Abstract
The equivalent porous medium (EPM) method is an efficient approximate processing method for calculating groundwater yield taking into account the equivalent permeability coefficient of a fractured geological medium. The EPM method finds wide application in addressing various hydrogeological problems ranging from local to regional scales; however, the adequacy of the EPM model for evaluating water head and velocity distributions has not been comprehensively assessed. This study quantitatively investigated the influence of fracture and matrix permeability on the EPM model’s suitability, defined by a 15% threshold for hydraulic-head prediction error, via numerical simulations. A fractured porous media system (fracture-matrix system) was considered as the prototype, and the EPM model simulation results were compared with those obtained using the discrete fracture-matrix (DFM) model. Results indicate a decrease in EPM suitability with larger fracture apertures. With a constant fracture aperture, the suitability of the EPM model increases as the matrix permeability increases. The size of the fracture aperture significantly affects the suitability of the EPM model, and it determines the point at which its suitability begins to increase and eventually stabilize. The fitted curve depicting the influence of matrix permeability on the suitability of the EPM model conforms to the Boltzmann formula, and the fracture aperture is linearly related to the parameter x0 in the formula. The derived empirical formula enables quantitative assessment of the impact of fracture and matrix permeabilities on the suitability of the EPM model in fractured porous media.
Résumé
La méthode du milieu poreux équivalent (MPE) est une méthode efficace de traitement approximatif pour calculer le rendement des eaux souterraines en tenant compte du coefficient de perméabilité équivalent d’un milieu géologique fracturé. La méthode MPE trouve une large application dans le traitement de divers problèmes hydrogéologiques allant de l’échelle locale à l’échelle régionale. Toutefois, l’adéquation du modèle MPE pour l’évaluation des distributions de la charge hydraulique et de la vitesse n’a pas fait l’objet d’une évaluation complète. Cette étude a étudié quantitativement l’influence de la perméabilité des fractures et de la matrice sur l’adéquation du modèle MPE, définie par un seuil de 15% d’erreur de prédiction de la charge hydraulique, à l’aide de simulations numériques. Un système de milieu poreux fracturé (système fracture-matrice) a été considéré comme le prototype, et les résultats de simulation du modèle MPE ont été comparés à ceux obtenus à l’aide du modèle matrice-fracture discrète (MFD). Les résultats indiquent une diminution de l’adéquation du MPE avec des ouvertures de fracture plus grandes. Avec une ouverture de fracture constante, l’adéquation du modèle MPE augmente à mesure que la perméabilité de la matrice augmente. La taille de l’ouverture de la fracture affecte considérablement l’adéquation du modèle MPE et détermine le point à partir duquel son adéquation commence à augmenter et finit par se stabiliser. La courbe ajustée décrivant l’influence de la perméabilité de la matrice sur l’adéquation du modèle MPE est conforme à la formule de Boltzmann, et l’ouverture de la fracture est linéairement liée au paramètre x0 dans la formule. La formule empirique dérivée permet une évaluation quantitative de l’impact de la perméabilité des fractures et de la matrice sur l’adéquation du modèle MPE dans les milieux poreux fracturés.
Resumen
El método del medio poroso equivalente (EPM) es un método eficaz de tratamiento por aproximación para calcular el rendimiento de las aguas subterráneas teniendo en cuenta el coeficiente de permeabilidad equivalente de un medio geológico fracturado. El método EPM encuentra una amplia aplicación en el tratamiento de diversos problemas hidrogeológicos que van desde la escala local a la regional. Sin embargo, la adecuación del modelo EPM para evaluar las distribuciones de altura y velocidad del agua no se ha evaluado de forma exhaustiva. Este estudio investiga cuantitativamente la influencia de la permeabilidad de la fractura y de la matriz en la adecuación del modelo EPM, definida por un umbral del 15% para el error de predicción de la altura hidráulica, mediante simulaciones numéricas. Se consideró como prototipo un sistema de medio poroso fracturado (sistema fractura-matriz), y los resultados de la simulación del modelo EPM se compararon con los obtenidos utilizando el modelo de fractura-matriz discreta (DFM). Los resultados indican una disminución de la adecuación del EPM con aperturas de fractura mayores. Con una apertura de fractura constante, la adecuación del modelo EPM aumenta a medida que lo hace la permeabilidad de la matriz. El tamaño de la apertura de fractura afecta significativamente a la adecuación del modelo EPM, y determina el punto en el que comienza a aumentar su adecuación y finalmente se estabiliza. La curva ajustada que representa la influencia de la permeabilidad de la matriz en la idoneidad del modelo EPM se ajusta a la fórmula de Boltzmann, y la apertura de la fractura está linealmente relacionada con el parámetro x0 de la fórmula. La fórmula empírica derivada permite evaluar cuantitativamente el impacto de las permeabilidades de la fractura y de la matriz sobre la adecuación del modelo EPM en medios porosos fracturados.
