.

Applying low-salinity water to alter wettability in carbonate oil reservoirs: an experimental study

Seawater has been widely used as an injection fluid for maintaining pressure in sandstone and carbonate reservoirs. In the literature related to EOR research, it was noted that diluted seawater (low-salinity water) can highly improve recovery due to the specific ions (such as Ca2+, Mg2+, and SO42−). Such conclusions lead to the application of “Smart Water” in which changing the ion composition of injected water alters wettability and enhances recovery. Although many theories have been established to explain the mechanism of this phenomenon, almost all of them are limited to sandstone rocks, and the impact of smart water on carbonated reservoirs has rarely been explored. This study experimentally investigates the impact of the injection of high- and low-salinity and smart water on the change of wettability and recovery improvement in an Iranian South oil reservoir. Two different sets of experimental work were conducted. In the first set of experiments, the effect of formation water, diluted formation water (from 223,969 to 5000 ppm and 2000 ppm), seawater (initially 51,400 ppm), and diluted seawater on wettability alteration is investigated by monitoring the contact angle and relative permeability variation. The results showed that dilution of seawater to 2000 ppm has the most impact on wettability alteration. The relative permeability changed, and the contact angle decreased by a significant value of 100°, and recovery increased by about 71%. In the second set, the effect of ion change on the result was studied. For this purpose, the sulfate ion of diluted seawater (2000 ppm) is substituted by phosphate ion (H2PO4−). The results show the wettability alteration similar to the sulfate one. This study sheds light on the possible mechanism of wettability alteration in the carbonate reservoir, and the result will help to design a better low-salinity injection scenario.

Correlation of cation distribution with structure, magnetic and electrical properties of ultrafine Ni2+-doped CoFe2O4

This work presents the estimation of cation distribution of ultrafine NixCo1−xFe2O4 nano-ferrites (x = 0, 0.25, 0.5, 0.75 and 1) from structure, magnetic and electrical investigations. “Sol gel” was the method used to synthesize Ni–Co ferrites. Ni2+-doped CoFe2O4 nanostructure has a broad range of electromagnetic applications due to its controllable magnetic and electrical properties. From XRD patterns, the formation of pure phase was obvious, and the crystallite size ranged from 9 to 12 nm. These sizes are small enough to achieve suitable signal-to-noise ratio for high-density recording media and have more opportunities for technological application, especially in medical fields. Besides, the variation of theoretical lattice parameter with x-content was in a good agreement with experimental ones. Field Emission Scanning Electron Microscope (FESEM) revealed the formation of small nanosphere particles with gradual size reduction on addition of Ni2+ ions. Energy-Dispersive X-ray analysis (EDX) confirmed the purity of prepared samples. Magnetic parameters (Ms, Mr and Hc) were determined from Vibrating Sample Magnetometer (VSM). It was found that both Ms and Hc have gradual decrease with increasing x-content. Curie temperature was determined from the variation of relative permeability with temperature and showed also a gradual decrease with addition of Ni2+ ions. Temperature dependence of both DC electrical resistivity and thermoelectric power coefficient were carried out in the temperature range 385–548 K. The resistivity showed a gradual decrease with increasing x-content. However, the negativity of thermoelectric power coefficient showed a gradual increase with addition of Ni2+ ions. Conduction mechanism was deduced for all studied samples from activation energy values. The cation distribution was proposed such that it could verify the lattice parameter and magnetization of each composition as well as the behavior of resistivity and thermoelectric power.

Dependence of the magnetic and optical properties of Ni-Co nanoferrite on the particle shape

This work presents the effect of particle shape on the magnetic and optical properties of Ni-Co nanoferrite (NCF). Two series of different preparation methods were used to obtain two different shapes of NixCo1-xFe2O4 where x=0, 0.25, 0.5, 0.75 and 1. Sol gel was the first method used to synthesize NCF with citric acid as a fuel and the second series was synthesized by hydrothermal route and the fuel was oxalic acid. Structural characterization for NCF nanomaterials was carried out by X-ray diffraction (XRD), Energy Dispersive Spectroscopy (EDS) and Transmission Electron Microscope (TEM). Magnetic parameters were determined from Vibrating Sample Magnetometer (VSM) for all samples and it is noticed that the values of Mr and Mr/Ms are greater in nanorod series than nanosphere series. RLC Bridge was used to obtain the Curie temperature of the polycrystalline nanomaterials by the variation of relative permeability with increasing temperature. Values of direct and indirect band gaps were determined from double beam spectrometer using Diffused Reflectance Spectroscopy (DRS) measurements. It was found that changing particle shape could change the cation distribution, magnetic and optical properties of NCF nanoparticles.

