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To date, efficient numerical simulation of contaminant transport in geologic porous media is challenged by parametric jumps resulting from stratification and the use of ideal initial/boundary conditions. Thus, to resolve some contaminant hydrology problems, this work presents the development of the Space-Time Conservation Element/Solution Element (CE/SE) scheme for __a__dvection-__d__ispersion-__r__eaction a-d-r transport in geologic media. The CE/SE method derives from the native form of Gauss conservation law. Therefore, it is able to effectively handle non-trivial discontinuities that may exist within the problem domain. In freshwater aquifer, stratification and other parametric jumps are examples of such discontinuity. To simulate the Nigerian experience of nitrate pollution of freshwater aquifers; the a-d-r contaminant transport model is herein solved under a time periodic nitrate fertilizer loading condition on farmlands. Results show that this approach is able to recover the well-known field pattern of nitrate profiles under farmlands. Cyclic loading impacts more on the dispersivity of an aquifer. Hence, dispersion coefficient modulates the response of aquifers to loading frequency. However, aquifers with conductivity less than 10^{-6} m/day are almost insensitive to periodic loads. The CE/SE method is able to sense slight (i.e. order of 10^{-3}) variation in hydrological parameters. Also, CE/SE computes contaminant concentration and its flux simultaneously. Thus, it facilitates a better understanding of some reported phenomena such as contaminant accumulation and localized reverse transport at the interface between fracture and matrix in geologic medium. Clearly, CE/SE is an efficient and admissible tool into the family of numerical methods available for tracking contaminant transport in porous media.

The Space-Time Conservation Element/Solution Element (CE/SE) method is gradually emerging as a reliable tool for tackling some problems in engineering and mathematical physics. In particular, it has been used to resolve some challenging problems in aero-acoustics and related shock flows. These include the shock tube and multicomponent combustion problems as treated in [

In approach, two basic concepts, namely; the percolation and hydro dispersive schemes are generally used to model transport of contaminants through geologic systems. However, in recent time, attempts have been made to hybridize these models. For example, [

On the other hand, the macroscopic framework of contaminant transport in geologic systems differentiates between coordinated and uncoordinated pore spaces. The coordinated pore spaces determine some macroscopic properties of the aquifer e.g. conductivity and permeability. However, uncoordinated pore spaces determine other properties such as storativity, adsorption and retardation. Under certain flow condition, an additional term is used to model geo-chemical reaction or physical transformation. For example, [

Traditionally, the Finite Difference, Finite Elements and Finite Volume methods are the well-known numerical tools that are used to solve contaminant transport problems. These tools take on the differential form of the conservation laws. Hence, they dissipate flux in the neighborhood of discontinuity. In such regions, the computational grid is usually refined to minimize flux dissipation. However, the need to satisfy the Courant, Friedrich, and Lewy (CFL) stability conditions sometimes overshoots the cost of this strategy. In the case of the FD, some robust schemes have been recently developed with enhanced flux conservation, stability and accuracy features. These schemes are built on the concepts of Total Variation Diminishing (TVD) and localized flux conservation, [

On the other hand, the Finite Element Method (FEM) obtains an approximate solution of the transport equation using the trial solution approach. The individual steps involved in FEM analysis are quite simple. However, the method becomes numerically tedious especially at the point where the elementary solutions are assembled to build the global solution. According to [

In addition to domain discontinuity, there are instances in contaminant hydrology problems when the model equation alternates between parabolic and hyperbolic states. This depends on the instantaneous order of magnitude of the advection and diffusion speeds. This instability in physics often leads to severe numerical oscillations. In such cases, standard numerical schemes that are built on the differential forms of conservation law are known to experience some difficulties. To handle such difficulties, extensive studies have been carried out either to customize standard methods, or design new ones. However, most of the emerging schemes are designed to address specific types of interface, or resolve definite numerical problems. Notable on this list is the Euler-Lagrangian based schemes. They include localized adjoint method and Eulerian-Lagrangian localized adjoint method developed in [

Spatial and temporal scale-effects are additional difficulties with numerical simulation of contaminant transport in geologic media. They induce errors in contaminants profiles in aquifers. In a finite element analysis, [

For the foregoing reasons, we herein present the Space-Time Conservation Element/Solution Element numerical method as an efficient tool for contaminant tracking in fractured porous media. The CE/SE method is a collection of innovative numerical tools originated by [13,14] for solving conservation laws. It was developed from a perspective that is quite different in concept and methodology from traditional numerical schemes. Its development was motivated by the desire to build a general and coherent numerical framework that avoids many of the limitations of traditional numerical methods.

