Porous media, particularly natural ones such as rock and soil, constitute a broad class of complex systems. Their pore space is highly chaotic, with pore sizes that vary across a broad range. But what makes porous media particularly complex is that the paths that they provide for fluid flow are not straight, but tortuous and meandering. A molecule often must traverse a path that is several times longer than the straight line between its original source and its destination. Not surprisingly, until the early 1970s pore-scale modeling of porous media and the flow and transport processes that take place in them was considered a hopeless task, precisely due to the complex morphology of the pore space. Fluid flow and transport processes were often modeled by averaging the microscopic conservation laws—the continuity and momentum equations—over a suitably selected segment of the pore space, such that on the scale of that segment a porous medium could be considered homogeneous. Such equations then contained effective flow and transport coefficients, such as the effective permeability, diffusivity, and electrical and thermal conductivities. These coefficients had to be measured experimentally for each medium—unless they could be predicted based on some hypothetical model of the pore space. Due to the complexity of pore space morphology, the most used model was the bundle of capillary tubes, in which the tubes, representing the pores, were arranged either in series or parallel (98). While such models were amenable to the analytical derivation of expressions for the effective flow and transport properties, the predictions that they provided rarely agreed with experimental data, simply because the structure (e.g., interconnectivity) of the pore space of almost any natural porous medium was nothing like a bundle of tubes.
多孔介质，尤其是像岩石和土壤这样的自然多孔介质，构成了一类复杂的系统。它们的孔隙空间高度无序，孔隙尺寸变化范围很广。但使多孔介质特别复杂的是，它们为流体流动提供的路径不是直的，而是曲折和蜿蜒的。分子往往必须穿越一条比其原始来源和目的地之间的直线长几倍的路径。不出所料，直到 20 世纪 70 年代初，多孔介质的孔隙尺度建模以及在其中发生的流动和传输过程被认为是一项无望的任务，这正是因为孔隙空间的复杂形态。流体流动和传输过程通常通过在一个适当选择的孔隙空间段上平均微观守恒定律——连续性和动量方程——来建模，这样在该段的尺度上，多孔介质可以被认为是均匀的。这些方程包含了有效流动和传输系数，如有效渗透率、扩散率以及电导率和热导率。 这些系数必须对每种介质进行实验测量——除非它们可以根据孔隙空间的某种假设模型进行预测。由于孔隙空间形态的复杂性，最常用的模型是毛细管束，其中代表孔隙的管子要么串联，要么并联（98）。虽然这些模型适用于推导有效流动和传输特性的表达式，但它们提供的预测很少与实验数据相符，这仅仅是因为几乎所有自然多孔介质的孔隙空间结构（例如，连通性）与管束大相径庭。

To address this glaring shortcoming, the concept of tortuosity was introduced by 12. The purpose (30) was to match the permeability, calculated based on the bundle of capillary tubes, to experimental data. In effect, all relevant length scales were divided by the tortuosity to make them longer than what was provided by the bundle of capillary tubes so as to approximate the actual length of the paths that fluid molecules travel through the pore space.
为了解决这一明显的缺陷，12 提出了曲折度的概念。目的是（30）将基于毛细管束计算的渗透率与实验数据相匹配。实际上，所有相关的长度尺度都除以曲折度，使它们比毛细管束提供的更长，以近似流体分子在孔隙空间中行走的实际路径长度。

While the concept of tortuosity seems straightforward, in practice tortuosity is not consistently defined, and its treatment in the literature is often misleading (107; 33; 88; 75). It is sometimes claimed that tortuosity depends on the type of flow or transport process being studied, and comparisons between predicted and measured fluid flow and transport are used to support that assertion (75; 113). Such a claim for the nature of tortuosity suggests, however, that its use in practice is often as an adjustable parameter—a “fudge factor”—hence implying that a clear understanding of tortuosity is lacking.
虽然曲折度的概念看似简单，但在实践中，曲折度的定义并不一致，文献中对它的处理往往具有误导性（107；33；88；75）。有时人们声称，曲折度取决于所研究的流动或传输过程类型，并且使用预测和测量的流体流动和传输之间的比较来支持这一说法（75；113）。然而，这种关于曲折度性质的声称表明，它在实际应用中通常被视为一个可调整的参数——“调整因素”，因此暗示了对曲折度的清晰理解不足。

A fundamental question, rarely addressed, is whether tortuosity is an intrinsic property of the medium, of a process within the medium, or neither, being simply an ad hoc parameter used to improve the agreement between theory and experiment. Because the tortuosity has a pronounced saturation dependence, tortuosity cannot simply be a property of the medium itself but must be derived from the actual paths of flow, conduction, or transport involved. This review provides a comprehensive discussion of the concept of tortuosity, its varying definitions in the literature, and their significance. It also provides a conceptual background to help the reader organize the various results discussed for tortuosity. The review was motivated by the observation that the many definitions of tortuosity do not fulfill their intended purpose in the calculation and prediction of flow and transport properties of porous media and may in fact contribute to further confusion. As already pointed out, there are several types of tortuosity in the literature; we begin by discussing them. Our discussion necessarily omits many details of the experimental and numerical procedures, but references are provided for those wanting more information.
一个基本问题，很少被解决，就是曲折率是否是介质的固有属性，是介质内部过程的一种，还是两者都不是，仅仅是一个用于提高理论和实验之间一致性的临时参数。由于曲折率具有明显的饱和度依赖性，曲折率不能简单地是介质本身的属性，而必须从实际流动、传导或传输路径中推导出来。本综述全面讨论了曲折率的概念，文献中对其的不同定义及其意义。它还提供了一个概念背景，以帮助读者组织讨论的曲折率的各种结果。本综述的动机是观察到许多曲折率的定义并没有在孔隙介质流动和传输性质的计算和预测中实现其预期目的，实际上可能进一步造成混淆。正如已经指出的那样，文献中存在几种类型的曲折率；我们首先讨论它们。 我们的讨论不可避免地省略了许多实验和数值方法的细节，但为那些想要更多信息的人提供了参考文献。

TYPES OF TORTUOSITY
扭曲类型

In the literature, tortuosity has been defined as either a geometric parameter or one that is related to the hydraulic, electrical, or diffusive properties (98; 88; 16; 69). Geometric tortuosity, τ_g, is the ratio of the average length, <L_g>, of the geometric flow paths through the medium to the straight-line length, L_s, across the medium (τ_g > 1):
在文献中，曲折度被定义为几何参数或与水力、电学或扩散性质相关的参数（98；88；16；69）。几何曲折度，τ _g ，是介质中几何流动路径的平均长度，，与介质横穿直线长度，L _s ，的比值（τ _g > 1）：

(1)

Tortuosity could instead be defined as the ratio of the shortest pathway (L_min) to the straight-line length L_s (1); however, the form presented in Eq. [1] is the more general definition. Note that L_min, the shortest pathway, is different from the chemical length, the length of the path corresponding to the minimal travel time of tracer particles that are dispersed in flow through a pore space (66). The geometric tortuosity coefficient may also be expressed as the inverse of the above definition (T_g = 1/τ_g), and it is always the case that T_g < 1 (50).
扭曲度也可以定义为最短路径（L _min ）与直线长度 L _s 的比值（1）；然而，公式[1]中给出的形式是更一般的定义。请注意，L _min ，最短路径，与化学长度不同，化学长度是指通过孔隙空间流动的示踪粒子对应的最小旅行时间的路径长度（66）。几何曲折度系数也可以表示为上述定义的倒数（T _g = 1/τ _g ），并且总是有 T _g < 1（50）。

There are two main alternatives for calculating the average length of the flow paths. One may average across the actual lengths of all the flow lines themselves, disregarding the fact that fluid particles move along these flow lines at different velocities. Alternatively, one may average the lengths of the flow lines of all fluid particles passing through a given cross-section during a given period of time, giving a flux-weighted average. The latter alternative appears more natural when fluid flow in porous media is considered (61). Other approaches have also been used, such as estimating the average path length from the pore size distribution (e.g., 39).
有两种主要方法用于计算流路径的平均长度。一种方法是对所有流线的实际长度进行平均，不考虑流体粒子沿这些流线以不同速度移动的事实。另一种方法是在给定时间段内，对通过给定横截面的所有流体粒子的流线长度进行平均，得到一个通量加权平均。当考虑多孔介质中的流体流动时，后者似乎更为自然（61）。还使用了其他方法，例如从孔径分布中估计平均路径长度（例如，39）。

The hydraulic tortuosity, τ_h, is the square of the ratio of the flux-weighted average path length for hydraulic flow, <L_h>, to the straight-line length (16). 5 defined the (hydraulic) tortuosity coefficient as the square of the inverse of that ratio, T_h = (L_s/<L_h>)², with reported literature values in the range of 0.56 to 0.8 for the ratio itself (5). In the definition of tortuosity given by 12, 13, which was followed by 5 and 33 among others, tortuosity is not simply a ratio of distances but also involves projecting the local potential gradient along a flow path onto a vector parallel to the macroscopic gradient; this is the origin of the square of the length ratio used by 5 and others. In other words, based on the Dupuit relation, for isotropically and randomly distributed pore space we might expect that the pore velocity parallel to the direction of flow is u/ϕ (where u is the Darcy flux and ϕ is the porosity), where the fractional free area is ϕ. If a random pore space does not exist, the Dupuit assumption should be modified. Therefore, the pore velocity must instead be taken as (u/ϕ)(<L_h>/L_s) because the actual flow path is tortuous (13). Carman (1956, p. 12) postulated that the time taken for a fluid particle to cross a tortuous pathway at a velocity (u/ϕ)(<L_h>/L_s) corresponds to the time taken to cross a distance L_s at a velocity u/ϕ. This tortuosity coefficient has often been used as an additional input, beyond the effects of the pore-size distribution, for predicting the hydraulic conductivity (30).
水力曲折率，τ _h ，是水力流动加权平均路径长度 与直线长度（ 16）的比率的平方。5 将（水力）曲折率系数定义为该比率的倒数的平方，T _h = (L _s /) ² ，文献报道的该比率的值在 0.56 到 0.8 之间（ 5）。在 12、13 给出的曲折率定义中，5 和 33 等人随后遵循，曲折率不仅仅是距离的比率，还涉及将局部势梯度沿流动路径投影到与宏观梯度平行的向量上；这是 5 和其他人使用长度比率平方的起源。换句话说，基于杜皮特关系，对于各向同性和随机分布的孔隙空间，我们可能期望孔隙速度平行于流动方向是 u/ϕ（其中 u 是达西通量，ϕ 是孔隙率），其中分数自由面积是 ϕ。如果不存在随机孔隙空间，杜皮特假设应进行修改。因此，孔隙速度必须取为 (u/ϕ)(/L _s )，因为实际的流动路径是曲折的（ 13）。 Carman（1956，第 12 页）假设流体粒子以速度（u/ϕ）（/L _s ）穿越曲折路径所需的时间与以速度 u/ϕ穿越距离 L _s 所需的时间相对应。这个曲折系数常被用作预测水力传导率（30）的额外输入，超出孔隙尺寸分布的影响。

When experimental results do not conform to predictions from 12 based on the aforementioned bundle of tubes model, the discrepancy is frequently interpreted in terms of tortuosity. But inferring the hydraulic tortuosity τ_h by comparing the Kozeny–Carman (63; 12) prediction with experiment is not supported by experimental data for all porous media: the Kozeny–Carman model does not hold for complex porous materials (122; 13; 7). For media whose solid particles deviate strongly from a spherical shape and/or whose particle size distributions are broad, and for consolidated materials, the Kozeny–Carman equation is not valid (13, p. 13; 30, p. 171). In addition, the Kozeny–Carman equation based on the assumption of conduit (pipe or tube) flow is not valid for porous media with high porosities (e.g., ϕ ≥ 0.95), where the frictional drag dominates the viscous shear and, consequently, the resistance to flow (30, p. 172). Thus, comparison of experimental data with the Kozeny–Carman prediction does not so much yield a tortuosity value (in the sense of a length ratio squared) as a rationale for invoking it as an adjustable parameter.
当实验结果不符合基于上述管束模型的预测时，这种差异通常被解释为曲折度。但通过将 Kozeny–Carman（63；12）预测与实验进行比较来推断水力曲折度τ _h ，并不适用于所有多孔介质的实验数据：Kozeny–Carman 模型不适用于复杂的多孔材料（122；13；7）。对于固体颗粒与球形形状偏差较大且/或颗粒尺寸分布较宽的介质，以及对于固结材料，Kozeny–Carman 方程无效（13，第 13 页；30，第 171 页）。此外，基于管道（管或管）流动假设的 Kozeny–Carman 方程对于孔隙率较高的多孔介质（例如，ϕ ≥ 0.95）也不适用，在这些介质中，摩擦阻力占主导地位，从而决定了流动阻力（30，第 172 页）。因此，将实验数据与 Kozeny–Carman 预测进行比较，并不主要产生一个曲折度值（即长度比平方），而是将其作为可调整参数的理由。

