non-Darcian Flow Research Articles

Evidence of non-Darcy flow and non-Fickian transport in fractured media at laboratory scale

Abstract. During a risk assessment procedure as well as when dealing with cleanup and monitoring strategies, accurate predictions of solute propagation in fractured rocks are of particular importance when assessing exposure pathways through which contaminants reach receptors. Experimental data obtained under controlled conditions such as in a laboratory allow to increase the understanding of the fundamental physics of fluid flow and solute transport in fractures. In this study, laboratory hydraulic and tracer tests have been carried out on an artificially created fractured rock sample. The tests regard the analysis of the hydraulic loss and the measurement of breakthrough curves for saline tracer pulse inside a rock sample of parallelepiped shape (0.60 × 0.40 × 0.08 m). The convolution theory has been applied in order to remove the effect of the acquisition apparatus on tracer experiments. The experimental results have shown evidence of a non-Darcy relationship between flow rate and hydraulic loss that is best described by Forchheimer's law. Furthermore, in the flow experiments both inertial and viscous flow terms are not negligible. The observed experimental breakthrough curves of solute transport have been modeled by the classical one-dimensional analytical solution for the advection–dispersion equation (ADE) and the single rate mobile–immobile model (MIM). The former model does not properly fit the first arrival and the tail while the latter, which recognizes the existence of mobile and immobile domains for transport, provides a very decent fit. The carried out experiments show that there exists a pronounced mobile–immobile zone interaction that cannot be neglected and that leads to a non-equilibrium behavior of solute transport. The existence of a non-Darcian flow regime has showed to influence the velocity field in that it gives rise to a delay in solute migration with respect to the predicted value assuming linear flow. Furthermore, the presence of inertial effects enhance non-equilibrium behavior. Instead, the presence of a transitional flow regime seems not to exert influence on the behavior of dispersion. The linear-type relationship found between velocity and dispersion demonstrates that for the range of imposed flow rates and for the selected path the geometrical dispersion dominates the mixing processes along the fracture network.

Approximate analytical solution for non-Darcian flow toward a partially penetrating well in a confined aquifer

In this study, non-Darcian flow to a partially penetrating well in a confined aquifer was investigated. The flow in the horizontal direction was assumed to be non-Darcian, while the flow in the vertical direction was assumed to be Darcian. The Izbash equation was employed to describe the non-Darcian flow in the horizontal direction of the aquifer. We used a linearization procedure to approximate the non-linear term in the governing equation enabling the mathematical model to be solved using a combination of Laplace and Fourier cosine transforms. Approximate analytical solutions for the drawdown were obtained and the impacts of different parameters on the drawdown were analyzed. The results indicated that a larger power index n in the Izbash equation leads to a larger drawdown at early times, while a larger n results in a smaller drawdown at late times. The drawdowns along the vertical direction z are symmetric if the well screen is located in the center of the aquifer, and the drawdown at the center of the aquifer is the largest along the vertical direction for this case. The length of the well screen w has little impact on the drawdown at early times, while a larger length of the well screen results in a smaller drawdown at late times. The drawdown increases with Kr at early times, while it decreases as Kr increases at late times, in which Kr is the apparent radial hydraulic conductivity. A sensitivity analysis of the parameters, i.e., the specific storage Ss, w, n and Kr, indicated that the drawdown is not sensitive to them at early times, while it is very sensitive to these parameters at late times especially to the power index n.

One-dimensional nonlinear consolidation of soft clay with the non-Darcian flow

Based on the non-Darcian flow law described by exponent m and threshold gradient i 1 under a low hydraulic gradient and the classical nonlinear relationships e-lgσ′ and e-lgk v (Mesri and Rokhsar, 1974), the governing equation of 1D nonlinear consolidation was modified by considering both uniform distribution of self-weight stress and linear increment of self-weight stress. The numerical solutions for the governing equation were derived by the finite difference method (FDM). Moreover, the solutions were verified by comparing the numerical results with those by analytical method under a specific case. Finally, consolidation behavior under different parameters was investigated, and the results show that the rate of 1D nonlinear consolidation will slow down when the non-Darcian flow law is considered. The consolidation rate with linear increment of self-weight stress is faster than that with uniform distribution one. Compared to Darcy’s flow law, the influence of parameters describing non-linearity of soft soil on consolidation behavior with non-Darcian flow has no significant change.

