A general unified expression for solute and heat dispersion in homogeneous porous media



[1] Perturbations of temperature or solute concentration in a porous medium spread out by heat or molecular diffusion, respectively. If the pore-filling medium (e.g., water in soil) flows, this causes additional spreading of the perturbation due to the variation of local flow velocities and the tortuous flow lines through pore space. Together, this is termed dispersion, which plays an important role in geothermal energy production, contaminant transport, and reactor beds. Numerous models have been proposed to describe the dispersion coefficient as a function of flow rates, diffusion rates and other parameters, such as pore geometry. These models are either for heat (thermal) or solute dispersion, and often only valid for a limited range of flow rates, typically expressed in terms of the Péclet number. Here we present a single, universal expression for both the heat and solute dispersion coefficient in homogeneous porous media, valid over a wide range of Péclet numbers. Only three parameters have to be determined, which depend mainly on the pore geometry of the material. The expression facilitates the physical understanding of dispersion and may be helpful for the interpretation of numerical microscopic modeling results. It has the practical advantage that the heat dispersion coefficient can easily be calculated from the solute dispersion coefficient (or vice versa) and that dispersion coefficients over a wide range of Péclet numbers can be estimated from measurements over only a limited range.

1. Introduction

[2] Dispersion in porous materials plays an important role in designing or regulating geothermal energy systems, fuel cell development, contaminant transport and groundwater remediation, flow in reactor beds, and other applications in porous or fractured media [Bear, 1972; Freeze and Cherry, 1979; de Marsily, 1986; Sahimi, 1995; Ma et al., 2005; Anderson, 2005; Molina-Giraldo et al., 2011]. Dispersion refers to the net transport of tracers, passively advected by fluid flowing through a porous material, while simultaneously subject to diffusion. This fluid can be a gas or a liquid (water, oil, etc.), while the tracers can be solutes (dissolved ions, molecules, or molecular complexes), microscopic particles, colloids or microbes. In this work, we concentrate on solute dispersion and refer to the different dispersive species as “particles” (in the mathematical sense, i.e., “tracers”) and their dispersion as “particle dispersion.” Interactive forces, multicomponent diffusion [e.g., Quintard et al., 2006], and other factors, such as chemotaxis [e.g., Valdés-Parada et al., 2009], are not considered here. Heat can also be treated as a tracer in porous materials, and heat dispersion is found to be closely analogous to particle dispersion [Koch and Brady, 1985; de Marsily, 1986; Anderson, 2005; Constantz et al., 2003; Rau et al., 2012]. One major difference with particle dispersion is that heat can diffuse both through the pore fluid and through the solid phase. About a century of research has resulted in a wide range of models to describe the dispersion coefficient (D, see Table 1 for a list of symbols) in homogeneous porous media as a function of flow rates, diffusion coefficients and other controlling parameters, such as pore geometry [Slichter, 1905; Saffman, 1959; Scheidegger, 1961; Bear and Bachmat, 1967; Fried and Combarnous, 1971; Levec and Carbonell, 1985a; Delgado, 2007]. These models are either for heat, particles, transverse or longitudinal dispersion [e.g., Bijeljic et al., 2004; Bijeljic and Blunt, 2007; Klenk and Grathwohl, 2002], and often only valid over a limited range of flow rates, expressed in terms of the Péclet number (Pe) [de Marsily, 1986]. Typically, four separate dispersion regimes are distinguished as a function of Pe in laminar flow: (i) the molecular diffusion regime (Pe < 0.1–0.3), (ii) the transition regime (0.1–0.3 <Pe < 5), (iii) the major regime (also known as the power-law regime, Bijeljic and Blunt [2007]) (5 < Pe < 250–4000), and (iv) the pure mechanical dispersion regime (250–4000 < Pe) [Fried and Combarnous, 1971; de Marsily, 1986; Delgado, 2007; Wood, 2007]. At very high Péclet numbers, inertial or turbulent effects set in (regime v), signaling the end of the mechanical dispersion regime. Typically, this occurs at Péclet numbers of the order of 105.

