Amro M. M. Elfeki

A Markov Chain Model for Subsurface Characterization: Theory and Applications

This paper proposes an extension of a single coupled Markov chain model to characterize heterogeneity of geological formations, and to make conditioning on any number of well data possible. The methodology is based on the concept of conditioning a Markov chain on the future states. Because the conditioning is performed in an explicit way, the methodology is efficient in terms of computer time and storage. Applications to synthetic and field data show good results.

Geostatistical analysis using GIS for mapping groundwater quality: case study in the recharge area of Wadi Usfan, western Saudi Arabia

The scarcity of water is one of the main issues in Saudi Arabia. In particular, the extreme climate in the form of less frequent rainfall affects the groundwater availability. Moreover, groundwater has been depleted by the increase in population. In this research, the spatial distribution of groundwater quality has been developed, and the prediction of groundwater chemical parameters has been made using geostatistical analysis in geographic information system (GIS) software. The study area is Madrakah village as the recharge area of Wadi Usfan located in the western region of Saudi Arabia. Ordinary kriging method was applied to map the spatial distribution of the groundwater chemistry. Most of the groundwater is not suitable for drinking purposes. Groundwater chemical parameters are decreasing toward the eastern part of Madrakah village. In predicting groundwater chemistry distribution maps, data transformation has been executed to reduce the skewness on most of the chemical parameters. The best semivariogram model for every parameter varies based on the root mean square error (RMSE) criterion. The groundwater chemical parameters, i.e., Na+, Mg2+, Cl−, conductivity, salinity, and total dissolved solid (TDS), have a strong spatial dependence, while, NO3− and temperature have a moderate and weak spatial dependence, respectively.

Simulation of solute transport under oscillating groundwater flow in homogeneous aquifers

In this paper, we focus on the influence of temporal variations in the regional hydraulic gradient, particularly, on the impact of temporal variations in the boundary conditions on the spreading of solute plumes in homogenous aquifers. We examined the problem by numerical simulations. Twodimensional fully implicit finite difference model TRANS_GW_2D for the unsteady groundwater flow and a random walk particle tracking model TRANS_RW_2D for solute transport have been developed to solve the governing equations without restriction on the values of aquifer storativity or on the magnitude of the temporal fluctuations. It has been shown that transient flow conditions (in terms of gradient magnitude variability) have a significant impact on contaminant transport only if the amplitude and period of the oscillations are relatively large. For relatively small oscillations, a steady state flow field can be justified. Transient conditions may be relevant in coastal aquifers with high tidal amplitudes. This tidal variation can have an effect on the spreading of solutes and on salt-water intrusion. Our numerical experiments demonstrate that in case of relatively high storativity values, the dispersion coefficient is amplified in time as the plume moves towards the fluctuating boundary.

Multi-Scale and Multi-Resolution Stochastic Modeling of Subsurface Heterogeneity by Tree-Indexed Markov Chains

A new methodology is proposed to handle multi-scale heterogeneous structures. It can be of importance in the field of hydrogeology and for petroleum engineers who are interested in characterizing subsurface heterogeneity at various scales. The framework of this methodology is based on a coarse to fine scale representation of the heterogeneous structures on trees. Different depths in the tree correspond to different spatial scales in representing the heterogeneous structures on trees. On these trees a Markov chain is used to describe scale to scale transitions and to account for the uncertainty in the stochastically generated images.We focus in this work on the description and application of the methodology to synthetic data that are geologically realistic. The methodology is flexible. Conditioning on field data and measurements is straightforward. Non-stationary and stationary fields, compound and nested structures can be addressed.

Solute Transport in Single and Multiple Scale Heterogeneous Formations: Numerical Experiments

The heterogeneity of natural formations has a significant impact on the spreading characteristics of groundwater contaminants. This study focuses on the influence of different geological settings and characterization methods on transport characteristics in heterogeneous porous formations. Several patterns of geological structures have been synthesized by different techniques and then the two dimensional form of flow and solute transport equations are solved numerically in these structures. The generated fields considered in these experiments are stationary Gaussian fields, Markovian fields developed by [Elfeki, Uffink and Barends, 1995] and combiend random fields (Gaussian-Markovian fields).

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.RésuméL’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.ResumenLa 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.摘要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。ResumoA 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.

