Influence of temporal fluctuations and spatial heterogeneity on pollution transport in porous media

The combined influence of temporal fluctuations and spatial heterogeneity on non-reactive solute transport mechanisms in porous media can be understood by performing simulations of steady and unsteady flow and transport in heterogeneous media. The study focuses on issues such as the degree of heterogeneity, correlation length, separation of the combined effects of temporal and spatial variations, and ergodicity conditions under unsteady flow conditions. It is shown that the effect of temporal variations on solute transport is masked by the strong effect of spatial heterogeneity. There is no obvious difference in plume shape between steady and unsteady flow conditions; the first and the second spatial moments of the plume of the unsteady-state flow condition fluctuate around the steady-state flow condition with the same period of oscillations as the input signal at small storage coefficient (S ≤ 0.001). At a relatively high standard deviation in hydraulic conductivity and a small storage coefficient, the unsteady flow condition sharpens the temporal variations in macrodispersion coefficients. The magnitude of the longitudinal macrodispersion coefficient under unsteady flow condition is almost doubled at the maximum values. However, the transverse macrodispersion coefficient fluctuates around zero. The Kubo number and Peclet number ranges are 1.2–64 and 10–250, respectively.

L’influence conjointe des variations temporelles et de l’hétérogénéité spatiale sur les mécanismes de transport non réactif de solutés en milieux poreux peut être comprise à l’aide de simulations numériques en régime permanent et transitoire du transport dans des milieux hétérogènes. L’étude se concentre sur les questions du degré de l’hétérogénéité, la longueur de corrélation, la séparation des effets combinés des variations spatio-temporelles ainsi que sur les conditions d’ergodicité pour des conditions d’écoulements en régime transitoire. L’effet des variations temporelles sur le transport de solutés est masqué par l’effet important de l’hétérogénéité spatiale. Il n’y a pas de différence significative dans la forme du panache en régime permanent et transitoire ; les moments spatiaux du panache en régime transitoire fluctuent autour de ceux en régime permanent avec la même période d’oscillations, pour un un faible coefficient d’emmagasinement (S ≤ 0.001). Pour un écart type relativement élevé pour la conductivité hydraulique et un faible coefficient d’emmagasinement, les conditions de régime transitoire ont pour conséquence de lisser les variations temporelles des coefficients de macrodispersion. L’ampleur du coefficient de macrodispersion longitudinale en régime transitoire est pratiquement le double pour les valeurs maximales. Cependant, le coefficient de macrodispersion transversale fluctue autour de zéro. Le nombre de Kubo et de Péclet sont compris entre 1.2 et 64, et entre 10 et 250, respectivement.

La influencia combinada de fluctuaciones temporales y heterogeneidad espacial sobre los mecanismos de transporte de soluto no reactivo en medios porosos puede ser entendida llevando a cabo simulaciones de flujo estacionario y no estacionario y transporte en medios heterogéneos. El estudio se enfoca en cuestiones tales como el grado de heterogeneidad, longitud de correlación, separación de los efectos combinados de las variaciones temporales y espaciales, y las condiciones de ergodicidad bajo condiciones de flujo no estacionario. Se demuestra que el efecto de las variaciones temporales en el transporte de soluto está enmascarado por el fuerte efecto de la heterogeneidad espacial. No hay ninguna diferencia obvia en la forma de la pluma entre las condiciones de flujo estacionario y no estacionario, los momentos espaciales primero y segundo de la pluma de la condición de flujo no estacionario fluctúan alrededor de la condición de flujo de estado estacionario con el mismo período de oscilación como señal de entrada a pequeños coeficientes de almacenamiento (S ≤ 0.001). En una desviación estándar relativamente alta en la conductividad hidráulica y un pequeño coeficiente de almacenamiento, la condición de flujo no estacionario agudiza las variaciones temporales en los coeficientes de macrodispersión. La magnitud del coeficiente de la macrodispersión longitudinal bajo condiciones de flujo no estacionario es casi duplicada en los valores máximos. Sin embargo, el coeficiente de macrodispersión transversal fluctúa alrededor de cero. Los intervalos del número de Kubo y del número de Peclet van de 1.2–64 y 10–250, respectivamente.