In general, geometric tortuosity and hydraulic tortuosity are not the same, and τ_g < τ_h. Imagine running a thread through a porous medium such that it lies precisely along a streamline (Fig. 1a). That is one possible hydraulic flow pathway L_h. If one then pulls the thread tight, without moving any solid particles, the result is the corresponding geometric flow pathway, L_g (Fig. 1b), which is shorter because it takes shortcuts that cross streamlines: hydraulic flow paths are smooth curves rather than straight lines or close tangents to the solid particles (16).
通常情况下，几何曲折率和水力曲折率并不相同，且τ _g < τ _h 。想象将一根线穿过多孔介质，使其精确地沿着流线（图 1a）运行。这是可能的水力流动路径 L _h 。如果此时将线拉紧，不移动任何固体颗粒，结果就是相应的几何流动路径，L _g （图 1b），因为它可以穿越流线的捷径而变得较短：水力流动路径是光滑的曲线，而不是直线或固体颗粒的切线（16）。

Consider the direct-current (DC) electrical conductivity. The electrical tortuosity, τ_e, in the spirit of Carman's definition, can be defined as (15; 104; 120)
考虑直流（DC）电导率。根据卡曼的定义，电迂曲度τ _e 可以定义为（15；104；120）

(2)

where <L_e> is the average path length for electrical flow.
式中 是电流量平均路径长度。

The electrical tortuosity can be also inferred by measuring the electrical resistivity of a medium as the product of its porosity ϕ and the formation factor F (121; 99; 30; 18):
电迂曲度也可以通过测量介质的电导率，将其孔隙率φ和形状因子 F 的乘积（121；99；30；18）来推断：

(3)

where F is the quotient of the electrical resistivity of the saturated porous medium ρ_p and the resistivity of the saturating liquid ρ_l. The formation factor is a dimensionless quantity whose value is always >1 in the absence of solid and/or surface conduction. In such cases, its value is supposedly determined uniquely by the pore geometry (30), although pore topology (connectivity) must also be relevant. A form of Eq. [3] with an empirical exponent η has also been reported: [τ_e|_sat = (ϕF)^η] (101). For example, 119 measured the transit time of ions flowing through saturated porous media under a sufficiently high electric potential gradient (to increase the rate of ionic migration relative to the speed of diffusing molecules), calculated the ratio <L_e>/L_s, and experimentally found that ϕF = (<L_e>/L_s)^1.67, which is not greatly different from ϕF = (<L_e>/L_s)², the combination of Eq. [2] and [3].
F 是饱和多孔介质电导率ρ _p 与饱和液体电导率ρ的商。形成因子是一个无量纲量，在没有固体和/或表面传导的情况下，其值总是>1。在这种情况下，其值据说由孔隙几何形状（30）唯一确定，尽管孔隙拓扑（连通性）也必须相关。还报道了一种具有经验指数η的方程[3]的形式：[τ _e | _sat = (ϕF) ^η ]（101）。例如，119 测量了在足够高的电势梯度下通过饱和多孔介质流动的离子的传输时间（相对于扩散分子的速度增加离子迁移率），计算了比< L _e >/L _s ，并通过实验发现ϕF = (< L _e >/L _s ) ^1.67 ，这与ϕF = (< L _e >/L _s ) ² ，即方程[2]和[3]的组合，没有很大差异。

Equation [3] has been used to estimate hydraulic tortuosity for the Kozeny–Carman equation. 123 used ϕF = (<L_h>/L_s), while 17 used ϕF = (<L_h>/L_s)². In the study of 123, the equivalent Kozeny–Carman pipe had a constant cross-sectional area ϕA (where A is the sample cross-sectional area), with an effective length L_eff, whereas 17 considered a pipe with constant cross-sectional area ϕAL/L_eff and the same effective length as 123 (noted by 122).
方程[3]已被用于估算 Kozeny-Carman 方程的水力曲折度。123 使用了ϕF = (/L _s )，而 17 使用了ϕF = (/L _s ) ² 。在 123 的研究中，等效的 Kozeny-Carman 管道具有恒定的横截面积ϕA（其中 A 是样品横截面积），有效长度为 L _eff ，而 17 考虑了一个具有恒定横截面积ϕAL/L _eff 且与 123 相同的有效长度（由 122 标注）。

68 demonstrated that the geometric tortuosity τ_g of flow lines accounting for both sinuosity and converging–diverging pore geometry was smaller than the electrical tortuosity: τ_g < τ_e = ϕF. Note that, in contrast to the Kozeny–Carman formulation for the hydraulic conductivity, Eq. [3] does not explicitly account for a pore size distribution influence on the electrical conductivity. This distinction is based on the argument that the electrical conductivity of a given pore is independent of its radius, a justification that has, however, been shown to be appropriate only in relatively homogeneous media (53).
68 表明，考虑了曲折度和收敛-发散孔隙几何形状的流线几何曲折率τ _g 小于电曲折率：τ _g < τ _e = ϕF。请注意，与用于水力导度的 Kozeny-Carman 公式不同，方程[3]没有明确考虑孔隙尺寸分布对电导率的影响。这种区别基于以下论点：给定孔隙的电导率与其半径无关，但这种说法仅在相对均匀的介质中是合适的（53）。

Analogously to electrical tortuosity, diffusive tortuosity, τ_d, can be defined (33) as
类似于电导率 tortuosity，扩散 tortuosity，τ _d ，可以定义为（33）：

(4)

where <L_d> is the average length of a chemical's diffusive pathway.
其中 是化学物质扩散路径的平均长度。

The diffusive tortuosity τ_d is sometimes defined as the diffusion coefficient of diffusing species in free fluid, d_f, relative to its value in a porous medium, d_p (96; 46; 16; 101; 88, 91):
扩散曲折率τ _d 有时被定义为扩散物种在自由流体中的扩散系数 d _f ，相对于其在多孔介质中的值 d _p （96; 46; 16; 101; 88, 91）：

(5)

which results in a tortuosity term that includes a direct porosity effect. An alternative relation, common in soil science and analogous to Eq. [3], removes porosity from the tortuosity value in Eq. [5] (21; 33; 75; 50; 58):
这导致了一个包含直接孔隙率效应的曲折率项。在土壤科学中常见的另一种关系，类似于方程[3]，从方程[5]的曲折率值中消除了孔隙率（21; 33; 75; 50; 58）：

(6)

Equation [6] is the theoretical result of 23 (cited in 105), who applied Fick's first law to describe diffusion in a tortuous channel. 75 substituted θ for ϕ to apply Eq. [6] to unsaturated conditions. The resulting parameter 1/τ_d is sometimes called pore continuity in the gas diffusion literature (75).
方程[6]是 23（在 105 中被引用）的理论结果，他将菲克第一定律应用于描述曲折通道中的扩散。75 用θ代替ϕ，将方程[6]应用于非饱和条件。由此产生的参数 1/τ _d 在气体扩散文献中有时被称为孔隙连续性（75）。

In contrast to Eq. [5], Eq. [6] accounts directly for the porosity (or water content in unsaturated media) contribution, interpreting d_p as depending on both the tortuosity of the medium and on the fractional volume (water content) or porosity (50). Tortuosity, however, is itself a function of porosity or water content. Equations [5] and [6] also illustrate how tortuosity has been defined and applied inconsistently in the literature.
与方程[5]相比，方程[6]直接考虑了孔隙率（或非饱和介质中的水分含量）的贡献，将 d _p 解释为取决于介质的曲折度和分数体积（水分含量）或孔隙率（50）。然而，曲折度本身是孔隙率或水分含量的函数。方程[5]和[6]还说明了在文献中如何不一致地定义和应用曲折度。

The diffusive tortuosity, τ_d, including the effects of chemical interactions, must depend on the molecular size of the diffusing particles (91) because some paths may become inaccessible to larger ones. If the molecular size becomes comparable to the pore size, the term d_f/d_p is multiplied by a dimensionless constrictivity factor δ (10). In a different interpretation of constrictivity, 111 used a constrictivity factor to correct for the cross-section of a pore varying over its length or, more specifically, for transport being restricted by the pore necks. Their δ is largely a function of the ratio of the pore's maximum and minimum cross-sections (111).
扩散曲折率，τ _d ，包括化学相互作用的影响，必须取决于扩散粒子的分子大小（ 91），因为某些路径可能对较大的粒子不可达。如果分子大小与孔径相当，则项 d _f /d _p 乘以一个无量纲收缩因子 δ（ 10）。在收缩性的不同解释中，111 使用收缩因子来校正孔横截面积随其长度变化的情况，或者更具体地说，校正由孔颈限制的传输。他们的 δ 主要取决于孔最大横截面积和最小横截面积的比率（ 111）。

For special conditions, e.g., steady-state diffusion under an established concentration gradient and volume or time-averaged flux in one-dimensional transport, 16 demonstrated that diffusive and electrical tortuosity are identical, implying that <L_e> = <L_d>. By using Einstein's relation (σ ∝ d, where σ is electrical conductivity and d is a diffusion coefficient), 89 and 120 obtained the same result. Others, e.g., 37, 60 and 43, found experimentally that τ_e and τ_d were almost the same. Nonetheless, sometimes measurements of electrical conductivity do not correctly predict diffusion (see, e.g., 8). In such cases, the disparity must stem from secondary phenomena that violate the analogy between diffusion and electrical conductivity, for example, (i) solid-phase conduction, (ii) reactions between the diffusing solute and the solid phase, (iii) a diffuse double layer in which diffusion is slowed by electrokinetic drag while electrical conductivity is enhanced by the higher charge density, or (iv) anion exclusion, which reduces diffusion while increasing electrical conductivity.
对于特殊条件，例如在已建立的浓度梯度下的稳态扩散和一维传输中的体积或时间平均通量，16 表明扩散曲折度和电曲折度相同，这意味着 = 。通过使用爱因斯坦关系（σ ∝ d，其中σ是电导率，d 是扩散系数），89 和 120 得到了相同的结果。其他人，例如 37、60 和 43，通过实验发现τ _e 和τ _d 几乎相同。然而，有时电导率的测量并不能正确预测扩散（例如，参见 8）。在这种情况下，差异必须源于违反扩散与电导率之间类比的第二性现象，例如：（i）固相传导，（ii）扩散溶质与固相之间的反应，（iii）扩散层，其中扩散由于电动力学阻力而减慢，而电导率由于更高的电荷密度而增强，或者（iv）阴离子排斥，这减少了扩散，同时增加了电导率。

UNIFYING CONCEPTS AND CONSTRASTS
统一概念与对比

To put the discussion on a firmer foundation, let us use an abstract, but nevertheless reasonable, model of a pore space, namely a network of interconnected bonds in which each bond represents a pore of a porous medium. Thus, imagine that we have a square network of cylindrical pores, with pore radii assigned at random from a given statistical distribution. Consider various types of transport, e.g., hydraulic or electrical, through the pores. Let the conductance g of an individual pore be proportional to its radius to the nth power (g ∝ rⁿ, where r is the pore radius). The geometric tortuosity would force a choice of n = 0, while electrical and diffusive tortuosities require n = 2 and hydraulic n = 4 (35; 53). As the exponent increases, flow is increasingly concentrated in fewer pathways, which become increasingly tortuous. Therefore, theoretically, τ_g < τ_e < τ_h. We use a Wheatstone bridge configuration, a simple model shown in Fig. 2, to investigate this proposition. The connection across the middle in a Wheatstone bridge is essentially a short circuit, corresponding in fluid flow to a pore of infinite radius. We define the tortuosity relative to the shortest possible path across the system, which is twice the length of an individual element. If one wanted to define the tortuosity relative to a straight-line length through the middle of the bridge, one would simply divide each of our values by the ratio √3/2 ≈ 0.866; such a constant factor would have no influence on our conclusions.
为了使讨论建立在更坚实的基础之上，让我们使用一个抽象但合理的孔隙空间模型，即由相互连接的键组成的网络，其中每个键代表多孔介质中的一个孔隙。因此，想象我们有一个由圆柱形孔隙组成的正方形网络，孔隙半径从给定的统计分布中随机分配。考虑通过孔隙的各种类型的传输，例如水力或电学传输。设单个孔隙的导纳 g 与其半径的 n 次方成正比（g ∝ r^0#，其中 r 是孔隙半径）。几何曲折性将迫使选择 n = 0，而电学和扩散曲折性需要 n = 2，水力需要 n = 4（35；53）。随着指数的增加，流动越来越集中在较少的路径上，这些路径变得越来越曲折。因此，从理论上讲，τ^1# < τ^2# < τ^3#。我们使用惠斯通电桥配置，如图 2 所示的一个简单模型，来研究这个命题。惠斯通电桥中间的连接本质上是一个短路，在流体流动中对应于无限半径的孔隙。 我们定义相对于系统中最短可能路径的曲折度，该路径长度为单个元素长度的两倍。如果想要定义相对于桥梁中间直线长度的曲折度，只需将我们的每个值除以比率的平方根 3/2 ≈ 0.866；这样的常数因子不会影响我们的结论。

