A fully implicit finite difference method with 2254 rectangular meshes and 2350 nodes is developed with relatively fine discretization near the collapsed cavern and relatively coarse discretization away from the salt dome. The initial value of salt concentration before the cavern collapse is selected based on an electrical log data (66011). The model is simulated to reach a steady-state solution and is used as an initial condition for the collapse. The temperature increases linearly from 20 °C at the top boundary to 53 °C at the bottom boundary based on well log information. The boundary conditions are as follows:• A Dirichlet boundary condition to total head, salt concentration, and temperature

distribution are assigned at the top, bottom and left boundaries.• No flow, salt flux, and heat flux are permitted across the right boundary.Figure 7(a)-(d) show the salt concentration, temperature, total head, and flow field pattern before cavern collapse. Figure 8(a)-(d) show the salt concentration, temperature, total head, and flow field pattern for 1-day simulation after cavern collapse. In this case, an over-pressure 4,000-meter equivalent freshwater head is assigned for over-pressured sediment in the cavern. This assumption is based on a sudden sedimentation in the cavern after collapse. It is assumed that after collapse a narrow band along the west flank of the dome has permeability one order of magnitude higher than sand (a zone with high permeability). It is seen that flow field pattern around the dome is upward and salt concentration is elevated around the sinkhole location. Figure 9(a)-(d) show the salt concentration, temperature, total head, and flow field pattern for 90-day simulation after cavern collapse. In this case, over-pressure in the cavern is considered only one day. The model run for 90 days without any overpressure. The figure shows that flow around the dome stays upward. The overpressure and permeability in the cavern has an important role in upward or downward flow around the dome. Figure 10(a)-(b) shows the changes in the salt concentration and the total equivalent freshwater head at a point beneath the sinkhole. At the first day after collapse, the total head is high and then decreases over time. Salt concentration increases over time.

3.1.Verification3.3.1. Henry problem The Henry problem has been discussed extensively in the literature as a seawater

intrusion benchmark for moderate density-dependent groundwater flow. The original boundary conditions used by Henry (1964) [4] have been modified by several studies to make it close to realistic conditions [5-9]. Using the Picard iteration method to solve the coupled flow and salt transport equations, the simulated isochlors shown in Figure 1(a) are compared well to the semi-analytical solution and previous published numerical solutions. Figure 1(b) shows the freshwater head distribution and flow field. It is noticed that the dispersive flux has an insignificant impact on the salt concentration solution in the Henry problem due to small density difference (2.5%), small diffusion coefficient value, and zero dispersivity value.

3.1.2. Brine Elder ProblemThe brine Elder problem is a modified Elder problem as a benchmark for strong

density-driven flow and salt transport. Figure 2(a)-(b) show the numerical solutions for 0.2 and 0.6 isochlor curves at different times in comparison with Voss and Souza (1987) [9]. In spite of small diffusion coefficient value and zero dispersivity value, the significant density difference (20%) produces significant dispersive flux and affects flow pattern and salt concentration distribution. Figure 2(c)-(d) show the different numerical solutions for 0.2, 0.4, 0.6 and 0.8 isochlor curves with and without the dispersive flux term in the flow equation.

3.1.3. Thermal Elder ProblemThe thermal Elder problem (1967) [10] is a free convection problem. The Elder

problem is a benchmark for strong density-driven flow and heat transport simulations. Figure 3(a)-(b) show the comparison of the numerical solution for 0.2 and 0.6 isothermal curves. The temperature distributions are satisfactory comparing to the Elder solution. Because the density difference is small, the dispersive flux does not significantly affects flow pattern and salt concentration distribution due to small diffusion coefficient value and zero dispersivity value.

3.2. Applications for Anisotropic and Non-Homogenous Porous Media Numerical experiments are conducted to study the influence of the dispersive flux on

the density-driven thermohaline flow and transport in non-homogeneous and anisotropic porous media. Two different cases are shown in Figure 4 to be studied. It is clear that the salt concentration and temperature distributions are strongly controlled by the arrangement of anisotropic and zonal permeability and influenced by the dispersive flux term.

Density-Driven Thermohaline Groundwater Flow and Brine Transport Near Salt DomesH51H | AGU Fall Meeting 2012, San Francisco, CA, USA | 3-7 December 2012

Zahra Jamshidzadeh1, Frank T.-C. Tsai2, S. Ahmad Mirbagheri1 , Hassan Ghasemzadeh1 1Department of Civil Engineering, K.N. Toosi University of Technology, Tehran, Iran, 2Department of Civil and Environmental Engineering, Louisiana State University, LA, USA

E-mail: z_jamshidzadeh@yahoo.com; ftsai@lsu.edu; mirbagheri@kntu.ac.ir; ghasemzadeh@kntu.ac.ir

[1] Abstract

In this study, we simulate density-driven thermohaline flow and transport in two-dimensional cross sections. The density-driven flow equation in terms of freshwater head is [1]

The salt transport equation is [2]