Table 1. List of Symbols and Subscripts Used, in Order of Appearance in Text
Dm2/sDiffusion or dispersion coefficient
PePéclet number
Vm/sMean velocity in pores
NExponent in the relation between D and Pe
A, BParameters in equation (3) from Bear and Bachmat [ 1967]
γExponent in equation (3) introduced by Chiogna et al. [ 2010]
Qm/sDarcy velocity
GmGrain size or typical length scale
κW/m/KThermal conductivity
CJ/kg/KHeat capacity
βProportionality factor associated with mechanical dispersion
NQuantifier: amount of solute in advective versus diffusive pore channels
δKronecker delta or delta function
RRetardation factor
PSolute concentration in numerical model (Section 6)
ΘStop criterion in numerical model (Section 6)
R2fitCoefficient of determination
ScSchmidt number
Subscripts and Superscripts
L Longitudinal
T Transverse
P (solute) “particles”
Ref Reference
M Molecular
H Heat
F Fluid
S Solid
V Advective
Loc Local (inside single pore channel)
C Critical
Tot Total

[3] As flow is directional, dispersion in an isotropic medium is usually described by two components: the longitudinal (DL) and transverse (DT) dispersion coefficient in the direction of mean flow and perpendicular to it, respectively [Bear, 1961]. In the limiting cases of either very slow (the molecular diffusion regime (i); Pe < 0.1–0.3) or fast flow (the pure mechanical dispersion regime (iv); 250–4000 < Pe < 200,000), the determination of DL/T is straightforward. In the limit of slow flow (v), DL and DT simply equal the effective heat or particle diffusion coefficient in the porous medium (D(v = 0)). In the limit of fast (but still laminar) flow, DL and DT should increase linearly with the flow rate: DL,TPe [Koch and Brady, 1985; Wood, 2007]. However, for laboratory experiments and groundwater flow, Péclet numbers typically range between 0.1 and 10,000 [Bijeljic and Blunt, 2006], hence these flow rates are often in the transition (0.1–0.3 <Pe < 5) or power-law (5 < Pe <250–4000) regimes between these two limiting regimes [Wood, 2007]. Thus, the accurate prediction of dispersion coefficients in this intermediate regime between the molecular and mechanical dispersion regimes (i.e., 0.1–0.3 < Pe < 250–4000) is of great practical interest.

[4] A simple approach to combine the two limiting regimes (i) and (iv) into a single expression is to sum the two expressions [de Josselin de Jong, 1958; Grathwohl, 1998; Guedes de Carvalho and Delgado, 1999; Ham et al., 2007]:

display math(1)

where the dispersivity α is a proportionality constant. However, this approach, which may be used to fit experimental data over a limited range of Pe numbers, ignores the complex, nonlinear behavior in the transition regime, clearly observed with particle dispersion [Sahimi, 1995; Bijeljic and Blunt, 2007]. In view of the poor results obtained when fitting equation (1) to experimental dispersion data, the data have been subdivided into up to five disjoint ranges of Pe numbers (as mentioned above), and in each range a separate form of dispersion relation was postulated to apply. Of particular interest is the power law regime (iii), where many studies propose using a power-law type expression with exponent (n):

display math(2)

[5] For example, Guedes de Carvalho and Delgado [2000] report an exponent of n = 0.57, Klenk and Grathwohl [2002] n = 0.5 and Olsson and Grathwohl [2007] n = 0.72 for transverse particle dispersion. Levec et al. [1985a, 1985b] apply the same equation to transverse heat dispersion and obtained n = 0.683. Higher values of n = 1.2 are reported for longitudinal particle dispersion [Bijeljic et al., 2004, and references therein; Wood, 2007], similar to n = 1.256 for longitudinal heat dispersion reported by Levec et al. [1985b]. Summarizing, the nonlinear equation (2) has been successfully used to fit experimental data in the power-law range of Péclet numbers, resulting in n < 1 for transverse and 1 < n < 2 for longitudinal dispersion. More complex models have also been proposed [Koch and Brady, 1985].

[6] Based on a geometrical argument involving a statistical averaging procedure of the microscopic flow, Bear and Bachmat [1967] proposed the following relationship:

display math(3)

with γ = 1, and where A and B are parameters of which A is purely geometric. The success of this equation in modeling both longitudinal and transverse dispersion has been limited, although it clearly prefigures the present work. Chiogna et al. [2010] introduced the exponent γ ≠ 1, postulating a potential, but unspecified nonlinearity in the underlying physical laws. Fits of the transverse dispersion coefficients to laboratory-derived data yielded γ = 0.5 [Chiogna et al., 2010; Rolle et al., 2012]. However, while the original model by Bear and Bachmat [1967] has the correct type of behavior in the mechanical dispersion limit (i.e., regime (iv) at high Pe, but still in the laminar flow region), the modified parametric model with γ ≠ 1 does not achieve the correct behavior in this limit.