Our findings are shown in Table 1. Note that for the case of no pore size variability, the geometric tortuosity is identical to both flux-based calculations. The hydraulic tortuosity increases more rapidly with increasing pore-size variability than does the electrical tortuosity, but in the limit of infinite pore-size variability, both reach the same maximum of 1.5. This is analogous to previous findings (35) for the saturation dependence of hydraulic conductivity K(θ) and electrical conductivity σ(θ), both of which follow universal results from percolation theory for homogeneous media (analogous to Kozeny–Carman and Archie's law); with increasing heterogeneity, however, the hydraulic conductivity develops a nonuniversal behavior earlier. Finally, in the limit of high heterogeneity, both conductivities exhibited nonuniversal behaviors predicted by relationship K(θ) = r_c²σ(θ) (where r_c is the critical pore radius for percolation, the smallest pore in the connected path on the sample-spanning [percolating] cluster that has the largest possible value of the smallest pore and controls the flow as a bottleneck) (42), implying that both forms of transport occurred along precisely the same pathways.
我们的发现显示在表 1 中。注意，对于没有孔隙尺寸变化的情况，几何曲折度与基于通量的计算相同。水力曲折度随着孔隙尺寸变化的增加而更快地增加，但电曲折度则不是这样，但在无限孔隙尺寸变化极限下，两者都达到相同的最大值 1.5。这与之前关于饱和度对水力传导率 K(θ)和电导率σ(θ)依赖性的发现（35）类似，这两者都遵循均匀介质渗透理论的普遍结果（类似于科兹尼-卡曼和阿奇定律）；然而，随着异质性的增加，水力传导率较早地表现出非普遍行为。 最后，在高度异质性的极限下，两种电导率都表现出非普遍行为，这种行为由关系式 K(θ) = r _c ² σ(θ) 预测（其中 r _c 是渗透的临界孔隙半径，它是跨越样本范围的连接路径上的最小孔隙，具有可能的最大值，并作为瓶颈控制流动）（42），这意味着两种形式的传输都沿着完全相同的路径发生。

At this point, we require some perspective on the expected and possible values of tortuosity. All conceptual formulations of tortuosity should be consistent with a value of 1 in the limit that porosity is 1. In contrast, at zero porosity, the system cannot percolate so defining a tortuosity would be meaningless; there is no path through the pore space across the system. In most cases, one expects the percolation transition (53) at a small but finite porosity value called the percolation threshold, ϕ_t; tortuosity is a meaningful quantity only for ϕ > ϕ_t.
在这一点上，我们需要对曲折度的预期和可能值有一些观点。所有关于曲折度的概念性表述都应该与孔隙率为 1 时的值 1 一致。相比之下，在零孔隙率时，系统无法渗透，因此定义曲折度将没有意义；在系统内没有通过孔隙空间的路径。在大多数情况下，人们期望在小的但有限的孔隙率值处发生渗透转变（53），这个值被称为渗透阈值 ϕ _t ；只有当 ϕ > ϕ _t 时，曲折度才有意义。

As a further consideration, the mathematics of fractals allows path lengths and their associated tortuosities to diverge (meaning path lengths get infinitely large), at least in principle. Consider the 118 fractal curve length definition where D_T is the tortuosity fractal dimension, varying in the range 1 to 3 in a three-dimensional medium, ε is the scale of measurement, L_s is the straight-line length, and L(ε) is the total length of a curve or path. The length of the pathway approaches infinity either when the system size is infinitely large or as the scale of measurement ε → 0. In fact, since the only connected pathways at the percolation threshold are fractal in nature, a calculation such as that presented by 118 should always apply at ϕ_t. In practice, however, the scale of measurement ε does not approach zero, nor does the sample size L_s approach infinity. Thus, while the tortuosity may become large in this limit, it remains bounded.
作为进一步考虑，分形数学允许路径长度及其相关的曲折度发散（意味着路径长度无限增大），至少在原则上如此。考虑 118 个分形曲线长度定义 ，其中 D _T 是曲折度分形维度，在三维介质中变化范围为 1 到 3，ε是测量尺度，L _s 是直线长度，L(ε)是曲线或路径的总长度。路径长度趋于无限大，要么是系统尺寸无限大，要么是测量尺度ε趋于 0。实际上，由于在渗透阈值处唯一相连的路径都是分形的，因此像 118 所展示的计算应该始终适用于ϕ _t 。然而，在实践中，测量尺度ε并不趋于 0，样本尺寸 L _s 也不趋于无限大。因此，虽然在这个极限下曲折度可能变得很大，但它仍然是有界的。

SPECIFIC TORTUOSITY MODELS FOR SATURATED POROUS MEDIA
饱和多孔介质的特定曲折度模型

Geometrical Models
几何模型

Models of geometric tortuosity have been commonly developed based on geometric (e.g., particle size, shape, and arrangement) properties, and also topological (e.g., the dimensionality and connectivity of the network, the coordination number—the number of pores connected to each other at each intersection—etc.) properties of a porous medium. These models, which approximately describe the geometric characteristics of the flow path, are generally applicable only to specific artificial porous media. For example, 124 proposed a tortuosity model for a porous medium consisting of two-dimensional square solid particles, given by
几何曲折度模型通常基于多孔介质的几何（例如，粒子大小、形状和排列）属性以及拓扑（例如，网络的维度和连通性，配位数——每个交点处相互连接的孔的数量等）属性而开发。这些模型大致描述了流动路径的几何特征，通常仅适用于特定的人工多孔介质。例如，124 提出了一种由二维正方形固体粒子组成的孔隙介质的曲折度模型，给出如下：

(7)

This model allows flow only between square particles in an equilateral-triangle arrangement with either (i) the flow path changing with a specific angle that depends on the particle size and pore space, or (ii) the streamline direction changes are limited to multiples of 90°. However ingenious its derivation, a porous medium with such underlying assumptions is far from any natural porous material.
此模型仅允许在等边三角形排列的正方形颗粒之间流动，流动路径要么（i）随颗粒大小和孔隙空间变化而改变特定角度，要么（ii）流线方向的变化仅限于 90°的倍数。然而，其推导方法再巧妙，具有这种基本假设的多孔介质也与任何自然多孔材料相去甚远。

125 developed geometric tortuosity models in three-dimensional porous media consisting of spherical, cubic, or rectangular solid particles. The particles were positioned in the form of equilateral triangle and square unit cell arrangements; however, 125 did not state explicitly how the unit cells were arranged in a network, i.e., whether the unit cells formed parallel planes or more complicated structures. Because the flow vector was implied to lie within the plane formed by the triangles (or squares), our interpretation is that the unit cells were arranged in a single sheet, and introducing additional sheets would not affect the final results. Note that in their model using spherical particles for tortuosity, any ϕ < 0.476 (calculated from their Eq. [8]) and 0.395 (calculated from their Eq. [5]) leads to a case of overlapping spheres in the square and triangular unit cells, respectively. But natural granular porous media are rarely monosized, and may have a porosity as small as 0.3 to 0.4 (gravel), 0.3 to 0.35 (gravel and sand), 0.1 to 0.2 (sandstone), or 0.01 to 0.1 (shale or limestone), as given by Bear (1972, p. 46). A poorly sorted material and a cubic arrangement of spherical grains of two sizes may have porosity values of 0.17 and 0.125, respectively (5, p. 47). Clearly, models based on monosize particles cannot represent natural sedimentary materials. Nonetheless, the results of 125 broadly agree with the results of 122 and 19, described below (Eq. [11]) with a logarithmic function of porosity.
125 个模型在三维多孔介质中开发了几何曲折度模型，该介质由球形、立方形或矩形固体颗粒组成。颗粒以等边三角形和正方形单元格排列的形式定位；然而，125 没有明确说明单元格在网络中的排列方式，即单元格是否形成平行平面或更复杂的结构。因为流动矢量被暗示位于由三角形（或正方形）形成的平面内，我们的解释是单元格以单层排列，引入额外的层不会影响最终结果。注意，在他们使用球形颗粒进行曲折度模型的模型中，任何小于 0.476（从他们的方程[8]计算得出）和 0.395（从他们的方程[5]计算得出）的ϕ值分别导致正方形和三角形单元格中球体的重叠。但自然颗粒多孔介质很少是单粒度的，孔隙率可能小至 0.3 到 0.4（砾石）、0.3 到 0.35（砾石和沙子）、0.1 到 0.2（砂岩）或 0.01 到 0.1（页岩或石灰岩），如 Bear（1972，第 46 页）所述。 杂乱无序的材料以及两种尺寸的球形颗粒的立方排列可能具有 0.17 和 0.125 的孔隙率值（5，第 47 页）。显然，基于单尺寸颗粒的模型无法代表自然沉积材料。尽管如此，125 号的结果与下面描述的 122 号和 19 号的结果基本一致（公式[11]），采用孔隙率的对数函数。

39 defined tortuosity in porous media by means of fractal geometry. They used the 118 fractal curve length definition and derived a fractal tortuosity model assuming a hierarchical structure in a saturated porous medium. Their model takes the form:
39 号通过分形几何学定义了多孔介质中的曲折度。他们使用了 118 号分形曲线长度定义，并假设饱和多孔介质中存在层次结构，推导出一个分形曲折度模型。他们的模型形式如下：

(8)

where τ(r) = (L_s/r) is the tortuosity for a pore pathway with radius r, with L_s being the straight-line length, f(r) = is the pore size probability density function, D is the fractal dimension of the pore space, r_min and r_max are the smallest and largest pore radii, respectively; and D_T is the tortuosity fractal dimension, restricted to the interval 1 < D_T < E (where E is the Euclidean dimension). The 39 approach (Eq. [8]) is restricted to saturated media because the full range of the pore sizes is used. Because Eq. [8] is based on concepts of geometric fractals applied to the structure of the medium itself, we argue that applying Eq. [8] to unsaturated conditions has no correspondence to the real flow path in porous media: it gives a cumulative pore length rather than the distance actually travelled by the fluid. Additionally, in Eq. [8] 39 assumed that pore length is related to pore size by the 118 fractal curve length given above, but this means that a pore's length L(r) is inversely proportional to its radius r. Although 36 considered this to be the case, in a self-similar fractal medium L(r) should be proportional to r (45). This direct proportionality is in agreement with the 80 approach, as well as with experiments on fracture networks (9).
τ(r) = (L _s /r) 是半径为 r 的孔隙路径的曲折度，其中 L _s 是直线长度，f(r) = 是孔隙尺寸概率密度函数，D 是孔隙空间的分形维度，r _min 和 r _max 分别是最小和最大孔隙半径；D _T 是曲折度分形维度，限制在 1 < D _T < E（其中 E 是欧几里得维度）的区间内。39 方法（方程[8]）仅限于饱和介质，因为使用了孔隙尺寸的全范围。由于方程[8]基于将几何分形概念应用于介质本身结构的理念，我们认为将方程[8]应用于非饱和条件与多孔介质中的实际流动路径没有对应关系：它给出的是累积孔隙长度，而不是流体实际行走的距离。此外，在方程[8]中，39 假设孔隙长度与孔隙尺寸通过上述 118 分形曲线长度相关联，但这意味着孔隙的长度 L(r)与其半径 r 成反比。尽管 36 认为这是正确的，但在自相似分形介质中，L(r)应与 r 成正比（45）。 这种直接比例关系与 80 方法一致，以及与断裂网络的实验（9）一致。

64 assumed that each tortuous path is represented by a sinuous tube with constant cross-sectional area and perimeter, and developed a theoretical model for the tortuosity of a fixed bed of randomly packed identical particles:
64 假设每条曲折路径都由一个横截面积和周长恒定的蜿蜒管表示，并针对随机堆积的相同颗粒的固定床的曲折度开发了一个理论模型：

