Darcian Velocity Research Articles

New Criteria to Estimate Local Thermal Nonequilibrium Conditions for Heat Transport in Porous Aquifers

AbstractA fundamental assumption in numerous studies of heat transfer in porous media is local thermal equilibrium (LTE), which assumes that the temperature of the porous media at the fluid and solid interface is in instantaneous equilibrium. Although significant efforts have been made to quantify the occurrence and consequences of local thermal nonequilibrium (LTNE), where the temperatures of the fluid and adjacent solid phases differ, there is no simple expression for quantifying the occurrence and effects of local thermal disequilibrium. Using a numerical model combining LTE and LTNE models, we develop here two simple general criteria based on Darcian velocities (q) and particle sizes (dp) of porous media for determining when LTNE effects occur (denoted as g(dp, q)) and when they become significant (denoted as f(dp, q)). Results show that using an LTE model can result in an underestimation of effective thermal diffusivity and the unaffected Darcian velocities when g(dp, q) > 0. It is possible that using the LTE model can result in an underestimation of the effective thermal diffusivity by more than 200 times within Darcian velocities ranging from 0 to 60 m/d. In the case of g(dp, q) < 0, the use of the LTE model can result in an overestimation of effective thermal diffusivity and Darcian velocities. The performances of the newly developed general criteria are demonstrated using three typical data sets and corresponding numerical models. These data sets include new heat tracer tests conducted in the laboratory and the field, as well as temperature‐time series collected in streambed sediments from a previous study by Shanafield et al. (2012, https://doi.org/10.5194/hessd‐9‐4305‐2012). The potential LTNE effects should be considered when using heat as a tracer to characterize flow and heat transport in porous media in the presence of Darcian velocities less than 2 m/d and particle sizes larger than 10 mm.

Analytical and HYDRUS solutions for exfiltration through inclined seepage faces

Steady, phreatic flows in wedge-shaped aquifers subtended by a tilted bedrock and discharged through a hillslope seepage face are studied by 2-D and 1-D analytical models for fully saturated and a FEM numerical model for saturated-unsaturated flows. The curvilinear triangle in the hodograph domain is conformally mapped onto a reference half-plane where the Hilbert problem is solved for two holomorphic functions. Free boundaries and sizes of seepage face segments are found for given flow rates of the incident flows seeping downhills. The approximate (Dupuit-Forchheimer) water table is conjugated with a saturated triangle where the exact 1-D solution gives a constant Darcian velocity as a function of the angles of the soil wedge. HYDRUS-2D model gives the distributions of the pressure head, volumetric moisture content, isotachs, vector-fields of velocity and other characteristics of seepage described by the Richards-Richardson equation. Applications to stability of exfiltrating zones near tailwaters of earth dams are discussed.

Turbulence and added drag over acoustic liners

We present pore-resolved compressible direct numerical simulations of turbulent flows grazing over perforated plates, that closely resemble the acoustic liners found on aircraft engines. Our direct numerical simulations explore a large parameter space including the effects of porosity, thickness and viscous-scaled diameter of the perforated plates, at friction Reynolds numbers $\textit {Re}_\tau = 500\unicode{x2013}2000$ , which allows us to develop a robust theory for estimating the added drag induced by acoustic liners. We find that acoustic liners can be regarded as porous surfaces with a wall-normal permeability and that the relevant length scale characterizing their added drag is the inverse of the wall-normal Forchheimer coefficient. Unlike other types of porous surfaces featuring Darcian velocities inside the pores, the flow inside the orifices of acoustic liners is fully turbulent, with a magnitude of the wall-normal velocity fluctuations comparable to the peak in the near-wall cycle. We provide clear evidence of a fully rough regime for acoustic liners, also confirmed by the increasing relevance of pressure drag. Once the fully rough asymptote is reached, canonical acoustic liners provide an added drag comparable to that of sand-grain roughness with viscous-scaled height matching the inverse of the viscous-scaled Forchheimer permeability of the plate.