[7] The variety of models and fit parameters shows that there is still no consensus on the equations that link DL/T to the flow rate, the properties of fluid and solid, the geometry of the porous medium and any other potentially significant parameters, especially in the intermediate regime (i.e., transition and power law regimes, (ii) and (iii)). Here we present a novel single, universal expression for both heat and particle dispersion in homogeneous porous media that is valid over the full relevant range of Péclet numbers. Our approach bears some similarity to the one proposed by Bear and Bachmat [1967] and the result is similar in form to equation (3) with γ = 1, although the derivation, based on scaling behavior, is more general and limits geometrical assumptions to a minimum, so that the resulting equations are expected to have broad validity. To achieve this, we follow the approach by van Milligen and Bons [2012], based on a statistical description of the competition between diffusion and mechanical dispersion in the pore channels, and generalize this by treating both heat and particles as tracers, i.e., noninteracting, massless, infinitesimally small entities, subject to diffusion and passively advected by the flow. For any given material, only three effective parameters related to the pore geometry have to be determined. The model explains the appearance of nonlinear growth of the dispersion coefficient as the flow rate is increased [Saffman, 1959, Sahimi, 1993; Olsson and Grathwohl, 2007; Bijeljic and Blunt, 2007]. The model only applies to homogeneous porous media, and does not incorporate additional factors such as heterogeneity and nonlocality, that play an additional role in larger-scale, nonhomogeneous natural aquifers [Sahimi, 1993; Fernàndez-Garcia and Gómez-Hernández, 2007; Zhang et al., 2009; Hunt et al., 2011; Engdahl et al., 2012]. Even so, a better understanding of dispersion in homogeneous media is expected to have an impact on the analysis, study and prediction of dispersion in aquifers and other fields. Finally, the relation between heat and solute transport established here may enable the use of heat measurements to study solute transport and vice versa [e.g., Vandenbohede et al., 2009; Ma et al., 2012].

2. Definitions

[8] The Darcy velocity (q) is the average velocity of the fluid traversing the matrix: the discharge volume per time, divided by the cross-sectional area. The mean velocity (v) in the pores (m/s), in the direction of flow, is higher as flow can only take place inside pores [Fried and Combarnous, 1971; Rau et al., 2012]:

display math(4)

where ϕ is the porosity of the material. It is customary to express flow velocity in terms of the Péclet number. In this paper we use the diffusional Péclet number, defined by Bear, [1972], de Marsily [1986], and Fetter [2008]:

display math(5)

where G (m) is the typical length scale (usually taken as the (average) grain size) and Dp,ref (m2/s) an appropriate mean diffusion coefficient of the tracer particle; it is customary to use the molecular diffusion coefficient (Dm) in the free fluid.

[9] In the case of heat dispersion, one merely substitutes the particle diffusion coefficient Dp,ref by the heat diffusion coefficient Dh,ref(m2/s) (commonly designated by α. We prefer the symbol Dh,ref for heat for reasons soon to be apparent). Thus:

display math(6)

[10] Here Dh,ref (m2/s) is the (mean) heat diffusion coefficient of the fluid + solid system. It may be related to the (mean) fluid and solid conductivity <κ> (W/m/K) by

display math(7)

where ρ is the density (kg/m3) and c is the heat capacity (J/kg/K). The angular brackets refer to a mean of the solid and fluid system, which can be expressed as follows [Diao et al., 2004; Metzger et al., 2004]:

display math(8)

where the subindices “f” refer to the fluid, and “s” to the solid. It is assumed the fluid fills the pore space completely. We note that there are other ways to define the mean heat diffusion coefficient of the fluid-filled porous material [Rau et al., 2012; Levec and Carbonell, 1985a, 1985b]. Alternatively, the heat diffusion coefficient for the fluid-filled porous material (under no-flow conditions) may be measured directly, thus removing any definitional uncertainties.

