non-Darcy Flow Behavior Research Articles

Evaluation of Forchheimer equation coefficients for non-Darcy flow in deformable rough-walled fractures

Summary This study focuses on experimental evaluation of the Forchheimer equation coefficients for non-Darcy flow in deformable rough-walled fractures. Water flow tests through twelve granite fracture samples with different roughness were conducted in a triaxial cell under confining stresses varying from 1.0 MPa to 30.0 MPa. A total of 2280 experimental data in the form of pressure gradient versus discharge were collected. Three representative types of nonlinear flow behaviors induced by inertial effect, fracture dilation and solid–water interaction, respectively, were observed. Regression analyses of the experimental data show that the Forchheimer equation adequately describes the non-Darcy flow behavior induced by significant inertial effect. Based on the experimental observations, two empirical equations were proposed for parametric expression of the Forchheimer’s nonlinear coefficient, one as a power function of hydraulic aperture and the other dependent on both hydraulic aperture and peak asperity of the fracture surface. A new criterion was presented for assessing the applicability of Darcy’s law, which relies on the ratio of discharge or pressure gradient predicted by the Forchheimer’s law incorporated with the single-parameter equation to that predicted by the Darcy’s law. A sensitivity analysis was performed using the double-parameter equation for examining the dependence of the Forchheimer’s nonlinear coefficient on peak asperity, demonstrating the importance of incorporating the fracture roughness in the development of non-Darcy flow models. The experimental results and the proposed models are useful for understanding and numerical modeling of the nonlinear flow behaviors in fractured aquifers.

Significance of non-Darcy flow effect in fractured tight reservoirs

Reservoir development is increasingly moving towards unconventional resources such as tight reservoirs due to the rapid decline in conventional reserves. Several researches have been done in the petroleum industry addressing significant influence of non-Darcy flow behavior on well deliverability and reservoir performance. In all of the previous works, it is generally believed that the non-Darcy flow in a reservoir, if occurs, becomes considerable within a few feet around the wellbores beyond which it is negligible. Furthermore, wherever non-Darcy effect is mentioned in the literature, attention has usually been focused on gas wells with high production rates (i.e.‚ rates higher than 10 MMSCF/day).In this paper, correlations that are typically used for determination of non-Darcy coefficient are reviewed. Then, it is shown that the correlations are distinct and lead to considerably different values of the non-Darcy coefficient for the same rock sample. A simple guideline is also presented for choosing the most appropriate correlation for a reservoir. Main body of this paper is directed at accurate description of non-Darcy flow in fractured tight reservoirs. This study evaluates the validity of a widely accepted assumption, which considers non-Darcy effect significant only within a few feet around wellbore of gas wells producing at high rates. A synthetic simulation model is made using the data of a well from one of the Iranian fractured tight reservoirs. A full feature compositional simulator is used for the computations in this study. The distance out of the wellbore is subdivided into a number of regions, and then the necessity of including the non-Darcy component in each of the regions for better performance predictions is investigated. The effect of accounting the non-Darcy term in each of the regions on the simulation results such as production rate, final recovery and pressure behavior is studied. Furthermore, three distinct reservoir fluid types including dry gas, gas condensate and black oil are used with the objective well model to evaluate dependence of non-Darcy effect on the type of flowing fluid. The study results are used to provide guidelines about the necessity of global consideration of the non-Darcy term in simulation of different fluid systems even at low production rates. The results highlight that contribution of the non-Darcy component to flow can be significant even far away from wellbores, thus it must be considered globally in the bulk of reservoirs. Additionally, the study demonstrates that the role of non-Darcy component can be crucial even at low production rates regardless of reservoir fluid type. Therefore, in order to have accurate modeling, design and successful implementation of projects, simulations of fractured tight reservoirs must be performed with the global inclusion of the non-Darcy flow formulations regardless of the type of flowing stream and levels of production rate.