The heat transport equation is [3]

This study introduces the dispersive flux of total fluid mass to the density-driven flow equations to improve thermohaline modeling of salt concentration and temperature in porous media. The dispersive flux is derived to account for fluid flow driven by density differentiation in space. The coupled flow, salt concentration, and temperature governing equations are solved simultaneously by a fully implicit finite difference method to investigate the influence of the additional dispersive flux term to the salt and heat transport. The numerical model is verified by the Henry problem and the thermal Elder problem under a moderate density effect and by the brine Elder problem for a strong density effect. The dispersive flux has an insignificant influence on the solutions of the Henry problem and the thermal Elder problem because the problems ignore mechanical dispersion, have small diffusion coefficient value, and have small density differentiation. However, in the brine Elder problem because of high density difference (20%), the dispersive flux term has a significant impact on salt transport. For a thermohaline Elder problem that considers mechanical dispersion and strong density difference, the salt and heat solutions show a significant influence by the dispersive flux term. The dispersive flux also shows influence on salt concentration and temperature distributions in anisotropic, non-homogeneous porous media. The model is applied to the Bayou Corne sinkhole near the Napoleonville salt dome, southeastern Louisiana. The spread of salt concentration and upward flow along the west flank of the salt dome is simulated to explain the elevated salinity in the sinkhole. We examine the hypothesis that a sudden overpressure in the collapsed cavern Oxy#3 in the salt dome causes the appearance of the sinkhole on the ground close to the dome. The results show that the flow pattern around the salt dome strongly depends on the magnitude of overpressure and permeability in the cavern.

[3] Benchmarks and Applications [4] Case Study: Bayou Corne Sinkhole near Napoleonville Salt Dome, Louisiana

[2] Governing Equations

[5] Results and Discussion

[6] Conclusions Improved density-driven thermohaline flow and transport models are suggested by

including the dispersive flux term of the total fluid mass to the flow equation. To illustrate the influence of the dispersive flux term, a fully implicit finite difference method is developed to solve the coupled flow, salt concentration and heat governing equations in non-homogeneous anisotropic porous media.

The Picard iteration method is used to handle the nonlinearity in the coupled system and to obtain numerical solutions. The numerical solutions are verified by the benchmarks and are compared well to the published solutions for the Henry problem [4], the thermal Elder problem [10], and the brine Elder problem [9].

The dispersive flux is influential for the thermohaline flow and transport in non-homogeneous anisotropic porous media.

In the general case that involve strong density-drive flow and transport modeling the dispersive flux of total fluid mass should be considered.

From the simulation results, a sudden overpressure in the collapsed cavern may be the main cause to the appearance of the Bayou Corne sinkhole near the Napoleonville salt dome.

Flow pattern around the dome can be upward or downward strongly depending on the overpressure value and permeability in the cavern.

[1] Bear, J., Cheng, A.H.D. Modeling Groundwater Flow and Contaminant Transport. Springer Dordrecht Heidelberg London. New York 2010, pp 834. [2] Bear J. Hydraulic of groundwater. McGraw-Hill. New York; 1979.[3] Younes A. On modeling the multidimensional coupled fluid flow and heat or mass transport in porous media. International Journal of Heat and Mass Transfer 2003; 46: 367-379.[4] Henry HR. Effects of dispersion on salt encroachment in coastal aquifers. Water supply paper 1613-C, US Geological Survey 1964; p. C71-84.[5] Smith L., Chapman S.D. On the Thermal Effects of Groundwater Flow 1. Regional Scale Systems. Journal of Geophysical Research 1983; 88:593-608.[6] Langevin C.D., Guo W. MODFLOW/MT3DMS- Based simulation of Variable-Density Ground Water Flow and Transport, Ground Water 2006; 44: 339-351.[7] Simpson M.J., Clement T.P. Theoretical Analysis of the Worthiness of Henry and Elder Problems as Benchmarks of Density-dependent Ground Water Flow Models, Advances in Water Resources 2003; 26:17-31. [8] Simpson M.J., Clement T.P. Improving the worthiness of the Henry problem as a benchmark for density-dependent groundwater flow models, Water Resources Research 2004; 40:1-11. [9] Voss C.I., Souza W.R. Variable density flow and solute transport simulation of regional aquifers containing a narrow fresh-water–saltwater transition zone. Water Resources Research 1987; 23(10):1851-66. [10] Elder J.W. Transient convection in a porous medium. Journal of Fluid Mechanics 1967; 27: 609-623.[11] Department of Natural Resources/Shaw, Assumption Parish Meeting Tuesday, August 7; 2012. [12] Louisiana Boat Disappears Into Sinkhole, Workers Rescued. website: abcnews.go.com.