Rather than the differential form of the conservation laws, the CE/SE method derives its solution from the native Gauss integral form of the conservation law. As a result, local and global flux conservation is intrinsic to CE/SE formulations. Therefore, it is able to overcome most limitations encountered by numerical schemes that are built on the differential form of conservation laws. In effect, CE/SE solutions are naturally compliant with the physical model. Like other numerical schemes, the CE/ SE method uses an approximating function i.e. the Taylor series to describe numerical approximation of the exact solution on a computational mesh. The CE/SE mesh consists of Conservation Elements (CE) and Solution Elements (SE). A CE is a small region in two-dimensional Euclidean space E_{2}, within which flux conservation is enforced. This ensures global flux conservation. On the other hand, a SE is another small region in E_{2} within which unknown variables are approximated.

In this work, we consider the hydro dispersive scheme of contaminant transport in two dimensions. Thus, three transport processes namely advection, dispersion and reaction are combined to describe the concentration profile i.e. C_{(y,z,t)} of an aqueous contaminant in natural freshwater aquifer. The effects of sorption and retardation are also considered to give the governing differential equation as:

where

Here, u_{z} and u_{y} are the longitudinal and transverse velocities of fluid. Similarly, D_{z} and D_{y} are the corresponding dispersion coefficients, while R_{y} and R_{z} are the coefficients of retardation of the contaminant along y and z directions respectively. Ã is the rate constant of the attendant geo-chemical and allied reactions. a_{y} and a_{z} are coefficients of dispersivity, while

Next we simplify Equation (1) by using the dual porosity concept to delineate contaminant pathways into two regions. The momentum transport zone called the channel. Flow in the channel is coupled with transport processes such as sorption, diffusion, and reaction in the adjoining matrix. Thus, using some simplifying assumptions as detailed in [

We consider channels of thickness D, bounded by vertical lines at −D and the origin. The centerline of the channel is at −D/2. The matrix separating two channels proceeds from the origin along the horizontal axis to 2x. Therefore, the matrix centerline is at y = x. Consequently, for this flow system, the equivalence of Equation (1) for transport in the channel and matrix is given by Equations (3) and (4) respectively.

where C_{(0,z,t)} = C_{(y,z,t)} y = 0

with

For further simplification, we normalize flow parameters in Equations (3) and (4) with their corresponding scale factors. Then we group the resulting parameters into standardized dimensionless forms. This is followed by order of magnitude analysis following the approach of

[15,16]. Hence, Equations (3) and (4) are reduced to Equations (5) and (6) respectively.

Equation (6) derives from Equation (4) under the assumption that the Peclet number only applies to the channel. In this case, the reaction term is treated as a first order geochemical reaction. In addition, the following non-dimensional variables apply

Next, we consider the boundary and initial conditions needed to complement Equations (5) and (6). Here, we attempt to model the Nigerian experience of nitrate pollution of groundwater resources. Thus, we introduce a realistic and coherent boundary condition for the associated groundwater pollution problem. Precisely, we simulate what obtains in normal agricultural practice. In this case, seasonal loading of inorganic fertilizer with an arbitrary nitrate concentration C_{o} is normally applied on topsoil. The surface behavior of the applied nitrate fertilizer is herein modeled as an integral Dirichelet/Neumann condition. This gives an exponentially decaying function of the form C_{o}. The function accounts for the progressive decay of the contaminant on topsoil as a result of some physico-chemical processes. With repeated seasonal (i.e. periodic) applications, the corresponding nitrate profile on topsoil can be represented as shown in

Here, β is the rate of disappearance of applied fertilizer on the soil surface. This may be due to percolation, evaporation, runoff etc. n is the number of cycles of fertilizer applications. On extension, the contaminant loading profile described above has widespread practical applications. With slight modification, it can be used to handle various types of physical loading conditions that are of interest to contaminant hydrologists and environmentalists. To be sure, the periodic, exponentially decaying function truly models actual physical conditions better than the impulsive, uniform and step function loadings that are well considered in literature. For instance, a leaking canister of toxic waste in a geological repository can be considered as having periodic effect on the surrounding aquifer, if for instance a similar canister has leaked in the region in the past or will most probably leak in the future. Such periodic loading profiles act to alter the hydrological properties of the aquifer and the initial concentration of the latest spill.

