Darcian Velocity Research Articles

Effective unsaturated hydraulic conductivity for one‐dimensional structured heterogeneity

Effective hydraulic conductivities of unsaturated soils are defined for one‐dimensional structured heterogeneity. The heterogeneity is defined using homogeneous sublayers forming repeated unit cells of length L. The effective conductivity is defined as the steady downward Darcian velocity in an infinitely deep profile. The dependence of the hydraulic conductivity upon the pressure head is found by computing the average value of pressure head within the repeating unit cells for each effective conductivity. For a cell of length L approaching zero, the effective conductivity becomes the harmonic average of the individual sublayer hydraulic conductivities weighted according to the cell fraction each occupies. In that case, as the profile approaches saturation, the effective conductivity is the same as the well‐known result for flow through an “array” of saturated layers. For a large cell length L, the effective hydraulic conductivity approaches an arithmetic average of the pressure heads which develop in each sublayer. Examples are computed for finite cell lengths L for both binary and tertiary systems. The effective hydraulic conductivity functions for the finite cell lengths fall within the envelope formed by the two limiting cases for small and large cell lengths. For the binary system, all of the hydraulic conductivity functions fall between the envelopes formed by the two hydraulic conductivity functions for the individual sublayer materials.

Capillary Fringe and Unsaturated Flow in a Porous Reservoir Bank

Steady, 2D Darcian seepage from a zero-depth reservoir into a homogeneous porous bank is studied analytically. Using the Green-Ampt assumption for hydraulic conductivity as a function of pressure and the Vedernikov model for the tension-saturated zone, a free boundary problem with the capillary fringe spread along the bank surface is solved. Accurate direct calculations are made for the extent of the capillary rise along a vertical and horizontal bank. Unsaturated flow from a vertical phreatic surface into a bank with either an impermeable or isobaric vertical soil slope is analyzed in terms of the Philip model. Explicit expressions for the Darcian velocity components, stream function, and/or Kirchhoff potential are presented. The flow topology and characteristics are shown to depend strongly on the boundary condition at the soil surface.

Experimental studies on pathway patterns of secondary oil migration

The oil migration patterns in porous medium are discussed based on the results of diphasic expulsing experiments. In the experiments, dyed kerosene was used as expulsing phase to expulse water saturated in a glass cylinder filled with sorted glass beads. Two series of experiments were done: one with different initial oil column heights and the other with variable oil injection rates. Three secondary migration patterns, including stable displacement pattern, fingering pattern and stringing pattern may be recognized in each series of experiments. Two classic dimensionless parameters, the modified Bond number and the Capillary number, were used to construct a phase diagram that may characterize the patterns. A new method was proposed to calculate the Darcian velocity of moving kerosene in the cylinder when only buoyancy force acts. It is found that when Darcian velocity is used to get the two dimensionless parameters, the data from both series of experiments, as well as those obtained from literatures, may be reasonably drawn in a phase diagram to identify the three migration patterns.

Metal foams as compact high performance heat exchangers

Open-cell metal foams with an average cell diameter of 2.3 mm were manufactured from 6101-T6 aluminum alloy and were compressed and fashioned into compact heat exchangers measuring 40.0 mm × 40.0 mm × 2.0 mm high, possessing a surface area to volume ratio on the order of 10,000 m 2/m 3. They were placed into a forced convection arrangement using water as the coolant. Heat fluxes measured from the heater-foam interface ranged up to 688 kW m −2, which corresponded to Nusselt numbers up to 134 when calculated based on the heater-foam interface area of 1600 mm 2 and a Darcian coolant flow velocity of approximately 1.4 m/s. These experiments performed with water were scaled to estimate the heat exchangers’ performance when used with a 50% water–ethylene glycol solution, and were then compared to the performance of commercially available heat exchangers which were designed for the same heat transfer application. The heat exchangers were compared on the basis of required pumping power versus thermal resistance. The compressed open-cell aluminum foam heat exchangers generated thermal resistances that were two to three times lower than the best commercially available heat exchanger tested, while requiring the same pumping power.

Analytical Determination of Seeping Soil Slopes of a Constant Exit Gradient

Soil slopes satisfying the condition of a constant exit gradient (constant Darcian velocity) and the seepage face (isobaricity) condition are found by complex analysis. For an empty drainage trench, soil is infinitely deep and the flow domain is bounded by two branches of a phreatic surface and the trench contour. The problem is solved by conformal mapping of the Zhukovskii domain (half-plane) onto the hodograph domain (lune). The ultimately stable seepage face and the inflow rate are determined as functions of a specified exit gradient. With decreasing of the gradient the trench flattens. If the gradient is 1, the depth-width aspect ratio of the trench reaches 0.21. Similar conformal mapping of a lune in the hodograph plane on a half-strip in the complex potential plane is used to solve the problem of a phreatic surface flow from a soil channel, whose equipotential contour satisfies the condition of constant entrance gradient. For a dam slope, the flow domain is underlain by an impermeable bottom and the hodograph is a circular triangle with a unit-gradient circle as the slope image. The Polubarinova-Kochina method of analytical differential equations is modified to reconstruct the complex coordinate and the complex potential as a function of an auxiliary variable. The resulting slope of unit exit gradient has a depth-width ratio of 1.22.

