Parameter Case 1 Case 2 Case 3 Case 4 Geometric mean hydraulic conductivity, Kg (m/day) 10 10 10 10 ln K variance 2.0 2.0 2.0 2.0 Correlation scale, λ (m) 2 2 2 2 Mean porosity, nm 0.25 0.25 0.25 0.25 Coefficient of variation – n and K NA 0.27 0.25 0.10 (ln n variance) Correlation – n and K NA Positive Negative None White noise, w NA 0 0 0 Local pore-scale dispersivity 0 0 0 0 Head difference (m) 1 1 1 1 Size of model (m) 200 x 50 200 x 50 200 x 50 200 x 50 Covariance function NA Exponential Exponential Exponential Approximate initial plume size (m) 10 10 10 10 Grid 401 x 407 401 x 407 401 x 407 401 x 407 Cell size, Δx (m) 0.5 x 0.5 0.5 x 0.5 0.5 x 0.5 0.5 x 0.5 Time step, Δt (days) 1 1 1 1 The model parameters and inputs are explained below: The ln K field is a normally distributed random field. Therefore, there is an equal probability of the plume encountering a ‘low’ K zone or a ‘high’ K zone. The heterogeneous conductivity field is uniquely characterized by the following set of statistical parameters: geometric mean (Kg) of K, correlation scale, variance and covariance. The geometric mean, ln K and correlation scale are constant for all cases. The covariance function for all cases is exponential. The covariance function for cases 2, 3 and 4 is exponential. The interactions between n and K are characterized by a coefficient of variation and the type of correlation. The cell size, Δx, is selected such that the correlation scale, λ, is resolved; typically, Δx is at least 3 to 4 times smaller than the correlation scale. The time step, Δt, is selected such that a particle of the plume travels a distance that is less than λ in one time step. Download model (To download, right click and select "Save Link As" )
INTERACTIONS BETWEEN CONDUCTIVITY AND POROSITY HETEROGENEITIES
Problem Statement
This video demonstrates the impact of interactions between conductivity and porosity heterogeneities on the spreading of a conservative solute plume. The modeling domain consists of constant head boundaries on the left and right extremes, and no-flow boundaries at the top and bottom. Details are provided in Table 1.3.
Key Observations
The following observations can be made from the video: Interaction between conductivity ( K ) and porosity ( n ) heterogeneities has a significant impact on plume spreading. In most situations n is positively correlated with K , which results in less plume spreading compared to when there is no heterogeneity in n . When n is negatively correlated with K , plume spreading is greater compared to when there is no heterogeneity in n . When n and K are independent of each other, plume spreading is less than case 1, although in general, this might not be the case.
Additional Observations
Case 1: Even though K varies continuously over space, n remains constant throughout the field. This is not a commonly encountered situation. In the absence of sufficient data, n is assumed to be constant throughout the field. Case 2: In a more common situation, n interacts positively with K . When n increases, the seepage velocity decreases. The seepage velocity is given by q/n , where q is the Darcy velocity. When n increases, the seepage velocity decreases, resulting in less plume spreading. Hence, the plume spreads lesser than in case 1. Case 3: In this case, n interacts negatively with K . This means that n decreases when K increases. This relatively uncommon situation results in enhanced spreading of the plume. The eventual size of the plume is greater in comparison to case 1. Case 4: When n varies with no relationship to K , velocity variability is random. The plume spreading is dependent on ln n realization. Giving n a single value ignores the variability, resulting in increasing or reducing plume spreading. The video shows that the effect on plume spreading is significant. The video shows only the variation of K over space. The variation of n over space is shown below. Notice that, even though the cases look very different from each other, the head contours are the same. The head is dependent only on K . Since the ln K realization is the same for all cases, the head contours are identical.