(9)

where ξ is the sphericity (or roundness) factor, equal to 1 for spheres and <1 for nonspherical particles (64). A shortcoming of this model is that as ϕ → 1, τ_g → 0; this is physically unrealistic because in this limit the tortuosity must be 1, and by definition the tortuosity cannot be <1.
ξ是球形度（或圆度）因子，对于球体等于 1，对于非球形颗粒小于 1（64）。该模型的不足之处在于，当ϕ → 1 时，τ _g → 0；这在物理上是不现实的，因为在这种情况下，曲折度必须是 1，并且根据定义，曲折度不能小于 1。

Recently, 67 developed a tortuosity model based on the hierarchical structure of the deterministic Sierpinski carpet; however, heterogeneous soils and fractured networks are better analyzed as statistical (random) fractals rather than deterministic Sierpinski (or Menger sponge) models. The Sierpinski carpet used by 67 is a pore fractal model that includes particles of different sizes and pores all of the same size at each iteration (84) (Fig. 3b). Although in the pore fractal model (Fig. 3b) only the pore phase is fractal (the solid phase is not geometrically fractal), the solid phase is scaled with a power-law function and the same exponent and fractal dimension scaling the fractal pore phase (84). In contrast to the pore fractal model, a solid fractal model has particles of the same size and pores of a range of discrete sizes at each iteration (Fig. 3a). The 67 model yields
最近，67 开发了一个基于确定性 Sierpinski 地毯分层结构的曲折度模型；然而，异质土壤和断裂网络更适合用统计（随机）分形而不是确定性的 Sierpinski（或 Menger 海绵）模型来分析。67 使用的 Sierpinski 地毯是一种孔隙分形模型，包括不同尺寸的颗粒和在每个迭代中所有孔隙尺寸相同的孔隙（图 3b）。尽管在孔隙分形模型（图 3b）中只有孔隙相是分形的（固体相不是几何分形的），但固体相与分形孔隙相具有相同指数和分形维度的幂律函数进行缩放（84）。与孔隙分形模型相反，固体分形模型在每个迭代中具有相同尺寸的颗粒和一系列离散尺寸的孔隙（图 3a）。67 模型产生

(10)

Characteristics, descriptions, and comments on different geometric models are summarized in Table 2. Comparing the predicted saturated geometric tortuosity τ_g|_sat (Fig. 4) from the 124, 64, and 67 models indicates that the power-law form (Eq. [9]) diverges more sharply than the 124 model (Eq. [7]). Equations [7], [9] and [10] all yield τ_g|_sat → ∞ in the limit of zero porosity. Figure 4 also shows that the 124 and 67 models predict the tortuosity similarly for porosities >0.60. Equation [8] is not written explicitly in terms of the porosity, so we have not graphically compared it with the other models.
特征、描述和不同几何模型的评论总结在表 2 中。比较 124、64 和 67 模型预测的饱和几何曲折率τ _g | _sat （图 4）表明，幂律形式（方程[9]）比 124 模型（方程[7]）发散得更尖锐。方程[7]、[9]和[10]在孔隙率为零的极限下都得到τ _g | _sat → ∞。图 4 还显示，124 和 67 模型在孔隙率>0.60 时预测的曲折率相似。方程[8]没有明确以孔隙率表示，因此我们没有将其与其他模型进行图形比较。

Table 2. The geometric tortuosity models and their characteristics proposed in the literature. Note that concavity is up for all models. See also the equations mentioned for variable definitions and details.
表 2. 文献中提出的几何曲折度模型及其特征。注意，所有模型均向上凹。另见所述方程，以了解变量定义和细节。

Eq. 等式	Input quantities† 输入量†	Threshold 阈值	Dimensions 尺寸	As ϕ→1, τg|sat‡ → 随 ϕ→1，τ g | sat ‡ →	Divergence for ϕ→0 ϕ→0 的散度	Remarks 注释	Description 描述	Reference 参考文献条目
[7]		0	two 二	1	power 次方	square particles 立方颗粒	analytical 分析	124
[8]	D, DT, Ls, rmin D，D T ，L s ，r min	–	two, three 二，三	1	–	fractal media 分形介质	analytical 分析的	39
[9]	ϕ, ξ	0	three 三个	0	power§ 功率 §	sinuous channel 弯曲的渠道	analytical 分析的	64
[10]	ϕ	0	two 两个	1	power¶ 功率	Sierpinski carpet 塞尔宾斯基地毯	analytical 分析的	67
–#	ϕ	–	three	NA††	none	spherical particles 球形颗粒	analytical 分析的	125
–#	ϕ	0	three 三个	1	power§ 功率 §	cubic particles 立方颗粒	analytical 分析的	125
–#	ϕ	0	three 三个	1	power¶ 功率 ¶	plate-like particles 板状颗粒	analytical 分析的	125

• † ϕ, porosity; D, pore space fractal dimension; D_T, tortuosity fractal dimension; L_s, straight-line length; ξ, sphericity factor.
  † ϕ，孔隙率；D，孔隙空间分形维度；D _T ，曲折度分形维度；L _s ，直线长度；ξ，球形因子。
• ‡ Saturated geometric tortuosity.
  满足几何曲折度。
• § Power equal to −1.
  力等于-1。
• ¶ Nonintegral power.
  非整数幂。
• # Corresponding equation is not presented in the text.
  对应的方程在文中未给出。
• †† NA, not available.
  †† 不适用，不可用。

Hydraulic Conductivity Models
水力传导率模型

Perhaps the first rough estimate of hydraulic tortuosity was proposed by 12 as <L_h>/L_s = sec(α), where α is the average angle between the flow pathway and the apparent direction of flow. For cubic packing of equal-sized spheres, α ≈ 45°; thus <L_h>/L_s ≈ √2 and τ_h ≈ 2 (12). A value of τ_h = 3 was then proposed by 29 for a cubic network built up from the parallel capillary model (more references given in 30, p. 225).
可能最早对水力曲折度的粗略估计是由 12 提出的，即 /L _s = sec(α)，其中α是流动路径与流动表观方向之间的平均角度。对于等大小球体的立方堆积，α ≈ 45°；因此 /L _s ≈ √2，τ _h ≈ 2（12）。随后，29 提出了τ _h = 3，用于由平行毛细管模型构建的立方网络（更多参考文献见 30，第 225 页）。

One of the most invoked models of tortuosity in the literature is a logarithmic function of porosity, given by
文献中经常引用的孔隙率曲折度模型之一是孔隙率的对数函数，表示为

(11)

where P is a constant found experimentally to be 1.6 (81) for wood chips (cited in 19). 19 found a range of 0.86 to 3.2 for plates with different height/side ratios. Using the results of 78, 70 reported that a value of P = 0.49 is appropriate for a capillary model of high-porosity beds composed of spheres and fibers.
其中 P 是一个实验上发现的常数，对于木屑来说，P=1.6（81），19 发现不同高度/边长比的板有 0.86 到 3.2 的范围。根据 78 的结果，70 报告说，对于由球体和纤维组成的高孔隙率床的毛细管模型，P=0.49 是合适的值。

79 proposed an empirical tortuosity–porosity power law for binary mixtures of spherical particles:
79 提出了一种用于球形颗粒二元混合物的经验曲折度-孔隙率幂律

(12)

where β = 0.4 was found by measuring the conductivity of porous media. A problem with empirical models is that by definition the coefficient(s), in this case the exponent, must be determined experimentally or numerically; however, two porous media with the same porosity may have different transport path lengths, as demonstrated by 109, so tortuosity cannot be a function of porosity only. Consequently, the exponent β in Eq. [12] cannot be universal, at least not without replacement of the porosity by some more general function, i.e., inclusion of a threshold. The variability of β is probably due to variations in pore connectivity, but this has not been examined. A similar power law, but in terms of water content rather than porosity, has been widely used in unsaturated hydraulic conductivity models and is discussed below in that context.
在测量多孔介质电导率时发现β=0.4。经验模型的问题在于，根据定义，系数（在这种情况下是指数）必须通过实验或数值确定；然而，具有相同孔隙率的两种多孔介质可能具有不同的传输路径长度，如 109 所示，因此曲折率不能仅是孔隙率的函数。因此，方程[12]中的指数β不能是通用的，至少在没有用更一般的函数替换孔隙率的情况下不是；即包括一个阈值。β的变化可能是由孔隙连通性的变化引起的，但这尚未得到检验。在非饱和水力传导率模型中，已经广泛使用了一个类似幂律，但它是关于含水率而不是孔隙率，以下将在此背景下进行讨论。

In addition to empirical equations, analytical models have also been developed to describe tortuosity in porous media. Using volume-averaging concepts, 31 derived an analytical model for isotropic granular media:
除了经验方程之外，还开发了分析模型来描述多孔介质中的曲折度。利用体积平均概念，31 推导出了一种各向同性颗粒介质的分析模型：

(13)

Note that the saturated hydraulic tortuosity τ_h|_sat in Eq. [13] ranges between 1 and 1.5 and does not include a percolation threshold, or the critical porosity for macroscopic connectivity, in the medium.
请注意，方程式[13]中的饱和水力曲折率τ _h | _sat 介于 1 和 1.5 之间，不包括渗透阈值或介质中宏观连通性的临界孔隙率。

2 also proposed an analytical derivation of tortuosity for monosized spheres using a volume-averaging approach:
2 还提出了一种使用体积平均方法对单尺寸球体进行曲折度分析的解析推导：

(14)

in which B is a constant equal to 1.209 for cubic packings (Fig. 5a, after 2) and 1.108 for tetrahedral (Fig. 5b, after 2). Note that Eq. [14] diverges as the −1/2 power of ϕ − ϕ_t at the threshold porosity ϕ_t = 1 − B^−3/2. For cubic and tetrahedral packings, the threshold porosity ϕ_t as calculated from Eq. [14] would therefore be 0.248 and 0.143, respectively, close to 0.2488 and 0.15, the bond percolation threshold values reported by 103, 89, and 83. Note that we have respected the original authors' choice in regard to the significant figures reported.
在其中 B 是一个常数，对于立方堆积（图 5a，之后 2）等于 1.209，对于四面体（图 5b，之后 2）等于 1.108。请注意，当达到临界孔隙率 ϕ _t = 1 − B ^−3/2 时，方程[14]会发散为 ϕ − ϕ _t 的-1/2 次幂。对于立方和四面体堆积，从方程[14]计算出的临界孔隙率 ϕ _t 分别为 0.248 和 0.143，接近 103、89 和 83 报道的键渗流阈值值 0.2488 和 0.15。请注意，我们已尊重原始作者报告的有效数字选择。

In the approach of Dullien (1979, p. 225), as followed by 48, a theoretical model was proposed to calculate τ_h|_sat from the intrinsic permeability, porosity, and pore size distribution. This model yields
在 Dullien（1979，第 225 页）的方法中，如 48 所遵循的，提出了一种理论模型，用于从固有渗透率、孔隙率和孔隙尺寸分布计算τ _h | _sat 。此模型得出

(15)

where k is the intrinsic permeability, f(r) is the pore size distribution function, and r_min and r_max are the smallest and largest pore radii of the medium, respectively. Because k is typically assumed to be given by an arithmetic average of the individual pore conductivities (i.e., the integral in Eq. [15]), this definition of tortuosity amounts to saying that tortuosity defines the discrepancy between the calculated value of k and experiment; however, the assumption regarding the arithmetic average is erroneous (59; 89; 103; 73; 53; 91), and one could less charitably refer to tortuosity so defined as a highly refined fudge factor.
k 为固有渗透率，f(r)为孔隙尺寸分布函数，r _min 和 r _max 分别为介质的最大和最小孔隙半径。因为 k 通常假设为单个孔隙电导率的算术平均值（即方程[15]中的积分），这种曲折度的定义相当于说曲折度定义了 k 的计算值与实验值之间的差异；然而，关于算术平均的假设是错误的（59；89；103；73；53；91），人们可以不那么慈善地称这种定义的曲折度为高度精细的调整系数。

Numerical methods have also been applied to model the hydraulic tortuosity in saturated porous media using lattice-gas cellular automata (127; 61, 62) and lattice-Boltzmann models (69; 28); however, hydraulic tortuosity modeling using numerical simulations includes many hidden problems that may lead to incorrect conclusions. For example, splitting and merging of a fluid stream as it hits a solid particle causes discontinuities in streamlines. Because the location of each streamline is a priori unknown, the discontinuity in a streamline may lead to ill-conditioned results (69).
数值方法也被应用于使用格子气体细胞自动机（127；61，62）和格子玻尔兹曼模型（69；28）来模拟饱和多孔介质的水力曲折度；然而，使用数值模拟进行水力曲折度建模包括许多可能导致错误结论的隐藏问题。例如，当流体流撞击固体颗粒时，流体流的分裂和合并会导致流线的间断。由于每条流线的位置事先未知，流线的间断可能导致病态结果（69）。