Determination of minimum permeability plateau and characteristic length in porous media with non-Darcy flow behavior

Abstract A mathematical model has been formulated to determine the minimum permeability plateau and characteristic length in porous media with non-Darcy flow behaviour. More specifically, two new parameters are first defined to respectively include effect of minimum permeability plateau and characteristic length on non-Darcy flow behaviour across the completion with consideration of wellbore storage and skin effects. The newly formulated mathematical model is solved by discretizing the convolution integral and material balance equation, and then validated by simplifying it to the traditional Forchheimer model. The pressure together with its corresponding derivative type curves for both buildup and drawdown tests can be reproduced to examine effects of both wellbore storage and non-Darcy flow across the completion. The pressure derivative response with non-Darcy flow across the completion has been found to have a steep slope since the non-Darcy effect is increased after the wellbore-storage-dominated period. Such steep derivatives resulting from the increased non-Darcy effect cannot be observed from the existing models based on Darcy׳s law. In particular, the non-Darcy parameter depending on flow rate is found to dominate the shape of the derivative curves. Compared to the drawdown derivative curves, there exists a larger gap in the buildup derivative curves, indicating that buildup tests are more appropriate than drawdown tests for type curve matching non-Darcy flow behaviour.

A Hybrid Analytical/Numerical Model for the Characterization of Preferential Flow Path with Non-Darcy Flow

Due to the long-term fluid-solid interactions in waterflooding, the tremendous variation of oil reservoir formation parameters will lead to the widespread evolution of preferential flow paths, thereby preventing the further enhancement of recovery efficiency because of unstable fingering and premature breakthrough. To improve oil recovery, the characterization of preferential flow paths is essential and imperative. In efforts that have been previously documented, fluid flow characteristics within preferential paths are assumed to obey Darcy's equation. However, the occurrence of non-Darcy flow behavior has been increasingly suggested. To examine this conjecture, the Forchheimer number with the inertial coefficient estimated from different empirical formulas is applied as the criterion. Considering a 10% non-Darcy effect, the fluid flow in a preferential path may do experience non-Darcy behavior. With the objective of characterizing the preferential path with non-Darcy flow, a hybrid analytical/numerical model has been developed to investigate the pressure transient response, which dynamically couples a numerical model describing the non-Darcy effect of a preferential flow path with an analytical reservoir model. The characteristics of the pressure transient behavior and the sensitivities of corresponding parameters have also been discussed. In addition, an interpretation approach for pressure transient testing is also proposed, in which the Gravitational Search Algorithm is employed as a non-linear regression technology to match measured pressure with this hybrid model. Examples of applications from different oilfields are also presented to illustrate this method. This cost-effective approach provides more accurate characterization of a preferential flow path with non-Darcy flow, which will lay a solid foundation for the design and operation of conformance control treatments, as well as several other Enhanced Oil Recovery projects.

Evaluation of the non-Darcy effect in coalbed methane production

The non-Darcy factor, an indicator for the non-Darcy effect, is dependent on the properties of porous media and pore fluid including permeability, viscosity, density, flow velocity and a coefficient named as β factor. Experimental results show that the β factor can be expressed as a power law of permeability. For conventional gas reservoirs, this β factor can be assumed as a constant as the permeability change is negligible. However, the constant β factor may not be suitable for coal seams with remarkable permeability change and a variable β factor as a function of coal permeability should be an alternative. Moreover, the coal permeability change is complex due to the competing effects of coal cleat compression and sorption induced coal shrinkage/swelling. Few studies have been done previously to incorporate the variable β factor as a function of coal permeability in reservoir simulations. In the present work, both the coal permeability change and the variable β factor are coupled in a dual porosity model to study the non-Darcy flow behavior in coal seams. The simulation results illustrate that the evolution of non-Darcy factor becomes tortuous by using a variable β factor, which differs from the monotonic behavior when constant β factors are applied. Furthermore, increasing the coal cleat compressibility and matrix shrinkage strain tends to intensify the tortuous behavior. The simulation results also indicate that using typical constant β factors, instead of the variable one, may significantly underestimate or overestimate the gas production rate for coalbed methane wells.

