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Hybrid Fixed-Point Fixed-Stress Splitting Method for Linear Poroelasticity

机译：线性多孔弹性的混合定点固定应力分裂方法

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摘要

Efficient and accurate poroelasticity models are critical in modeling geophysical problems such as oil exploration, gas-hydrate detection, and hydrogeology. We propose an efficient operator splitting method for Biot’s model of linear poroelasticity based on fixed-point iteration and constrained stress. In this method, we eliminate the constraint on time step via combining our fixed-point approach with a physics-based restraint between iterations. Three different cases are considered to demonstrate the stability and consistency of the method for constant and variable parameters. The results are validated against the results from the fully coupled approach. In case I, a single iteration is used for continuous coefficients. The relative error decreases with an increase in time. In case II, material coefficients are assumed to be linear. In the single iteration approach, the relative error grows significantly to 40% before rapidly decaying to zero. This is an artifact of the approximate solutions approaching the asymptotic solution. The error in the multiple iterations oscillates within 10 ? 6 before decaying to the asymptotic solution. Nine iterations per time step are enough to achieve the relative error close to 10 ? 7 . In the last case, the hybrid method with multiple iterations requires approximately 16 iterations to make the relative error 5 × 10 ? 6 .
机译：高效，准确的孔隙弹性模型对于模拟石油勘探，天然气水合物探测和水文地质等地球物理问题至关重要。我们基于定点迭代和约束应力，为Biot的线性多孔弹性模型提出了一种有效的算子拆分方法。在这种方法中，我们通过将定点方法与迭代之间基于物理的约束相结合来消除时间步长的约束。考虑了三种不同的情况，以证明恒定和可变参数方法的稳定性和一致性。对照完全耦合方法的结果验证了结果。在情况I中，单个迭代用于连续系数。相对误差随着时间的增加而减小。在情况II中，假设材料系数是线性的。在单迭代方法中，相对误差在迅速衰减为零之前显着增长到40％。这是逼近渐近解的近似解的伪像。多次迭代中的误差在10？衰减到渐近解前的6。每个时间步长9次迭代足以使相对误差接近10？ 7。在最后一种情况下，具有多次迭代的混合方法需要大约16次迭代，以使相对误差5×10？ 6。

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