non-Darcian Flow Model Research Articles

Analysis of One-Dimensional Consolidation Considering Non-Darcian Flow Described by Non-Newtonian Index Incorporating Impeded Drainage Boundaries

The nonlinear flow law and soil boundaries greatly affect the dissipation process of soil consolidation. Thus, to study the impact of nonlinear flow under impeded drainage boundaries, the classical non-Darcian flow model described by non-Newtonian index was introduced. The numerical solutions are derived in detail by the finite difference method (FDM) for one-dimensional (1-D) consolidation incorporating the impeded boundaries, and the computer program is compiled. Then, comparing two analytical solutions based on Darcy’s law and a numerical case of Forchheimeer’s flow, the validity of the present method was verified. The numerical results indicate that there is a critical depth phenomenon for the non-Darcian flow incorporating impeded drainage boundaries. The excess pore water pressure of the soil below the critical depth dissipates more slowly than that of Darcy’s law, whereas the pore pressure of the soil above the critical depth dissipates more quickly than that of Darcy’s law. Moreover, considering that the non-Darcian flow with the non-Newtonian index will still delay the overall consolidation rate of the soft ground, the greater the nondimensional parameter I0 is, the more obvious the lagging phenomenon of the overall dissipation of pore pressure is.

Two-region Darcian and non-Darcian flow towards a well with exponentially decayed rates considering time-dependent critical radius

Field pumping tests are frequently encountered with variable pumping rates, and the high-velocity non-Darcian region near the pumping well is often time-dependent, leading to special drawdown behaviors that could not be soundly explained by the classical analytical well hydraulics models. Considering the commonly encountered situation of exponential decay in rates, a two-region Darcian and non-Darcian flow model with time-dependent critical radius, which separates the Darcian and non-Darcian regions, is constructed and the corresponding numerical method is applied to analyze the characteristics of aquifer drawdown and the critical radius. Simulation results revealed that the critical radius and drawdown may reach their local maximum value at the intermediate times, and then decrease for a period of time. A larger decay coefficient leads to an earlier deviation in the drawdown and critical radius. For the near-well non-Darcian region, the time-drawdown curves for different decay coefficients have a common critical point (i.e., the early deflection point), while the curves for different inertial force coefficients have different critical points and a larger inertial force coefficient results in a larger critical time. Besides, a larger inertial force coefficient may result in larger extreme points of drawdowns and experience a shorter time from the critical point to the extreme points. Considering the special deflection characteristics of drawdown, a new deflection point method is proposed to estimate the inertial force coefficient, storage coefficient, and hydraulic conductivity of the confined aquifer. The applicability and robustness of this new method were demonstrated by a synthetic pumping test.

Identification of non-Darcian flow effect in double-porosity fractured aquifer based on multi-well pumping test

Well testing in double-porosity fractured aquifers or oil and gas reservoirs is one of the long-lasting research problems in subsurface hydrology and petroleum engineering, where the double-porosity implies that the media of concern can be approximated as a two interrelated continuums (fracture network and rock matrix) with two distinctively different porosities. However, most of those studies in double-porosity fractured aquifers only concern Darcian flow in the fractured continuum. In this study, we will expand such studies from Darcian flow regime to non-Darcian flow regime, which is most likely to be the case for the field well testing site reported in this investigation. New solutions based on power law and a linearization approximation are obtained for hydraulic heads in Laplace domain and subsequently inverted to give spatiotemporal distributions in real-time domain. The non-Darcian flow model in single porosity media is obtained by setting the leakage coefficient (C) to 0 or the storage coefficient of the matrix (Sm) to 0, where C is the rate of fluid transfer between the fracture and rock matrix. Parameter analysis is conducted using dimensionless formats. A larger dimensionless quasi conductivity coefficient KqD, a larger drawdown in the early stage, and a smaller drawdown in the late stage. The dimensionless parameters CD and ϕ influence the drawdown in transitional state, and the values get larger, the drawdown in transitional state drop smaller. A pumping test conducted in Huangtun, Anhui province of China has been applied to test the non-Darcian flow effect in a double-porosity aquifer, in which a particle swarm optimization (PSO) algorithm will be applied here to seek the optimal hydraulic parameters. As the result shows, the observed drawdown data fit well with the new model in the pumping stage. The predicted recovery drawdown curve with the calculated parameters of the new model also performs well with the field drawdown data, which supports the applicability of the new model to interpret the field data.