3. Mechanical Dispersion

[11] In the absence of diffusion, tracers advected with fluid flowing through a porous medium will spread out from an initial small concentrated volume, due to the fact that the flow rates within pore channels are variable and fluid flow lines meander around the grains of the medium, which is assumed to be approximately random in structure [Koch and Brady, 1985]. While this motion appears random, it clearly is not (at least, while the flow is sufficiently slow to remain laminar and not transit into the turbulent regime). In fact, the laminar flow is fully deterministic and can be calculated from the applied hydraulic head by solving the flow equations for an incompressible fluid with the appropriate boundary conditions [e.g., Ovaysi and Piri, 2011]. The whole fluid-flow problem is linear in the hydraulic head, so increasing the hydraulic head will simply increase the velocity everywhere by a linearly proportional factor, without altering the geometry of the tracer trajectories. Thus, the mean mechanical tracer dispersion coefficient, calculated as

display math(9)

is also proportional to the hydraulic head or the flow velocity, as <d2>, the variance of the tracer positions after a time t, is dictated by the geometry and does not change, whereas the tracer travel time t through the finite porous medium is inversely proportional to the hydraulic head, and hence the flow velocity v. One can thus write:

display math(10)

where βL,T is a dimensionless number, dependent on the geometry of the system (shape of pores, tortuosity, etc). In the mechanical dispersion limit, dispersion is independent from molecular or thermal diffusion. The microscale G is included to satisfy dimensional requirements. It is customary to normalize the dispersion coefficient to a reference diffusion coefficient and express the result as a function of Pe, or:

display math(11)

4. Tracer Statistics in a Porous Medium

[12] The local flow velocities vary within and between pore channels, depending on factors such as channel orientation and width. To understand how the statistics of tracers moving with the flow vary as a function of the flow speed, we classify the pore channels according to their mean local flow velocity (vloc) and the diffusion coefficient (Dm) of solute particles, taken constant for the system under consideration. For this purpose of classification, the local Péclet number, vlocG/Dm, is suited, as it is proportional to the ratio vloc/Dm. If it exceeds an arbitrary threshold value (typically taken equal to 1), then we will call transport in the channel dominantly advective, and otherwise dominantly diffusive.

[13] Particle (solute) tracers that are released at a point in the system will spread out in a cloud as the tracers travel through the various pore channels. A single tracer will alternately traverse pores classified as advective or diffusive along its path. After a time that is sufficiently long to avoid initial transients, but not so long as to lose tracers through the system boundaries, we take a snapshot of the location of individual tracers, and count how many are in dominantly advective (Nv) or diffusive (Nf) channels (alternatively, a steady state situation with continuous injection can be used for the same purpose). The total ratio of tracers in advective and diffusive channels in the snapshot will directly reflect the mean ratio of time spent in advective (tv) and diffusive (tf) channels by any tracer:

display math(12)

[14] If the global flow rate (v) is increased, local flow rates increase in all channels, and more and more channels will therefore become advection dominated. Also, at any channel junction, the tracer particles will be distributed over the outgoing channels according to the ratio of fluxes of the outgoing channels. If v is increased, the flux into advective channels increases in proportion to v, while the flux into diffusive channels is not affected much. These two effects (reclassification of channels and redistribution of flux ratios at every channel junction) contribute to raise the number of particles in advective channels with respect to those in diffusive channels. We note that the precise choice of the criterion for assigning the label “advective” or “diffusive” to the channels is not very important—the important issue is the robust increase of the mentioned ratio as v is raised. It follows that the ratio of mean time spent in dominantly advective and diffusional channels must increase with v:

display math(13)

where f(x) is a monotonically increasing function of x, such that f(0) = 0 and f(1) = 1. The precise form of this function will of course depend on the topology of the network. In the following, we will assume f(x) = x for simplicity. In section 6, we will show that this behavior does indeed occur in a computer simulation of solute transport in a simplified pore network. Physically, vc is the global flow rate at which solute tracers are equally distributed among dominantly advective and diffusive channels. Expressing vc in terms of the Péclet number, we get:

display math(14)

[15] Pec is the “critical Péclet number,” a dimensionless parameter that depends mainly on the geometry of the porous material. Pec is the Péclet number at which solute tracers are equally distributed among dominantly advective and diffusive channels (cf. section 6). Combining equations (13) and (14) with f(x) = x gives:

display math(15)

