Many subsurface engineering applications involve tight-coupling between fluid flow, solid deformation, fracturing, and similar processes. To better understand the complex interplay of different governing equations, and therefore design efficient and safe operations, numerical simulations are widely used. Given the relatively long time-scales of interest, fully-implicit time-stepping schemes are often necessary to avoid time-step stability restrictions. A major computational bottleneck for these methods, however, is the linear solver. These systems are extremely large and ill-conditioned. Because of the wide range of processes and couplings that may be involved – e.g. formation and propagation of fractures, deformation of the solid porous medium, viscous flow of one or more fluids in the pores and fractures, complicated well sources and sinks, etc. – it is difficult to develop general-purpose but scalable linear solver frameworks. This challenge is further aggravated by the range of different discretization schemes that may be adopted, which have a direct impact on the linear system structure. To address this obstacle, we describe a flexible strategy based on multigrid reduction (MGR) that can produce purely algebraic preconditioners for a wide spectrum of relevant physics and discretizations. We demonstrate that MGR, guided by physics and theory in block preconditioning, can tackle several distinct and challenging problems, notably: a hybrid discretization of single-phase flow, compositional multiphase flow with complex wells, and hydraulic fracturing simulations. Extension to other systems can be handled quite naturally. We demonstrate the efficiency and scalability of the resulting solvers through numerical examples of difficult, field-scale problems.

许多地下工程应用涉及流体流动、固体变形、压裂和类似过程之间的紧密耦合。为了更好地理解不同控制方程之间复杂的相互作用，从而设计高效和安全的操作，数值模拟被广泛使用。鉴于感兴趣的时间尺度相对较长，通常需要完全隐式时间步长方案来避免时间步长稳定性限制。然而，这些方法的主要计算瓶颈是线性求解器。这些系统非常庞大且条件不佳。由于可能涉及的过程和耦合范围广泛——例如裂缝的形成和扩展、固体多孔介质的变形、孔隙和裂缝中一种或多种流体的粘性流动、复杂的井源和汇等。– 很难开发通用但可扩展的线性求解器框架。可能采用的不同离散化方案的范围进一步加剧了这一挑战，这对线性系统结构有直接影响。为了解决这个障碍，我们描述了一种基于多重网格缩减 (MGR) 的灵活策略，该策略可以为广泛的相关物理和离散化生成纯代数预处理器。我们证明 MGR 在区块预处理的物理和理论指导下，可以解决几个独特且具有挑战性的问题，特别是：单相流的混合离散化、复杂井的组合多相流和水力压裂模拟。可以很自然地处理对其他系统的扩展。