Porous flow is of major importance in the shallow subsurface, since it directly impacts on reservoir-scale processes such as waste fluid sequestration or oil and gas exploration. Coupled and non-linear hydro-mechanical processes describe the motion of a low-viscous fluid interacting with a higher viscous porous rock matrix. This two-phase flow may trigger the initiation of solitary waves of porosity, further developing into vertical high-porosity pipes or chimneys. These preferred fluid escape features may lead to localised and fast vertical flow pathways potentially problematic in the case of for instance CO₂ sequestration. Constraining the porosity and the non-linearly related permeability distribution in such environments is a major challenge. Although seismic imaging methods accurately localise the high-porosity chimneys in the inverted wave-speed field, the conversion to porosity is not straightforward. We develop an inversion framework to reconstruct the unknown porosity field using relevant observable quantities such as subsurface fluid fluxes. We introduce the adjoint framework for the two-phase flow equations, which allows for efficient calculations of the pointwise gradients of the flow solution with respect to the porosity. We then use the gradients in a gradient descent method to invert for the pointwise porosity. We solve the forward and the adjoint equations using an iterative matrix-free pseudo-transient approach with the finite difference method. The proposed parallel solving procedure executes optimally on the latest many-core hardware accelerators such as GPUs. Numerical results show that an inversion for porosity is challenging if data is sparse since the porosity is very locally sensitive to the fluid flux. We introduce the concepts of sensitivity kernels as employed in seismology for the set of two-phase equations and suggest this approach as a standard for future studies.

多孔渗流在浅层地下非常重要，因为它直接影响储层规模的过程，例如固存废液或油气勘探。耦合的非线性流体力学过程描述了低粘度流体与较高粘性的多孔岩石基质相互作用的运动。这种两相流可能触发孔隙孤波的产生，并进一步发展成垂直的高孔隙度管道或烟囱。这些优选的流体逸出特征可能导致局部和快速的垂直流动路径，例如在CO ₂的情况下可能会出现问题。隔离。在这样的环境中限制孔隙率和非线性相关的渗透率分布是一个重大挑战。尽管地震成像方法可以将高孔隙度烟囱准确地定位在反向波速场中，但转换为孔隙度并非易事。我们开发了一个反演框架，使用相关的可观测量（例如地下流体通量）来重建未知孔隙率场。我们引入了两相流方程的伴随框架，该框架允许高效地计算流动解决方案相对于孔隙度的逐点梯度。然后，我们使用梯度下降法中的梯度来反演逐点孔隙度。我们使用带有限差分法的无迭代矩阵伪暂态方法求解正向方程和伴随方程。所提出的并行求解过程可以在最新的多核硬件加速器（例如GPU）上最佳执行。数值结果表明，如果数据稀疏，则孔隙度的反演将具有挑战性，因为孔隙度对流体通量非常局部敏感。我们介绍了地震学中用于两相方程组的敏感性核的概念，并建议将此方法作为将来研究的标准。