Determination of Fracture Conductivity in Tight Formations With Non-Darcy Flow Behavior

Summary In this paper, a mathematical model has been developed and successfully applied to accurately determine the fracture conductivity in tight formations with non-Darcy flow behavior. A new non-Darcy flow number is first defined to account for the effect of characteristic length in a hydraulic fracture. A semianalytical method is then applied to solve the newly formulated mathematical model by discretizing the fracture into small segments, assuming that there exists unsteady flow between the adjacent segments. The newly developed model has been validated by simplifying it to the traditional Forchheimer (i.e., non-Darcy) model and by performing numerical simulation with a reservoir simulator as well. The pressure response and its corresponding derivative type curves have been reproduced to examine non-Darcy flow behavior under different fracture conductivities. Both relative minimum permeability and characteristic length are found to impose a negative effect on the fracture conductivity. Compared with relative minimum permeability, characteristic length is a strong function dominating the non-Darcy flow behavior in the fractures. It is obvious that the fracture conductivity can be accurately determined when non-Darcy flow behavior in the fracture network is taken into account.

Geological Characteristics and Waterflooding Strategy for Yuanshi Low Permeability Sandstone Reservoir

Core test data from 10 wells indicate that the sandstone reservoir in Yuanshi oilfield is characterized by low permeability, thin pore throat, high clay mineral content, and strong sensitivity, which are likely to cause reservoir damage and affect development result. Based on the characteristics of water-sensitive, velocity-sensitive low permeability reservoir as well as non-Darcy flow behavior, the development strategy of high pressure water injection and cyclic water injection has been formulated with the assistance of engineering measures such as fracturing and reducing well spacing.

Numerical modeling of non-Darcy flow in near-well region of a geothermal reservoir

Non-Darcy flow behavior has been recognized as an important factor in fluid flow through porous and fractured rock, especially for the near-well region during high-flow-rate injection/production in a reservoir formation. The non-Darcy effect is mostly reported to exist in gas/oil fields due to the high flow rate of a gaseous fluid, but it has been increasingly suggested that liquid flow in a geothermal reservoir may also be influenced by the non-Darcy effect. In this study, non-Darcy well flow behavior is described using the Forchheimer equation, and a nonlinear finite element model is used to simulate the near-well performance in a geothermal reservoir with non-Darcy effects. Results show that by considering the Forchheimer flow in the near-well region, both the fluid velocity and the wellbore pressure are much different from the Darcy flow simulation and the production well pressure is sensitive to the non-Darcy flow coefficient. A series of pressure drawdown tests are simulated on a geothermal reservoir with liquid water, vapor and two phase mixture as working fluids. Thermodynamic properties of water are calculated by the latest industry steam table functions for determining the pressure/temperature-dependent fluid properties. The simulation results suggest that the ratio of fluid density to viscosity (ρ/μ) has significant impact on non-Darcy flow behavior in reservoir performance. The pressure drawdown increases 60% due to a 247% increase of ρ/μ in one of the well tests.

Non-Darcy Porous-Media Flow According to the Barree and Conway Model: Laboratory and Numerical-Modeling Studies

Summary This paper presents the results of our new experimental studies conducted for high flow rates through proppant packs, which show that the Barree and Conway (2004) flow model is capable of overcoming limitations of the Forchheimer non-Darcy equation at very high flow rates. To quantify the non-Darcy flow behavior using the Barree and Conway model, a numerical model is developed to simulate non-Darcy flow. In addition, an analytical solution is presented for steady-state linear non-Darcy flow and is used to verify the numerical-simulation results. The numerical model incorporates the Barree and Conway model into a general-purpose reservoir simulator for modeling multidimensional, single-phase non-Darcy flow in porous and fractured media and supplements the laboratory findings. The numerical model is then used to perform sensitivity analysis of the Barree and Conway flow model's parameters and to investigate transient behavior of non-Darcy flow at an injection well.