Modelling of Droplet Capture in an Open-Cell Metal Foam at the Pore and Macroscopic Scales

Open-cell metal foams are often used in applications where particulate and/or droplet capture is important. Here a Computational Fluid Dynamics (CFD) modelling approach is described which models the metal foam at both the pore-scale and the macroscopic scale. At the pore-scale, the detailed internal geometry of the foam is included and the flow field and droplet tracking and capture is modelled explicitly. At this scale, a coefficient is found for each metal foam that relates the distance a droplet can freely travel through the foam to both the droplet diameter and the Darcian velocity in the porous medium. Then, at the macroscopic scale, the coefficient from the pore-scale droplet capture simulations is used in a novel stochastic particle extinction model. Here, the droplets travel through a porous zone and are removed from the model, the probability of which is determined by the coefficient from the pore-scale modelling. A test case is described in which the macroscopic model is verified against the pore-scale model with acceptable levels of accuracy.

The Saint‐Venant type isoperimetric inequalities for assessing saturated water storage in lacunary shallow perched aquifers

AbstractThe Dupuit‐Forchheimer (DF) approximation for unconfined groundwater flows is reduced to 2D Poisson's equation (PE), the right hand side of which involves an intensive evapotranspiration (ET) rate. A shallow water table dips inward from a constant piezometric head boundary (closed curve) such that a “dry gap,” demarcated by another closed curve (unknown front), may emerge. Apriori estimates of the volume of the saturated zone and of the “dry gap” area are important for water resources management in drylands. We use the results from the theory of linear elasticity, torsion of elastic bars, for which PE is solved for the Prandtl function. Using the Poincaré metrics with the Gaussian constant curvature , isoperimetric inequalities are obtained for steady‐state DF flows. Conformal moments and confromal radii of the domains and 2‐D Hardy's type inequalities in domains, modeling the Saint Venant bar torsion problem, are involved in the obtained estimates. The studied boundary value problems (BVPs) are nonlinear for ET rates depending on the depth of the water table. In the DF model, the vadose zone (VZ) is considered as a “distributed sink” (similar to a standard “distributed source,” which models recharge to the water table in humid climates). The analytical VZ is collated with one obtained from a numerical solution to BVP for Richards’ equation in a 3‐D saturated‐unsaturated flow. BVPs for Richards’ equation in cylindrical domains, solved by HYDRUS software, give the pressure head, moisture content, Darcian velocity fields, and streamlines.

Finite Element Modeling of Diffusion in Fractured Porous Media by Using Hierarchical Material Properties

AbstractFractured porous media challenge modeling approaches due to high computational costs and excessive mesh refinement imposed by the extreme scale variability of fractures and the heterogeneity of the surrounding porous rock. To overcome such difficulties, we utilize the hierarchical finite element method (Hi‐FEM) that has been developed previously to simulate the electrical potential distribution in complex geologic environments. The method employs the hierarchical basis functions in classical finite element analysis to enable representation of material properties on each dimensional component of a given 3D unstructured finite element, thereby inherently allowing for interactions at the boundary between fracture and a host rock. In this study, we extend its application to transient fluid flow and heat conduction in the Laplace domain. Time‐domain flow solutions are obtained by numerical inverse Laplace transform. We evaluate the accuracy of the method using different flow models and demonstrate its robustness for large‐scale, rock mass models featuring complex fracture networks. Moreover, for the computation of nodal Darcian velocity fields in fractured porous media where the fractures are represented as 2D features, a new approach that employs the Yeh's Galerkin model for both volume and facet elements is proposed. Results show that Hi‐FEM can produce accurate flow solutions for fractured porous media without any need of coupling or transfer mechanism while still being computationally economical and numerically robust, even for large‐scale simulations.

Analytical traveling-wave solutions and HYDRUS modeling of wet wedges propagating into dry soils: Barenblatt's regime for Boussinesq's equation generalized

The classical Barenblatt solution of an initial-boundary value problem (IBVP) to the parabolic Boussinesq equation, which gives a rectangular triangle of full saturation, propagating from a reservoir into an adjacent porous bank with a vertical slope, is shown to coincide with a solution of IBVP to the elliptic Laplace equation with a phreatic surface along which both isobaricity and kinematic conditions are exactly met. For an arbitrary bank slope, a saturated wedge, which propagates (translates) into dry soil, is also explicitly found. The analytical solutions favorably compare with the results of HYDRUS-2D modeling, i.e., with the FEM solutions of the same IBVPs to the Richards equation. Applications to geotechnical engineering of dykes subject to the impact of flash floods are discussed by comparisons of phreatic lines, loci of the fronts, isobars, equipotential contours, vector fields of Darcian velocity, isotachs, and streamlines in the three models. For example, it is shown that a rapid drawup of the reservoir level induces hydraulic gradients, which may cause seepage-induced erosion of the porous medium, in particular, lessivage.

