A Highly Implicit Steamflood Model

We present a model of mixed-wet triangular tubes that calculates three-phase capillary pressure and relative permeability curves. Several fluid configurations may occur in triangular pore cross-sections, and capillary displacements may either occur as piston-like displacements of the fluids occupied in the bulk, or as piston-like displacements of the fluids in layers. To our knowledge, this latter type of displacement has not been analyzed before in mixed-wet pores. Using minimization of Helmholtz free energy, we derive accurate three-phase capillary entry pressures for such layer displacements, accounting for contact-angle hysteresis. Numerical examples are presented to illustrate how the entry pressures for the different possible displacements relate to each other during gas and water invasion into pores with a specific fluid configuration. It turns out that the entry pressures for related displacements are consistent. This implies that pores occupied by the same fluid in the bulk portion must have the same fluid configuration for a constant value of capillary pressure. With this model we calculate three-phase capillary pressure and relative permeability, and explore how the saturation-dependencies of these quantities change according to saturation-reversal points. We simulate the sequence of processes primary drainage, imbibition and gas invasion, for different maximum capillary pressures Powmax er primary drainage. In the simulation results presented here, we find that the oil and gas relative permeability, and their saturation-dependencies, are sensitive to variations of Powmax, While the water relative permeability is less sensitive. Such effects are absent in cylindrical tubes. This is caused by the capillary entry pressures, which are strongly affected by hinging interfaces in the corners of angular pores when contact-angle hysteresis is assumed. Thus the choice of pore geometry is important if hysteretic capillary pressure and relative permeability relationships are simulated using network models. With respect to these findings, relative permeability and capillary pressure correlations should be formulated with parameters that strongly depend on saturation-reversal points such that different saturation-dependencies can be accounted for in subsequent invasion processes.

This paper summarizes evidence of gas effects in several Athabasca SAGD projects and shows that these effects can be easily reproduced with simple 1D column simulations. Solution gas probably has a material impact on SAGD rise, and therefore production rates in general, especially where the operating pressure is significantly below the initial pressure. Introduction Following the UTF Phase A SAGD test, it was reported that: "No significant accumulations [of non-condensable gas] were observed in the steam chamber. This was fortunate because such accumulations appear to be harmful because they inhibit steam transport to the front. " "Current simulators incorrectly predict that even small amounts of gas???will accumulate in a steam chamber and substantially impair drainage???more work is required to account for gas transport mechanisms adequately so that reasonable predictions can be made for reservoirs containing greater amounts of gas(1). " Simulators still do not properly account for known amounts of solution gas in the McMurray formation in the sense that it is not generally possible to produce a reasonable history match of field behaviour if the full amount of solution gas is included(2). The question of how the gas is being removed from the chamber, and therefore what is missing from the simulators, remains unresolved, and is beyond the scope of this paper. Meanwhile, two important points have been passed over. The first is that, even at UTF, some insignificant gas accumulations can be found in the data, for example as reported by Ito et al.(3). The second is that "reservoirs containing greater amounts of gas " than UTF or Dover, include most of the current McMurray SAGD projects. In the case of Foster Creek, at the deep end, saturation pressure is 2,800 kPa and the solution GOR is estimated to be about five times that of Dover. Since the gas removal mechanism(s) is unknown, it has always been possible that it was strong enough under UTF circumstances to prevent "significant " accumulations, but not strong enough to do this in deeper Athabasca reservoirs. However, the difficult nature of the SAGD gas problem, combined with the apparent success of UTF history matching without it, has understandably encouraged a situation where it has become standard practice to ignore solution gas in SAGD simulations. Standard practice is also to arbitrarily lower absolute and/or relative permeabilities in order to match field performance. In the case of the UTF project, there was good correspondence between measured core and history match permeabilities using "open " relative permeabilities. This correspondence does not seem to apply to deeper, thicker reservoirs such as Surmont or Foster Creek, particularly with respect to steam rise rates. Field Evidence Hangingstone Ito et al.(3,4) conducted detailed analyses and history matching of observation well temperature data from the UTF Phase B and Hangingstone Phase I projects. Although the Hangingstone situation is complicated by the high operating pressure of 5 MPa at a depth of 280 m (i.e. 18 kPa/m, which they demonstrated by cold water injection was enough to begin deformation of the sand matrix), it show anomalies in the rise rates at some wells where steam rise would "suddenly stop " for a while and then resume.