Comparison of Klinkenberg-corrected gas permeability and water permeability in sedimentary rocks

We measured intrinsic permeability of sedimentary rocks from the western foothills of Taiwan by using nitrogen gas and distilled water as pore fluids in effective-pressure cycling tests at room temperature. The observed difference in gas and water permeabilities was analyzed in view of the Klinkenberg effect. This effect is due to the slip flow of gases at pore walls, which enhances gas flow when pore sizes are very small. Our experimental results showed that (1) gas permeability was larger than water permeability by several times to one order of magnitude, (2) gas permeability increased with increasing pore pressure, and (3) water permeability increased slightly as the pore-pressure gradient across the specimen increased. Results (1) and (2) can be explained quantitatively by an empirical power law in relation to the Klinkenberg constant b that is applicable in low permeable range. This correlation enables us to estimate water permeability from gas permeability. The Klinkenberg effect is important when permeability is lower than 10 −18 m 2 and at low pore-pressure differentials, and correction for the effect is essential to estimate water permeability from gas permeability measurement data. A simplified Bingham flow model for water can partially explain the trend of result (3), although non-Darcy flow behavior or inertial forces of water–rock interaction are needed to account for the observed deviation from Darcy's law.

Gas well production analysis with non-Darcy flow and real-gas PVT behavior

A rigorous semi-analytical model is developed to study the production rate behavior of wells in gas reservoirs with Forchheimer's non-Darcy flow under constant or varying bottomhole pressure conditions. Rigorous modeling of gas property variation in reservoirs is integrated in this model. Three dimensionless parameters, normalized gas viscosity and compressibility product, Forchheimer number, and normalized viscosity, are introduced to quantify their effects on non-Darcy flow in the reservoir. The variation of viscosity and compressibility product in the gas reservoir depletion process leads to a smaller production decline rate than that of exponential decline. The parameter b in Arps' decline equation under Darcy flow is defined as the base value, b ⁎, that helps identify the production decline caused by reservoir non-Darcy flow. Under the condition of constant pressure drop the more severe the non-Darcy flow, the larger the Forchheimer number. Generally, non-Darcy results in a smaller production rate, a larger decline rate in the boundary-dominated period, and a longer transition period between these two periods. The viscosity variation enhances non-Darcy flow, which lowers the initial production rate but has very little effect on the production decline rate. A larger production decline rate during the boundary-dominated period may help petroleum engineers identify the non-Darcy flow effect from the production data. Analyses show that the traditional quadratic equation is a good approximation only if non-Darcy flow is not severe and that for non-Darcy flow in the reservoir, the traditional Fetkovich's type curves may underestimate reservoir permeability, overestimate well skin factor, and misinterpret reservoir drainage area. A method of using the proposed model to identify the non-Darcy flow from production data is presented. Two examples from the literature are analyzed, and good type curve matches and more reliable results are obtained.

A PORE NETWORK MODEL FOR THE CALCULATION OF NON-DARCY FLOW COEFFICIENTS IN FLUID FLOW THROUGH POROUS MEDIA

A computational method was developed to determine the correlation between permeability and the Forchheimer (non-Darcy) coefficient and to investigate numerically the factors that affect this correlation. The method is based on the construction of a 2-D random pore network. The porous medium is represented as a network of cylindrical pipes with randomly generated size, orientation, and connectivity. Probability density functions are used to characterize the orientation, length, diameter, and connectivity properties of the pipes that form the pore space. In this article the development and validation of the computational method are discussed. The computational procedure provides an expected value for the medium properties and a variation around this expected value, which is a measure of the uncertainty associated with the calculation of porous media properties. It was found that the porous medium structure has a strong effect on the flow properties. Specifically, the splitting of the pores is mostly responsible for non-Darcy flow behavior. Examination of the behavior of the medium properties under compaction shows that there is a unifying correlation for the behavior of the permeability. However, each type of porous medium exhibits different non-Darcy flow behavior under compaction. Finally, a statistical model, which correlates the structure characteristics of the porous medium to the measured properties, is presented.