Recharge and groundwater models: an overview

Recharge is a fundamental component of groundwater systems, and in groundwater-modeling exercises recharge is either measured and specified or estimated during model calibration. The most appropriate way to represent recharge in a groundwater model depends upon both physical factors and study objectives. Where the water table is close to the land surface, as in humid climates or regions with low topographic relief, a constant-head boundary condition is used. Conversely, where the water table is relatively deep, as in drier climates or regions with high relief, a specified-flux boundary condition is used. In most modeling applications, mixed-type conditions are more effective, or a combination of the different types can be used. The relative distribution of recharge can be estimated from water-level data only, but flux observations must be incorporated in order to estimate rates of recharge. Flux measurements are based on either Darcian velocities (e.g., stream baseflow) or seepage velocities (e.g., groundwater age). In order to estimate the effective porosity independently, both types of flux measurements must be available. Recharge is often estimated more efficiently when automated inverse techniques are used. Other important applications are the delineation of areas contributing recharge to wells and the estimation of paleorecharge rates using carbon-14.

Salinity distribution in paddy root zone under subsurface drainage

One-dimensional convective dispersion solute transport model was solved by Crank–Nicholson finite difference scheme and was applied over the flow domain of subsurface drained paddy fields laid with tile drain at different spacings. The flow domain was divided into a number of stream tubes to predict salinities at different distances from drain centre, at different depths from the ground surface and at different times after the initiation of the operation of the subsurface drainage system. The sub division of the flow domain into a number of stream tubes was done for two purposes, viz. (i) to enable estimation of pore water flow velocity more appropriately with respect to the flow area within a stream tube and (ii) to enable comparison of predicted salinities with the observed values which were available within some of the stream tubes. The initial and the boundary conditions in solving one dimensional equation were based on the field investigated salinity values. In the solute transport model there are two essential input parameters, viz. the pore water velocity and the dispersion coefficient. The pore water velocity was calculated by dividing the Darcian velocity by the drainable porosity. The dispersion coefficient was fitted by trial and error till the average absolute deviation between the predicted and the observed salinities became minimum.

Analysis of solute transport in a heterogeneous aquifer: the Krauthausen field experiment

A field-scale natural gradient tracer experiment with bromide, uranin and lithium was conducted in a heterogeneous aquifer at Krauthausen, Germany. The temporal and spatial evolution of the solute plumes was monitored over 398 days for bromide and 449 days for uranin and lithium. The spatial variability of basic aquifer parameters, hydraulic conductivity, and sorption parameters of the aquifer material was investigated in detail. Local Darcian velocities were measured using NH 4 Br 82 as a radioactive tracer in 33 observation wells. Vertical and horizontal correlation lengths of hydraulic properties were determined using variogram analysis. The magnitude of the local Darcian velocities was found to be lognormally distributed. In addition, the groundwater flow direction showed a clear trend in the vertical direction. The horizontal correlation length of the magnitude of Darcian velocity agreed with the correlation length of K estimated from grain size data. Batch experiments on aquifer sediments showed that sorption of uranin and lithium could be described by a Freundlich isotherm. The Freundlich n parameter of uranin sorption showed little variation with depth. The time evolution of the bromide plume was quantified in terms of spatial moments. The longitudinal effective dispersivity estimated from spatial moment analysis was within the range of calculated effective dispersivities, taking into account the uncertainty of estimates of the ln K statistics.

Hydrodynamics is a major determinant of streambed biofilm activity: From the sediment to the reach scale