Besides appropriate temperature and pressure ranges, sufficient gas supply is necessary for gas hydrate crystallization within gas hydrate stability zone. In this article we formulate the calculation of methane solubility assuming hydrate plus‐aqueous solution plus free gas, hydrate plus‐aqueous solution, or aqueous solution plus free gas coexist at equilibrium. There are four fields of methane phase (free gas, dissolved gas, hydrate, and hydrate + free gas) in the diagram of methane solubility versus depth. The contents of hydrate and free gas at Sites 1249, 1250 and 997 are computed. Calculated hydrate contents are 10%~61% (13.5~72.4 mbsf) and 0.7%~1.9% (35~106.5 mbsf) in sediment pores at Site 1249 and at Site 1250, respectively. The layer of free gas, 22 m in thickness and ~2.4% in sediment pores, immediately occurs below the gas hydrate zone at these two sites. At Site 997, hydrate occurs from 202.4 to 433.3 mbsf occupying about 5%~7% of porosity. The free gas layer is ~80 m thick below the hydrate layer methane gas occupying about 0.2%~28% of sediment pore space.

This paper describes the development of a fully implicit general thermal model (ISCOM) for simulating in situ combustion or steam processes. The model includes four phases, a variable number of oil components, a variable number of chemical reactions, gravity and capillary pressure terms, and the possibility of modeling emulsions. The finite-difference formulation of ISCOM operates in one, two or three dimensions and contains implicit and sequential implicit solution techniques as options. It is highly stable, averaging 30% saturation or molar fraction changes during each time step. The model includes the use of pseudo-phase equilibrium ratios which allows a single formulation, even though phases may disappear. The computer program was designed to emphasize clarity and to program was designed to emphasize clarity and to optimize storage. The matrix solution technique is a modified form of Gaussian elimination. The variables and fundamental equations have been aligned so that pivoting for numerical stability is not necessary. pivoting for numerical stability is not necessary. Other special features include Newtonian iteration with numerically calculated derivatives, embedding of the coke equation, two-point upstream and harmonic mobilities as options to reduce truncation error and the grid orientation effect, and an automatic time step selector. Results are presented showing the sensitivity to changes in number of grid blocks and time step size and showing the influence of harmonic weighting on the grid orientation effect. Introduction Thermal methods occupy a prominent position in the recovery of low API crudes. The processes taking place during thermal oil recovery are of a complex place during thermal oil recovery are of a complex nature. Experience has shown that in order to model such processes, any numerical simulator must be extremely stable to produce practical results. The trend, therefore, has been towards increasing degrees of implicitness in the construction of such simulators. This demands, however, the utilization of large computer storage and run times, especially for multi-dimensional problems. This paper describes the development of a fully implicit, finite-difference, general purpose thermal simulator (ISCOM) for modeling in situ combustion and steam processes. The storage and computation time problems have been resolved by the development of a problems have been resolved by the development of a special matrix hand reducing option. Another feature of thermal recovery simulations in heavy oils is the severity of the grid orientation effect. In ISCOM this effect is alleviated by an option which introduces harmonic mobility weighting. The steam drive/soak options are subsets of the in situ combustion model, whereby unnecessary reactions and equations are removed. The steam versions are described here only briefly. Therefore this paper concentrates primarily on the in situ combustion option. The formulation of the model, physical property package, and method of solution will be described, package, and method of solution will be described, together with the results of some test calculations. FORMULATION OF THE MODEL Basic Assumptions The mathematical model is based on the following assumptions:The model can operate in one, two or three dimensions with variable grid spacing.The number of components is variable, and components are distributed among four phases: water (w), oil (o), gas (g), and solid (c).Water and oil components may be present in both oil and water phases, allowing for an approximate treatment of emulsions.The number and type of chemical reactions is variable.

Experimental investigation of the live oil-water relative permeability and displacement efficiency on Kingfisher waxy oil reservoir

