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Spatial Scaling of Effective Modulus and Correlation of Deformation Near the Critical Point of Fracturing

机译:有效模量的空间尺度和压裂临界点附近的变形相关性

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摘要

Many observations point to the lithosphere being metastable and close to a critical mechanical point. Exercises in modelling deformation, past or present, across subsurface reservoirs need to take account of this criticality in an efficient way. Using a renormalization technique, the spatial scaling of effective elastic modulus is derived for 2-D and 3-D bodies close to the critical point of through-going fracturing. The resulting exponent, d μ , of spatial scaling of effective modulus with size, , takes the values ~ −2.5 and −4.2 in two- and three-dimensional space, respectively. The exponents are compatible with those for scaling of effective modulus with fracture density near the percolation threshold determined by other workers from numerical experiments; the high absolute values are also approximately consistent with empirical data from a) fluctuations in depth of a seismic surface; b) `1/k' scaling of heterogeneities observed in one-dimensional well-log samples; c) spatial correlation of slip displacements induced by water injection. The effective modulus scaling modifies the spatial correlation of components of displacement or strain for a domain close to the critical point of fracturing. This correlation function has been used to geostatistically interpolate components of the strain tensor across subsurface reservoirs with the prime purpose of predicting fracture densities between drilled wells. Simulations of strain distributions appear realistic and can be conditioned to surface depths and observations at wells of fracture-related information such as densities and orientations, welltest permeabilities, changes in well-test permeabilities, etc.
机译:许多观察表明岩石圈是亚稳的,并且接近临界力学点。对地下储层过去或现在的变形进行建模的练习需要以有效的方式考虑到这一关键性。使用重归一化技术,可以得出接近于贯通压裂临界点的2-D和3-D体的有效弹性模量的空间比例。有效模量随大小变化的空间缩放比例的最终指数dμ在二维和三维空间中分别取值〜-2.5和-4.2。指数与有效模量缩放的指数兼容,其断裂密度接近其他工人根据数值实验确定的渗流阈值。高绝对值也与来自a)地震表面深度波动的经验数据大致一致; b)在一维测井样本中观察到的异质性的“ 1 / k”缩放; c)注水引起的滑移位移的空间相关性。有效模量缩放可在接近断裂临界点的区域修改位移或应变分量的空间相关性。此相关函数已用于对地下张储层之间的应变张量的分量进行地统计学插值,其主要目的是预测钻井之间的裂缝密度。应变分布的模拟似乎是现实的,并且可以根据表面深度和在裂缝相关信息的井中观察到的条件进行调节,例如密度和方向,试井渗透率,试井渗透率的变化等。

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