The effects of hydrodynamics and organic matter on sediment biofilm esterase activity and [3H]thymidine incorporation were investigated at several depths in the streambed and riparian zone of the alpine stream Oberer Seebach (Austria). On the sediment scale, microbial activity increased with both upwelling and downwelling Darcian velocities of interstitial water and was minimal when the surface/subsurface water exchange decreased. As revealed by stepwise multiple regression analyses, sediment carbohydrates and chlorophyll a explained a considerable percentage (up to 97% in summer) of the variance of biofilm esterase activity and [3H]thymidine incorporation, whereas pore‐water temperature and concentrations of dissolved organic carbon (DOC) and dissolved oxygen (DO) were poor predictors of biofilm activity. DOC and DO fluxes, however, accounted for a considerable percentage of the variance in [3H]thymidine incorporation, which points toward the importance of flow in biofilm functioning. On the reach scale, the ratio of surface water travel length to the subsurface travel length, which is an index of the surface/subsurface water exchange frequency, determined average reach biofilm activity. Downstream routing along predominantly subsurface flow paths increased streambed esterase activity, whereas elevated surface travel lengths reduced esterase activity. Esterase activity and [3H]thymidine incorporation were also positively related to the streambed DOC retention efficiency, which underscores the role of biofilms in solute retention. These results highlight the coupling between streambed hydrodynamics and microbial activity and link sediment‐scale and reach‐scale processes.

Laboratory measurements anomalous 0.1–0.5 Hz streaming potential under geochemical changes: Implications for electrotelluric precursors to earthquakes

Streaming potentials resulting from flow of various salt solutions in rock were measured on saturated sediments (Fontainebleau sandstones). The streaming potential ΔV was found to be proportional to the driving pore pressure ΔP. Pulses of amplitude 15–40 mV in the frequency range of 0.1 to 0.5 Hz were observed when the conductivity of the injected water was decreased and the fluid flow rate was relatively low, corresponding to a Darcian velocity of 17 to 30 cm/h. The amplitudes of these pulses are 47% to 133% of the corresponding steady components of the ΔV values. Such geochemically induced effects may possibly be responsible for the frequency signals from 0.1 to 0.5 Hz that were sometimes observed before an earthquake.

Coupled surface—subsurface hydrologic model of a nuclear subsidence crater at the Nevada test site

A series of computer models describing overland flow, ponding and infiltration was dynamically linked to simulate the movement of water during seepage beneath a nuclear subsidence feature. The coupled surface-subsurface hydrologic model contains a terrain analysis module, an overland flow simulator using the kinematic wave equation and a numerical Richards' equation solver to simulate moisture migration within the vadose zone. Simulations resulted in successful approximations to measured surface and subsurface data. The surface water overland flow component was able to reach an agreement between observed and simulated water levels following a precipitation event. Long-term simulations underpredicted the volumetric water contents and depth of moisture fronts. The flux (Darcian velocity) at 10 m below the surface was temporally stable with a temporal mean value of 0.75 m year −1, which is in agreement with an earlier estimate of 0.60 m year −1. The mean flux at 5 m below the surface was 1.4 m year −1. Long-term simulations suggest that significant recharge does exist beneath surface collapse structures owing to periodic ponding.

Non-Newtonian Fluid Flow and Heat Transfer in a Porous Medium. 1st Report. Two-Dimensional Numerical Modeling of Fluid Flow.

Numerical calculations have been carried out to simulate comlex non-Newtonian fluid motion within a porous medium. A collection of square rods placed regularly in a region of infinite extent has been proposed as a two-dimensional numerical model of porous structure. The numerical simulation results obtained at a microscopic level with the power-law non-Newtonian model are processed to extract the macroscopic hydrodynamic characteristics in terms of the Darcian velocity and intrinsic average pressure. The macroscopic results in terms of the apparent permeability agree well with the results of the semi-empirical formula proposed by Christopher and Middleman.

Darcy velocity computations in the finite element method for multidimensional randomly heterogeneous porous media

For numerical modeling of transport of contaminants through porous media, an accurate determination of the velocity field is a prerequisite. Some previous schemes of obtaining the Darcian velocity field through numerical solution of the flow equation result in a physically inconsistent velocity distribution in a heterogeneous medium. A scheme that is consistent with the physics of velocity variation near material interfaces is examined and compared with previous schemes. Numerical simulations are used to demonstrate the capability and accuracy of the proposed scheme for randomly heterogeneous porous media.

A cubic‐spline technique to calculate nodal Darcian velocities in aquifers

A continuous and reliable velocity field is required for the simulation of mass and heat transport in an aquifer system. In this paper, a cubic‐spline technique is proposed to calculate nodal Darcian velocities after the flow field is obtained. With this technique, a continuous velocity field is produced, and mass conservation is satisfied to an acceptable degree. Its requirements for computer storage are very small, and the calculation speed is very fast; it can be implemented in parallel. A synthetic diffusion problem is used to examine the accuracy of this technique. The results agree very well with the analytical solution. Application to another problem shows that this technique can be applied to any heterogeneous and anisotropic aquifer. The results of the two sample problems are also compared with the Galerkin finite element method [Yeh, 1981] and the dual mesh method [Batu, 1984]. It is observed that its agreement with the Galerkin method is much better than the dual mesh method.