Modeling Non-Darcian Single- and Two-Phase Flow in Transparent Replicas of Rough-Walled Rock Fractures

While it is generally assumed that in the viscous flow regime, the two-phase flow relative permeabilities in fractured and porous media depend uniquely on the phase saturations, several studies have shown that for non-Darcian flows (i.e., where the inertial forces are not negligible compared with the viscous forces), the relative permeabilities not only depend on phase saturations but also on the flow regime. Experimental results on inertial single- and two-phase flows in two transparent replicas of real rough fractures are presented and modeled combining a generalization of the single-phase flow Darcy’s law with the apparent permeability concept. The experimental setup was designed to measure injected fluid flow rates, pressure drop within the fracture, and fluid saturation by image processing. For both fractures, single-phase flow experiments were modeled by means of the full cubic inertial law which allowed the determination of the intrinsic hydrodynamic parameters. Using these parameters, the apparent permeability of each fracture was calculated as a function of the Reynolds number, leading to an elegant means to compare the two fractures in terms of hydraulic behavior versus flow regime. Also, a method for determining the experimental transition flow rate between the weak inertia and the strong inertia flow regimes is proposed. Two-phase flow experiments consisted in measuring the pressure drop and the fluid saturation within the fractures, for various constant values of the liquid flow rate and for increasing values of the gas flow rate. Regardless of the explored flow regime, two-phase flow relative permeabilities were calculated as the ratio of the single phase flow pressure drop per unit length divided by the two-phase flow pressure drop per unit length, and were plotted versus the measured fluid saturation. Results confirm the dependence of the relative permeabilities on the flow regime. Also the proposed generalization of Darcy’s law shows that the relative permeabilities versus fluid saturation follow physical meaningful trends for different liquid and gas flow rates. The presented model fits correctly the liquid and gas experimental relative permeabilities as well as the fluid saturation.

Validation of non-Darcian flow effects in slug tests conducted in fractured rock boreholes

Summary A series of rising and falling head slug tests with different initial applied head differentials (Δ H o ) were conducted in open fractured dolostone and sandstone boreholes using straddle packers isolating specific depth intervals (1.5 m length) to examine the influence of non-Darcian flow. The open holes were developed and inspected using video and acoustic televiewing (ATV) to ensure that evidence of skin effects due to drilling were absent. The transmissivity ( T ) values obtained from both the rising and falling head slug tests were very similar at low initial applied head; however, the T values were progressively smaller at larger Δ H o , suggesting error due to non-Darcian flow. Non-Darcian flow behavior was confirmed by constant head step tests conducted in the same test intervals where the injection rate ( Q ) vs. applied head (d H ) relationship became non-linear at relatively low injection rates, and the non-Darcian data also resulted in lower T values. For a series of slug tests conducted at different Δ H o , non-Darcian flow effects gradually increased as Δ H o increased, consistent with the trends for constant head step tests conducted in the same test intervals. To maintain Darcian flow conditions in the fractured dolostone and sandstone tested in this study, Δ H o must be kept small, generally less than 0.2 m. This study demonstrates that by conducting both “stepped” slug tests and constant head step tests, the Darcian flow assumption for both types of tests can be rigorously validated. However, when only slug tests are conducted, it is necessary to conduct a series of “stepped” slug tests, including tests with small applied head differentials, to avoid errors due to non-Darcian flow.

Vertical drain consolidation with non-Darcian flow and void-ratio-dependent compressibility and permeability

Vertical drains increase the rate of consolidation in soft soils by facilitating faster dissipation of excess pore water pressure through short, horizontal drainage paths. This paper presents an analytical solution for non-linear radial consolidation under equal-strain conditions incorporating smear but ignoring well resistance. Three aspects of non-linearity are considered: (a) non-Darcian flow, (b) a log-linear void-ratio–stress relationship; and (b) a log-linear void-ratio–permeability relationship. The analytical solution to non-linear radial consolidation can explicitly capture the behaviour of both overconsolidated and normally consolidated soils. For non-linear material properties, consolidation may be faster or slower when compared with the cases with constant material properties. The difference depends on the compressibility/permeability ratios (Cc/Ck and Cr/Ck), the preconsolidation pressure and the stress increase. If Cc/Ck < 1 or Cr/Ck < 1 then the coefficient of consolidation increases as excess pore pressures dissipate, and the corresponding rate of consolidation is greater.