Numerical investigation of natural convection in an inclined porous enclosure using non-Darcian flow model

PurposeA thoroughly literature review reveals that considerable attention have been given only to the two common cases, i.e. enclosure heated from below and heated from the side. For the inclined layer, on the other hand, the numbers of investigations are relatively small. Therefore, this paper aims to investigate the natural convective heat transfer in an inclined porous cavity using non-Darcian flow model, including the boundary surface and inertia effects.Design/methodology/approachThe flow characteristics have been assumed to be two-dimensional, steady, incompressible flow, whereas the properties of porous media have been considered to be homogeneous and isotropic properties solid matrix. The non-Darcian flow model, including the boundary surface and inertia effects, has been numerically solved using finite difference method.FindingsThe initiation of multicellular flow and counter-rotating cell are strongly dependent on the aspect ratio A and the inclination angle θ. The orientation of the porous cavity, for a given Ra*, Fs/Pr* and A, has a significant effect on the heat transfer rate. The results also indicated that A has a dominant effect on the Nusselt number. The Nusselt number is strongly dependent on the Ra*, Fs/Pr*, A and θ. Therefore, operating conditions and geometry of the porous enclosure are required to be properly designed to achieve the desired objective.Originality/valueThe developed model can reveal the non-Darcian effects on the fluid flow and heat transfer in inclined porous media under natural convection case.

Combined role of leaky and non-Darcian effects on the flow to a pumping well with a non-uniform flux well-face boundary

In this study, a non-Darcian flow model was developed for a constant-rate test of a partially penetrating well with a non-uniform flux boundary in a leaky confined aquifer. The Izbash equation was applied to describe non-Darcian flow in the radial direction. Both analytical and numerical methods were employed to solve this model. It was found that the analytical solution with linearization was adequate only when non-Darcian effects were relatively small. With the finite difference method, the radial and vertical fluxes along the well screen were analyzed under four different cases involving leaky and non-Darcian effects. The leaky and non-Darcian effects on the drawdowns with non-uniform flux (NUF) boundary were compared with those with uniform flux (UF) boundary. The results indicate that both leaky and non-Darcian effects can reduce the fluxes along the well screen, while non-Darcian effects can diminish the leakage-induced differences of flux and drawdown between the two well screen ends. Non-Darcian effects can reduce the differences between UF and NUF drawdowns at any elevation with the greatest reduction difference at the elevation of the top end of the screen. The UF solution can replace the NUF solution when the distance is far from the well, as this distance is smallest at the elevation of the screen midpoint and can be distinctly reduced by non-Darcian effects. In general, the results are most sensitive to the power index n, moderately sensitive to the other aquifer parameters and least sensitive to the parameters of the aquitard.

Catastrophic Evolution of Water Inrush from A Water-Rich Fault in Front of Roadway Development: A Case Study of The Hongcai Coal Mine

A fault can be both the inrush channel and the water source. In order to better understand how such a mine disaster develops, a fluid–solid coupling analysis model and a Forchheimer, nonlinear, non-Darcian flow model were used to study an inrush in the Hongcai coal mine. Numerical simulation was used to reconstruct the dynamic process of the inrush. Then the coupled evolution of the stress, displacement, and seepage fields were analyzed and the precursor information characteristics were summarized. The results showed that the inrush was the result of excavation disturbance and hydraulic pressure in the fault fracture zone. Before the inrush occurs, the stress of the surrounding rock drops after continually increasing, the displacement of the surrounding rock increases sharply after a stable build-up, and the gradually increasing pore water pressure of the surrounding rock decreases rapidly just before the inrush. During this process, the multi-field information in the aquiclude continuously changes. During the inrush, groundwater flows into the roadway from top to bottom along the fault fracture zone. The pressure in the runoff path continuously decreases and the flow velocity generally increases. However, when the water initially enters the channel and roadway, the flow velocity first decreases due to the change in flow direction, and then increases again, and the velocity and pressure gradually stabilizes after the water finally enters the roadway. These results help explain how flooding caused by a water-rich fault inrush evolves and provide a scientific basis for determining the characteristics of appropriate inrush precursors.

