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How Darcy Flow and Darcy Velocity work

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Darcy Flow and Darcy Velocity, in conjunction with Particle Track and Porous Puff, can be used to perform rudimentary advection–dispersion modeling of constituents in groundwater. This methodology models two-dimensional, vertically mixed, horizontal, and steady state flow, where head is independent of depth.

Darcian flow calculations

The equations used in the calculation of Darcian flow are detailed in the following sections.

Calculating flow and velocity

  • Darcy's Law states that the Darcy velocity q in a porous medium is calculated from the hydraulic conductivity K and the head gradient Head gradient (the change in head per unit length in the direction of flow in an isotropic aquifer) as:

    • q = -K Head gradient

      where K may be calculated from the transmissivity T and thickness b as K = T/b.

    This q, with units of volume/time/area, is also known as the specific discharge, the volumetric flux, or the filtration velocity. Bear (1979) defines it as the volume of water flowing per unit time through a unit cross-sectional area normal to the direction of flow.

  • Closely related to this volumetric flux is the aquifer flux U, which is the discharge per unit width of the aquifer (with units of volume/time/length):

    • U = -T Head gradient

    This construction assumes that the head is independent of depth so that flow is horizontal.

  • The average fluid velocity within the pores, called the seepage velocity V, is the Darcy velocity divided by the effective porosity of the medium:

    • Formula for seepage velocity (V)
  • In the Darcy Flow implementation, it is this seepage velocity V that is calculated on a cell-by-cell basis. For cell i,j, the aquifer flux U is calculated through each of the four cell walls, using the difference in heads between the two adjacent cells and the harmonic average of the transmissivities Ti+1/2,j (Konikow and Bredehoeft, 1978), which are assumed to be isotropic.

    For example, for the x component of Head gradient, the equation between cells i,j and i+1,j would be:

    • δh/δx ≈ (hi+1 - hi) / Δx
  • This scheme is illustrated in the following graphics.

    Illustration of seepage velocity (V) calculated on a cell-by-cell basis

Calculating residual volume

In the cell wall calculation that follows, the aquifer flux between cell i,j and cell i+1,j flows parallel to the x direction and is calculated as:

  • Formula for aquifer flux flowing parallel to the x direction

To determine a groundwater volume balance, the groundwater discharge through the cell wall must be calculated. This discharge Qx(i+1/2) is calculated from the aquifer flux U and the width of the cell wall Δy by:

  • Qx(i+1/2,j) = Ux(i+1/2,j) Δy

Similar values are obtained for all four cell walls. These values are used to calculate the groundwater volume balance residual Rvol for the cell, which is written to the output raster. This value represents the surplus (or, in the case of a negative number, the deficit) of water in each cell given the net flow into the cell, calculated as:

  • Formula for volume balance residual Rvol

This residual Rvol should ideally be zero for all cells. When examining the output raster containing the residuals, look for deviations from zero. Large positive or negative residuals indicate a production or loss of mass, which violates the principle of continuity and suggests inconsistent head and transmissivity data. Consistent patterns of positive or negative residuals suggest that unidentified sources or sinks are present. Reduce the residuals before any further modeling. Typically, adjustments are made to the transmissivity field to reduce the residuals.

Calculating flow vectors

The actual equations used in Darcy Flow for calculating the flow vectors for each cell are condensed from the arithmetic average of Ux(i-1/2,j) and Ux(i+1/2,j), divided by the center cell's porosity ni,j and thickness bi,j to give a value for the seepage velocity Vx at the center:

  • Formula for seepage velocity Vx

and a similar equation is used to calculate Vy at the center:

  • Formula for seepage velocity Vy

This centering is done to conform to the convention that stored values represent values at the center of the cell. These values are converted to direction and magnitude in geographic coordinates for storage in the output direction and magnitude rasters.

In the case of the bounding cells of the raster in which the information is incomplete, values for velocity are simply copied from the nearest interior cell.

Porosity values

The following tables summarize some values for porosity and hydraulic conductivity for a variety of geologic media.

Table 1: Hydraulic conductivities of unconsolidated media

MediumK (m/s)

Coarse gravel

10-1 - 10-2

Sand and gravel

10-1 - 10-5

Fine sand, silts, loess

10-5 - 10-9

Clay, shale, glacial till

10-9 - 10-13

Hydraulic conductivities of consolidated media, Marsily (1986).

Table 2: Hydraulic conductivities of consolidated media

MediumK (m/s)

Dolomitic limestone

10-3 - 10-5

Weathered chalk

10-3 - 10-5

Unweathered chalk

10-6 - 10-9

Limestone

10-5 - 10-9

Sandstone

10-4 - 10-10

Granite, gneiss, compact basalt

10-9 - 10-13

Hydraulic conductivities of consolidated media, Marsily (1986).

Table 3: Porosities of geologic media

MediumTotal porosity

Unaltered granite and gneiss

0.0002 - 0.018

Quartzite

0.008

Shale, slate, mica schist

0.005 - 0.075

Limestone, primary dolomite

0.005 - 0.125

Secondary dolomite

0.10 - 0.30

Chalk

0.08 - 0.37

Sandstone

0.035 - 0.38

Volcanic tuff

0.30 - 0.40

Sand

0.15 - 0.48

Clay

0.44 - 0.53

Swelling clay, silt

up to 0.90

Tilled arable soil

0.45 - 0.65

Porosities of geologic media, Marsily (1986).

Additional tabulated values for porosity and hydraulic conductivity are provided in Freeze and Cherry (1979). Gelhar et al. (1992) present a summary of porosity and transmissivity of various specific formations reported in the literature. A detailed discussion of porosity in sedimentary materials appears in Blatt et al. (1980). A complete discussion of advection–dispersion modeling using these functions is presented in Tauxe (1994).

Examples

The typical sequence for groundwater dispersion modeling is to perform Darcy Flow, then Particle Track, then Porous Puff.

  • An example of the settings to make in the tool dialog box for Darcy Flow follows:

    Input groundwater head raster : head

    Input effective formation porosity raster : poros

    Input saturated thickness raster : thickn

    Input formation transmissivity raster : transm

    Output groundwater volume balance raster : resid1

    Output direction raster : dir1

    Output magnitude raster : mag1

  • An example of the settings to make in the tool dialog box for Darcy Velocity follows:

    Input groundwater head raster : head

    Input effective formation porosity raster : poros

    Input saturated thickness raster : thickn

    Input formation transmissivity raster : transm

    Output direction raster : dir1

    Output magnitude raster : mag1

References

Bear, J. Hydraulics of Groundwater. McGraw-Hill. 1979

Blatt, H., G. Middleton, and R. Murray. Origin of Sedimentary Rocks, 2nd Ed. Prentice-Hall. 1980

Freeze, R. A., and J. A. Cherry. Groundwater. Prentice-Hall. 1979

Gelhar, L. W., C. Welty, and K. R. Rehfeldt. "A Critical Review of Data on Field-Scale Dispersion in Aquifers". Water Resources Research, 28 no. 7: 1955-1974. 1992.

Konikow, L. F. and J. D. Bredehoeft. "Computer Model of Two-Dimensional Solute Transport and Dispersion in Ground Water", USGS Techniques of Water Resources Investigations, Book 7, Chap. C2, U.S. Geological Survey, Washington, D.C. 1978.

Marsily, G. de. Quantitative Hydrogeology. Academic Press. 1986.

Tauxe, J. D. "Porous Medium Advection-Dispersion Modeling in a Geographic Information System". Doctoral Dissertation in Civil Engineering. The University of Texas at Austin, 1994.

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