硬币型裂缝介质的频散与衰减

Frequency dispersion and attenuation of porous rocks with penny shaped fractures

为了快速、准确地模拟硬币型裂缝的频散和衰减,同时避免求解Galvin硬币型裂缝频变弹性模量表达式中存在的奇异性问题,基于Gauss-Lobatto等效离散积分的求解方法,将求解的第二类Fredholm积分方程转变为有限区间内的离散积分,并利用高阶近似方法将远场多重散射方程在零极限的求解问题与离散积分过程统一;进一步利用各向异性Gassmann方程,结合线性滑移理论与Hudson硬币型裂缝模型,分别给出了裂缝介质在高、低频极限条件下各个弹性模量的解析式。相速度和黏弹性反射系数数值模拟表明:随裂缝密度变大,衰减峰值变大;随裂缝尺度变大或流体黏度增强,衰减峰值频率向低频移动;相对于平行裂缝方向,垂直方向的速度和衰减变化最大,且当介质饱气时,反射系数明显大于含水、含油介质。数值模拟结果表明裂缝密度影响衰减,裂缝尺度和流体黏度影响介质由弛豫向非弛豫过渡的频带位置。

Conventional integral methods usually have the singularity problem in solving frequency-dependent elastic modulus of penny shaped fractures,and can only obtain a single normal frequency-dependent modulus.In order to quickly and accurately perform numerical simulation of the penny shaped fracture,based on the Gauss-Lobatto discrete integral,a new approach is proposed.First the second Fredholm integral equation is transformed into a discrete integral in a finite interval.Then,the solving the zero-limit of far field multiple scatter equation and the discrete integral procedure are unified with the high-order approximation.To obtain the frequency-dependent elastic tensor,analytic elastic moduli equations at the high and low frequency limits are respectively obtained with the anisotropic Gassmann equation,the linear slip theory,and the Hudson crack model.Based on the analysis of phase velocity and viscoelastic reflection coefficient,the following understanding are obtained:A.The greater the fracture denty is,the graeter the attenuated peak amplitude is;B.The attenuated peak amplitude has lower frequency when the fracture size or the fluid viscosity becomes greater;C.The incident velocity and attenuation in the vertical direction have greater change than that in the horizontal direction;D.The reflection coefficient of gas-bearing media is significantly larger than that of oil-bearing or water-bearing media.