Understanding the Non-Gaussian Nature of Linear Reactive Solute Transport in 1D and 2D: From Particle Dynamics to the Partial Differential Equations

In the present study, we examine non-Gaussian spreading of solutes subject to advection, dispersion and kinetic sorption (adsorption/desorption). We start considering the behavior of a single particle and apply a random walk to describe advection/dispersion plus a Markov chain to describe kinetic sorption. We show in a rigorous way that this model leads to a set of differential equations. For this combination of stochastic processes, such a derivation is new. Then, to illustrate the mechanism that leads to non-Gaussian spreading, we analyze this set of equations at first leaving out the Gaussian dispersion term (microdispersion). The set of equations now transforms to the telegrapher’s equation. Characteristic for this system is a longitudinal spreading that becomes Gaussian only in the longtime limit. We refer to this as kinetics-induced spreading. When the microdispersion process is included back again, the characteristics of the telegraph equations are still present. Now, two spreading phenomena are active, the Gaussian microdispersive spreading plus the kinetics-induced non-Gaussian spreading. In the long run, the latter becomes Gaussian as well. Another non-Gaussian feature shows itself in the 2D situation. Here, the lateral spread and the longitudinal displacement are no longer independent, as should be the case for a 2D Gaussian spreading process. In a displacing plume, this interdependence is displayed as a ‘tailing’ effect. We also analyze marginal and conditional moments, which confirm this result. With respect to effective properties (velocity and dispersion), we conclude that effective parameters can be defined properly only for large times (asymptotic times). In the two-dimensional case, it appears that the transverse spreading depends on the longitudinal coordinate. This results in ‘cigar-shaped’ contours.

A Coupled Markov Chain Model for Quantification of Uncertainty in Transport in Heterogeneous Formations

Field tracer tests show that aquifer heterogeneity leads to irregularly shaped contaminant plumes. Our inability to characterize this heterogeneity deterministically suggests that predictions of plume sizes must be expressed in probabilistic terms. In this paper the uncertainty in predicting solute transport is addressed. Different kinds of uncertainty are distinguished: geological, parameter and a combination of both. Geological uncertainty is handled with coupled Markov chains, while parameter uncertainty is treated in the classical Gaussian way. Calculations have been performed for 100 realizations. It is demonstrated that the coupled Markov chains model is successful in quantifying geological uncertainty in systems with discrete features. A fixed geological structure (a single realization) gives a significantly different dispersion behavior from the ensemble mean. It has been found that the contaminated area can be indicated with some certainty, but the local concentration at a given point is highly uncertain. Outside the envelope of all possible plumes the concentration is practically zero, with certainty. Deterministic information about the geological structure reduces the uncertainty in the local concentration.

Development of storm hyetographs for flood forecasting in the Kingdom of Saudi Arabia

This paper presents the derivation of the design storm hyetograph patterns for the Kingdom of Saudi Arabia based on real rainfall events from meteorological stations distributed throughout the Kingdom. Two thousand twenty-seven rainfall storms for a 20–28-year period were collected and analyzed covering 13 regions of the Kingdom. Four distinct dimensionless rainfall hyetograph patterns have been obtained over the Kingdom, while two patterns have been obtained for each individual region because of the lack of data for long-duration storms in individual regions. The resulting dimensionless rainfall patterns for each region can be used to develop storm hyetographs for any design duration, total rainfall depth and return period. It has been shown that the developed storm hyetographs have different features from other storm patterns that are commonly used in arid zones. The study recommends using these curves for the design of hydraulic structures in Kingdom of Saudi Arabia and regions alike.

Prediction of contaminant plumes (shapes, spatial moments and macrodispersion) in aquifers with insufficient geological information

The first Macrodispersion Experiment (MADE1) at Columbus Air Force Base in northern Mississippi is utilized to perform numerical simulations of solute transport in an aquifer. The purpose is to illustrate the capability of the coupled Markov chain (CMC) model in delineating the complex geometrical configuration at the site for solute transport simulations under the lack of geological information. The CMC model is also used to study the effect of reducing geological information on the transport predictions in terms of plume configuration, first and second spatial moments and macrodispersion. The results show the power of the coupled Markov chain methodology in delineating aquifer heterogeneity at the site. Conditional simulations on 16, 9 and 6 boreholes show reasonably the same plume behavior in terms of average longitudinal and vertical extensions, especially in the far-field; 9 boreholes seem to provide practically acceptable results in terms of the global plume shape. This indicates more reliability on the use of the CMC model for subsurface characterization. Comparison of CMC model results, in terms of aquifer characterization, with the model used by Eggleston and Rojstaczer [Water Resour. Res. 34 (1998) 2155] (polynomial regression trend, Kalman filter trend, hydrofacies trend and Kriging) shows that the CMC model behaves better. The CMC model conditioned on 9 and 16 boreholes with mid-range conductivities for each lithology captures both the plume shape and the observed plume spatial moments at the MADE site.