Our analysis is now reduced to solving simultaneous partial differential equations posed in Equations (5) and (6) to track the breakthrough curves of the contaminants in time and depth of an aquifer, under the normalized boundary condition:

Equation (8a) is a Fourier series representation of the loading pattern in

b. Contaminant concentration decays with depth in the aquifer as defined by

c. Boundary layer between the fracture and matrix has uniform concentration

d. Symmetry along the y = x line suggests that

Next, we split the numerical domain of the model; this is to reduce the problem in the fracture into one where one may solve alternately a relatively simpler pair of one dimensional problem in the vertical and horizontal directions. In preparation for such a simple pair of one dimensional numerical solution we introduce the dimensional decomposition;

where

Using this decomposition, the fracture flow Equation (5) splits into the form;

and

By invoking Gauss divergence theorem, the conservation law expressed in Equation (11a) can also be written as;

where

and

To be sure, the integral equations expressed in Equations (12) and (13) are different from the Eulerian Lagrangian integral form derived from Reynolds transport theorem. In contrast to Reynolds transport theorem, CE/ SE method is built around the concept of unified space and time, proposed in [_{2}). Consequently, a quadrilateral conservation element in Chang’s space has two parallel space and time boundaries.

To obtain CE/SE solution to Equations (11a) and

(11b), we define

In _{+} and CE_{−} respectively. The common/adjoining boundaries of these conservation elements for the mesh point ·(0, k, n) define their solution element as in

Flux conservation demands that the total flux across a conservation element e.g. ABCD (CE_{+}) and AFED (CE_{−}) in

Flux leaving CE_{+} through

On line

There is no temporal change on

As a result,

In this regard,

Using this procedure, the total flux across the boundaries of CE_{+} is given as;

Therefore, the condition for flux balance on this conservation element is given in Equation (17) below.

Normalizing Equation (17) with

The complementary part of Equation (18a) is obtained by enforcing flux conservation criterion on the contaminant across the boundaries of CE_{−}. This evaluates to

Thus the solution;

where

Equation (18) is a one-dimensional CE/SE advectiondispersion (ad) scheme. a is the advection speed and

To build a CE/SE solution for the reaction dispersion equation in the matrix, and to streamline development, we recall Equation (6) and introduce an arbitrary advection term with speed

Using the transformation

In Equation (21a)

In view of Equations (19) and (21), Equation (6) has the CE/SE solution

Setting

with

where

and

Similarly the concentration distribution across the fracture-matrix interface is given as;

and the concentration distribution in the matrix satisfies

where

and

with

Here, we have chosen

To perform an indirect check on the accuracy and validity of CE/SE algorithms developed in Section 3, we consider as a test case the problem of the transient distribution of charge carriers q(x, t) on the normalized base length of an electrical transistor satisfying the following transport model;

with

Here

Comparative performance of the CE/SE scheme relative to the exact solution and the quickest five point’s finite difference scheme are shown on Figures 4 and 5. The chosen values of a, k, Dx, and Dt are identical to the parameters employed for simulations in this work.

In ^{−4} and 7.50 ´ 10^{−3} respectively. However, in ^{±}^{n} implies b e ± n.

Numerical simulations of fracture/porous matrix flow conditions are performed over an average of 20,000 static conservation elements. Except for redefining the computational step size to stretch the fracture matrix interface, no other mesh refinement algorithm is implemented throughout the physical domain. Our simulation results are the time dependent concentration profiles of contaminant in the fracture and matrix. Simultaneously, we also obtain the time dependent flux variation of the contaminant. This flux variation result is new and it explains some phenomena identified in literature. The CE/SE schemes described in this work are explicit solvers, hence it requires minimal computational resources. This is because it does not require global matrix assembling and inversion processes.

Figures 6 and 7 show the concentration profile of the contaminant in silty clay and fissured limestone aquifers. These distributions are due to one, two and three cycles of fertilizer applications per annum for a period of 30 years.

As shown in

that the sensitivity of the aquifers to periodic loading is more of a direct function of their dispersivity. Conversely, it can be deduced that the dispersion coefficient of a contaminant in an aquifer depends on its loading history. These figures further show that an increase in the dispersion coefficients in the fracture will accelerate solute arrival time.