Non-Darcy flow through anisotropic porous media

Non-Darcy flow occurs in the near-wellbore region of high-capacity gas and condensate reservoirs. Forchheimer’s equation describes this flow in isotropic media. Many naturally occurring porous media are, however, anisotropic. This study is directed at understanding the non-Darcy flow behavior in porous media that are anisotropic at the pore-scale. Three structural models of anisotropy are proposed: size-induced, connectivity-induced and spatial correlation-induced. Non-Darcy flow has been computed in these netwoks, as well as the tortuosity and porosity of the media. A tensorial form of the Forchheimer equation has been proposed and verified. The non-Darcy term has been found to be tensorial and proportional to the square of the superficial velocity at the macroscopic scale. As the pore-scale anisotropic parameter increases, non-Darcy and tortuosity components increase, but permeability components decrease. An approximate correlation is derived between the non-Darcy coefficient, the permeability, and the tortuosity tensors.

Network modeling of non-Darcy flow through porous media

Darcy's law is inadequate for describing high-velocity gas flow in porous media, which occurs in the near well-bore region of high capacity gas and condensate reservoirs. This study is directed at understanding the non-Darcy flow behavior. A pore-level network model has been developed to describe high velocity flow. The inputs to the model are pore size distributions and network coordination numbers. The outputs are permeability, non-Darcy coefficient, tortuousity and porosity. The additional pressure gradient term is found to be proportional to the square of the velocity in accordance with the Forchheimer's equation. The correlation between the non-Darcy coefficient and other flow properties (the permeability, the porosity and the tortuousity) is found to depend on the morphological parameters being changed. General correlations are derived between these flow properties.

Non-Darcy Gas Flow Through Propped Fractures: Effects of Partial Saturation, Gel Damage, and Stress

Summary The objective of this work was to identify and quantify important parameters affecting gas production through propped fractures under a non-Darcy gas flow regime. The gas flow capacity of a simulated propped fracture was studied systematically to determine the effects of partial saturation, gel damage, and stress conditions. The flow-capacity response of the 20/40-mesh sand tested throughout this project was affected significantly by variations in the effective gas porosity of the proppant pack. Permeability and non-Darcy flow characteristics were correlated to effective gas porosity. Partial saturation was found to be a key parameter influencing the permeability and non-Darcy gas flow behavior of a proppant pack. Partially saturated fractures may result from incomplete removal of fracturing fluid, mobility of formation waters, or production of condensates. The partial saturation of the proppant pack, in effect, changes the open porosity available for gas flow, which adversely affects gas permeability and non-Darcy flow parameters. The results from this investigation demonstrate that non-Darcy gas flow behavior through propped fractures in which a saturation phase is present cannot be estimated from results using dry-proppant-pack tests.

Electrokinetic flow through porous media composed of fine cylindrical capillaries

Electrokinetic flow through fine cylindrical capillaries was investigated experimentally, using dilute solutions of potassium iodide in purified water as the electrolyte. Two types of porous media were studied: a fused array of borosilicate glass capillaries each 2 μm in diameter and 1.91 μm long with a moderatly high zeta potential; and a polycarbonate filter membrane containing nearly uniform circular pores each about 0.1 μm in diameter and 12 μm long, with a relatively low zeta potential. The electrokinetic radius was varied by varying the concentration of potassium iodide in the solution, the conductivity of which was continuously monitored. Streaming potentials were measured using platinum electrodes for the glass disc and silver iodide electrodes for the polycarbonate membrane. An apparent effect of pressure drop on the ratio of streaming potential to pressure drop for the polycarbonate membrane, which could serve as an explanation for the non-Darcy flow behavior reported by others for low liquid velocities through certain porous media, was rationalized as fortuitous by invoking arguments recently presented by Ball and Fuerstenau [7]. The electroviscous retardataion effect was measured for the glass disc by comparing the product of pressure drop and displacement time for a fixed volume of liquid with the corresponding product obtained under conditions of high electrokinetic radius (small double layer thickness). The results for both streaming potential and electroviscous retardation effect are compared with the theory of Rice and Whitehead [1] for low zeta potentials, as well as with the more recent theory of Levine et al. [2] for all zeta potentials. The data support the latter theory more than the former.

