We develop a new multiscale model to compute effective properties such as relative permeability, contact angle and partition coefficients in low salinity enhanced oil recovery processes for two-phase flow in sandstones containing reactive surfaces of kaolinite clay. In this setting, we construct a three-scale approach which entails the local nanoscale description ruled by the electro-chemistry of a confined electrolyte solution containing Na+, Ca2+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ca^{2+}$$\end{document}, H+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^+$$\end{document}, Cl-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Cl^-$$\end{document} and OH-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$OH^-$$\end{document} ions residing between bounded crude-oil droplets at residual saturation and clay substrate. Our analysis focuses on the case of surface complexation geochemical reactions between the ionic species of the invading water and the electrically charged kaolinite and oil–water interfaces. In this scenario, we construct a local electric double layer problem for the electric potential based on a non-symmetric Poisson–Boltzmann equation supplemented by nonlinear boundary conditions with the magnitude of the surface charge strongly dictated by the geochemical reactions. By invoking the local mechanical equilibrium of the electrolyte solution and solving numerically the nonlinear problem using the finite element method, we compute the local ionic profiles and reconstruct numerically the disjoining pressure and adsorption isotherms for each ionic species for a wide range of brine compositions and pH of the water phase. Furthermore, combining the disjoining pressure results with the Frumkin/Derjaguin wetting theory allows to compute the dependence of the contact angle on wettability, pH and salinity. Subsequently, the formal homogenization procedure is adopted to upscale the pore-scale flow and ion transport to the macroscale giving rise to a new Darcy scale coupled flow/transport model. The hyperbolic part of the nonlinear homogenized model is solved analytically in an 1D example of enhanced oil recovery.

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含粘土砂岩低盐度注水的新多尺度计算模型

我们开发了一种新的多尺度模型，用于计算含高岭石粘土反应表面的砂岩中两相流的低盐度提高采收率过程中的有效属性，例如相对渗透率、接触角和分配系数。在这种情况下，我们构建了一个三尺度方法，该方法需要由包含 Na+、Ca2+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} 的受限电解质溶液的电化学控制的局部纳米级描述\usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ca^{2+}$$ \end{文档}, 我们的分析侧重于入侵水的离子物质与带电高岭石和油水界面之间的表面络合地球化学反应的情况。在这种情况下，我们基于非对称 Poisson-Boltzmann 方程构建了电位的局部双电层问题，并辅以非线性边界条件，表面电荷的大小由地球化学反应强烈决定。通过调用电解质溶液的局部机械平衡并使用有限元方法数值求解非线性问题，我们计算局部离子分布并数值重建每个离子种类的分离压力和吸附等温线，适用于各种盐水成分和 pH 值的水相。此外，将分离压力结果与 Frumkin/Derjaguin 润湿理论相结合，可以计算接触角对润湿性、pH 值和盐度的依赖性。随后，采用正式的均质化程序将孔隙尺度流动和离子传输放大到宏观尺度，从而产生新的达西尺度耦合流动/传输模型。非线性均质模型的双曲线部分在提高采收率的一维示例中解析求解。