The Gulf Coast region of the United States is underlain by deeply buried sandstone reservoirs containing low salinity water at abnormal pressures and elevated temperatures. In addition, the water is believed to contain significant amounts of dissolved natural gas. This geothermal resource, if present in sufficient recoverable quantities, has great energy producing potential and is located in one of the nation's major energy consuming areas. Dorfman and Kehle estimate the energy contained within the geopressured aquifers of Texas may be as much as 20,000 MW centuries excluding the natural gas. A comprehensive theoretical formulation is presented for the important thermomechanical processes presented for the important thermomechanical processes operative in a geopressured geothermal reservoir. The formulation includes the effects of four major drive mechanisms (pore fluid compressibility, reservoir rock compaction, the evolution of dissolved natural gas and the influx of water from adjacent shale formations) expected to be operative during the productive life of the reservoir. Finite difference productive life of the reservoir. Finite difference techniques were used to solve the governing equations describing mass conservation, momentum and energy transport for two flowing phases in a multidimensional heterogeneous reservoir. Constitutive equations were used to describe the changes of fluid properties and reservoir parameters with changes in reservoir pressure and temperature. A series of test calculations pressure and temperature. A series of test calculations were performed to assess the sensitivity of reservoir performance to the gas in solution, sediment compaction performance to the gas in solution, sediment compaction and the reinjection of waste fluids. Based on these calculations, it is concluded that sediment compaction and water from interbedded shales can be significant depletion drive mechanisms in geopressured aquifers. The natural gas drive will probably not exceed the water expansion unless there is a significant initial gas saturation. Costs permitting, reinjection of produced fluids into geopressured geothermal aquifers will be desirable to both increase the recovery of thermal energy and natural gas. Introduction Geothermal energy is presently a small but viable contributor to the United States' energy supply. In addition to known areas of geothermal reserves in the western United States, a unique form of potential geothermal energy exists at moderate to great depths in geopressured aquifers underlying the United States Gulf Coast. Water from such geopressured aquifers often contains natural gas in solution. The geothermal water may be converted into electrical energy after lowering the pressure to extract the natural gas content. Moreover, the fact that temperatures in geothermal reservoirs change very little with time makes geothermal geopressured reservoirs a potentially attractive source of geothermal energy, as there may exist sufficient pressure in these reservoirs to deliver substantial amounts of fluid. The northern shoreline of the Gulf of Mexico extends more than 1,000 miles, from the Rio Grande River to the Florida panhandle. Underlying a large portion of this shoreline area, both onshore and offshore, in a strip 200 to 300 miles wide axe clastic sedimentary deposits of great thickness. The penetration of sands into underlying muds as a result of continuing deposition or faulting resulted in isolation of large sand members from continuous permeability channels to the overlying strata. Above the intervals thus isolated, pressure throughout the basin approximate 0.465 psi/ft. This is considered normal hydrostatic pressure based on the fluid pressures exerted by a column of saline water. However, beneath the normally pressured zones the isolated units of shales and sands contain pressures far greater than normal. These abnormally pressures far greater than normal. These abnormally pressured zones are now commonly referred to as pressured zones are now commonly referred to as geopressured zones. Dickey and Dorfman and Kehle state that the generation of geopressure is primarily the result of compaction phenomena. Newly deposited sediments have high porosities and axe saturated with water.

A multi-dimensional parametric study of variability in multi-phase flow dynamics during geologic CO2 sequestration accelerated with machine learning