Water table rise in urban shallow aquifer with vertically-heterogeneous soils: Girinskii’s potential revisited

ABSTRACT Seepage through an aquifer, the hydraulic conductivity of which varies vertically, is studied using the Dupuit-Forchheimer approximation (Girinskii’s potential) and numerically by FDM-MODFLOW and FEM-HYDRUS-2D. In urban water hydrology, the effect of compaction of the top stratum of an aquifer on the flow rate and the position of the water table (characterized by an integral quantity of the saturated/dry area) is analysed. The area of the saturated zone is evaluated as a function of the conductivity ratio of the two strata, and their thicknesses. Conductivity varying linearly and exponentially is also analytically studied. Boundary-value problems are solved for a linear (nonlinear) ordinary differential equation if the evaporation rate is constant (decreasing exponentially with depth). The MODFLOW and HYDRUS-2D simulations give consistent fields of the piezometric head, pressure head, and Darcian velocity, as well as the water table position, which may have a global minimum (watershed).

Analysis of gravel back-filled borehole heat exchanger in karst fractured limestone aquifer at local scale

In designing and sizing of borehole thermal energy system, natural groundwater movement and temperature driven flow have a great importance on the borehole heat exchanger efficiency. The efficiency of double U – tube arrangement in gravel – backfilled borehole installed in a fractured limestone aquifer has been analyzed by means of three – dimensional numerical simulations. The numerical model is representative of 1 m deep of gravel back – filled borehole surrounded by the fractured aquifer. Several simulations have been carried out in order to evaluate the effect of aquifer parameters and boundary conditions on heat exchange efficiency by varying the mean temperature within the double U - tube. The fractured limestone aquifer of the industrial area of Bari (Italy) has been chosen as field site in order to identify the aquifer parameter range and the respective combinations. The results highlight that borehole thermal energy system efficiency is strictly dependent on aquifer transmissivity and groundwater Darcian velocity. The conducted analysis shows that, under lower Darcian groundwater flow and lower aquifer transmissivity, heat transfer efficiency increases at least by 25% compared to stagnant water, whereas heat transfer in the aquifer is governed by heat conduction. The increase of aquifer transmissivity induces the thermosiphon effect enhancing heat transfer processes both in the gravel back-filled borehole and aquifer. At higher values of groundwater Darcian velocity (> 0.1 m/d) advection due to groundwater flow is not negligible and mixed with free convection enhancing heat transfer further. Based on the results, discussion on the performance and environmental constraint of gravel back – filled borehole at field site has been presented.

An improved VOF-DEM model for soil-water interaction with particle size scaling

This study presents an improved VOF-DEM model where the Darcian velocity and a compound variable are treated as unknowns in the pressure-velocity calculation procedure such that the use of interpolated porosity at cell faces is minimised and stability is ensured even if the porosity field is not smooth or even ragged. A higher-order porosity estimation method is also used such that the porosity and interaction force are evaluated correctly when the CFD cell size is of the same order as the DEM particle size. Additionally, a particle size scaling technique is proposed to let the DEM particle size different from the real soil particle size and soil-water interaction forces are the same as when the real soil particle size is used. This is achieved by modifying the calculation of drag force. The solution scheme is verified in two case where analytical solutions exist. Particle size scaling technique is also used and tested in permeability tests and wave interaction with porous structure. Subsequently, the settling and collision of particles in water, dambreak of soil-water mixture and submerged landslides are simulated. With the present improvements and the particle size scaling, the capability of the VOF-DEM is extended in soil-water interaction problems.