Theoretical basis for the wavy patterns of the concentration profiles is provided in [

However, ^{−6}, cyclic loading has a negligible effect on the contaminant profile. On

Adapted from [

the other hand, when compared with

Similarly,

The profiles in the matrix indicate storage in regions adjacent to the fracture. This delays the progress of the

contaminant in the fracture. As shown on Figures 10 and 11, the storing process continues until the matrix saturation limit at the depth is attained. Beyond this saturated zone, the contaminant proceeds through a non-linear diffusion process. Consequently, solute concentration in layers adjacent to the fracture in the matrix at certain depths in the aquifer tends to be slightly higher than the concentration in the fracture, especially in formations with very low conductivity.

In addition to the breakthrough curves, CE/SE now facilitates a more illuminating analysis of the effect of matrix diffusion through an investigation of the pattern of solute flux across the fracture/matrix interface. The flux corresponding to the profiles in Figures 10 and 11 are shown in Figures 12 and 13 respectively. Close to the wall of the fracture, the flux of the solute changes from positive values to negative values as we go down the

depth of the aquifer. This suggests that the matrix has the tendency to feed the fracture at certain locations where advective transport is slower than diffusive transport or when flow in the fracture is off season.

Thus, Figures 12 and 13 explain the observation made by [

The presence of an accumulation zone at the interface between the two strata is evident. This effect is pronounced when the ratio of the conductivities or dispersion coefficients of the upper to the lower formation is of the order of 10^{p} with p ³ 1.0.

Another important feature of the breakthrough curves in Figures 14 and 15 is the relative degree of refraction

of the profiles at the interface between two strata. Clearly, the CE/SE method has an exceptional ability to handle stratification. The angle of refraction depends on the relative conductivity and dispersion coefficient of the neighboring strata. Also, mild variation in either the conductivity or dispersion coefficient of a geological system is sufficient to modulate stratification.

On the other hand,

Similarly, in the 80A/20B composition the minimum depth to safe water is 60.0 meters if nitrate concentrations in excess of 50.0 ppm on topsoil. Clearly, the composition of a stratified aquifer determines its susceptibility to pollution. Next, we look at the effects of ordering of layers on relative susceptibility of stratified aquifers. The breakthrough curves corresponding to 80A/20B and 20B/80A arrangement of layers in a binary system are shown in

place the 80A/20B and the 20B/80A systems with an equivalent aquifer whose hydraulic parameters are the weighted averages and mean hydrological parameters of constituents. Clearly, this approach is misleading due to sharp variations in properties. Similar results are obtained even when the conductivities are comparable but with significant difference in coefficients of dispersion.

Similarly, ^{2}/day) underlain with fissured limestone. The figure shows that mean value or weighted average approach to system’s simplification in stratified aquifers can be used to optimize computational cost provided the conductivities and dispersion coefficients of adjacent layers are relatively close.

In recognition of the importance of protecting freshwater

aquifers from aqueous phase liquid contaminants, this work presents the development and application of an advection-dispersion-reaction a-d-r Space-Time Conservation Element/Solution Element (CE/SE) numerical scheme for contaminant tracking in fractured porous media. Using flux splitting approach, a two-dimensional CE/SE scheme is developed for simulating the evolution and fate of aqueous phase liquid contaminant in stratified fractured porous media. Thus, the CE/SE scheme is herein extended to handle geological time scale problems. The developed scheme is able to resolve some of the issues associated with numerical solution of groundwater contaminant hydrology problems. Such difficulties include the use of ideal initial/boundary conditions and parametric jumps that may be due to stratification or scale effects. To simulate the Nigerian experience of nitrate pollution of freshwater aquifers; we have deployed the developed scheme to solve the advection-dispersionreaction a-d-r equation in geologic media under a time periodic Dirichelet type boundary condition. This models the actual pattern of aqueous phase contaminant on arable agricultural lands. It is established that the CE/SE method is a viable and efficient tool for tracking contaminant transport in geologic profiles. The scheme is able to sense order of 10^{−3} variation in hydrological properties of aquifers. It is also shown that attempts at systems simplification in heterogeneous reservoirs through the use of mean or weighted average values of hydrological parameters are in error unless the hydro-geological properties of neighboring geological formations are very close. The developed model was validated with analytical and the quickest five points finite difference scheme. CE/SE is computationally inexpensive and easily programmed.