A Study of Scrap Heating by Burners: Part II—Numerical Modeling

A computational fluid dynamics code was developed to model the heating of a bed of porous steel scrap by combustion gases from a burner. The code accounted for nonuniform void fraction in the bed; turbulent, non-Darcian flow, heat transfer from the gas to the scrap; and radiation. The measured bed porosity values were used in the code. Because steel scrap pieces are very irregular in shape and size, the effective particle diameter was fitted to measurements made in a 1-m3 capacity furnace, as reported in part I. It was found that the lower porosity of the scrap was the most beneficial in increasing the efficiency of heat transfer to the scrap bed because the interfacial area is larger. The effect of particle size was much smaller. It was found that the configurations that increased the residence time or path length of the gases increased the efficiency. The measured porosity of the bed approached unity at the walls, so this provided an easy path for the gas to short-circuit the bed, which limited the effectiveness of decreasing the porosity to increase heat-transfer efficiency. Similarly, simulations of nonuniform scrap distributions reduced efficiency of heat transfer due to short circuiting. The implications of the findings for industrial operations are discussed.

On the relationship between water flux and hydraulic gradient for unsaturated and saturated clay

Summary Experimental results indicate that the traditional form of Darcy’s law is not adequate for describing water flow processes in clay media because the observed relationship between water flux and hydraulic gradient can be highly non-linear. To capture this non-Darcian flow behavior, we propose a new relationship between water flux and hydraulic gradient by generalizing the currently existing relationships. The new relationship is shown to be consistent with experimental observations for both saturated and unsaturated conditions. In this paper, we also develop an empirical relationship between permeability and threshold hydraulic gradient, an important measure of non-Darcian behavior. The latter relationship is practically useful because it can reduce the number of parameters whose values need to be determined from experiment data in order to model non-Darcian behavior. How to incorporate the impacts of temperature and electrolyte concentrations into the proposed relationships needs further research.

Analysis of one-dimensional consolidation of soft soils with non-Darcian flow caused by non-Newtonian liquid

Based on non-Darcian flow caused by non-Newtonian liquid, the theory of one-dimensional (1D) consolidation was modified to consider variation in the total vertical stress with depth and time. The finite difference method (FDM) was adopted to obtain numerical solutions for excess pore water pressure and average degree of consolidation. When non-Darcian flow is degenerated into Darcian flow, a comparison between numerical solutions and analytical solutions was made to verify reliability of finite difference solutions. Finally, taking into account the ramp time-dependent loading, consolidation behaviors with non-Darcian flow under various parameters were analyzed. Thus, a comprehensive analysis of 1D consolidation combined with non-Darcian flow caused by non-Newtonian liquid was conducted in this paper.

Approximate analytical and numerical solutions for radial non‐Darcian flow to a well in a leaky aquifer with wellbore storage and skin effect

SUMMARYThis study investigated non‐Darcian flow to a well in a leaky aquifer considering wellbore storage and a finite‐thickness skin. The non‐Darcian flow is described by the Izbash equation. We have used a linearization procedure associated with the Laplace transform to solve such a non‐Darcian flow model. Besides, the Stehfest method has been used to invert the Laplace domain solutions for the drawdowns. We further analyzed the drawdowns inside the well for different cases. The results indicated that a smaller BD results in a smaller drawdown at late times and the leakage has little effect on the drawdown inside the well at early times, where BD is a dimensionless parameter reflecting the leakage. We have also found that the flow for the negative skin case approaches the steady‐state earlier than that for the positive skin. In addition, the drawdown inside the well with a positive skin is larger than that without skin effect at late times, and a larger thickness of the skin results in a greater drawdown inside the well at late times for the positive skin case. A reverse result has been found for the negative skin case. Finally, we have developed a finite‐difference solution for such a non‐Darcian flow model and compared the numerical solution with the approximate analytical solution. It has been shown that the linearization procedure works very well for such a non‐Darcian flow model at late times, and it underestimates the drawdowns at early times. Copyright © 2012 John Wiley & Sons, Ltd.

Recent advances in modeling of well hydraulics

Well hydraulics is a discipline to understand the process of flow to the well in an aquifer which is regarded as a source of groundwater. A variety of analytical and numerical models have been developed over the last few decades to provide a framework for understanding and quantifying the flow behavior in aquifer systems. In this review, we first briefly introduce the background of the theory of well hydraulics and the concepts, methodologies, and applications of analytical, semi-analytical, numerical and approximate methods in solving the well-hydraulic problems. We then address the subjects of current interests such as the incorporation of effects of finite well radius, wellbore storage, well partial penetration, and the presence of skin into various practical problems of groundwater flow. Furthermore, we also summarize recent developments of flow modeling such as the flow in aquifers with horizontal wells or collector wells, the capture zone delineation, and the non-Darcian flow in porous media and fractured formations. Finally, we present a comprehensive review on the numerical calculations for five well functions frequently appearing in well-hydraulic literature and suggest some topics in groundwater flow for future research.