Evaluation of a Magnetic Dipole Model in a DC Magnetic Flux Leakage System

One of the most common methods for performing non-destructive testing in steel tank floors is DC magnetic flux leakage (MFL). The magnetic dipole method is the most widely used mathematical technique to predict the MFL from defects in such structures. However, due to the complexity of an exact analytical description of an MFL system, researchers often make coarse approximations for the profile of the magnetic surface charge density $\sigma _{m}$ , orientation of the magnetic field H, and variation of relative permeability $\mu _{r}$ . In this paper, the validity of these approximations is evaluated for 2-D rectangular defects in a steel plate, by comparing model predications with finite element results. The primary sources of deviation between the approximate solutions and true MFL profiles were found to be caused by assumptions that 1) $\sigma _{m}$ on the specimen surface adjacent to a flaw is zero. This assumption is equivalent to treating the orientation of H to be parallel to the specimen surface, even at locations in close proximity to a flaw and 2) local variation in permeability around the defect can be ignored. This approximation was found to cause an underestimation of $\sigma _{m}$ and, consequently, the predicted MFL. In contrast, approximating $\sigma _{m}$ to be zero at the bottom of a flaw, and approximating uniform distribution for $\sigma _{\mathbf {m}}$ on the vertical defect sides of a slot defect was found to generate only minor errors in an estimate of flux leakage.

Relative permeability variations during low salinity water flooding in carbonate rocks with different mineral compositions

This article presents an extensive relative permeability study for low salinity water flooding (LSWF) in minerally different carbonate rocks. The experiments were performed with unsteady state method using artificial cores, crude oil and synthetic brines. The effect of water salinity and rock composition on relative permeability was obtained. The results show that the oil (water) relative permeability increases (decreases) first and then decreases (increases) with the decrease of the salinity of brines. Actually, there exists an optimal salinity (2384.6 mg/L) which maximizes the oil relative permeability and minimizes water relative permeability. When low salinity water is kept the same, the higher calcite content in the carbonate rock, the bigger oil relative permeability and the smaller water relative permeability will be.

Effect of sedimentary heterogeneities in the sealing formation on predictive analysis of geological CO2 storage

Numerical models of geologic carbon sequestration (GCS) in saline aquifers use multiphase fluid flow-characteristic curves (relative permeability and capillary pressure) to represent the interactions of the non-wetting CO2 and the wetting brine. Relative permeability data for many sedimentary formations is very scarce, resulting in the utilisation of mathematical correlations to generate the fluid flow characteristics in these formations. The flow models are essential for the prediction of CO2 storage capacity and trapping mechanisms in the geological media. The observation of pressure dissipation across the storage and sealing formations is relevant for storage capacity and geomechanical analysis during CO2 injection.This paper evaluates the relevance of representing relative permeability variations in the sealing formation when modelling geological CO2 sequestration processes. Here we concentrate on gradational changes in the lower part of the caprock, particularly how they affect pressure evolution within the entire sealing formation when duly represented by relative permeability functions.The results demonstrate the importance of accounting for pore size variations in the mathematical model adopted to generate the characteristic curves for GCS analysis. Gradational changes at the base of the caprock influence the magnitude of pressure that propagates vertically into the caprock from the aquifer, especially at the critical zone (i.e. the region overlying the CO2 plume accumulating at the reservoir-seal interface). A higher degree of overpressure and CO2 storage capacity was observed at the base of caprocks that showed gradation. These results illustrate the need to obtain reliable relative permeability functions for GCS, beyond just permeability and porosity data. The study provides a formative principle for geomechanical simulations that study the possibility of pressure-induced caprock failure during CO2 sequestration.