Non-Darcian Flow to a Partially Penetrating Pumping Well in a Leaky Aquifer Considering the Aquitard–Aquifer Interface Flow

AbstractIn this study, non-Darcian flow to a partially penetrating pumping well in a leaky aquifer was investigated. The aquifer system is composed of a main aquifer with an aquitard bounded on the top of the aquifer. The storage of the aquitard was considered in this study, which is the main contribution of this study. The horizontal flow in the main aquifer is assumed to be non-Darcian, while both the vertical flows in the main aquifer and the aquitard were assumed to be Darcian due to the relative low velocities. The non-Darcian flow was described by the Izbash equation. A linearization procedure associated with Laplace transform and separate variable method were used to solve the non-Darcian flow model. Semianalytical solutions (Laplace-domain solutions) were obtained then inverted to time domain by using the Stehfest method. The results indicated that the power index n in the Izbash equation results in a smaller drawdown at late times, the flow approaches quasi steady state earlier, and the leakage h...

Non-Darcian flow to a partially penetrating well in a confined aquifer with a finite-thickness skin

Non-Darcian flow to a partially penetrating well in a confined aquifer with a finite-thickness skin was investigated. The Izbash equation is used to describe the non-Darcian flow in the horizontal direction, and the vertical flow is described as Darcian. The solution for the newly developed non-Darcian flow model can be obtained by applying the linearization procedure in conjunction with the Laplace transform and the finite Fourier cosine transform. The flow model combines the effects of the non-Darcian flow, partial penetration of the well, and the finite thickness of the well skin. The results show that the depression cone spread is larger for the Darcian flow than for the non-Darcian flow. The drawdowns within the skin zone for a fully penetrating well are smaller than those for the partially penetrating well. The skin type and skin thickness have great impact on the drawdown in the skin zone, while they have little influence on drawdown in the formation zone. The sensitivity analysis indicates that the drawdown in the formation zone is sensitive to the power index (n), the length of well screen (w), the apparent radial hydraulic conductivity of the formation zone (Kr2), and the specific storage of the formation zone (Ss2) at early times, and it is very sensitive to the parameters n, w and Kr2 at late times, especially to n, while it is not sensitive to the skin thickness (rs).

Non-Darcian flow toward a larger-diameter partially penetrating well in a confined aquifer

In this study, non-Darcian flow to a larger-diameter partially penetrating well in a confined aquifer was investigated. The flow in the horizontal direction was assumed to be non-Darcian and described by the Izbash equation, and the flow in the vertical direction was assumed to be Darcian. A linearization procedure was used to approximate the nonlinear governing equation. The Laplace transform associated with the finite cosine Fourier transform was used to solve such non-Darcian flow model. Both the drawdowns inside the well and in the aquifer were analyzed under different conditions. The results indicated that the drawdowns inside the well were generally the same at early times under different conditions, and the features of the drawdowns inside the well at late times were similar to those of the drawdowns in the aquifer. The drawdown in the aquifer for the non-Darcian flow case was larger at early times and smaller at late times than their counterparts of Darcian flow case. The drawdowns for a partially penetrating well were the same as those of a fully penetrating well at early times, and were larger than those for a fully penetrating well at late times. A longer well screen resulted in a smaller drawdown in the aquifer at late times. A larger power index n in the Izbash equation resulted in a larger drawdown in the aquifer at early times and led to a smaller drawdown in the aquifer at late times. A larger well radius led to a smaller drawdown at early times, but it had little impact on the drawdown at late times. The wellbore storage effect disappears earlier when n is larger.

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.