Phreatic seepage flow through an earth dam with an impeding strip

New mathematical models are developed and corresponding boundary value problems are analytically and numerically solved for Darcian flows in earth (rock)–filled dams, which have a vertical impermeable barrier on the downstream slope. For saturated flow, a 2-D potential model considers a free boundary problem to Laplace’s equation with a traveling-wave phreatic line generated by a linear drawup of a water level in the dam reservoir. The barrier re-directs seepage from purely horizontal (a seepage face outlet) to purely vertical (a no-flow boundary). An alternative model is also used for a hydraulic approximation of a 3-D steady flow when the barrier is only a partial obstruction to seepage. The Poisson equation is solved with respect to Strack’s potential, which predicts the position of the phreatic surface and hydraulic gradient in the dam body. Simulations with HYDRUS, a FEM-code for solving Richards’ PDE, i.e., saturated-unsaturated flows without free boundaries, are carried out for both 2-D and 3-D regimes in rectangular and hexagonal domains. The Barenblatt and Kalashnikov closed-form analytical solutions in non-capillarity soils are compared with the HYDRUS results. Analytical and numerical solutions match well when soil capillarity is minor. The found distributions of the Darcian velocity, the pore pressure, and total hydraulic heads in the vicinity of the barrier corroborate serious concerns about a high risk to the structural stability of the dam due to seepage. The modeling results are related to a “forensic” review of the recent collapse of the spillway of the Oroville Dam, CA, USA.

Free-surface laminar flow of a Herschel–Bulkley fluid over an inclined porous bed

The current work presents a mathematical solution for the flow of a Herschel--Bulkley fluid over a porous bed. To obtain the velocity profile, a dimensional analysis is firstly conducted for the Darcian-like flow, showing that the ratio between pore and flow scales impose the domain of validity for the Beaver-like kinematic boundary condition. A friction velocity dependent only on fluid and porous medium properties was found to identify the scale of the Darcian velocity. After applying a modified kinematic boundary condition between free flow and Darcian-like flow, free-surface flow velocity profile was obtained, which can effectively predicts non-porous solutions for less rheological complex fluids. Dimensional analysis was then performed for the velocity profile solution, which allowed to identify the effect of the porous bed on flow properties. Experimental results from the literature were employed to identify the necessary conditions for yield stress fluids to have Darcian-like flow in natural open-channel flows. Sensitivity analysis pointed out that the porous medium permeability and non-Newtonian fluid parameters have greater influence on the velocity profile for pseudoplastic fluids than for dilatant ones.

Modelling of 2-D seepage from aquifer towards stream via clogged bed: The toth-trefftz legacy conjugated

Baseflow-type interaction between a river and adjacent/subjacent aquifer across a thin clogging layer of fine sediments controls the dynamics of surface-pore water resources, quality of both waters, seepage induced erosion of the river bed and other hydrological phenomena. Transient 2D phreatic flow from an unconfined aquifer into a river with a thin low-permeable cake (hydraulic skin) is approximated by a sequence of steady states, each of which assumes the water table to be horizontal. The scalar and vector fields of piezometric head, stream function and Darcian velocity are found from analytical solution of the Dirichlet and Robin (linear combination of the velocity potential and its normal derivative) boundary value problems for the piezometric head (harmonic function). The time-shrinking Tothian “unit basins” are a half-strip, half-plane or rectangle. Stream banks are assumed to be horizontal or vertical segments. Cross-flow from the aquifer into the stream is controlled by the aquifer-skin conductivity ratio and the stages of the river and adjacent aquifer. The head and cross-flux on the interface (Robin's boundary) is shown to vary along this line and therefore even for vertical river banks the Dupuit–Forchheimer approximation is not strictly valid. Numerical simulations in HYDRUS2D are reasonably close to the analytical results. Early-stage drawdown of a rectangular cake due to a sudden drop of the water level on the river side and formation of a seepage face is analysed with potential applications to stability of earth dams.

New finite volume multiscale finite element model for simultaneously solving groundwater flow and darcian velocity fields in porous media

This paper proposes a new finite volume multiscale finite element model (NFVMSFEM) for groundwater flow and velocity simulation in porous media. The main goal of this method is to simultaneously obtain the heads and continuous x,y direction Darcian velocities, thereby simplifying the simulation process and improving the efficiency. This goal is accomplished by introducing a novel velocity matrix in the control volume boundary flux estimation of the finite volume multiscale finite element method (FVMSFEM) framework. The velocity matrix allows for the NFVMSFEM to directly represent continuous fine-scale velocities by coarse-scale heads, thus merging the two kinds of unknowns into one. Through a modified Yeh’s Galerkin model, the velocity matrix is constructed in each local coarse element before the head computation; thus, the computational cost is very little. Furthermore, the NFVMSFEM inherits the advantageous features of the FVMSFEM, which ensures the local conservation of heads and further improves the efficiency. The results of the comparison between the NFVMSFEM and certain classical methods for head and velocity computation demonstrate its effectiveness and high efficiency.