K-Values-Based Upscaling of Compositional Simulation

Summary Enhanced-oil-recovery (EOR) processes involve complex flow, transport, and thermodynamic interactions; as a result, compositional simulation is necessary for accurate representation of the physics. Flow simulation of compositional systems with high-resolution reservoir models is computationally intensive because of the large number of unknowns and the strong nonlinear interactions. Thus, there is a great need for upscaling methods of compositional processes. The complex multiscale interactions between the phase behavior and the heterogeneities lie at the core of the difficulty in constructing consistent upscaling procedures. We use a mass-conservative formulation and introduce upscaled phase-molar-mobility functions for coarse-scale modeling of multiphase flow. These upscaled flow functions account for the subgrid effects caused by the absolute permeability and relative permeability variations, as well as the effects of compressibility. Upscaling of the phase behavior is performed as follows. We assume that instantaneous thermodynamic equilibrium is valid on the fine scale, and we derive coarse-scale equations in which the phase behavior may not necessarily be at equilibrium. The upscaled thermodynamic functions, which represent differences in the component fugacities, are used to account for the nonequilibrium effects on the coarse scale. We demonstrate that the upscaled phase-behavior functions transform the equilibrium phase space on the fine scale to a region of similar shape, but with tilted tie-lines on the coarse space. The numerical framework uses K-values that depend on the orientation of the tie-lines in the new nonequilibrium phase space and the sign of upscaled thermodynamic functions. The proposed methodology is applied to challenging gas-injection problems with large numbers of components and highly heterogeneous permeability fields. The K-value-based coarse-scale operator produces results that are in good agreement with the fine-scale solutions for the quantities of interest, including the component overall compositions and saturation distributions.

Influence of Stator Topologies on Average Torque and Torque Ripple of Fractional-Slot SPM Machines With Fully Closed Slots

This paper investigates the influence of concentrated winding configurations and stator core structures on torque performance of fractional-slot surface-mounted permanent magnet (SPM) machines. From analyzing the separated torque components of prototype SPM machines with 12-slot/10-pole (12S/10P) combination by frozen permeability (FP) method, it can be found that the torque ripple is closely related with local saturation in fully closed slot (FCS) machines. Besides, the heavier local saturation will also jeopardize the merit of using alternate teeth wound windings, such as obtaining higher average torque than electrical machines with all teeth wound windings. The further analysis of stator tooth relative permeability variation over one electrical period demonstrates that the major reason for torque ripple difference among these electrical machines with different stator topologies is the asymmetric saturation between the adjacent teeth. Both the subharmonic due to armature reaction (determined by winding connection and current value) and the asymmetry of stator core structure contribute to such asymmetric saturation. In order to more clearly verify this, the complementary slot/pole number combination (12S/14P) is also analyzed. The conclusion is effective for electrical machines with other slot/pole number combinations as well. Experiments have been carried out to validate the predictions.

A new and simple model for the prediction of horizontal well productivity in gas condensate reservoirs

Horizontal well drilling is a well-established technology to enhance well productivity by increasing reservoir contact compared to that of vertical wells under the same conditions. In gas condensate reservoirs, in addition to the three-dimensional (3D) nature of the flow geometry around the horizontal well, the flow behaviour is further complicated by the phase change and the variation of relative permeability (kr) due to the coupling (increase in kr by an increase in velocity or decrease in IFT) and inertia (a decrease in kr by an increase in velocity) effects. There are no practically attractive simple methods for well productivity calculations that account for these effects. Therefore, as an alternative, numerical simulation of such a complex 3D flow using commercial numerical simulators is usually adopted. This approach requires a 3D fine grid compositional approach which is very demanding, cumbersome and often associated with convergence problems due to numerical instability. Consequently, the introduction of a quick and reliable tool for long term well productivity calculations for gas-condensate systems is the main objective of the present work.An in-house simulator was developed to realistically simulate the multiphase flow of gas and condensate around horizontal wells. Using this model, a large data bank was then generated covering the impact of a wide range of pertinent geometric and flow parameters on well performance including: well and reservoir geometries, reservoir and bottom-hole pressure, fluid velocity, gas oil ratio and fluid composition.Based on the results of these simulations, a new method has been proposed to predict the productivity of horizontal wells for gas and condensate systems. In this approach, the flow behaviour of gas and condensate around the well is quantified in terms of the effective wellbore radius of an equivalent open hole that replicates flow around the actual 3D system. The effective wellbore radius varies with fluid properties, velocity and interfacial tension (IFT), reservoir and wellbore conditions. The integrity of the new methodology has also been verified for various fluids and flow conditions.This approach, included in a simple spreadsheet, can predict the horizontal well performance, significantly facilitating engineering and management decisions on the application of costly horizontal well technology.

