GEO-SLOPE International Ltd, Calgary, Alberta, Canada Salt Flow Example

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1 1 Intoduction Salt Flow Example Total hydaulic head consists o elevation head and pessue head. Pessue head is dependent on the density o the poe luid, which is a unction o both tempeatue and chemical composition. Fo densitydependent goundwate low poblems, the diving oces o low include the hydaulic gadient, as well as a buoyancy oce tem caused by vaiations in density. Consequently, spatial vaiations in salinity geneate a total hydaulic head gadient that govens density-dependent goundwate low. The obective o this detailed example is to veiy, by means o some simple calculations, the hydaulic head and goundwate luxes calculated by CTRAN/W o coupled one-dimensional density-dependent goundwate low poblems. 2 Backgound Only a bie summay o the theoy o density-dependent low is povided hee. A moe compehensive eview can be ound in the CTRAN/W Engineeing book and the liteatue (e.g. Fette, 1998; Find, 1982a, b). As noted in the intoduction, the poe luid density aects the pessue head component o total hydaulic head. Conside the two point measuements o hydaulic head shown in Figue 1. The actual hydaulic pessue at P 1 in the saline-wate aquie is P gh [1] 1 s s whee s = saline poe luid density; g = acceleation due to gavity; and, h s = height o saline wate in the piezomete. Figue 1 Equivalent esh wate head (ate Fette, 1988) Assume that anothe piezomete was completed in the saline-wate aquie at P 2, but that it was illed with esh wate. The hydaulic pessue at P 2 is CTRAN Example File: Salt Flow Example (pd)(gsz)page 1 o 7

2 P gh [2] 2 whee = eshwate density; and, h = height o esh wate in the piezomete. The piezometes ae completed at the same elevation, so P 1 must equal P 2. Setting Eq. [1] equal to Eq. [2] esults in the ollowing elationship between the esh wate pessue head and saline-wate pessue head: h s hs [3] The equivalent esh wate total head can now be calculated as the esh wate pessue head plus the elevation head z : h s s z [4] Use o the equivalent eshwate total head o density-dependent goundwate low poblems equies the geneal Dacy equation to be ecast as qi Ki n x [5] whee K i = hydaulic conductivity tenso; n 1 indicates the vetical diection and n 0 indicates the hoizontal diection; and, = elative density deined as: s 1 [6] The indices i and ae vaied as i 0, which coesponds to the hoizontal low component (note: the hoizontal low component is unaected by the second tem because n 0 ), and i 1 which coesponds to the vetical low diection. In this om, Dacy s equation contains two dynamic diving oces: the hydaulic gadient calculated om the equivalent eshwate total head and the buoyancy oce ( tem). The buoyancy oce is oten eeed to as the body oce and epesents the additional gadient caused by vaiations in density. The implementation o Eq. [5] equies knowledge o the poe luid density s, o moe speciically the atio o s. The poe luid density is a unction o the concentation o the dissolved solute. Fo CTRAN Example File: Salt Flow Example (pd)(gsz)page 2 o 7

3 isothemal conditions and a ange o concentations up to that o seawate, thee is essentially a linea elationship between luid density and concentation (Figue 2) that can be witten as (1 ) [7] s c whee c is the concentation nomalized to the maximum concentation (C max ) and anging om 0 to 1.0, and is the contaminant density contast equal to max 1 [8] Maximum Density 1020 s c (non-dimensional) Figue 2 Relationship between density and concentation Substituting Eq. [7] into [6] yields the ollowing o elative density ( ): c [9] By substituting Equation [9] into [5], Dacy s Law can be witten as qi Ki cn x [10] In this om, the contaminant density contast ( ) and nomalized concentation (c) ae equied to compute seepage luxes. In SEEP/W, the use is equied to speciy a elative density at a eeence concentation unde the KeyIn Analysis dialogue box. The elative density (not to be conused with Eq. [6]) is simply the atio o / max in Eq. [8] at a maximum concentation (i.e. the eeence concentation). Accodingly, CTRAN can compute the nomalized concentation (c) using the eeence concentation (C/C e ). Fo example, assume that the use input elative density was (i.e. seawate) at a eeence concentation o 10,000 g/m 3. I the concentation at a point in the model domain was 5000 g/m 3, the elative concentation is c 0.5 and = CTRAN Example File: Salt Flow Example (pd)(gsz)page 3 o 7