Dipolic Flows Relevant to Aquifer Storage and Recovery: Strack’s Sink Solution Revisited

Steady, 2-,3-D Darcian flows generated by a dipole (a pair of horizontal or vertical injection–abstraction wells closely placed one above another), with circulation of fresh water inside an interface confined lens or “bubble” underneath an impermeable caprock, surrounded by a static saline groundwater, are analytically studied. For 2-D dipole, the complex potential domain is a plane with a horizontal cut. This domain is conformally mapped onto a reference half-plane where the Keldysh–Sedov formula is used to obtain the complex physical coordinates. Explicit closed-form expressions for the vase-shaped interface, flow net, isohypses, magnitudes of the Darcian velocity and Riesenkampf’s resultant force are obtained, depending on the dipole moment, its position with respect to the caprock, and the ratio of densities of the two fluids. It is shown that for sufficiently small injection-pumping rates the fresh water “vase” separates from the caprock and becomes a circle, inside which streamlines are Newtons’ loops of monodiametral degenerate hyperbolae (cubics). Two numerical codes, MT3DMS and SEAWAT, are also used for delineation of isoconcentric lines, which qualitatively corroborate the analytical solutions in delineation of the “bubble” in the part where the sharp interface model predicts stable free boundaries and evidencing “dimples” on the boundary of the “bubble” where the saline water overlies the fresh one. For 3-D dipole not bounded by the caprock, the analytical fresh water “bubble” is a sphere and solution follows, mutatis mutandis, from the textbook formulae for flow of an ideal fluid past an impermeable sphere. The Stokes streamlines inside the sphere are sixtics; isotachs are plotted in an axial section. Stability of the soil matrix near the wells is also discussed.

A domain decomposed finite element method for solving Darcian velocity in heterogeneous porous media

This paper proposes a domain decomposed finite element method (DDFEM) for groundwater flow velocity simulation, which can effectively and efficiently deal with arbitrarily oriented or intersected material interfaces in heterogeneous porous media. The main idea of this problem is to employ domain decompose technique to break the velocity problem down to subproblems by material interfaces, thereby achieving high accuracy at interface nodes and improving the velocity computation efficiency. By solving the global flow field by FEM before velocity computation, this method can capture the global information of the whole study region so as to ensure the subproblem solution accuracy by Yeh’s model. After one subproblem has been solved, the DDFEM employs refraction law to obtain the Dirichlet boundary velocities of the adjacent subdomains. Numerical examples have been done to demonstrate the applicability and accuracy of the proposed method. Comparison between the DDFEM and two classical methods shows that the DDFEM can indeed save much computational cost, while achieving high accuracy.

Analytical solution for tension‐saturated and unsaturated flow from wicking porous pipes in subsurface irrigation: The Kornev‐Philip legacies revisited

AbstractThe Russian engineer Kornev in his 1935 book raised perspectives of subsurface “negative pressure” irrigation, which have been overlooked in modern soil science. Kornev's autoirrigation utilizes wicking of a vacuumed water from a porous pipe into a dry adjacent soil. We link Kornev's technology with a slightly modified Philip (1984)'s analytical solutions for unsaturated flow from a 2‐D cylindrical pipe in an infinite domain. Two Darcian flows are considered and connected through continuity of pressure along the pipe‐soil contact. The first fragment is a thin porous pipe wall in which water seeps at tension saturation; the hydraulic head is a harmonic function varying purely radially across the wall. The Thiem solution in this fragment gives the boundary condition for azimuthally varying suction pressure in the second fragment, ambient soil, making the exterior of the pipe. The constant head, rather than Philip's isobaricity boundary condition, along the external wall slightly modifies Philip's formulae for the Kirchhoff potential and pressure head in the soil fragment. Flow characteristics (magnitudes of the Darcian velocity, total flow rate, and flow net) are explicitly expressed through series of Macdonald's functions. For a given pipe's external diameter, wall thickness, position of the pipe above a free water datum in the supply tank, saturated conductivities of the wall and soil, and soil's sorptive number, a nonlinear equation with respect to the total discharge from the pipe is obtained and solved by a computer algebra routine. Efficiency of irrigation is evaluated by computation of the moisture content within selected zones surrounding the porous pipe.