Experimental study on the variation of relative permeability due to clay minerals in low salinity water-flooding

This study presents experimental results on the variation of relative permeability due to clay types and contents in low salinity water-flooding. To analyze the effect of clay types and contents on relative permeability, core-flooding using unconsolidated sand core samples was conducted. Relative permeability curves were measured from the experiments using the Johnson, Bossler, and Naumann (JBN) method. From the results, the moving range (Sor-Swi) of the relative permeability curve was proportional to the clay contents of kaolinite. The moving range for illite also showed a relative permeability curve proportional to the clay contents, but it was less than that for kaolinite. To extend the equation of the Honarpour end-point relative permeability curves in low salinity water-flooding, end-point relative permeability curves were matched with the experimental relative permeability curves by changing the Honarpour index No and Nw. Then, the difference of No and Nw between the conventional water-flooding and the low salinity water-flooding was correlated with the clay contents. Based on the results, extended functions of index No and Nw were proposed to describe the relative permeability curves applied to low salinity water-flooding. It is believed that end-point relative permeability curves can be determined by the proposed No and Nw instead of the conventional exponent index in low salinity water-flooding with reservoirs containing kaolinite.

Phase behavior and interfacial tension evaluation of a newly designed surfactant on heavy oil displacement efficiency; effects of salinity, wettability, and capillary pressure

This work aims to discuss the results of wide ranges of laboratory investigations to evaluate the performance of a newly-formulated surfactant for heavy oil reservoirs in order to improve the microscopic sweep efficiency after water flooding processes. In the first part, the specific behavior of the formulated surfactant including its salinity tolerance, interfacial tension, and optimum performance window was determined. Then, the application of surfactant solutions in real sandstone reservoir rocks was assessed for both oil-wet and water-wet cases. Besides, the effect of changing the capillary and viscous forces and interfacial tension on the residual phase saturations were characterized. According to the results, the newly formulated surfactant provides middle phase micro-emulsion and low IFT value (i.e., about 0.01mN/m) in a salinity window around 150,000ppm which indicates its proper performance at high salinity ranges. Both the relative volume and solubilization ratios demonstrate the proper values to be applied for real case applications. Monitoring the differential pressure response and the effluent states in both water-wet and oil-wet Berea sandstone cores represent an incremental residual heavy oil production after water flooding through the formation of the emulsified oil droplets in the effluent solutions by chemical slug flooding. Plotting the capillary desaturation curves for both wetting and non-wetting phases through different capillary numbers demonstrate a normal variation trend for the non-wetting phase (approximately a semi-log relationship), while the wetting phase is greatly affected by the capillary end effect and relative permeability variations in the porous medium. This useful information could be extended for the other similar cases of chemically enhanced heavy oil recovery processes.

Investigating the Temperature Dependency of Oil and Water Relative Permeabilities for Heavy Oil Systems

A look into the literature on the temperature dependency of oil and water relative permeabilities reveals contradictory reports. There are some publications reporting shifts in the water saturation range as well as variations in the relative permeability curves by temperature. On the other hand, some authors have blamed the experimental artifacts, viscous instabilities and fingering issues for these variations. We have performed core flooding experiments to further investigate this issue. Glass bead packs and sand packs were used as the porous media, and Athabasca bitumen with varying viscosities was displaced by hot water at differing temperatures. The unsteady-state method of relative permeability measurement was applied and the experimental data were history matched by a simulator that is tailor made to predict the relative permeabilities. The matches were obtained by varying the relative permeability correlation parameters. The results indicated that the initial water saturation has a direct relation with temperature, while residual oil saturation generally drops at higher temperatures. Although the water saturation range shifts, no direct and unique trend for either oil or water relative permeability is justified. The spread in relative permeabilities especially in the case of higher permeable cores suggests that viscous instabilities are present. As the same saturation shift happens by only changing the oil viscosity, the relative permeability variations with temperature can be attributed to oil to water viscosity ratio changes with temperature. Temperature dependency of relative permeabilities is more related to experimental artifacts, viscous fingering and viscosity changes than fundamental flow properties.