Using a lattice-gas cellular automaton method, 61 investigated flow in a two-dimensional porous medium, constructed by randomly distributed solid rectangles of equal size and with unrestricted overlap. They found that the hydraulic tortuosity was linearly related to the porosity:
使用格子气体细胞自动机方法，61 研究了由随机分布的相同大小的固体矩形组成的二维多孔介质中的流动，这些矩形可以无限制地重叠。他们发现，水力曲折度与孔隙率呈线性关系：

(16)

Equation [16] indicates that when ϕ → 1, the hydraulic tortuosity approaches 1 (τ_h → 1). As ϕ → 0, however, τ_h → 1.8.
方程[16]表明，当ϕ→1 时，水力曲折度趋近于 1（τ _h → 1）。然而，当ϕ→0 时，τ _h → 1.8。

62 used lattice-gas cellular automata and numerically modeled the hydraulic tortuosity for flow of a Newtonian uncompressible fluid in a two-dimensional porous medium with a threshold porosity of 0.33 (ϕ_t = 0.33). In contrast to 61, 62 used periodic boundary conditions in the lattice to decrease the finite-size and boundary effects. In addition, to include the effect of nonconducting pores, they considered the effective porosity, the ratio of the volume of the conducting pores to the total volume (ϕ_eff), rather than the total porosity (ϕ) of the medium. 62 proposed the porosity-dependent tortuosity function
62 使用了格子气体细胞自动机，并数值模拟了牛顿不可压缩流体在二维多孔介质中的水力曲折度，该介质的阈值孔隙率为 0.33（ϕ _t = 0.33）。与 61 相比，62 在格子中使用了周期性边界条件以减少有限尺寸和边界效应。此外，为了包括非导电孔隙的影响，他们考虑了有效孔隙率，即导电孔隙体积与总体积之比（ϕ _eff ），而不是介质的总孔隙率（ϕ）。62 提出了与孔隙率相关的曲折度函数。

(17)

where a = 0.65 and γ = 0.19 are constant parameters, ϕ_t is the threshold porosity, and ϕ is the total porosity. Note that as ϕ → 1, the saturated tortuosity approaches 1. Note also that a power law with a small exponent is often difficult to distinguish from a logarithmic law.
在 a = 0.65 和γ = 0.19 为常数参数的情况下，ϕ _t 是阈值孔隙率，ϕ是总孔隙率。注意，当ϕ → 1 时，饱和曲折率趋近于 1。还请注意，小指数的幂律往往难以与对数律区分。

69 used a lattice-Boltzmann model to simulate the porosity dependence of tortuosity for laminar fluid flow in a two-dimensional porous medium composed of freely overlapping solid squares that had a percolation threshold of 0.367. In contrast to 61, who weighted the tortuosities of each streamline by local fluid velocities, 69 weighted with local fluxes and found a value of 0.77 for the constant P in Eq. [11].
69 使用格子玻尔兹曼模型模拟了二维多孔介质中层流流体流动的曲折度与孔隙率的关系，该多孔介质由自由重叠的固体正方形组成，具有 0.367 的渗透阈值。与 61 不同，61 通过局部流体速度对每条流线的曲折度进行加权，而 69 通过局部通量进行加权，并发现方程[11]中的常数 P 的值为 0.77。

Recently, 28 presented a brief overview of the aforementioned three numerical approaches, namely, those of 127, 61, and 69, as well as a new method that enables calculation of the hydraulic tortuosity without the need to determine streamlines. Determining streamlines is often an ill-conditioned numerical problem, especially if only the approximate fluid velocity field is available. 28 assumed a percolation threshold of 0.367 and provided an analysis of their numerical results as a function of system size that would support a divergence of the tortuosity at the percolation threshold. Because they were particularly interested in the analytical limit ϕ→1, they concentrated their analysis of the porosity dependence in the region ϕ > 0.8, where they found that the empirical function
最近，28 对上述三种数值方法进行了简要概述，即 127、61 和 69 的方法，以及一种新的方法，该方法可以在不确定流线的情况下计算水力曲折度。确定流线通常是一个病态的数值问题，尤其是在只有近似的流体速度场可用时。28 假设渗透阈值为 0.367，并提供了系统尺寸作为函数的数值结果分析，这支持了在渗透阈值处的曲折度发散。因为他们特别关注分析极限ϕ→1，所以他们集中在孔隙率依赖性在ϕ>0.8 的区域分析，在那里他们发现经验函数

(18)

best described the results of their simulations. Note that the value of χ was not reported by 28.
最好地描述了他们的模拟结果。请注意，28 没有报告χ的值。

The saturated hydraulic tortuosity models (Table 3) make some similar predictions (Fig. 6). For example, Eq. [11] with P = 0.49, Eq. [12] with β = 0.4, and Eq. [14] with B = 1.209 all predict similar τ_h|_sat for ϕ > 0.4. Differences are greater for ϕ < 0.4. In particular, τ_h|_sat predicted by Eq. [14] with B = 1.209 increases rapidly as ϕ decreases, and diverges at ϕ = 0.248. Figure 6 also indicates that the shape of the saturated hydraulic tortuosity predicted by the 31 model (Eq. [13]) is convex, whereas all other models predicted a concave shape for τ_h|_sat. In fact, given the suggested bounds of τ → 1 for ϕ → 1, and τ → ∞ for ϕ → 0, the concave shape is required. Experimental (27; 4) and numerical (62; 69; 28) results also support a concave curve for this relationship.
饱和水力曲折度模型（表 3）做出一些相似的预测（图 6）。例如，方程[11]中 P=0.49，方程[12]中β=0.4，以及方程[14]中 B=1.209 都预测了当ϕ>0.4 时相似的τ _h | _sat 。当ϕ<0.4 时，差异更大。特别是，方程[14]中 B=1.209 预测的τ _h | _sat 随着ϕ的减小而迅速增加，并在ϕ=0.248 时发散。图 6 还表明，31 模型（方程[13]）预测的饱和水力曲折度形状是凸的，而所有其他模型都预测τ _h | _sat 的形状是凹的。事实上，考虑到τ → 1 时ϕ → 1 和τ → ∞时ϕ → 0 的建议界限，凹形状是必需的。实验（27；4）和数值（62；69；28）结果也支持这种关系的凹曲线。

Table 3. The hydraulic tortuosity models and their characteristics proposed in the literature. See also the equations mentioned for variable definitions and details.
表 3. 文献中提出的孔隙介质水力曲折度模型及其特性。请参阅文中提到的方程式，以了解变量定义和详细信息。

Eq. 等式	Input quantities† 输入量†	Adjustable parameters 可调参数	Threshold 阈值	Dimensions 尺寸	Concavity 凹度	As ϕ→1, τh|sat‡ → 随 ϕ→1，τ h | sat ‡ →	Divergence for ϕ→0 or ϕt ϕ→0 或 ϕ t 的发散	Remarks 注释	Description	Reference
[11]	ϕ	P = 1.6	0	three 三个	up 上	1	logarithmic 对数	wood chips 木屑	empirical	81
[11]	ϕ	P = 0.86–3.2	0	three 三个	up 上	1	logarithmic 对数	platy particles 扁平颗粒	empirical	19
[11]	ϕ	P = 0.49	0	three 三个	up 上	1	logarithmic 对数	high-porosity beds 高孔隙度床	empirical	70
[12]	ϕ	β = 0.40	0	three 三个	up 向上	1	power 功率	binary mixtures of spherical particles 球形颗粒的二元混合物	empirical	79
[13]	ϕ	–	–	three	down	1	none	isotropic granular media	analytical	31
[14]	ϕ, B = 1.209	–	0.248	three	up	1	power§	cubic packing	analytical	2
[14]	ϕ, B = 1.108	–	0.143	three	up	1	power§	tetrahedral packing	analytical	2
[15]	ϕ, k, f(r)	–	–	three	up	–	none	–	analytical	30
[16]	ϕ	–	–	two	none	1	none	randomly distributed squares with unrestricted overlap	numerical	61
[17]	ϕ, ϕt	a = 0.65, = 0.19	0.33	two	up	1	power¶	randomly distributed squares with unrestricted overlap	numerical	62
[11]	ϕ	P = 0.77	0.367	two	up	1	logarithmic	freely overlapping squares	numerical	69
[18]	ϕ	χ	0.367	two	up	1	power§	porosity value >0.8	numerical	28

• † ϕ, porosity; B, constant coefficient; k, intrinsic permeability; f(r), pore size distribution function; ϕ_t, threshold porosity.
  ϕ，孔隙率；B，常数系数；k，固有渗透率；（r），孔径分布函数；ϕ，阈值孔隙率。
• ‡ Saturated hydraulic tortuosity.
  满足水力曲折度。
• § Nonzero threshold.
  非零阈值。
• ¶ Nonintegral power.
  非整次幂。

Electrical Conductivity Models
电导率模型

Based on the solution of Laplace's equation for steady-state conduction, 71 proposed a formula for the electrical conductivity of a conducting medium having a dilute suspension of nonconducting spheres, indicating that the electrical tortuosity, τ_e, satisfies
基于拉普拉斯方程稳态导热的解，71 提出了具有稀悬浮非导电球体的导电介质的电导率公式，指出电曲折率τ _e 满足

(19)

Equation [19] implies that as ϕ → 0, τ_e|_sat → 1.5. Note that Maxwell's derivation was for the dilute limit (i.e., when the volume fraction of the nonconducting particles is small) and was never intended to be extrapolated to zero porosity.
方程[19]表明，当ϕ → 0 时，τ _e | _sat → 1.5。请注意，麦克斯韦的推导是在稀疏极限下进行的（即非导电粒子的体积分数很小时），并且从未打算外推到零孔隙率。

If the empirical Archie's law (F = ϕ^−m) (3) is combined with Eq. [3], the resulting equation would be
如果将经验阿奇定律（F = ϕ ^−m ）（3）与方程[3]相结合，得到的方程将是

(20)

which is similar to Eq. [12] but with an exponent that generally lies between −3.4 and −0.2 (41). The larger the m value, the more tortuous the path (120). The exponent m may depend on a variety of factors, as discussed in detail by 90, including the shape and polarization of the particles.
这与方程[12]类似，但指数通常介于-3.4 和-0.2 之间（41）。m 值越大，路径越曲折（120）。指数 m 可能取决于多种因素，如 90 所详细讨论的，包括粒子的形状和极化。

Using Einstein's equation (32) that expresses the relationship between electrical conductivity and the diffusion coefficient, 18 applied an approach similar to that of 86, defined electrical tortuosity as Eq. [3], and consequently found a power-law divergence of τ_e as a function of porosity under saturated conditions:
利用爱因斯坦方程（32）表达了电导率和扩散系数之间的关系，18 应用了类似于 86 的方法，将电迂曲度定义为公式[3]，从而在饱和条件下发现τ _e 作为孔隙率的幂律发散：

(21)

where D_w is the random walk fractal dimension, i.e., 2 divided by the exponent that defines the power-law dependence of the mean-square displacement of a diffusant with time (D_w > 2; D_w = 2 refers to Fickian diffusion), D_m is the mass fractal dimension of the Sierpinski carpet (or the Menger sponge in three dimensions) (0 < D_m < E), and E is the Euclidean dimension. One should substitute θ for ϕ to apply Eq. [21] to unsaturated conditions. 18 reported a range of −0.4 to −2 for the exponent in Eq. [21]. Note that electrical tortuosity in Eq. [21] tends to 1 as porosity approaches 1 and diverges at zero porosity. In addition, in Eq. [21] the percolation threshold is zero: the system remains connected down to zero porosity or saturation.
其中 D _w 是随机游走分形维度，即 2 除以定义扩散质随时间平均平方位移幂律依赖的指数（D _w > 2；D _w = 2 指的是菲克扩散），D _m 是 Sierpinski 地毯（或三维的 Menger 海绵）的质量分形维度（0 < D _m < E），E 是欧几里得维度。在非饱和条件下应用公式[21]时，应将θ替换为ϕ。18 报告了公式[21]中指数的范围为-0.4 到-2。请注意，在公式[21]中，电迂曲度随着孔隙率接近 1 而趋于 1，在零孔隙率时发散。此外，在公式[21]中，渗透阈值是零：系统在零孔隙率或饱和度下仍然保持连通。