Non-Darcy Transient Radial Gas Flow Through Porous Media

Abstract A computer program was developed for solving the nonlinear partial differential equations describing transient radial gas flow through porous media. Specifically, the program can handle the four sets of boundary conditions encountered in a finite radial system. The program has been employed to obtain pressure distributions and mass fluxes as functions of time and position for the constant terminal rate case and the constant terminal pressure case, in porous media having different values of permeability, inertial resistance coefficient, slip coefficient, and rock compressibility; the program takes into account changes of gas properties with pressure. Results indicate that each of these rock properties can, under certain conditions, appreciably affect the transient response. For instance, it was observed that slip, rather than being a laboratory curiosity, can affect the transient response in a typical tight reservoir. Introduction The flow of gases through porous media has been a subject of investigation for many years. The method of attack has consisted of developing simplified models describing steady flow and combining them with the continuity equation in order to describe transient behavior. The simplified steady-flow models were gradually made more sophisticated and were in turn used to develop models which were better able to describe transient behavior in real systems. Because of its relative simplicity, linear flow has usually been studied first, with the results obtained being then used to solve the radial case. Thus, the nonlinear partial differential equations describing the transient flow of an ideal gas through a linear porous medium were first solved by means of finite-difference methods. This was followed by a more detailed treatment of the linear case and a treatment of the radial system. Others proposed solutions for the radial case. The manner in which molecular streaming or slip altered the transient in linear systems was examined theoretically by Collins and Crawford and by Aronofsky and verified experimentally by Wallick and Aronofsky. The influence of nonideal gas behavior on the transient response for the linear case was examined by Aronofsky and Ferris. The effect of such behavior on the radial case was initially investigated by Aronofsky and Porter and later by Al-Hussainy et al. Non-Darcy How and its effect on the transient response was examined by Swift and Kiel and Tek et al. To this point the relaxations regarding the initial simplified model had consisted of adding one effect; viz, slip, real gas behavior, or non-Darcy flow. The introduction of two complicating effects was first applied to the transient response by Eilerts et al, who studied condensate reservoirs by allowing non-Darcy flow and real gas behavior. The manner in which inertial and slip effects together influenced steady linear gas flow was studied by Dranchuk and Sadiq and by Dranchuk and Kolada. These same two effects were incorporated in the transient model for the linear case by Dranchuk and Chwyl. Recently, a model was developed describing steady radial gas flow in which slip, inertial effects, and real gas behavior are permitted simultaneously; the model has been successfully applied to laboratory data. On the basis of this model and the realization that the interactions between the various effects in a nonlinear partial differential equation can be unpredictable, this study was undertaken. The object was to develop a transient radial gas flow model in which slip, inertial effects, real gas behavior, and rock compressibility are permitted to operate simultaneously; to solve the resulting model for various boundary conditions; and to examine the influence on the resulting transient response of these complicating effects. MATHEMATICAL MODEL The purpose of this study was accomplished by observing the behavior of some hypothetical gas reservoirs. SPEJ P. 129

Non-Darcy flow behavior in liquid-saturated porous media

Velocity-gradient data for water-saturated porous media were considered in some detail. Deviations from Darcy's law were found in data originally presented in support of the law. Recent data were analyzed on the basis of a postulated three-parameter equation, in contrast with the one-parameter Darcy proportionality. The equation contains Darcy's law as a special case and in general fits the data quite well for water and electrolyte solutions flowing in sandstones and clays. The fitted parameters behave consistently, and are capable of reasonable interpretation. Possible mechanisms giving rise to non-Darcy behavior are suggested, including that of non-Newtonian liquid effects caused by clay-water interaction.