Unsaturated properties for non-Darcian water flow in clay

Clay rock formations, and compacted clay (e.g., bentonite) used as backfill within disposal drifts, have been considered as natural and engineered barriers, respectively, for isolating high-level nuclear wastes in mined geologic repositories. Accurately modeling unsaturated flow in those clay materials is important for assessing the performance of a geological repository. While the non-Darcian behavior of water flow in clay materials has been demonstrated in the literature, a systematic study of modeling unsaturated non-Darcian flow is still lacking. Based on a hypothesis that pore water in clay becomes non-Newtonian as a result of water–clay interaction, we propose new constitutive relationships for unsaturated flow, including a relationship between water flux and hydraulic gradient and those among capillary pressure, water saturation, and hydraulic conductivity. An evaluation based on a set of laboratory experimental observations supports the usefulness of the proposed relationships. More experimental studies are desirable for further confirming the non-Newtonian water flow behavior in clay materials and evaluating the proposed relationships.

One-dimensional consolidation of double-layered soil with non-Darcian flow described by exponent and threshold gradient

Based on non-Darcian flow law described by exponent and threshold gradient within a double-layered soil, the classic theory of one-dimensional consolidation of double-layered soil was modified to consider the change of vertical total stress with depth and time together. Because of the complexity of governing equations, the numerical solutions were obtained in detail by finite difference method. Then, the numerical solutions were compared with the analytical solutions in condition that non-Darcian flow law was degenerated to Dary’s law, and the comparison results show that numerical solutions are reliable. Finally, consolidation behavior of double-layered soil with different parameters was analyzed, and the results show that the consolidation rate of double-layered soil decreases with increasing the value of exponent and threshold of non-Darcian flow, and the exponent and threshold gradient of the first soil layer greatly influence the consolidation rate of double-layered soil. The larger the ratio of the equivalent water head of external load to the total thickness of double-layered soil, the larger the rate of the consolidation, and the similitude relationship in classical consolidation theory of double-layered soil is not satisfied. The other consolidation behavior of double-layered soil with non-Darcian flow is the same as that with Darcy’s law.

MHD non-Darcian flow through a non-isothermal vertical surface embedded in a porous medium with radiation

The effect of MHD on steady two-dimensional laminar mixed flow about a vertical porous surface is numerically analyzed. Also the effects of radiation and heat generation and absorption are considered. A power law variation of temperature along the vertical wall is assumed. The nonlinear boundary-layer equations were transformed and the resulting differential equations were solved by an implicit finite difference scheme (Keller box method). Numerical results for the velocity distribution and the temperature distribution are presented for various values of Prandtl number Pr, magnetic parameter, porous medium parameter and internal heat generation or absorption coefficient. Further validation with previous works is carried out.

Applicability of the ‘cubic law’ for non-Darcian fracture flow

The influence of inertial effects on the flow of fluids through fractured media is a topic of interest for a number of engineering applications, particularly within the energy sector (e.g. production from gas and oil reservoirs, heat extraction from enhanced geothermal systems, coal-bed methane production). The ‘cubic law’ can provide a simple relationship between the hydraulic aperture of a fracture and its permeability, for Darcian flow of a given fluid. At significant flow velocities, fracture flow becomes non-Darcian, the velocity–permeability relationship becomes non-linear and the cubic law becomes invalid. However, for transitional flow, which retains a component of linearity, the cubic law may still be applicable for initial determination of superficial velocities from flow rate data. Here, the results of fluid flow experiments carried out for air flow through a fractured granite sample under various pressures of confinement are presented. Analysis of the flow data using the cubic law produced valid results for non-Darcian flow cases where inertial effects were modest. The cubic law appears to be applicable for flow cases that, following analysis using the cubic law, return a Forcheimer number < 1.30.