Elucidating the Role of Interfacial Tension for Hydrological Properties of Two-Phase Flow in Natural Sandstone by an Improved Lattice Boltzmann Method

We investigated the interfacial tension (IFT) effect on fluid flow characteristics inside micro-scale, porous media by a highly efficient multi-phase lattice Boltzmann method using a graphics processing unit. IFT is one of the most important parameters for carbon capture and storage and enhanced oil recovery. Rock pores of Berea sandstone were reconstructed from micro-CT scanned images, and multi-phase flows were simulated for the digital rock model at extremely high resolution (3.2 $$\upmu $$ m). Under different IFT conditions, numerical analyses were carried out first to investigate the variation in relative permeability, and then to clarify evolution of the saturation distribution of injected fluid. We confirmed that the relative permeability decreases with increasing IFT due to growing capillary trapping intensity. It was also observed that with certain pressure gradient $$\Delta P$$ two crucial IFT values, $$\sigma _{1}$$ and $$\sigma _{2}$$ , exist, creating three zones in which the displacement process has totally different characteristics. When $$\sigma _{1}< \sigma < \sigma _{2}$$ , the capillary fingering patterns are observed, while for $$\sigma < \sigma _{1}$$ viscous fingering is dominant and most of the passable pore spaces were invaded. When $$\sigma > \sigma _{2}$$ the invading fluid failed to break through. The pore-throat-size distribution estimated from these crucial IFT values ( $$\sigma _{1 }$$ and $$\sigma _{2})$$ agrees with that derived from mercury porosimetry measurements of Berea sandstone. This study demonstrates that the proposed numerical method is an efficient tool for investigating hydrological properties from pore structures.

Optimization of hydraulic fracture geometry in gas condensate reservoirs

An optimized design for hydraulic fracturing is of great importance especially with the growing demand for this method as a means of production enhancement from unconventional gas reservoirs. The first Optimum Fracture Design (OFD) approach, which maximizes well productivity for a given fracture volume, was introduced by Prats in 1960 for single-phase Darcy flow systems. This was then further developed and presented in the form of Unified Fractured Design (UFD) charts by Valko et al. (1998), which is applicable to Pseudo-steady state conditions. Later on, some methodologies have been proposed to make UFD applicable to gas condensate systems assuming the distribution of the condensate phase around the fracture as a rectangular damage zone with constant thickness and reduced permeability. These latter methods are generally oversimplified as they neglect different possible shapes of the two phase region around the fracture and the variation of relative permeability with interfacial tension (IFT) and velocity for these low IFT systems. They also require data that are not readily available, in particular the pressure profile (required to identify the two-phase boundary) around the wellbore.In this paper, we introduce an explicit formulation and a more general methodology for OFD that is applicable to both Steady state and Pseudo-steady state single-phase gas and two-phase gas condensate flow systems and includes the important flow parameters in both the matrix and fracture. The optimum fracture dimensions are obtained by maximizing the effective wellbore radius, using the recently developed correlation by Mahdiyar et al. (2011). This formulation accounts for the mechanical and flow skins based on quite readily available information at wellbore conditions.The integrity of the introduced formulation has been verified for many different prevailing conditions, whilst highlighting the errors of using conventional approaches with some important practical guidelines. In this exercise, the maximum productivity calculated using the proposed formulation is compared with results of the literature or our in-house simulator. This program, using a fine grid approach, simulates gas condensate flow around a hydraulically fractured well for various fracture length–width ratios and identifies the optimum fracture dimensions, for a given fracture volume, providing maximum mass flow rate.