Correlation of the two-phase flow coefficients of porous media with the rheology of shear-thinning fluids

The macroscopic equations of the two-phase flow in porous media are generalized for the case that the displacing non-wetting phase (NWP) is a shear-thinning fluid, and a numerical scheme of inverse modeling is developed to estimate simultaneously the capillary pressure, P c( S nw), non-wetting phase, k rnw( S nw), and wetting phase, k rw( S nw), relative permeability curves, from unsteady-state experiments. The parameter estimation procedure is demonstrated by using transient experimental data of the immiscible displacement of a Newtonian wetting phase (WP) by a shear-thinning NWP from a glass-etched planar pore network. Scaling laws are developed for the width of the frontal region, by extending concepts of the gradient invasion percolation to the rate-controlled displacement of a Newtonian fluid by a power-law fluid. These scaling laws are coupled with the macroscopic fractional flow equations in order to express the width of the frontal region and macroscopic parameters of P c( S nw), k rnw( S nw) and k rw( S nw) as functions of the capillary number ( Ca), in terms of universal scaling exponents. Two parameter estimation procedures are adopted: (i) in the first approach, the NWP is treated as a mixed Meter-and-power-law fluid and a non-Darcian flow model is used; and (ii) in the second approach, the NWP is treated as a pseudo-Newtonian fluid and classical Darcy law with a constant apparent viscosity are used. The variation of the front width with Ca follows a power law, the exponent of which agrees satisfactorily with that predicted by scaling relationships. The estimated P c( S nw) and k rw( S nw) are increasing functions of Ca. The estimated P c( S nw) is a strongly increasing function of Ca, even at high values of this parameter, and almost independent on the flow model considered. The shape of k rnw( S nw) changes from concave at low Ca values, where the width of the frontal region is large, to convex at increasing Ca values, where the front width is small and the displacement is dominated by viscous forces at gradually larger scales. The variation of the parameters of P c( S nw), k rnw( S nw), k rw( S nw) with Ca agrees semi-quantitatively with the scaling relations only in the first approach, where the non-Darcian flow model is used.

A FULL 3-DIMENSIONAL ADAPTIVE FINITE VOLUME SCHEME FOR TRANSPORT AND PHASE-CHANGE PROCESSES, PART II: APPLICATION TO CRYSTAL GROWTH

Application of the full three-dimensional (3-D) adaptive finite volume scheme as presented in a companion paper (Part I) to hydrothermal and two variants of Czochralski (Cz) crystal growth processes is presented. A composite fluid-superposed porous layer formulation with non-Darcian flow model is employed to model the hydrothermal system to demonstrate the versatility of this formulation. The unified formulation is then employed to model the Cz process, which is the main focus here. The model takes into account the irregular geometries, nonplanar and moving interfaces, and multiple driving forces that typically characterize a generic class of crystal growth processes. Simulation of the low and high pressure Cz growth of silicon and indium phosphide, respectively, reveal complex three dimensionality in flow, heat transfer, and interface dynamics, which have a profound effect on other quality affecting phenomena such as defect generation, segregation, and so on.

Fully developed laminar mixed convection through a vertical annular duct filled with porous media

The fully developed laminar mixed convection through a vertical annular duct embedded in a porous medium has been solved by using the non-Darcian flow model, where thermal boundary conditions on inner and outer walls are prescribed as isothermal-isothermal, isothermal-isoflux, and isoflux-isothermal, separately. The analytical solution has been derived to obtain velocity and temperature profiles, mass flow rate, wall friction factor and heat carried out by fluid. Finally, the parametric zones for flow characteristics of velocity distribution with the upward or downward flow are demonstrated.

Transient Natural Convection on a Vertical Flat Plate Embedded in a High-Porosity Medium

This paper provides an analysis of the non-Darcian effect on transient natural convection of a vertical flat plate embedded in a high-porosity medium. The plate surface is either maintained at an uniform wall temperature (UWT), or subjected to an uniform heat flux (UHF), and convective, boundary and inertia effects are considered. The local volume-averaged principles and certain empirical relations have been utilized to establish the governing equations. The coupled nonlinear partial differential equations are solved with a numerical integration technique using a cubic spline. Along with transient mean and local Nusselt numbers at the plate, representative transient velocity and temperature profiles are presented. Both effects for non-Darcian flow model are shown to be more pronounced in high-porosity medium and, hence, reduce the heat transfer rate.