摘要
等效多孔介质(EPM)方法是一种用考虑裂缝地质介质的等效渗透系数来计算地下水可开采量有效的近似处理方法。EPM方法被广泛应用于从局部到区域尺度解决各种水文地质问题。然而,EPM模型在评估水头和流速分布方面的适用性还没有得到全面评估。本研究通过数值模拟定量研究了裂缝和基质渗透性对EPM模型适用性的影响, 该适用性由水头预测误差的15%阈值定义。考虑了一个裂缝多孔介质系统(裂缝-基质系统)作为原型, 并将EPM模型的模拟结果与使用离散裂缝-基质(DFM)模型得到的结果进行了比较。结果表明, 裂缝孔径越大, EPM的适用性下降。在裂缝孔径不变的情况下, 随着基质渗透性的增加, EPM模型的适用性增强。裂缝孔径的大小显著影响EPM模型的适用性, 并决定了其适用性开始增加并最终到达稳定。描述基质渗透性对EPM模型适用性影响的拟合曲线符合波尔兹曼公式, 而裂缝孔径与公式中的参数x0线性相关。推导的经验公式使得能够定量评估裂缝和基质渗透性对裂缝多孔介质中EPM模型适用性的影响。
Resumo
O método do meio poroso equivalente (MPE) é um método eficiente de processamento aproximado para calcular a vazão de água subterrânea, levando em consideração o coeficiente de permeabilidade equivalente de um meio geológico fraturado. O método MPE encontra ampla aplicação na resolução de vários problemas hidrogeológicos, abrangendo desde escalas locais até regionais. No entanto, a adequação do modelo MPE para avaliar as distribuições de carga hidráulica e velocidade não foi avaliada de forma abrangente. Este estudo investigou quantitativamente a influência da permeabilidade das fraturas e da matriz na adequação do modelo MPE, definida por um limiar de 15% para o erro de previsão da carga hidráulica, por meio de simulações numéricas. Foi considerado um sistema de meios porosos fraturados (sistema fratura-matriz) como protótipo, e os resultados das simulações do modelo MPE foram comparados com aqueles obtidos utilizando o modelo de fratura-matriz discreto (FMD). Os resultados indicam uma diminuição na adequação do modelo MPE com aberturas de fraturas maiores. Com uma abertura de fratura constante, a adequação do modelo MPE aumenta à medida que a permeabilidade da matriz aumenta. O tamanho da abertura da fratura afeta significativamente a adequação do modelo MPE, determinando o ponto em que sua adequação começa a aumentar e eventualmente se estabiliza. A curva ajustada que representa a influência da permeabilidade da matriz na adequação do modelo MPE segue a fórmula de Boltzmann, e a abertura da fratura está linearmente relacionada ao parâmetro x0 na fórmula. A fórmula empírica derivada permite a avaliação quantitativa do impacto das permeabilidades das fraturas e da matriz na adequação do modelo MPE em meios porosos fraturados.
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The authors would like to acknowledge the editors and anonymous reviewers for their invaluable discussion and suggestions.
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This work was supported by the National Natural Science Foundation of China (Nos. U2267218 and 42072276).
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Appendix: Nomenclature
Appendix: Nomenclature
Symbol | Description | Units |
q | Unit width of discharge | L2T−1 |
Q | Source-sink term at fracture | L2T−1 |
N′ | Degree of node i | |
N | Number of nodes | |
M | Number of line elements | |
A | An aggregation matrix reflecting the aggregation relationship between the line elements and nodes | |
q | q = (q1,q2,…,qN)T | L2T−1 |
Q | Q = (Q1,Q2,…,QN)T | L2T−1 |
μ | Dynamic viscosity coefficient of the fluid | ML−1 T−1 |
γ | Specific gravity of groundwater | |
Jf | Hydraulic gradient of fracture water flow | |
b | Fracture width | L |
V | Average velocity | LT−1 |
K | Hydraulic conductivity | LT−1 |
l | Fracture length | L |
ΔH | Head difference | L |
Rj | Hydraulic conductivity of the j line element | LT−1 |
h | Head | L |
ΔH | ΔH = (ΔH1, ΔH2,…,ΔHM)T | L |
R | R = diag(R1,R2,…,RM) | LT−1 |
H | H = (h1, h2,…, hM)T | L |
W | Source-sink term | LT−1 |
mi | Number of units with common nodes i, j | |
K | Permeability coefficient matrix | LT−1 |
F | Known constant term matrix | L2T−1 |
w | Infiltration or evaporation water | LT−1 |
m | Number of observations | |
a | Aperture | L |
\({\delta }_{i}\) | Relative error | |
η | Suitability of the EPM model |
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Han, D., Ma, L., Qian, J. et al. The effect of fracture aperture and matrix permeability on suitability of the equivalent porous medium model for steady-state flow in fractured porous media. Hydrogeol J (2024). https://doi.org/10.1007/s10040-024-02781-3
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DOI: https://doi.org/10.1007/s10040-024-02781-3