非均质性的时空变化对孔隙介质中非反应性溶质的综合影响可以通过对非均质介质中稳定与非稳定流运移进行模拟来理解。本文主要关注以下几个问题,如非均质的程度、相关长度、时空变化双重影响的分离和非稳定流条件下的遍历性条件。结果表明,时间变化对溶质运移的影响被空间非均质性的强烈影响所掩盖。在稳定流与非稳定流条件下,辐射羽形状没有明显的差别。在小贮存系数的背景下(S ≤ 0.001),非稳定流条件下辐射羽的第一和第二部分波动与稳定流条件下同期波动相近。在渗透系数标准偏差相对较大且贮水系数较小的情况下,非稳定流更加突出弥散系数的瞬时变化。非稳定流条件下,径向弥散系数的量级达到最大值时几乎可以翻番。然而,横向弥散系数则在0左右波动。Kubo数和Peclet数的范围分别是1.2–64 和 10–250。

A influência combinada das flutuações temporais e da heterogeneidade espacial sobre os mecanismos de transporte de contaminantes não reactivos em meios porosos pode ser compreendida através da realização de simulações de fluxo e transporte em regimes permanente e transitório em meios heterogéneos. Este estudo foca questões como o grau de heterogeneidade, a distância de correlação, a separação dos efeitos combinados das variações temporais e espaciais e as condições de ergodicidade sob condições de fluxo variável. Mostra-se como o efeito de variações temporais em transporte de solutos é mascarado pelo forte efeito da heterogeneidade espacial. Não há nenhuma diferença óbvia na forma da pluma entre as condições de fluxo constante e variável; o primeiro e o segundo momentos espaciais da pluma em condições de fluxo variável flutuam em torno da condição de fluxo constante com o mesmo período de oscilações que o sinal de entrada para pequenos valores de coeficiente de armazenamento (S ≤ 0.001). Para um desvio padrão relativamente elevado da condutividade hidráulica e um coeficiente de armazenamento pequeno, a condição de fluxo variável aumenta as variações temporais dos coeficientes de macrodispersividade. A magnitude do coeficiente de macrodispersividade longitudinal sob a condição de fluxo variável quase duplicou para os valores máximos. No entanto, o coeficiente de macrodispersividade transversal oscila em torno de zero. Os números Kubo e Peclet variam entre 1.2–64 e 10–250, respectivamente.

Authors and Affiliations

1. Department of Hydrology and Water Resources Management, Faculty of Meteorology, Environment and Arid Land Agriculture, King Abdulaziz University, P.O. Box 80208, Jeddah, Kingdom of Saudi Arabia

   Amro M. M. Elfeki

2. Geo-Engineering Section, Delft University of Technology, P.O. Box 5048, 2600, Delft, The Netherlands

   Gerard Uffink

3. Water Resources Section, Delft University of Technology, P.O. Box 5048, 2600, Delft, The Netherlands

   Sophie Lebreton

Corresponding author

Correspondence to Amro M. M. Elfeki.

Amro M. M. Elfeki is on sabbatical leave from Faculty of Engineering, Mansoura University, Egypt.

Appendix A: derivation of the numerical groundwater flow model

A numerical scheme with a five-point operator is used. The solution is based on the backward difference approximation in order to get a stable solution whatever the size of the time step. The following equation must then be transformed

The finite difference analogue of the derivatives is given by:

with the harmonic mean of the transmissivity between neighboring nodes as \( \mathop{T}\nolimits_{{\mathop{{xx}}\nolimits_{{i + 1/2,j}} }} = \frac{{\mathop{{2T}}\nolimits_{{\mathop{{xx}}\nolimits_{{i + 1,j}} }} \mathop{T}\nolimits_{{\mathop{{xx}}\nolimits_{{i,j}} }} }}{{\mathop{T}\nolimits_{{\mathop{{xx}}\nolimits_{{i + 1,j}} }} + \mathop{T}\nolimits_{{\mathop{{xx}}\nolimits_{{i,j}} }} }} \) and \( h_{{_{{i,j}}}}^{{k + 1}} \) is the hydraulic head at the node (i,j), at time (k + 1).

The derivative analogue with respect to y, is obtained similarly. Further evaluation leads to the following expressions of the second derivatives

Substitution of these equations into the governing equation, Eq. 14, leads to the finite difference approximation of the governing equation as

with

The boundaries and initial conditions are given by

where Г is the lateral boundary and n is the unit vector normal to Г pointing out forward from the boundary.

Appendix B: solution algorithm

The procedure for solving the model equations involves the following steps between two successive time steps k and k + 1.