4 Fo density-dependent poblems, the constitutive elationship (i.e. Eq. [5]), and theeoe the goundwate velocity, is a unction o concentation via the luid density ( s ). Similaly, the govening equation o solute tanspot (ee to the CTRAN Engineeing book) is a unction o the goundwate velocity though the advective tanspot tem. The equations ae theeoe coupled though density and velocity. In the CTRAN/W and SEEP/W omulation, SEEP/W computes the velocity and passes this inomation to CTRAN/W. CTRAN/W then computes the concentation and then passes the concentation values back to SEEP/W. The two pogams pass this inomation back and oth until thee is no uthe change in H and C, o in othe wods, until the solution has conveged. 3 Bounday Conditions and Mateial Popeties The Salt Flow gsz ile includes a density-dependent SEEP/W analysis and a density-dependent CTRAN/W analysis. Adding a density-dependent SEEP/W analysis automatically geneates the associated CTRAN/W analysis (o vice vesa). The time step inomation and convegence citeia ae speciied in the CTRAN/W KeyIn Analysis. A sceen captue o the KeyIn Analyses dialogue box is pesented in Figue 3. The elative density was speciied as (i.e. seawate) at a eeence concentation c 1.0 g/m 3. Figue 3 Model stuctue o the Salt Flow Example The example ile includes two model domains: a vetical column and a hoizontal column. Each column is 1 m in length and 0.1 m in width with a mesh that consists o 20 elements and 22 nodes. The hydaulic conductivity and poosity o the soil is 1 m/sec and 0.35, espectively. Fo the vetical column, the top bounday is set as a constant head o 1 m (H = 1 m). The let and ight boundaies o the hoizontal column ae set to total hydaulic heads o 0.2 m and 0.1 m, espectively. The soil is assigned a coeicient o diusion (D) o m 2 /sec. The longitudinal and tansvese dispesivity values wee abitaily set to a low value ( m 2 /sec). A unit concentation o 1 g/m 3 is applied to both columns. Accodingly, the concentation thoughout the model domain is equal to the use-deined eeence concentation (i.e. c = 1.0) and the body oce tem will equal The initial CTRAN Example File: Salt Flow Example (pd)(gsz)page 4 o 7

5 Height (m) GEO-SLOPE Intenational Ltd, Calgay, Albeta, Canada poe wate conditions ae speciied using a piezometic line, while the initial concentations ae deined using the activation concentation unde KeyIn Mateials. The analyses wee un o an elapsed time o 100 seconds with one time incement. 4 Results and Discussion 4.1 Vetical Column Figue 4 pesents a poile o the equivalent esh wate head computed by SEEP/W o the vetical column. The esh wate head at the bottom o the column is m, which is equal to the hydaulic pessue ( kpa) divided by the unit weight o esh wate (9.81 kpa/m). Although thee is an upwads hydaulic gadient o x in the column, the vetical goundwate velocity is zeo. This occus because the upwad hydaulic gadient is countebalanced by the downwad body oce c in Eq. [10] Fesh Wate Head Fesh Wate Head (m) Figue 4 Vetical Poile o Fesh Wate Head 4.2 Hoizontal Column Figue 5 pesents the contous o equivalent esh wate head along with values at the ou cones o the hoizontal column. The esh wate hydaulic heads computed at points A, B, C, and D ae consistent with Eq. [3]. Fo example, the salt wate head along the let bounday was set to 0.2 m, so the poe-wate pessue at point A is equal to kpa (i.e. sghs = (1025 kg/m 3 )(9.81 m/s 2 )( )m). The equivalent eshwate head is calculated as m using Eq. [3]. Thee is an upwad esh wate hydaulic gadient o m/m acoss the entie column, as was the case with the vetical column. CTRAN Example File: Salt Flow Example (pd)(gsz)page 5 o 7

6 Figue 5 Contous o Equivalent Fesh Wate Total Head Results o a Gauss egion in the model domain ae shown in Figue 6. The upwad esh wate hydaulic gadient is countebalanced by the downwad body oce, so the vetical gadient and the liquid y-velocity thoughout the model domain ae zeo. The esh wate hydaulic gadient in the hoizontal diection acoss the column at all elevations is Given a hydaulic conductivity o 1.0 m/s and a column length o 1.0 m, the calculated goundwate lux is m/s. The cumulative lux ove an elapsed time o 100 seconds is m 3, as shown on the lux section in Figue 5. Figue 6 Results o a Gauss Region 5 Concluding Remaks This example illustates the esh wate hydaulic heads and luxes calculated by SEEP/W o a coupled density-dependent goundwate low poblem. All hydaulic heads ae conveted to an equivalent esh wate total head based on the density o the poe luid. Poe luid density is a unction o the solute concentation, so the solution is coupled and theeoe SEEP/W must be integated with CTRAN/W. Intepeting the computed hydaulic heads and goundwate luxes equies an undestanding o the undelying theoy o density-dependent goundwate low poblems. This simple example demonstates that SEEP/W and CTRAN/W have been coectly coded o densitydependant low and that the pocedue o passing velocity and concentation between the two pogams is an acceptable pocedue o solving this coupled pocess, even though the two patial dieential equations ae not solved simultaneously. CTRAN Example File: Salt Flow Example (pd)(gsz)page 6 o 7

7 6 Reeences GEO-SLOPE Intenational Ltd, Calgay, Albeta, Canada Fette, Applied Hydogeology, 3 d Edition. Pentice Hall, Englewood Clis, NJ. Find, 1982a. Simulation o long-tem tansient density-dependent tanspot in goundwate. Advances in Wate Resouces, Vol. 5, pp Find, 1982b. Seawate instuction in continuous coastal aquie-aquitad systems. Advances in Wate Resouces, Vol. 5, pp CTRAN Example File: Salt Flow Example (pd)(gsz)page 7 o 7