
ln K legend



EFFECTS OF HETEROGENEITY OF HYDRAULIC CONDUCTIVITY 

Problem Statement  
This video demonstrates the effect of heterogeneous hydraulic conductivity media on the transport of a conservative solute in the absence of porescale dispersion. The initial size of the plume is much larger than the scale of heterogeneity. The modeling domain consists of constant head boundaries on the left and right extremes, and noflow boundaries at the top and bottom. Case 1 has a homogeneous field; and cases 2 through 4 show increasing degrees of heterogeneity. Details are provided in Table 1.1 

Key Observations  
The following observations can be made from the video:


Additional Observations  
The pattern of plume spreading shows a clear trend. The plume does not spread at all when the ln K variance is 0.0, and spreads the most when the ln K variance is 3.0. Even though the plume spreading is enhanced, the mean plume displacement is largely unaffected. The fingers of the plume contain the maximum mass of the plume, but are concentrated over a small spatial area. The tails have less mass, but are elongated. Consequently, the center of mass, and thus mean displacement, remains the same, irrespective of the ln K variance. The effect of the symmetric ln K field on plume spreading is asymmetric. The fingers (leading edge) of the plume are shorter than the tails (trailing edge) of the plume, which results in a skewed plume. This can be explained very simply. High K zones are convergent zones, through which the plume prefers to travel, and in which it converges. Convergence is the opposite of spreading. Therefore, shorter fingers are found. In contrast, low K zones are zones of divergence, which impede the movement of the plume. When a low K zone is encountered, the plume travels around it rather than through it and this results in spreading. If a part of the plume reaches the low K zones, it gets 'trapped' due to the extremely low velocities in that zone. The bulk of the plume continues moving, while the portion trapped in the low K zone is released slowly; thus contributing to tail elongation. 

Mathematical Interpretation  
Mathematically, these processes can be explained using the following equations. In a heterogeneous field, in the absence of porescale dispersion, the conservative solute concentration at any point for any irregular plume realization is given by:
where: C = Concentration; u_{i} = Seepage velocity. To determine large scale behavior, the method of perturbation is used, and on performing local spatial averaging, we get the following:
where: = Mean concentration; = Perturbation to mean concentration; = Mean seepage velocity; = Perturbation to mean seepage velocity. This additional term on the RHS of (1.1.2) reflects the effects of heterogeneity. It can be shown [ Gelhar and Axness, 1983a ] that:
The coefficient A_{ii}is called macrodispersivity and is given by [ Gelhar, 1993 ]:
where: = ln K variance; λ = Correlation scale; A_{11} = Longitudinal macrodispersivity; A_{22} = Transverse macrodispersivity; α_{L}= Longitudinal porescale dispersivity; α_{T}= Transverse porescale dispersivity. Substituting (1.1.3a) into (1.1.2), yields:
This is the classical advectiondispersion equation. The additional term on the RHS in (1.1.4) represents macrodispersion. From (1.1.3b), it is evident that with increasing ln K variance, the longitudinal component of macrodispersion will increase. However, there will be no effect on transverse macrodispersion, since no porescale dispersion is present. The mean velocity ( ) controls the mean displacement of the plume. Darcy's equation gives the seepage velocity of the flow as:
where: K = Mean conductivity; n = Porosity. The mean velocity is given by:
Gelhar ( 1993, equation 4.1.48 ) showed that
where: K ' = Perturbation to mean conductivity; = Mean hydraulic head; = Perturbation to mean hydraulic head; K_{g}= Geometric mean of hydraulic conductivity. Substituting (1.1.6) into (1.1.5), we get:
Since the geometric mean hydraulic conductivity and mean hydraulic gradient are the same for all cases, is the same. As a result, the mean plume displacement remains largely unaffected. 