A mathematical model for dispersion with a moving front in groundwater

A new, non‐Fickian, constitutive equation is presented for the mass flux due to macroscopic dispersion in porous media. This constitutive equation contains the classical term of the diffusive type as well as a new term that may be viewed as an inertia term. Combination of the constitutive equation with the mass balance equation for a conservative contaminant yields a set of four first‐order partial differential equations in terms of the three components of the vector of dispersive mass flux and the concentration. For the case of longitudinal dispersion only, this system is reduced to a set of two first‐order partial differential equations in terms of the convective and dispersive mass fluxes as the two unknown functions. The sets of differential equations are hyperbolic and can be used to simulate and predict the movement of the front of a contaminant plume at finite velocity. An expression of the velocity of the front is presented in terms of the Darcian velocity and the ratio of the two dispersion coefficients that enter into the model. The effect of the non‐Fickian terms decreases for increasing times; the model reduces to the classical one as time increases. An exact solution is presented for the case of one‐dimensional flow and compared with the results of simulated column experiments.

Interpretation of temperature measurements from the Kaiko-Nankai cruise: Modeling of fluid flow in clam colonies

During the Kaiko-Nankai detailed submersible survey, numerous measurements of the temperature gradient inside the sediment were performed on the deepest active zone of fluid venting, which is situated on the anticline related to the frontal thrust, using the Ifremer T-Naut temperature probe operated from the submersibleNautile. We thus obtained the temperature structure below different types of clam colonies associated with fluid venting. We used the finite element method to model the thermal structure and fluid flow pattern of these vents and to determine the velocity of upward fluid flow through the colonies. On a biological basis, four types of clam colonies are defined. Each biological type has distinctive thermal characteristics and corresponds to a particular fluid flow pattern. Darcian flow velocity in the most active type of colony (type A) is of the order of 100 m/a. The total amount of fluid flowing through colonies in the studied area is estimated to be 200 m3 a−1 per metre width of subduction zone. Most of the flow is vented through type A colonies. This value is more than one order of magnitude too high to be compatible with the amount of water available from steady-state compaction of sediments in the whole wedge. Thermal arguments suggest that downwelling of seawater occurs around type A colonies and that seawater is then mixed with upcoming fluids at a depth of 1 or 2 metres. Furthermore, finite element modeling shows that a salinity difference of a few parts per mil between the upcoming fluids and seawater is sufficient to drive convection around the colonies. As water samples from a few vents indicate that the fluid source should actually be significantly less saline than seawater, we propose that the very high fluid flows measured are a consequence of the dilution of the fluid of deep origin with seawater by a factor of 5 to 10.

Groundwater munition residues and nitrate near Grand Island, Nebraska, U.S.A.

Munition residues from waste disposal on ordnance property have resulted in a defined plume of RDX contaminated groundwater stretching 6.5 km and underlying an area of 6.5 km2. A smaller plume of TNT was detected near the plant's boundary. The relative positions of the plumes combined with an historical review of total plant output of RDX and TNT indicates that RDX is much more persistent than TNT. The estimated RDX transport velocity of 0.5 m day−1 closely approximates the calculated Darcian velocity. The RDX plume sinks with recharge at a rate of about 0.5 m yr−1.Nitrate is associated primarily with adjacent upgradient landuse and is not related to plant manufacture of ammonium nitrate. The average δ15N of the Nitrate was about + 10% and strongly suggests that animal wastes are the predominant source.

An improved conductiometric standpipe technique for measuring interstitial seepage velocities

The measurement of the seepage velocity through river-bed gravels is required in many biological and chemical investigations concerned with chemical fluxes through the gravel interstices. An improved conductiometric technique is described which is based on an earlier electrolytic method. The apparatus is calibrated for a range of gravel types to yield a spatially-averaged seepage velocity rather than a Darcian bulk velocity. Problems of stratification of the electrolyte and the need for temperature monitoring have been eliminated. The apparatus is compact and, incorporating a micro-computer-based data aquisition system, is easy and rapid to use even in adverse field conditions.

On the computation of Darcian velocity and mass balance in the finite element modeling of groundwater flow

Darcian velocity has been conventionally calculated in the finite element modeling of groundwater flow by taking the derivatives of the computed pressure field. This results in discontinuities in the velocity field at nodal points and element boundaries. Discontinuities become enormous when the computed pressure field is far from a linear distribution. It is proposed in this paper that the finite element procedure that is used to simulate the pressure field or the moisture content field also be applied to Darcy's law with the derivatives of the computed pressure field as the load function. The problem of discontinuity is then eliminated, and the error of mass balance over the region of interest is much reduced. The reduction is from 23.8% to 2.2% by one numerical scheme and from 29.7% to –3.6% by another for a transient problem.