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

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

The utilization of hydrocarbon gas in enhanced oil recovery (EOR) processes offers two significant advantages: an increase in the recovery factor and a reduction in net emissions. Through core-scale experimental and numerical investigations, effective diffusion coefficients for single-phase and cross-phase behavior were determined by Fu et al. (2021), enabling their application in larger-scale predictions. [1] The primary objectives of this study are to 1) better understand the impact of upscaling from core-scale to field-scale simulations; 2) verify the effect of diffusion mechanism during huff-n-puff by history matching a model for a single well pilot; and 3) conduct a comprehensive sensitivity analysis and optimization of the recovery factor for huff-n-puff schedule, taking into account fracture spacing and injection-production patterns in both the dead and live oil windows of the Eagle Ford formation. The fluids in place in the Eagle Ford shale show a wide range of GORs, with hydrocarbon maturities ranging from black oil to lean gas condensates, [2] therefore, both live and dead oil regions are investigated in this study. Two compositional models, incorporating dual porosity and dual permeability characteristics, were constructed using the Petrel software. The first model replicated a huff-n-puff field pilot study reported by Orozco et al. (2020) in the Eagle Ford [3] and consisted of one well with the well length of 6,240 ft and 26 hydraulic fracture stages. The second model encompassed a single stage of eleven horizontal wells, designed according to the field blueprint reported by Baldwin et al. (2020). [4] Within this model, six wells were allocated for injection and production during the huff-n-puff cycles, four were used as containment wells, and one functioned as a monitoring well at the center of all eleven wells. The well spacing was set at 1000 ft, with the first stage of each well measuring 220 ft in length, and each well containing 10 hydraulic fractures. These fractures were spaced 20 ft apart (cluster spacing), with a height of 100 ft, and a half-length of 500 ft. Once the pilot well's primary and huff-n-puff oil production rate was history matched, the same reservoir properties, including matrix and natural-fracture porosity, permeability, natural fracture spacing, and relative permeability, were applied to the eleven-well model. Both models employed history-matched effective diffusion coefficients and a tuned equation of state fluid model to fluid samples collected and analyzed for the Eagle Ford formation. [5, 1] Results show that models including the diffusion mechanism had a 2.2% higher oil recovery factor compared to those that did not include diffusion after five cycles of huff-n-puff. The sensitivity analysis on hydraulic fracture spacing showed that smaller fracture spacing creates larger contact surface area between the matrix and fracture, promoting the diffusion mechanism and facilitating higher oil recoveries. The sensitivity analysis also revealed that depletion level on the producer before starting Huff n Puff also had an impact on recovery efficiency. Producing a well on primary production for 6 years and then implementing huff-n-puff yielded the most oil cumulative produced. If the huff-n-puff cycle was delayed to 10 years after initial production, cumulative values were lower than at the 6-year mark due to depletion effects and difficulties in re-pressurizing the formation. The sensitivity analysis on the "puff" production period suggested that longer production times delayed the speed of oil production, but resulted in higher oil production after completing six cycles of huff-n-puff. Further sensitivity analysis on the length of the soaking period suggested that longer soaking times delayed oil production and did not contribute significantly to oil production. These parameters’ effects on cumulative oil production and reservoir pressure were analyzed to determine the optimal approach for field application. Investigations on using different injection gases such as CO2, y-grade, and lean gas for dead oil and live oil systems rank the best injectants for maximizing oil production in the following order: y-grade > CO2 ≈ hydrocarbon gas > lean gas. The findings of this study provide a deeper understanding of upscaling considerations and offer recommendations for huff-n-puff pilot designs in the Eagle Ford formation.

A three-dimensional pore-level network model is developed to calculate three-phase capillary pressure and relative permeability curves. The model combines a description of pore-space morphological features and three-phase displacement physics to model capillarity-controlled gas invasion into a water-wet medium containing oil and water. The gas-water capillary pressure curves in three-phase systems depend on the fluid saturations, the spreading coefficient, and the oil-water capillary pressure. Water relative permeability for the strongly water-wet system modeled, is a function of its saturation alone. Gas and oil relative permeability curves depend on the gas and oil saturation, respectively, the saturation history and the spreading coefficient. The displacement of oil as a response to changing the capillary pressure in a three-phase system cannot be described by existing models for multiphase flow. A modification of Darcy's law is proposed to describe capillary pressure induced oil mobilization in three-phase systems. Introduction Simultaneous flow of gas, oil and water in porous media is prevalent in many applications such as oil recovery and subsurface contaminant transport. Numerous experiments show that higher oil recovery is possible following gravity drainage or gas-injection of reservoirs than by water-flooding alone. However, little success has been achieved in the mechanistic modeling of oil mobilization and the rate of oil recovery in three-phase systems at the pore scale or the continuum scale. Unlike three-phase flow models, continuum-scale models (usually based on Darcy's law) have had much success in describing two-phase flow in porous media. Darcy's law defines the superficial velocity of each phase ui, as a function of the phase pressure gradient Pi, viscosity i, relative permeability Kri, and the medium absolute permeability K as follows: (1) The pressures in the two phases are related by the capillary pressure which is the difference between the pressure in the non-wetting phase and the wetting phase. Therefore to model multiphase flow, in addition to system parameters such as the medium permeability and the phase viscosities, the capillary pressure and relative permeability values are required. To develop better mathematical models of three-phase systems, identification of the pore-level mechanisms involved has been the focus of many studies. The amount of oil that can be recovered following gravity drainage or gas injection is shown to be closely related to the presence of continuous oil layers in the system, which in turn depends on the capillary pressure and the spreading coefficient. The spreading coefficient, S, is defined as (2) where, , and are the gas-water, gas-oil and oil-water interfacial tensions, respectively. Three pore-level mechanisms have been identified in three-phase gas invasion processes by etched-glass micromodel experiments in water-wet media:direct water drainage,direct oil drainage, anddouble drainage. Direct water drainage results when gas displaces water and the displaced water is recovered at the outlet. Direct oil drainage occurs in a similar fashion when oil is directly displaced by gas out of the medium. In the double drainage mechanism, gas displaces oil, which in turn displaces water. It is possible to relate the gas-water capillary pressure required for the occurrence of each of the three displacement mechanisms to the fluid-fluid interfacial tensions, local pore geometry and the oil-water capillary pressure. A number of pore-level models have been proposed based on the displacement mechanisms of three-phase flow. Capillarity-controlled gas invasion is typically modeled by increasing the gas-water capillary pressure in small steps. P. 405^