Using constant head step tests to determine hydraulic apertures in fractured rock

The initial step in the analysis of contaminant transport in fractured rock requires the consideration of groundwater velocity. Practical methods for estimating the average linear groundwater velocity ( v ¯ ) in fractured rock require determination of hydraulic apertures which are commonly calculated by applying the cubic law using transmissivity (T) values and the number of hydraulically active fractures in the test interval. High-resolution, constant-head step injection testing of cored boreholes in a 100 m thick fractured dolostone aquifer was conducted using inflatable packers to isolate specific test intervals from the rest of the borehole. The steps in each test interval were gradually increased from very low to much higher injection rates. At smaller injection rates, the flow rate vs. applied pressure graph projects through the origin and indicates Darcian flow; non Darcian flow is evident at higher injection rates. Non-Darcian flow results in significantly lower calculated T values, which translates to smaller hydraulic aperture values. Further error in the calculated hydraulic aperture stems from uncertainty in the number of hydraulically active fractures in each test interval. This estimate can be inferred from borehole image and core logs, however, all of the fractures identified are not necessarily hydraulically active. This study proposes a method based on Reynolds number calculations aimed at improving confidence in the selection of the number of active fractures in each test interval.

A well deliverability model for non-Darcian flow in geothermal reservoirs

A new numerical algorithm is proposed to estimate the deliverability of a geothermal well. The mathematical model assumes a single-component fluid. The model accounts for both the phase change phenomenon and the non-Darcian behavior of the flow. The contemporaneous presence of both effects has to the authors’ knowledge never been fully taken into account. Provided that the numerical solution is based on a finite differences or finite volumes method, the numerical code we developed (called DNDmultiphase) can be plugged as an additional module into existing codes devoted to simulations of a whole reservoir (such as TOUGH2 or HYDROTHERM).

Non-Darcian flow and heat transfer along a permeable vertical surface with nonlinear density temperature variation

The non-Darcy free convection flow on a vertical flat plate embedded in a fluid-saturated porous medium in the presence of the lateral mass flux with prescribed constant surface temperature is considered. The coupled nonlinearities generated by the density variation with temperature, inertia, and viscous dissipation are included in the present study. In particular, we analyze a system of nonlinear ODEs describing self-similar solutions to the flow and heat transfer problem. These transformed equations are integrated numerically by a second-order finite difference scheme known as the Keller box method. Furthermore, some analytical results are provided to establish relationships between the physical invariants in the problem, and also to validate the numerical method. One of the important findings of our study is that an increase in the Rayleigh number increases the velocity boundary layer thickness, while the opposite is true for the thermal boundary layer thickness.

Solutions for Non-Darcian Flow to an Extended Well in Fractured Rock

We have investigated non-Darcian flow to a vertical fracture represented as an extended well using a linearization procedure and a finite difference method in this study. Approximate analytical solutions have been obtained with and without the consideration of fracture storage based on the linearization procedure. A numerical solution for such a non-Darcian flow case has also been obtained with a finite difference method. We have compared the numerical solution with the approximate analytical solutions obtained by the linearization method and the Boltzmann transform. The results indicate that the linearized solution agrees generally well with the numerical solution at late times, and underestimates the dimensionless drawdown at early times, no matter if the fracture storage is considered or not. When the fracture storage is excluded, the Boltzmann transform solution overestimates the dimensionless drawdown during the entire pumping period. The dimensionless drawdowns in the fracture with fracture storage for different values of dimensionless non-Darcian hydraulic conductivity β approach the same asymptotic value at early times. A larger β value results in a smaller dimensionless drawdown in both the fracture and the aquifer when the fracture storage is included. The dimensionless drawdown is approximately proportional to the square root of the dimensionless time at late times.

Non-Darcian flow to a well in a leaky aquifer using the Forchheimer equation

Non-Darcian flow to a well in a leaky aquifer was investigated using a finite difference method. Flow in the leaky aquifer is assumed to be non-Darcian and horizontal, while flow in the aquitard is assumed to be Darcian and vertical. The Forchheimer equation was employed to describe the non-Darcian flow in the aquifer. The finite difference solution was compared with the solution of Birpinar and Sen (2004). The latter overestimates the drawdown at early times and underestimates the drawdown at late times; also, the impact of βD on the drawdown depends on the value of BD, where βD is a dimensionless turbulent factor in the Forchheimer equation and BD is the dimensionless leakage parameter. The impact of leakage on drawdown is similar to that of Darcian flow. A sensitivity analysis indicated that the drawdown is very sensitive to the change in the dimensionless well radius rcD when BD is relatively large, while it is sensitive to the change in BD when BD is relatively small. The numerical solution has been applied to analyze the pumping test data in Chaj-Doab area of Pakistan. Birpinar ME, Sen Z (2004) Forchheimer groundwater flow law type curves for leaky aquifers. J Hydrol Eng 9(1):51–59