Darcian flow under/through a leaky cutoff wall: Terzaghi–Anderson's seepage problem revisited

SummaryAn analytical solution is obtained for 2‐D steady Darcian flow under and through a cutoff wall partially obstructing a homogeneous isotropic foundation of a dam. The wall is leaky; that is, flow across it depends on the ratio of hydraulic conductivity of the wall and the wall thickness that results in the third‐type (Robin) boundary condition along the wall, as compared with the Terzaghi problem for an impermeable wall. The Laplace equation for the hydraulic head is meshlessly solved in a non‐standard flow tube. A Fredholm equation of the second kind is obtained for the intensity of leakage across the wall. The equation is tackled numerically, by adjusted successive iterations. Flow characteristics (total Darcian discharge and its components through the wall and the window between the wall top and horizontal bedrock, stream function, head distribution, and Darcian velocity along the wall and tailwater bed) are obtained for various conductivity ratios, head drops across the structure, thicknesses of the foundation, and the degree of its blockage by the wall. Comparisons with the Terzaghi limit of an impermeable wall show that for common wall materials and thicknesses, the leakage may constitute tens of percent of the discharge under the dam. The through‐flow hydraulic gradients on a vertical wall face (Robin's boundary condition) as well as the exit gradients along a horizontal tailwater boundary (Dirichlet's boundary condition) acting for decades have deleterious impacts on dam stability because of potential heaving, piping, and mechanical–chemical suffusion. Copyright © 2017 John Wiley & Sons, Ltd.

Combination of Multiscale Finite-Element Method and Yeh’s Finite-Element Model for Solving Nodal Darcian Velocities and Fluxes in Porous Media

AbstractBased on the multiscale finite-element method (MSFEM) and Yeh’s finite-element model, this paper proposes a MSFEM–Yeh model (MSFEM-Y) for solving nodal Darcian velocities in porous media. The main idea of this method is to solve Darcy’s law directly by MSFEM so as to obtain continuous coarse-scale velocities with high efficiency, while ensuring the mass conservation is satisfied to an acceptable degree. The MSFEM features allow this method to obtain fine-scale velocities directly by an interpolation equation, which only consists of coarse-scale velocities and base functions. Because the base functions have been constructed in head computation, the computation of fine-scale velocities does not need much cost. Furthermore, MSFEM-Y also ensures the continuity of fluxes, so that the MSFEM-Y fluxes are more accurate than those obtained by the original MSFEM. Numerical experiments indicate that the MSFEM-Y can achieve more accurate heads, velocities, and fluxes with high efficiency.

Tension‐saturated and unsaturated flows from line sources in subsurface irrigation: Riesenkampf's and Philip's solutions revisited

AbstractRiesenkampf's (1938), R‐38 (referred to here as R‐38), analytical solution for steady 2‐D flow from a buried line source in a homogeneous Green‐Ampt soil, with a wetting plume bounded by a free surface (capillary fringe), is compared with Philip's (1969), (P‐69), one for genuinely unsaturated wetting of Gardner's infinite‐extension soil. Conformal mappings are used in R‐38, from which we derived the flow net, pore‐water isobars, isochrones, fields of Darcian velocity and resultant force acting on saturated porous skeleton, fine geometry (shape and size) of the constant‐head contour encompassing a mole‐emitter or leaky‐pipe, as well as the dependence of the total discharge per unit pipe length on uniform pressure in the pipe, capillarity of the soil, radius of the pipe, and saturated hydraulic conductivity. An ovalic “water table” isobar, encompassing P‐69 source, is compared with one of R‐38 for a fixed discharge and saturated conductivity but adjusted sorptive numbers. The Whisler and Bouwer (1970) relation between the static height of capillary rise and sorptive number is shown to give a good match between R‐38 and P‐69 isobars. This allows to use R‐38 in the source vicinity and P‐69 in the far‐field zone. Computer algebra (Mathematica) routines are used for visualization of the known and extended R‐38 and P‐69 solutions.