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    基于迭代求解的储层时移大地电磁法三维有限元正演方法

    Iterative finite element-based 3D time-lapse magnetotelluric forward modeling of reservoir structures

    • 摘要: 储层时移大地电磁监测需要在不同开发阶段对同一区域反复进行三维正演计算,由于模型规模大、频率点多,传统直接求解方法内存消耗随自由度急剧增长,难以满足大规模时移正演的实际需求。针对这一问题,采用灵活广义最小残差法(flexible generalized minimal residual method,FGMRES)作为外层迭代求解器,结合辅助空间Maxwell求解器(auxiliary-space maxwell solver,AMS)预条件替代直接求解器。将复系数线性方程组转化为等价的实对称形式,构造2×2分块对角预条件矩阵,内层利用AMS对每个实对称子块进行预条件求解,以加快Krylov子空间收敛。通过层状模型验证算法的正确性,视电阻率与一维解析解吻合良好,最大相对误差小于2.5%;在含储层时移电阻率异常体的三维模型中,算法基本在20次迭代内收敛,求解结果与Pardiso直接求解器得到的视电阻率和相位响应特征一致。对于10989659条棱边的模型,与Pardiso直接求解器相比,求解时间由47196.8 s降至9780.2 s,峰值内存由320.3GB降至122.2GB,分别降低约79.3%和61.9%。该迭代求解方案能够支撑多期次时移正演的计算需求,为大规模储层时移大地电磁三维正演提供了高效可行的技术途径。

       

      Abstract: Time-lapse magnetotelluric (MT) monitoring of reservoirs requires repeated 3D forward modeling over the same region at different development stages. Due to the large model size and numerous frequencies involved, conventional vector finite element methods based on direct solvers such as LU decomposition suffer from memory consumption that grows rapidly with the number of degrees of freedom, making them impractical for large-scale time-lapse forward modeling. To address this issue, we employ the flexible generalized minimal residual (FGMRES) method as the outer iterative solver, combined with the auxiliary-space Maxwell solver (AMS) as a preconditioner to replace the direct solver. The complex-valued linear system is reformulated into an equivalent real-valued symmetric form, and a 2×2 block-diagonal preconditioner is constructed, in which AMS is applied to each real symmetric sub-block as the inner preconditioner, effectively accelerating the convergence of the Krylov subspace iteration. The algorithm is first validated against a layered model, where the apparent resistivity agrees well with the 1D analytical solution, with a maximum relative error of less than 2.5%. For a 3D model containing a reservoir-related time-lapse resistivity anomaly, convergence is generally achieved within 20 iterations over the tested frequency range, and the iterative results exhibit apparent-resistivity and phase responses consistent with those obtained by the Pardiso direct solver. For the model with 10989659 edges, compared with the Pardiso direct solver, the computational time is reduced from 47196.8 to 9780.2 s, and the peak memory usage is reduced from 320.3 to 122.2 GB, corresponding to reductions of approximately 79.3% and 61.9%, respectively. These results demonstrate that the proposed iterative solution scheme can support the computational demands of multi-vintage time-lapse forward modeling and provides an efficient and viable approach for large-scale 3D time-lapse MT forward modeling of reservoirs.

       

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