Comparison of Eq. [20] and [21] with Eq. [12] (τ_h|_sat = ϕ^−β) would imply that τ_e > τ_h in a saturated medium; however, Eq. [12] was proposed empirically for binary mixtures of spherical particles. 30, 26, and 127 found τ_h > τ_e. As we explained above, the geometric, electrical (or diffusive), and hydraulic tortuosities require the pore conductances to be proportional to their radius to the zeroth, second, and fourth powers, respectively, meaning that the hydraulic flow is concentrated in fewer (and, accordingly, more tortuous) pathways than the electrical (or diffusive) and geometric flows. As a result, one can theoretically assume that τ_g < τ_e (≈ τ_d) < τ_h, as pointed out above.
比较方程[20]和[21]与方程[12]（τ _h | _sat = ϕ ^−β ）表明，在饱和介质中τ _e > τ _h ；然而，方程[12]是针对球形颗粒的二组分混合物经验提出的。30、26 和 127 发现τ _h > τ _e 。正如我们上面所解释的，几何、电学（或扩散）和液压曲折度要求孔隙导纳分别与它们的半径的零次、二次和四次方成正比，这意味着液压流动集中在更少（因此，相应地，更曲折）的路径上，比电学（或扩散）和几何流动。因此，理论上可以假设τ _g < τ _e （≈ τ _d ）< τ _h ，如上所述。

Using the data presented in 122, and calculating electrical tortuosity from Eq. [3], 19 found P = 0.41 for monosized and polydisperse spheres and P = 0.63 for cubes in Eq. [11].
利用 122 中提供的数据，并从公式[3]计算电迂曲度，19 发现单分散和多分散球体的 P 值为 0.41，立方体的 P 值为 0.63（公式[11]）。

65 introduced three distinct definitions of the tortuosity in terms of the electrical conductivity and percolation properties of random conductance networks. These definitions are: (i) the ratio of p, the conducting volume fraction, and the conductivity fraction of the network, the ratio of the conductivity of a lattice with a random fraction p of identical conducting bonds to the conductivity of the same lattice with all bonds conducting, (ii) the ratio of percolation probability—the conducting volume fraction in the sample-spanning cluster—and the conductivity fraction, and (iii) the ratio of the backbone fraction—the conducting volume fraction of in the backbone of the network, i.e., the current-carrying part (some parts are dead-end and carry no current)—and the conductivity fraction of the network. All three definitions yield a tortuosity that diverges at the percolation threshold (critical porosity) and approaches 1 in the limit of unit porosity.
65 提出了三种关于电导率和随机导纳网络渗透特性的迂曲度定义。这些定义是：（i）网络导电体积分数 p 与导电分数的比值，即具有随机分数 p 的相同导电键的晶格导电率与所有键导电的相同晶格导电率的比值；（ii）渗透概率的比值——贯穿样品的导电体积分数与导电分数的比值；（iii）骨架分数的比值——网络骨架中的导电体积分数（即电流承载部分，一些部分是死胡同，不承载电流）与网络导电分数的比值。这三种定义得到的迂曲度在渗透阈值（临界孔隙率）处发散，在单位孔隙率极限下趋近于 1。

4 measured the porosities and formation factors of porous media composed mostly of spherical particles and calculated the tortuosity using Eq. [3]. They found that Eq. [11] with P = 0.49 reported by 70 was in agreement with their calculated electrical tortuosity in a monosized glass bead medium.
4 测量了主要由球形颗粒组成的孔隙介质的孔隙率和形成因子，并使用公式[3]计算了曲折度。他们发现，70 报道的公式[11]（P = 0.49）与他们在单尺寸玻璃珠介质中计算的电曲折度一致。

The various electrical tortuosity models (Table 4) give similar predictions at high porosity values (Fig. 7). Specifically, the predicted saturated electrical tortuosity τ_e|_sat using the 71 model and the 4 model (with P = 0.49) are quite close for porosity >0.6. This result was mathematically anticipated by 117, as we explain below.
各种电导率模型（表 4）在高孔隙率值（图 7）下给出相似的预测。具体来说，使用 71 模型和 4 模型（P = 0.49）预测的饱和电导率τ _e | _sat 在孔隙率>0.6 时相当接近。这一结果正如我们下面所解释的那样，在数学上是可以预见的。

Table 4. The electrical tortuosity models and their characteristics proposed in the literature. For all models, as porosity ϕ→1, the saturated electrical tortuosity τ_e|_sat → 1. See also the equations mentioned for variable definitions and details.
表 4. 文献中提出的电导率模型及其特征。对于所有模型，当孔隙率ϕ→1 时，饱和电导率τ _e | _sat → 1。请参阅文中提到的方程式，以了解变量定义和详细情况。

Eq. 等式	Input quantities† 输入量†	Adjustable parameters 可调参数	Threshold 阈值	Dimensions 尺寸	Concavity 凹凸性	Divergence for ϕ→0 当 ϕ→0 时的发散	Remarks 注释	Description	Reference
[19]	ϕ	–	–	three 三个	none	none	dilute suspension of nonconducting spheres 稀释的非导电球体悬浮液	analytical	71
[20]	ϕ	m	0	three 三个	up 上	power‡ 功率†	petroleum industry 石油工业	empirical	3
[21]	ϕ, Dw, Dm, E	–	0	two, three 两个，三个	up 上	power‡ 功率	fractal media 分形介质	analytical	18
[11]	ϕ	P = 0.41	0	three 三个	up 上	logarithmic 对数	spherical particles 球形颗粒	empirical	19
[11]	ϕ	P = 0.63	0	three 三个	up 上	logarithmic 对数	cubic particles 立方颗粒	empirical	19
[11]	ϕ	P = 0.49	0	three 三个	up 上	logarithmic 对数	monosized glass beads 单一尺寸玻璃珠	empirical	4

• † ϕ, porosity; D_w, random walk fractal dimension; D_m, Sierpinski carpet mass fractal dimension; E, Euclidean dimension.
  † ϕ，孔隙率；D _w ，随机游走分形维度；D _m ，Sierpinski 地毯质量分形维度；E，欧几里得维度。
• ‡ Nonintegral power.
  非整次幂

Liquid-Phase Diffusion-Based Models
基于液相扩散的模型

Solute species spreading due to the thermal energy of molecules is called diffusion. Net transport of solutes by diffusion occurs in response to spatial differences in concentration. Diffusion may happen in the solid, liquid, and gas phases. It has been asserted that diffusion characteristics in each phase should be unique because the pore geometry and connectivity are different (75; 40). This statement, while apparently obvious, tends to obscure certain unifying characteristics in, e.g., the saturation dependence of the tortuosity in the different phases, which can be predicted using percolation theory (88; 52). Insofar as descriptions of the air permeability (44) and electrical conductivity (35) follow universal scaling from percolation theory, the associated tortuosities should also.
溶质种类因分子热能而扩散称为扩散。溶质通过扩散的净运输是对浓度空间差异的反应。扩散可能发生在固体、液体和气相中。有人断言，由于孔隙几何形状和连通性不同，每个相的扩散特性应该是独特的（75；40）。这个说法虽然表面上明显，但往往掩盖了某些统一特征，例如不同相中曲折度的饱和度依赖性，这可以使用渗透理论进行预测（88；52）。在空气渗透率（44）和电导率（35）的描述遵循渗透理论的普遍尺度的情况下，相关的曲折度也应该如此。

82 was a pioneer of the study of gas diffusion in soils. He experimentally found a constant tortuosity coefficient of 0.66; however, it was later acknowledged (see, e.g., 21, 22; 24; 74, 76) that it was unlikely for the diffusive tortuosity coefficient to take on the same value for all types of soils and under all conditions because the tortuous path length should increase as the air-filled pore volume decreases (50). In this section, we focus on liquid-phase diffusive tortuosity models.
82 是土壤中气体扩散研究的先驱。他通过实验发现了一个恒定的曲折系数为 0.66；然而，后来人们公认（参见，例如，21，22；24；74，76）对于扩散曲折系数在所有类型的土壤和所有条件下都取相同值是不太可能的，因为曲折路径长度应该随着空气填充孔隙体积的减小而增加（50）。在本节中，我们重点关注液相扩散曲折模型。

117 analytically derived a result similar to Eq. [11] with P = 0.5 for diffusive tortuosity in porous media, applicable to randomly overlapping spheres of either uniform or nonuniform sizes. 117 also demonstrated that for dilute media where ϕ → 1, Eq. [11] with P = 0.5 would be identical to the 71 model (Eq. [19]) up to the first term (1 − ϕ).
117 通过解析推导出了一个类似于方程[11]的结果，其中 P = 0.5，适用于孔隙介质中的扩散曲折性，适用于大小均匀或不均匀的随机重叠球体。117 还证明了对于稀释介质，当ϕ → 1 时，方程[11]中的 P = 0.5 将与 71 模型（方程[19]）在第一项（1 − ϕ）上相同。

Equation [11] with P = 1 or ⅔ has been also reported for diffusive tortuosity and various arrangements of cylinders (106). 6 developed an analytical approach to model the nature of highly interconnected pores in a heterogeneous catalyst in two- and three-dimensional systems. His theory is based on a travel process through the catalyst and enables the properties of the pore space to be predicted. 6 found that
方程[11]在 P=1 或⅔的情况下也已被报道用于扩散曲折度以及各种圆柱排列（106）。6 提出了一种分析方法来模拟异相催化剂中高度互联孔隙在二维和三维系统中的性质。他的理论基于催化剂中的旅行过程，能够预测孔隙空间的性质。6 发现，

(22)

As ϕ → 0, τ_d|_sat → 3, meaning that τ_d|_sat varies in the range 1 to 3. Although for networks with the usual range of coordination numbers of four and more, τ_h|_sat ≈ 3 is the upper limit for hydraulic tortuosity (30, p. 225), the range 1 to 3 is unlikely to suffice for describing tortuosity in natural porous media such as soils and rocks. In such materials, hydraulic and electrical conductivities vary across many orders of magnitude; hence the tortuosity will probably also vary widely.
当ϕ趋近于 0 时，τ _d | _sat → 3，意味着τ _d | _sat 在 1 到 3 的范围内变化。尽管对于具有通常范围的四和更多协调数的网络，τ _h | _sat ≈ 3 是水力曲折度的上限（30，第 225 页），但对于描述土壤和岩石等自然多孔介质的曲折度，1 到 3 的范围可能不足以描述。在这些材料中，水力和电导率跨越许多数量级；因此，曲折度也可能广泛变化。

54 measured diffusion coefficients of SO₄ and CH₄ in seawater and sediments. They determined the diffusive tortuosity τ_d using Eq. [5], finding a linear relationship between τ_d and ϕ similar to Maxwell's formula:
54 测量了 SO ₄ 和 CH ₄ 在海水沉积物中的扩散系数。他们使用方程[5]确定了扩散曲折度τ _d ，发现τ _d 与ϕ之间存在类似于麦克斯韦公式的线性关系：

(23)

where q = 2 for sandy sediments and 3 for clay and silt sediments.
q = 2 对沙质沉积物，3 对粘土和粉砂沉积物。

27 determined the diffusive tortuosity using Eq. [5] after measuring diffusion in free fluid, d_f, and in packed beds of silica sand with a narrow range of particle sizes, d_p. The diffusive tortuosity in packed beds agreed well with the empirical model of 79 (Eq. [12]), with exponent β = 0.4, and with Eq. [11] with P = 0.5 (117); it also agreed with the theoretical model proposed by 126 for the case of porosities >0.4.
27 使用公式 [5] 在测量自由流体中的扩散 d _f 和具有狭窄粒径范围的石英砂填充床中的扩散 d _p 后确定了扩散曲折度。填充床中的扩散曲折度与 79 的经验模型（公式 [12]）相吻合，指数 β = 0.4，与公式 [11] 的 P = 0.5（117）相吻合；它还与 126 提出的孔隙率 >0.4 的情况下的理论模型相吻合。

The characteristics of different diffusive tortuosity models are summarized in Table 5. Equation [11] predicts divergence of the tortuosity at a critical porosity (in particular ϕ = 0), but its logarithmic form diverges more slowly than power laws do (Fig. 8). In many of the comparisons with simulations, the tortuosity varies over less than an order of magnitude, even when a critical porosity is approached, probably providing the chief rationale for using Eq. [11]. Figure 8 also indicates that prediction of τ_d|_sat using Eq. [11] with P = 0.5 is consistent with Eq. [12] with β = 0.4 as porosity decreases from 1 to 0.35.
不同扩散曲折度模型的特点总结在表 5 中。公式 [11] 预测在临界孔隙率（特别是 ϕ = 0）时曲折度会发散，但其对数形式比幂律发散得慢（图 8）。在许多与模拟的比较中，即使接近临界孔隙率，曲折度变化也小于一个数量级，这可能是使用公式 [11] 的主要原因。图 8 还表明，使用 P = 0.5 的公式 [11] 预测 τ _d | _sat 与孔隙率从 1 降至 0.35 时公式 [12] 的 β = 0.4 一致。

