Abstract Hydraulic fracturing treatment in rocks creates surfaces that are not smooth but rough in general. Accurate characterization of surface roughness is necessary to relate fracture deformation to fluid flow. In this study, we analyze the surface of an experimentally generated hydraulic fracture using a practical fractal approach which is capable of modeling applications. The hydraulic fracturing test is conducted using a nearly homogeneous siltstone cube in a true triaxial cell, and a fracture is created showing a perfectly radial pattern. To evaluate roughness, each surface profile is decomposed into large-scale fracture waviness and localized surface roughness considering various length scales. Despite the waviness, estimated roughness amplitudes follow a power-law relation up to a length-scale, showing a fractal nature. Unlike ideal brittle materials with an exponent of 0.5, the roughness exponent is found to vary in a narrow range of 0.1 but exceeds 0.5. The fractal dimension (box dimension) of the hydraulic fracture surface is estimated to be 1.4 showing a good match with roughness exponents. An increase in roughness exponent may indicate an increasing difficulty in fracture propagation and fluid and proppant transport along the fracture. As such, the topology of a hydraulic fracture surface is essential to hydraulic fracture growth to assess fracturing performance.

中文翻译：

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实验室水力裂缝表面粗糙度分析的分形方法

摘要 岩石中的水力压裂处理产生的表面通常不光滑但粗糙。表面粗糙度的准确表征对于将裂缝变形与流体流动联系起来是必要的。在这项研究中，我们使用能够模拟应用的实用分形方法分析实验生成的水力压裂的表面。水力压裂试验是在真正的三轴单元中使用几乎均质的粉砂岩立方体进行的，并且产生的裂缝显示出完美的径向图案。为了评估粗糙度，考虑到各种长度尺度，每个表面轮廓被分解为大尺度断裂波纹和局部表面粗糙度。尽管有波纹，但估计的粗糙度幅度在长度尺度上遵循幂律关系，显示出分形性质。与指数为 0.5 的理想脆性材料不同，粗糙度指数在 0.1 的狭窄范围内变化，但超过 0.5。水力裂缝表面的分形维数（框维数）估计为 1.4，显示出与粗糙度指数的良好匹配。粗糙度指数的增加可能表明裂缝扩展以及流体和支撑剂沿裂缝传输的难度增加。因此，水力压裂面的拓扑结构对于水力压裂生长评估压裂性能至关重要。粗糙度指数的增加可能表明裂缝扩展以及流体和支撑剂沿裂缝传输的难度增加。因此，水力压裂面的拓扑结构对于水力压裂生长评估压裂性能至关重要。粗糙度指数的增加可能表明裂缝扩展以及流体和支撑剂沿裂缝传输的难度增加。因此，水力压裂面的拓扑结构对于水力压裂生长评估压裂性能至关重要。