First iteration, denoted by (0), for the time step (k + 1):

• An initial estimation of the heads \( h_{{i,j}}^{{^{{k + 1(0)}}}} \) is given for all the nodes

• The residual \( r_{{ij}}^{{^{{k + 1(0)}}}} \) is computed as follows:

  $$ r_{{ij}}^{{k + 1(0)}} = {A_{{ij}}}h_{{i + 1,j}}^{{k + 1(0)}} + {B_{{ij}}}h_{{i,j - 1}}^{{k + 1(0)}} + {C_{{ij}}}h_{{i - 1,j}}^{{k + 1(0)}} + {D_{{ij}}}h_{{i,j + 1}}^{{k + 1(0)}} + {E_{{ij}}}h_{{i,j}}^{{k(0)}} - {F_{{ij}}}h_{{i,j}}^{{k + 1(0)}} $$

  (21)

• A matrix \( P_{{ij}}^{{^{{k + 1(0)}}}} \) is introduced as

  $$ P_{{ij}}^{{k(0)}} = r_{{ij}}^{{k(0)}} $$

  (22)

• Another matrix \( {Q_{{ij}}}^{{k + 1(0)}} \) is introduced as

  $$ Q_{{_{{ij}}}}^{{k + 1(0)}} = {F_{{ij}}}r_{{i,j}}^{{k + 1(0)}} - {A_{{ij}}}r_{{i + 1,j}}^{{k + 1(0)}} - {B_{{ij}}}r_{{i,j - 1}}^{{k + 1(0)}} - {C_{{ij}}}r_{{i - 1,j}}^{{k + 1(0)}} - {D_{{ij}}}r_{{i,j + 1}}^{{k + 1(0)}} + {E_{{ij}}}r_{{i,j}}^{{k(0)}} $$

  (23)

• The residual is summed over all the nodes

  $$ {\left| {{r^{{k + 1(0)}}}} \right|^2} = \sum\limits_i {\sum\limits_j {{{\left( {r_{{ij}}^{{k + 1(0)}}} \right)}^2}} } $$

  (24)

• The inner product of \( {P_{{ij}}}^{{k + 1(0)}} \) and \( {Q_{{ij}}}^{{k + 1(0)}} \) is computed to evaluate the parameter \( {\alpha^{{k + 1(0)}}} \),

  $$ {\alpha^{{k + 1(0)}}} = \frac{{{{\left| {{r^{{k + 1(0)}}}} \right|}^2}}}{{\left( {{P^{{k + 1(0)}}},{P^{{k + 1(0)}}}} \right)}} $$

  (25)

• The heads and the residuals are updated

  $$ \begin{gathered} h_{{ij}}^{{k + 1(1)}} = h_{{ij}}^{{k + 1(0)}} + {\alpha^{{k + 1(0)}}}{P_{{ij}}}^{{k + 1(0)}} \hfill \\ r_{{ij}}^{{k + 1(1)}} = r_{{ij}}^{{k + 1(0)}} - {\alpha^{{k + 1(0)}}}{Q_{{ij}}}^{{k + 1(0)}} \hfill \\ \end{gathered} $$

  (26)

• A parameter \( {\beta^{{k + 1(0)}}} \) is computed

  $$ {\beta^{{k + 1(0)}}} = \frac{{{{\left| {{r^{{k + 1(1)}}}} \right|}^2}}}{{{{\left| {{r^{{k + 1(0)}}}} \right|}^2}}} $$

  (27)

• For the next iterations (m + 1), the matrix P and Q are updated as follows

  $$ \begin{array}{*{20}{c}} {\mathop{P}\nolimits_{{ij}}^{{k + 1(m + 1)}} = \mathop{r}\nolimits_{{ij}}^{{k + 1(m + 1)}} + {\beta^{{k + 1(m)}}} \mathop{P}\nolimits_{{ij}}^{{k + 1(m)}} } \hfill \\ {\mathop{Q}\nolimits_{{ij}}^{{k + 1(m + 1)}} = \left[ {{F_{{ij}}}r_{{i,j}}^{{k + 1\left( {m + 1} \right)}} - {A_{{ij}}}r_{{i + 1,j}}^{{k + 1\left( {m + 1} \right)}} - {B_{{ij}}}r_{{_{{i,j - 1}}}}^{{k + 1\left( {m + 1} \right)}} - {C_{{ij}}}r_{{_{{i - 1,j}}}}^{{k + 1\left( {m + 1} \right)}} - {D_{{ij}}}r_{{_{{i,j + 1}}}}^{{k + 1\left( {m + 1} \right)}} + {E_{{ij}}}r_{{_{{i,j}}}}^{{k + 1\left( {m + 1} \right)}}} \right]} \hfill \\ { + {\beta^{{k + 1(m)}}} \mathop{Q}\nolimits_{{ij}}^{{k + 1(m)}} } \hfill \\ \end{array} $$

  (28)

• The iteration counter is incremented for a new and better estimation of the heads, until the sum of the residual is less than the convergence criterion ε, which has been chosen a priori.

  $$ {\left| {{r}^{{k + 1(m)}}} \right|^2} < \varepsilon $$

  (29)

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