Table 5. The diffusional tortuosity models and their characteristics proposed in the literature. For all models, as porosity ϕ→1, the saturated diffusional tortuosity τ_e|_sat →1; all models are in three dimensions and ϕ is the sole input quantity. See also the equations mentioned for variable definitions and details.
表 5. 文献中提出的扩散曲折度模型及其特征。对于所有模型，当孔隙率ϕ→1 时，饱和扩散曲折度τ _e | _sat →1；所有模型均为三维，且ϕ是唯一的输入量。有关变量定义和详细情况，请参阅所提到的方程。

Eq. 等式	Adjustable parameters 可调参数	Threshold 阈值	Concavity 凹凸性	Divergence for ϕ→0 ϕ→0 的发散	Remarks 注释	Description 描述	Reference 参考文献条目
[11]	P = 0.50	0	up 上	logarithmic 对数	randomly overlapping spheres 随机重叠的球体	analytical 分析的	117
[11]	P = 1 or ⅔ P = 1 或 ⅔	0	up 上	logarithmic 对数	cylinders 柱体	empirical 经验的	106
[22]	–	–	down 向下	none	catalyst 催化剂	analytical 分析的	6
[23]	q = 2 (3)	–	none	none	sandy (clay–silt) sediments 砂质（粘土-粉砂）沉积物	empirical 经验的	54
[11]	P = 0.50	0	up 上	logarithmic 对数	packed beds of silica sand 硅砂填充床	empirical 经验的	27
[12]	β = 0.40	0	up 上	power† 功率†	packed beds of silica sand 硅砂填充床	empirical 经验的	27

• † Nonintegral power.
  非整次幂

EXISTING TORTUOSITY MODELS FOR UNSATURATED POROUS MEDIA
现有非饱和多孔介质曲折度模型

To be truly useful, a tortuosity relationship should work across the complete range of saturation, but developing such a model is a formidable task. Many expressions for the saturation-dependent tortuosity are given without specific reference to the property addressed. Consequently, we discuss all saturation-dependent expressions together: geometric, hydraulic, electrical, and diffusive. Where the researchers have made specific mention of the form of the tortuosity intended, our notation will indicate their choice.
要真正有用，曲折率关系应该在饱和度的完整范围内都适用，但开发这样的模型是一项艰巨的任务。许多关于饱和度依赖性曲折率的表达式都没有具体说明所涉及的属性。因此，我们将所有饱和度依赖性表达式一起讨论：几何、水力、电学和扩散。当研究人员具体提到所期望的曲折率形式时，我们的符号将表明他们的选择。

38 proposed an empirical microscopic representation for the hydraulic tortuosity τ_h as a function of pore size
38 提出了一种关于水力曲折率 τ _h 的经验性微观表示，该表示是孔隙尺寸的函数

(24)

where τ_hr is the relative hydraulic tortuosity (the subscript r denotes relative), τ_h|_sat is the hydraulic tortuosity of the saturated medium, τ_h(r) is the hydraulic tortuosity of the unsaturated medium, r is the radius of the largest water-filled pore in the unsaturated medium, and r_max is the medium's largest pore radius. The empirical exponent j was estimated to be 0.5 by 38, but they found that j values varied in soils. Specifically, they mentioned that j tended to take on values <½ for Basal Tuscaloosa sand and equal to ½ for Gatchell sand. They also found that for some types of sand j values >½ led to more accurate hydraulic conductivity predictions but did not provide upper and lower limits. 38 may have been the first to use this power-law formulation for tortuosity modeling in porous media.
τ _hr 是相对水力曲折度（下标 r 表示相对），τ _h | _sat 是饱和介质的液压曲折度，τ _h (r) 是非饱和介质的液压曲折度，r 是非饱和介质中最大水填充孔隙的半径，r _max 是介质的最大孔隙半径。经验指数 j 被估计为 0.5，但他们在土壤中发现 j 值有所不同。具体来说，他们提到 j 对于 Basal Tuscaloosa 砂倾向于小于 ½，而对于 Gatchell 砂等于 ½。他们还发现对于某些类型的砂，j 值大于 ½ 可以导致更准确的水力传导率预测，但没有提供上限和下限。38 可能是第一个在多孔介质曲折度建模中使用这种幂律公式的。

11 also presented a relative hydraulic tortuosity model as a function of effective saturation (cited in 20):
11 还提出了一种相对水力曲折度模型，作为有效饱和度的函数（在 20 中引用）：

(25)

where effective saturation S_e = (θ − θ_r)/(ϕ − θ_r) in which θ is water content, θ_r is residual water content (the volume fraction of water that remains in a pore space after displacement because it is disconnected), ϕ is porosity, and λ is a constant suggested to be equal to 2. 80 proposed a similar power function, but in his unsaturated hydraulic conductivity equation, the factor S_e^–λ (with λ = 0.5) was intended to correct simultaneously for both pore correlation and flow path tortuosity, with no attempt to distinguish between the two. Equation [25] (and also Eq. [24], in the case that r is given by the 108 model) is consistent with a power-law divergence of the tortuosity as saturation approaches a critical value (the residual water content). Fitting the Mualem–van Genuchten (80; 112) hydraulic conductivity function to measured unsaturated conductivity values, 97 and 102 found exponents −1 and −0.77, respectively. Exponents less than zero, however, result in τ_h(θ)/τ_h|_sat < 1, which is physically nonsensical: it implies that removing connections (by draining pores) makes flow less tortuous rather than more.
有效饱和度 S _e = (θ − θ _r )/(ϕ − θ _r )，其中θ是含水率，θ _r 是残余含水率（在孔隙空间中由于断开而剩余的水的体积分数），ϕ是孔隙率，λ是一个建议等于 2.80 的常数。提出了一个类似的幂函数，但在他的非饱和水力传导率方程中，因子 S _e ^–λ （λ = 0.5）旨在同时校正孔隙相关性和流动路径曲折性，没有试图区分两者。方程[25]（以及当 r 由 108 模型给出时的方程[24]）与饱和度接近临界值（残余含水率）时的曲折性幂律发散一致。将 Mualem–van Genuchten（80；112）水力传导率函数拟合到测量的非饱和传导率值，97 和 102 分别找到了指数-1 和-0.77。然而，小于零的指数会导致τ _h (θ)/τ _h | _sat < 1，这在物理上是说不通的：它意味着通过排水孔去除连接会使流动变得不那么曲折，而不是更曲折。

The widely used 72 saturation-dependent tortuosity model is
广泛应用的 72 饱和度相关曲折度模型

(26)

Although 72 derived Eq. [26] to predict unsaturated hydraulic conductivity, they also proposed its use to estimate diffusion in unconsolidated media, implying that hydraulic and diffusive tortuosities are equivalent. Equation [26], with tortuosity diverging as the water content approaches zero, has been widely used to estimate the ratio d_p/d_f in either gas (replacing θ by ε) or liquid phases (see, e.g., 74, 77; 49).
尽管他们推导出了方程[26]来预测非饱和水力传导率，但他们还提出了将其用于估计非固结介质的扩散，这意味着水力 tortuosities 和扩散 tortuosities 是等效的。当含水量接近零时，方程[26]中的 tortuosities 会发散，该方程已被广泛用于估计 d _p /d _f 的比率，在气相（用 ε 替换 θ）或液相中（例如，参见 74、77；49）。

If the medium is saturated, Eq. [26] reduces to τ_h|_sat = ϕ^−4/3, which differs markedly from the 79 and 27 results (Eq. [12] with β = 0.4). Dividing Eq. [26] by the saturated tortuosity τ_h|_sat = ϕ^−4/3, we have
如果介质饱和，方程[26]简化为 τ _h | _sat = ϕ ^−4/3 ，这与 79 和 27 的结果（β = 0.4 的方程[12]）明显不同。将方程[26]除以饱和 tortuosities τ _h | _sat = ϕ ^−4/3 ，我们得到

(27)

Note that in the absence of the second factor, (θ/ϕ)^−4/3, Eq. [27] would reduce to the Burdine model, Eq. [25].
注意，如果没有第二个因子（θ/ϕ） ^−4/3 ，方程[27]将简化为 Burdine 模型，方程[25]。

18 developed a power-law expression for electrical tortuosity as a function of water content in unsaturated porous media:
18 提出了一个幂律表达式，将非饱和多孔介质的电 tortuosities 作为含水量的函数进行了描述：

(28)

If Eq. [28] is divided by Eq. [21], the relative electrical tortuosity is given by
如果将公式[28]除以公式[21]，则相对电迂曲度由以下公式给出

(29)

Characteristics of tortuosity models proposed for unsaturated media are summarized in Table 6. Comparison (Fig. 9) shows that the 11 model (Eq. [25]) with λ = 2 and the 72 model (Eq. [27]) predict τ_r values that are one to two orders of magnitude greater at small porosity values than values from the 80 model (Eq. [25] with λ = 0.5). The discrepancy between the 11 and 72 models is due to the (θ/ϕ)^−4/3 factor in the 72 model.
总结了为非饱和介质提出的迂曲度模型的特征，见表 6。比较（图 9）显示，当孔隙率较小时，λ = 2 的 11 模型（公式[25]）和λ = 0.5 的 80 模型（公式[25]）预测的τ _r 值比 72 模型（公式[27]）大一个到两个数量级。11 模型和 72 模型之间的差异是由于 72 模型中的(θ/ϕ) ^−4/3 因子。

94 assumed that the ratio of the air–water interfacial area in a real porous medium to that in an idealized porous medium is equal to the square of <L_h>/L_s and proposed a new definition of tortuosity for saturated and unsaturated porous media based on the interfacial area ratio. For unsaturated media, their tortuosity is
94 假设真实多孔介质中空气-水界面面积与理想化多孔介质中空气-水界面面积的比值为/L _s 的平方，并提出了基于界面面积比率的饱和和非饱和多孔介质的新的曲折率定义。对于非饱和介质，他们的曲折率为

(30)

where a_aw is the air–water interfacial area of the unsaturated medium and a_aw,0 is the same parameter for the corresponding, idealized capillary bundle. This phenomenological relationship does not fit readily into any useful classification scheme and has been used to estimate both unsaturated hydraulic conductivity and unsaturated diffusion in porous media (94; 57; 56).
其中 a _aw 是非饱和介质的空气-水界面面积，a _aw,0 是对应的理想化毛细管束的相同参数。这种现象学关系难以纳入任何有用的分类方案，并已被用于估计多孔介质中的非饱和水力传导率和非饱和扩散（94；57；56）。

TORTUOSITY OF POROUS CATALYSTS
孔隙催化剂的曲折度

Due to the immense industrial significance of porous catalyst pellets, mass transfer phenomena in catalyst pellets have been intensively studied since the early 1960s. The goal was to estimate the tortuosity or (equivalently) the diffusion coefficient of the diffusing species in a given catalyst (34; 100; 114, 115; 55). Diffusion is the most significant, and often the only, reactant transport mechanism in porous catalysts, so tortuosity of a catalyst pellet is usually defined via Eq. [6]. The pore structure of most catalyst pellets is as chaotic as that of natural porous material, so advances in modeling pore structures in these two different systems have advanced in parallel.
由于多孔催化剂颗粒在工业上的巨大意义，自 20 世纪 60 年代初以来，对催化剂颗粒中的传质现象进行了深入研究。目标是估算给定催化剂中的曲折度或（等价地）扩散物种的扩散系数（34；100；114，115；55）。扩散是孔隙催化剂中最重要，通常是唯一的反应物传输机制，因此催化剂颗粒的曲折度通常通过公式[6]定义。大多数催化剂颗粒的孔结构就像天然多孔材料一样混乱，因此在这两个不同系统中建模孔结构的进展是并行的。

Similar to early studies of flow through porous media, early studies of the tortuosity of porous catalysts used a one-dimensional model of pore space: a bundle of capillary tubes. 115, for example, proposed a random pore model in a porous catalyst with both micropores and macropores. Diffusion in their model took place through three parallel paths: macropores, micropores, and composite pores consisting of macro- and micropores in series, with diffusion through macropores assumed proportional to ϕ². In a similar spirit, 55 integrated the diffusion flux over the entire pore size distribution of the pellet, arriving at a tortuosity value of 3. Due to the wide range of pore sizes in many catalyst pellets, many studies considered molecular, Knudsen, and surface diffusion; the derived tortuosity value depended on both molecular and Knudsen diffusion. For example, using a one-dimensional model, 116 derived the following equation for d_p and, hence, the tortuosity:
与早期对多孔介质中流体流动的研究类似，早期对多孔催化剂曲折度的研究使用了孔隙空间的一维模型：一束毛细管。例如，115 提出了一个具有微孔和宏孔的多孔催化剂的随机孔隙模型。在他们的模型中，扩散通过三条平行路径进行：宏孔、微孔以及由宏孔和微孔串联组成的复合孔，假设通过宏孔的扩散与ϕ ² 成正比。在类似的精神下，55 在整个颗粒孔隙尺寸分布上集成了扩散通量，得到了曲折度值为 3。由于许多催化剂颗粒孔隙尺寸范围广泛，许多研究考虑了分子、克努森和表面扩散；推导出的曲折度值取决于分子和克努森扩散。例如，使用一维模型，116 推导出了以下关于 d _p 的方程，从而得到了曲折度：

