高阶自适应有限元三维直流电阻率正演方法及其在沁水盆地煤气层压裂监测中的应用

Forward modeling of 3D DC resistivity based on high-order adaptive finite element and its application in Qinshui Basin

自适应有限元解的精度主要受单元大小和形函数阶数的影响,为了得到高精度的有限元解,需要将网格自适应加密到足够的密度或者在网格中应用较高阶数的形函数,但这会大大增加计算时间,同时消耗大量计算资源。为提高有限元解的精度并节约计算资源,应用h型自适应加密算法并结合高阶形函数实现了三维直流电阻率模型的正演。在程序实现过程中,通过一维多项式的张量积生成三维空间中任意阶数的形函数,应用Kelly后验误差估计指导网格自适应加密。数值算例表明,形函数阶数p=3时,算法具有较高的精度,并且比p=1、2时有限元解的收敛更快,即实现了用最少的自由度个数得到最精确的有限元解。最后通过对沁水盆地南部某煤气层压裂监测区的三维合成数据模型进行直流电阻率模拟,验证了程序的有效性。

Considering the fact that the accuracy of an adaptive finite element solution is mainly affected by the cell size(h)and the order of a shape function(p),we should adaptively refine the meshes to be dense enough or apply a higher-order shape function in the meshes to acquire a high-accuracy finite element solution.However,this will greatly increase the time burden and require sufficient memory space of the computer.In light of these problems,in this paper,we combine an h-adaptive refinement algorithm with a higher-order shape function for the forward modeling of a 3DDC resistivity model.In the program,we use the tensor product of 1D polynomials to generate a shape function of any order in the 3D space and apply Kelly posteriori error estimation to guide the adaptive refinement of meshes.Numerical examples show that in the case of p=3,our program has high accuracy,and the convergence of the finite element solutions is faster than that in the case of p=1 or 2.This means that the most accurate finite element solution with the fewest degrees of freedom can be obtained when the order of the shape function is 3.Finally,the simulation of 3 DDC resistivity is performed on the 3D modeling of a coalbed fracturing monitoring region in the southern Qinshui Basin,verifying the effectiveness of our program.