Synchrotron x-ray microtomography combined with sensitive pressure differential measurements were used to study flow during steady-state injection of equal volume fractions of two immiscible fluids of similar viscosity through a 57-mm-long porous sandstone sample for a wide range of flow rates. We found three flow regimes. (1) At low capillary numbers, Ca, representing the balance of viscous to capillary forces, the pressure gradient, $\mathbf{\nabla }P$, across the sample was stable and proportional to the flow rate (total Darcy flux) $q_t$ (and hence capillary number), confirming the traditional conceptual picture of fixed multiphase flow pathways in porous media. (2) Beyond ${\text{Ca}}^{*}\approx {10}^{-6}$, pressure fluctuations were observed, while retaining a linear dependence between flow rate and pressure gradient for the same fractional flow. (3) Above a critical value $\text{Ca}>{\text{Ca}}^i\approx {10}^{-5}$ we observed a power-law dependence with $\mathbf{\nabla }P\sim q_t^a$ with $a\approx 0.6$ associated with rapid fluctuations of the pressure differential of a magnitude equal to the capillary pressure. At the pore scale a transient or intermittent occupancy of portions of the pore space was captured, where locally flow paths were opened to increase the conductivity of the phases. We quantify the amount of this intermittent flow and identify the onset of rapid pore-space rearrangements as the point when the Darcy law becomes nonlinear. We suggest an empirical form of the multiphase Darcy law applicable for all flow rates, consistent with the experimental results.

中文翻译：

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孔隙尺度动力学和多西达西定律

通过同步加速器X射线显微断层照相术结合灵敏的压差测量，研究了通过57毫米长的多孔砂岩样品在稳态下以相等的体积分数注入两种相似粘度的两种不混溶流体时的流动情况，并获得了宽范围的流速。我们发现了三种流动状态。（1）在毛细管数低的情况下，Ca代表粘性与毛细管力之间的平衡，压力梯度，$\mathbf{\nabla }P$，整个样品稳定且与流速成正比（总达西通量） $q_{Ť}$（以及毛细管数），这证实了多孔介质中固定多相流路的传统概念图。（2）超越${\text{钙}}^{*}\approx {10}^{-6}$观察到压力波动，同时对于相同的分流，流量与压力梯度之间保持线性关系。（3）超过临界值$\text{钙}>{\text{钙}}^{\mathrm{一世}}\approx {10}^{-5}$ 我们观察到 $\mathbf{\nabla }P〜q_{Ť}^{\mathrm{一种}}$ 与 $\mathrm{一种}\approx 0.6$与压力差的快速波动有关，该波动幅度等于毛细管压力。在孔隙尺度上，捕获了孔隙空间部分的瞬时或间歇性占据，其中局部流动路径被打开以增加相的电导率。我们量化这种间歇流动的量，并确定快速的孔隙空间重排的发生是达西定律变为非线性时的点。我们建议适用于所有流速的多相达西定律的经验形式，与实验结果一致。