(31)

where f(r) is the pore size distribution of the catalyst, d_A and d_KA are the molecular and Knudsen diffusivities of species A, respectively, y_A is the mole fraction of A, α is a constant that depends on the molecular weight of A, and r_min is the lower limit of the pore size (determined by the molecular size). Early work in this field was well summarized by 95, while 14 reviewed the range of tortuosity values found in industrial catalyst pellets. The early pore space models yielded tortuosity values that were roughly consistent with experimental values determined by Eq. [6], but because the models were oversimplified and unrealistic, predictions were generally considered unreliable until more realistic catalyst models were developed in the 1980s.
f(r)是催化剂的孔径分布，d _A 和 d _KA 分别是物种 A 的分子和 Knudsen 扩散率，y _A 是 A 的摩尔分数，α是依赖于 A 的分子量的常数，r _min 是孔径的下限（由分子大小决定）。该领域的早期工作被 95 很好地总结，而 14 则回顾了在工业催化剂颗粒中发现的曲折度值范围。早期的孔隙空间模型得到的曲折度值与通过公式[6]确定的实验值大致一致，但由于模型过于简化和不现实，预测通常被认为不可靠，直到 20 世纪 80 年代开发了更现实的催化剂模型。

93 were the first to use a pore network model and percolation concepts to model diffusion and reaction in porous catalysts. Pore network models were soon recognized as the most realistic models of catalyst pellets (92). This recognition enabled researchers to carry out precise and large-scale computations and determine the effective diffusivities and, hence, tortuosity values via Eq. [6]. Advances in instrumentations, such as pulsed-gradient spin echo nuclear magnetic resonance (PGSE-NMR) (110), also enabled precise measurement of the diffusivities. These advances were reviewed by 85.
93 年首次使用孔隙网络模型和渗透理论来模拟多孔催化剂中的扩散和反应。孔隙网络模型很快被认定为催化剂颗粒最真实的模型（92）。这一认识使研究人员能够进行精确和大规模的计算，并通过公式[6]确定有效扩散率和因此曲折度值。仪器技术的进步，如脉冲梯度回波核磁共振（PGSE-NMR）（110），也使得扩散率的精确测量成为可能。这些进展由 85 综述。

Typical of catalyst pore network models is that of 51, building on the earlier work of 87. They varied the connectivity of the pore network, built on a simple cubic lattice and randomized so that the average coordination number could be varied, and showed that the tortuosity was not too sensitive to the connectivity. Most of the tortuosity values that they computed were around 4, remarkably close to what 55 had computed using a one-dimensional model. But when they used a network with a completely random topology, the resulting tortuosities sometimes differed from those computed on regular networks of the same coordination numbers, although they were still around 4.
典型的是催化剂孔网络模型，即 51 号模型，它建立在 87 号早期工作的基础上。他们改变了孔网络的连通性，基于简单的立方晶格并随机化，以便可以改变平均配位数，并表明曲折度对连通性不太敏感。他们计算的大多数曲折度值约为 4，与 55 号使用一维模型计算的结果非常接近。但当他们使用具有完全随机拓扑的网络时，得到的曲折度有时与具有相同配位数的规则网络计算的结果不同，尽管它们仍然约为 4。

Mass transfer in catalyst pellets is always accompanied by chemical reactions, with either linear or nonlinear kinetics. Over the years, a question of great interest has been whether the effective diffusivity of a species in a catalyst is the same with and without reaction. If the two diffusivities are not the same, then the tortuosity value defined by Eq. [6] will be greatly influenced by reaction. The same question is relevant in soil, where transport is typically accompanied by chemical reactions such as sorption and degradation. Experimentally, the subject was controversial for years, with conflicting results abounding in the literature, as summarized by 25. There was thus a real need for precise and large-scale computations to address the issue.
催化剂颗粒中的传质总是伴随着化学反应，具有线性或非线性动力学。多年来，一个备受关注的问题是催化剂中某物种的有效扩散率在有反应和无反应的情况下是否相同。如果两种扩散率不同，那么由公式[6]定义的曲折度值将受到反应的很大影响。同样的问题在土壤中也很相关，因为传输通常伴随着如吸附和降解等化学反应。实验上，这个问题多年来一直存在争议，文献中充斥着相互矛盾的结果，如 25 所述。因此，确实需要精确和大规模的计算来解决这一问题。

A first step toward addressing this question was taken by 87, who used a variety of network models, an effective-medium approximation, and numerical simulation to study the problem in the presence of a first-order reaction. He reported that effective diffusivities with and without first-order reaction do not differ much unless the pore space is poorly connected (is near its critical porosity or percolation threshold); in that case the two diffusivities (and therefore the tortuosity values) may differ greatly. A more refined simulation study (51) varied the rate of reaction relative to diffusion and also reported that the tortuosity value is different under reactive and nonreactive conditions if the reaction is first order. Most reactions in industrial applications of porous catalysts—as well as in soil—follow nonlinear kinetics, such as second-order reactions and Michaelis–Menten kinetics. 25 performed extensive numerical simulations using pore network models, as well as analytical analysis, and showed that the effective diffusivities in a pore space in the absence and presence of nonlinear reactions are fundamentally different. The diffusive tortuosity of catalyst pellets and soil therefore depends on reaction kinetics, as well as the other factors we have discussed in this review.
第一步解决这个问题的方法是 87，他使用各种网络模型、有效介质近似和数值模拟来研究一阶反应存在下的问题。他报告说，有和无一阶反应的有效扩散率差异不大，除非孔隙空间连接不良（接近其临界孔隙率或渗透阈值）；在这种情况下，两种扩散率（因此是曲折度值）可能差异很大。更精细的模拟研究（51）改变了反应速率相对于扩散的速率，并报告说，如果反应是一阶的，则在反应和非反应条件下，曲折度值是不同的。在多孔催化剂的工业应用中，以及土壤中的大多数反应——都遵循非线性动力学，如二阶反应和米歇尔-门滕动力学。25 使用孔隙网络模型进行了广泛的数值模拟，以及分析分析，并表明在孔隙空间中，无非线性反应和有非线性反应的有效扩散率在本质上是有区别的。 催化剂颗粒和土壤的扩散曲折度因此取决于反应动力学，以及我们在本综述中讨论的其他因素。

CONCLUSIONS
结论

In the vast literature on porous media, four types of tortuosity—geometric, hydraulic, electrical, and diffusive—have been defined and used. Although the basic concept and definition of tortuosity seem simple, we find that analytical, empirical, and numerical models have been used to estimate different kinds of tortuosity interchangeably, in both saturated and unsaturated media. Because the geometric, electrical (or diffusive), and hydraulic tortuosities require a pore's conductance to be proportional to its radius to the zeroth, second, and fourth power, respectively, we postulated that for higher powers flow is concentrated in fewer pathways; consequently, hydraulic flow becomes more tortuous than electrical (or diffusive) flow, which is more tortuous than geometric “flow”. Using a Wheatstone bridge model, we demonstrated the result that τ_g < τ_e (≈τ_d) < τ_h. In practice, however, it seems to us that it would be better to restrict definitions of tortuosity to the purely geometric in the sense of path lengths.
在关于多孔介质的大量文献中，已经定义并使用了四种类型的曲折度——几何、水力、电学和扩散——。尽管曲折度的基本概念和定义看似简单，我们发现分析、经验和数值模型被交替用来估计不同类型的曲折度，无论是在饱和介质还是非饱和介质中。由于几何、电学（或扩散）和水力曲折度要求孔隙的导纳与其半径分别成正比于零次、二次和四次方，我们推测对于更高的幂次，流动会集中在更少的路径中；因此，水力流动比电学（或扩散）流动更曲折，而电学（或扩散）流动比几何“流动”更曲折。使用惠斯通电桥模型，我们证明了τ _g < τ _e （≈τ _d ）< τ _h 的结果。然而，在实践中，我们认为最好将曲折度的定义限制在纯粹几何意义上的路径长度。

Given the remarkable diversity of definitions and concepts of tortuosity in the literature, the subject would benefit from a unifying concept. Experimental inferences regarding tortuosity to date have largely been confounded by inconsistently modeled effects of pore size variability. But some properties, such as the saturation dependence of liquid- and gas-based diffusion, may be related to tortuosity by the universal scaling of percolation theory. Some models concentrate on the geometry of the individual pores for calculating tortuosity, rather than on the geometry of the flow paths. We suggest that approaches that do not recognize this fundamental distinction should be avoided, which means also that experimentally determined values of the tortuosity should not be expected to yield information regarding the structure of the medium.
鉴于文献中关于曲折度的定义和概念的显著多样性，该主题将受益于一个统一的概念。迄今为止，关于曲折度的实验推断在很大程度上被孔隙尺寸变化的不一致建模效应所混淆。但某些性质，如基于液体和气体的扩散的饱和度依赖性，可能通过渗透理论的普适尺度与曲折度相关。一些模型专注于单个孔隙的几何形状来计算曲折度，而不是流动路径的几何形状。我们建议避免不承认这一基本区别的方法，这意味着实验确定的曲折度值不应期望提供有关介质结构的信息。

Many proposed tortuosity functions diverge at a porosity of (or near) zero, while others either do not diverge or diverge only logarithmically. Power-law divergences can probably be related to the relevance of percolation theory. When applicable, scaling relationships from percolation theory, expressed in terms of power laws, can yield useful expressions both for fluid flow properties, such as the air permeability, and for diffusion. In these cases, the property vanishes at the percolation threshold and the power is positive. Percolation also delivers power laws for the tortuosity that should be applicable whenever its expressions for flow or diffusion are verified. In these cases, however, the power is negative and the tortuosity diverges at the percolation threshold. If the purpose were just to find the limiting behavior of such functions for systems with large (or divergent) tortuosity, we would propose that the framework should be percolation theory. We have found, however, that some researchers find the limit of porosity equal to 1 more interesting than the opposite, small porosity limit. Further, many of the numerical studies are confined to large values of the porosity or small system sizes. In either of these cases, tortuosity values never grow very large, and the concept of a divergent porosity might appear to be extraneous. It should be possible, however, to apply concepts of finite-size scaling to address the limitations of the relatively small systems treated in models and simulations.
许多提出的曲折度函数在孔隙率为（或接近）零时发散，而其他函数要么不发散，要么仅以对数形式发散。幂律发散可能与渗流理论的相关性有关。当适用时，以幂律形式表达的渗流理论中的尺度关系可以为流体流动特性，如空气渗透率，以及扩散提供有用的表达式。在这些情况下，该属性在渗流阈值处消失，幂为正。渗流还为曲折度提供了幂律，只要其流动或扩散的表达式得到验证，就应该适用。然而，在这种情况下，幂为负，曲折度在渗流阈值处发散。如果目的仅仅是找到具有大（或发散）曲折度的系统的此类函数的极限行为，我们建议框架应该是渗流理论。然而，我们发现一些研究人员认为孔隙率等于 1 的极限比相反的小孔隙率极限更有趣。 此外，许多数值研究仅限于孔隙率较大或系统尺寸较小的值。在这两种情况下，曲折度值都不会变得非常大，发散孔隙率的概念似乎显得多余。然而，应该可以应用有限尺寸缩放的概念来解决模型和模拟中处理的相对较小系统所面临的局限性。

The remaining difficulty in proposing percolation theory as a unifying concept will probably be that so much of the research involves detailed comparisons of values of the tortuosity in systems far from the percolation threshold (near ϕ = 1), where asymptotic formulations from percolation theory are not likely to be accurate. If the community evolves to attach less importance to the porosity limit of 1, which is not particularly applicable to natural porous media, it is likely that we will indeed be able to apply the framework of percolation theory to develop a conceptual understanding of tortuosity.
将渗流理论作为统一概念提出的剩余困难可能在于，大量研究涉及远离渗流阈值（接近 ϕ = 1）的系统中曲折度的详细比较，在这种情况下，渗流理论的渐近公式可能不太准确。如果社区逐渐降低对孔隙率极限 1 的重要性，而这个极限对自然多孔介质并不特别适用，那么我们确实有可能应用渗流理论的框架来发展对曲折度的概念理解。