Summary A crucial component of all multiphase flow models is the relationship among relative permeabilities, fluid saturations, and capillary pressures. Relative permeability and capillary pressure parametric models can be very useful for predicting fluid behavior in porous media. However, relative permeabilities and capillary pressures used in oil reservoir simulators are commonly determined through interpolation between laboratory measurements. A problem with this approach is that the relations are valid only for the specific saturation path measured. Therefore, simulations of oil production using different saturation paths from those measured are likely to be in error and can limit the investigation of alternative production scenarios. In this paper, saturation-history-dependent relative permeability and capillary pressure functions for two-phase flow in mixed-wet rocks are discussed. Relative permeabilities are predicted by integrating a pore-distribution model between limits that reflect how oil and water are distributed in mixed-wet porous media. The proposed model was tested against mixed-wet capillary pressure data. The model then was incorporated in the U. of Texas Chemical Compositional Simulator (called UTCHEM) to compare waterflood simulations in water- and mixed-wet reservoirs. The simulation results agree qualitatively with laboratory core and field observations. The model and its implementation also were validated against a sandpack experiment.

A statistical mechanical model of gas solubility in glass is presented. For equilibrium, the Gibbs free energy of the free gas is equated to that of the gas in solution. Expressions for the Gibbs energies are obtained from assumptions of the atomic motion in the free gas and dissolved states. The resulting solubility equations express the solubility (in atoms per cubic centimeter of glass) as a function of gas pressure, temperature, fundamental constants, and material parameters. An initial evaluation of the model is made by comparison with literature data for helium and neon in fused silica.

In this paper, techniques have been developed to interpret three-phase relative permeability and water–oil capillary pressure simultaneously in a tight carbonate reservoir from numerically simulating wireline formation tester (WFT) measurements. A high-resolution cylindrical near-wellbore model is built based on a set of pressures and flow rates collected by dual packer WFT in a tight carbonate reservoir. The grid quality is validated, the effective thickness of the WFT measurements is examined, and the effectiveness of the techniques is confirmed prior to performing history matching for both the measured pressure drawdown and buildup profiles. Water–oil relative permeability, oil–gas relative permeability, and water–oil capillary pressure are interpreted based on power-law functions and under the assumption of a water-wet reservoir and an oil-wet reservoir, respectively. Subsequently, three-phase relative permeability for the oil phase is determined using the modified Stone II model. Both the relative permeability and the capillary pressure of a water–oil system interpreted under an oil-wet condition match well with the measured relative permeability and capillary pressure of a similar reservoir rock type collected from the literature, while the relative permeability of an oil–gas system and the three-phase relative permeability bear a relatively high uncertainty. Not only is the reservoir determined as oil-wet but also the initial oil saturation is found to impose an impact on the interpreted water relative permeability under an oil-wet condition. Changes in water and oil viscosities and mud filtrate invasion depth affect the range of the movable fluid saturation of the interpreted water–oil relative permeabilities.

In this paper, pragmatic and robust techniques have been developed to simultaneously interpret absolute permeability and relative permeability together with capillary pressure in a naturally fractured carbonate formation from wireline formation testing (WFT) measurements. By using two sets of pressure and flow rate field data collected by a dual-packer tool, two high-resolution cylindrical near-wellbore numerical models are developed for each dataset on the basis of single- and dual-porosity concepts. Then, simulations and history matchings are performed for both the measured pressure drawdown and buildup profiles, while absolute permeability is determined and relative permeability is interpreted with and without considering capillary pressure. Compared to the experimentally measured relative permeability curves for the same formation collected from the literature, relative permeability interpreted with consideration of capillary pressure has a better match than those without considering capillary pressure. Also, relative permeability obtained from dual-porosity models has similar characteristics to those from single-porosity models especially in the region away from the endpoints, though the computational expenses with dual-porosity models are much larger. Absolute permeabilities in the vertical and the horizontal directions of the upper layer are determined to be 201.0 mD and 86.4 mD, respectively, while those of the lower layer are found to be 342.9 mD and 1.8 mD, respectively. Such a large vertical permeability of the lower layer reflects the contribution of the extensively distributed natural fractures in the vertical direction.