References

H51H-1449

4.1. Bayou Corne SinkholeThe Napoleonville salt dome is located in Assumption, Iberville and Ascension

parishes, southeastern Louisiana. On August 3, 2012 in the town of Bayou Corne, Assumption Parish, a sinkhole with 324 feet in diameter and up to 422 feet at its deepest part formed in a swamp near the Napoleonville salt dome. Researchers have been trying to determine the cause of this sinkhole. The research done by Louisiana Department of National Resources (LaDNR) indicated that some theories may cause sinkhole at the Bayou Corne area such as salt dome movement, regional tectonic activity, collapse of Texas Brine salt cavern wall inside the Napoleonville dome, and salt or cap-rock instability. In this research, the effect of collapsed cavern in making the sinkhole and salt migration into this sinkhole is considered. Figure 5 (a)-(c) shows the Napoleonville salt dome area, the Bayou Corne sinkhole, and the conceptual model of current situation for the Napoleonville salt dome.

4.2. Hydrogeological StructureParameter Symbol Value Unit

Sand Permeability ksand 10-12 m2

Shale Permeability kshale 10-13 m2

Cap-rock Permeability kcaprock 10-13 m2

Salt dome Permeability ks 10-26 m2

Molecular diffusion

coefficient Dm 1.5×10-6 m2s-1

Specific storage Sp 8.6×10-10 m-1

Sand Porosity sand 0.35 ---

Shale Porosity sand 0.45 ---

Salt dome porosity s 1×10-4 ---

Longitudinal dispersivity αL 100 m

Longitudinal dispersivity αT 10 m

Freshwater Density f 998.2 kg m-3

Brine Density max 1210 kg m-3

Maximum salt concentration Cmax 300 ---

Freshwater dynamic viscosity µf 10-3 Kgm-1s-1

Maximum dynamic viscosity µmax 2.39×10-3 Kgm-1s-1

Salt concentration coefficient β 8.407×10-4 ---

Thermal conductivity of fluid f 0.65 kgms-3oc-1

Thermal conductivity of solid s 1.59 kgms-3oc-1

Thermal conductivity of

dome salt 7 kgms-3oc-1

Thermal coefficient α -3.5×10-4 oc-1

Maximum temperature Tmax 53 oc

Reference temperature T0 20 oc

Solid density s 2650 kgm-3

Solid heat capacity cs 2.52×103 m2s-2 oc-1

Fluid heat capacity cf 4.18×103 m2s-2 oc-1

Figure 2: Boundary conditions for the brine Elder problem. Comparison of 0.2 and 0.6 isochlor curves with the brine Elder solution for time at (a) 2.63 years and (b) 10.63 years. Comparison of numerical solution with and without considering the dispersive flux term after for time at (c) 2.63 years and (d) 10.63 years.

Figure 1: The solution for the Henry problem (Henry, 1964): (a) 10%, 25%, 50%, 75% and 90% isochlor curves, and (b) freshwater head distribution and flow field.

Figure 4: Fifteen-year simulation for (a) case 1, (b) 0.2, 0.4, 0.6, and 0.8 isochlor curves for case 1, and (c) 0.2, 0.4, 0.6, and 0.8 isothermal curves for case1. Fifteen-year simulation for (d) case 2, (e) 0.2, 0.4, 0.6, and 0.8 isochlor curves for case 2, and (f) 0.2, 0.4, 0.6, and 0.8 isothermal curves for case2. Solid lines are solutions with dispersive flux of the total mass. Dash lines are solutions without dispersive flux of the total mass.

Figure 3: Comparison of 0.2 and 0.6 isothermal curves with the Elder solution (1967) for time at (a) 1 year, (b) 3 years, and (c) 8 years.

(a)

(b) (c)Figure 5: (a) Napoleonville salt dome area [11], (b) Bayou Corne sinkhole [12], and (c) conceptual model of the current situation [11].

The sediments surrounding the salt dome are dated from Miocene to Recent alluvial deposits. The aquifer structure consist of sand and shale. Top of the salt dome is approximately 700 feet in depth. The location of the cavern is shown in Figure 5(c). Based on the data from two borehole logs (66011 and 191131) in the study area, the simplified sand and shale units are shown in Figure 6.

Figure 6: The hydrostratigraphy of sand and shale units around the salt dome .

Table 1: Model parameters

Figure 7: Before collapse: (a) salt concentration (b) temperature, (c) total head and flow pattern, and (d) total head and flow pattern near the dome.

Figure 8: 1-day after collapse: (a) salt concentration (b) temperature, (c) total head and flow pattern, and (d) total head and flow pattern near the top of the dome.

Figure 9: 90-day after collapse: (a) salt concentration (b) temperature, (c) total head and flow pattern, and (d) total head and flow pattern near the top of the dome.

AcknowledgementsZahra Jamshidzadeh was supported by the Iran Ministry of Science, Research and Technology to conduct research at Louisiana State University. Frank Tsai was supported by the U.S. Geological Survey under grant no. G10AP00136 to conduct saltwater intrusion simulation.

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Figure 10: (a) Salt concentration and (b) the total equivalent freshwater head at a point beneath the sinkhole over 90 days after collapse.

Dispersive Flux

K.N. ToosiUniversity