5. General Dispersion in Homogeneous Porous Media

[16] To derive the dispersion coefficient for flow through homogeneous, isotropic, saturated porous media we follow the method by van Milligen and Bons [2012] for particle dispersion. In general, the mean square displacement of a tracer after time t is the product of its diffusivity and t, so if a tracer experiences various (possibly overlapping) phases of motion (lasting times ti) with different diffusion coefficients Di, the total effective diffusion is

display math(16)

where ttot is the total time considered. If all diffusive mechanisms affect tracer motion during the whole tracer trajectory, all ti's are equal to the total time ttot of the tracer trajectory and equation (16) simplifies to math formula. The latter assumption underlies the simplest of all expressions for dispersion in porous media, cf. equation (1), in which the total dispersion coefficient is written as the sum of the molecular diffusion and mechanical dispersion coefficients [de Josselin de Jong, 1958; Grathwohl, 1998; Ham et al., 2007; Delgado, 2007]. However, most studies agree that this approach is unsatisfactory and only matches the experimental data in the limits of very low or very high Péclet numbers. Literature provides ample evidence for apparently complex behavior in the transition and power-law regimes (ii) and (iii), which has motivated the present work.

[17] Now consider the dispersion of heat. The transport of heat (diffusion and advection) can be modeled using “heat tracers,” which are infinitesimal “packets of energy” either moving randomly (in the case of diffusion) or with the fluid flow [Emmanuel and Berkowitz, 2007; Hecht-Méndez et al., 2010]. A heat tracer in the porous medium may experience three phases of motion (Figure 1): molecular diffusion inside the fluid with diffusion coefficient Dh,f, diffusion inside the solid with a diffusion coefficient Ds, and advection by the laminar fluid flow, associated with an effective diffusion due to the tortuous path of the flow through the pore channels, characterized by the mechanical dispersion coefficient Dv. It is evident that the tracer is only affected by Ds while residing in the solid, and by a combination of Dh,f and Dv while residing in the fluid.

Figure 1.

Sketch of transport in a porous medium with blue arrows indicating fluid flow. Particle transport is a combination of advection with the flow for a period tv and molecular diffusion during period tf. Heat advection has an additional period ts, where heat diffuses through the solid.

[18] Following the ideas set out in the preceding section [cf. van Milligen and Bons, 2012], we make a distinction between dominantly diffusive and dominantly advective channels. In the case of transverse dispersion, the three phases with diffusion coefficients Dh,f, Ds, and Dv affect the tracer during time periods tf+ tv, ts, and tv, respectively. In other words, (a) diffusion in the fluid, with coefficient Dh,f, acts as long as the tracer stays in the fluid, (b) Ds acts as long as the tracer resides in the solid, and (c) Dv acts whenever the tracer resides in a dominantly advective channel. In the case of longitudinal dispersion, the three phases with diffusion coefficients Dh,f, Ds, and Dv affect the tracer during time periods tf, ts, and tv, respectively. This is similar to the case for transverse dispersion, except that Dh,f only acts whenever the tracer resides in a dominantly diffusive channel, and is considered negligible whenever the tracer resides in a dominantly advective channel. The reason for this assumption is that longitudinal dispersion will be mainly affected by the flow in pore channels at a small angle with respect to the mean flow vector, in which advection readily becomes the dominant transport mechanism (contrary to transverse dispersion, more sensitive to transport in channels at nearly right angles to the mean flow vector, in which advection is less easily dominant). This simplified statistical accounting allows calculating the effective transport due to the various transport mechanisms acting on the tracer as it travels through the medium.

[19] The total (longitudinal or transverse) dispersion coefficients of the tracer with respect to the mean tracer position is obtained by inserting the diffusion coefficients and times in equation (16):

display math(17)

[20] Here, δT = 1 for transverse dispersion and δT = 0 for longitudinal dispersion. Next, these time periods are expressed in terms of physically relevant parameters.

[21] In most cases, temperature of fluid and solid equilibrate very quickly [de Marsily, 1986] and we therefore assume thermal equilibrium at the solid-pore interfaces (Ts = Tf, steady state). The ratio between the time the heat tracer resides in the solid and in the fluid, ts/(tf + tv), then equals the retardation factor, which is the volumetric ratio of heat capacities of the solid and fluid [Bodvarsson, 1972; Koch and Brady, 1985]:

display math(18)

[22] Using equation (18) and recalling from section 3 that tv/tf = v/vc, equation (17) can now be rewritten as

display math(19)

[23] This is the main outcome of this study, as it summarizes the generic behavior of tracer dispersion in porous media. It is customary to restate this equation in terms of the Péclet number. In any given homogeneous system with fixed parameters, v is proportional to Peh, hence we may substitute v/vcPeh/Pehc (cf. equation (15)) to obtain:

display math(20)

[24] Next, we use δL = 1–δT as one has either longitudinal or transverse dispersion and set Dv = βhL,T(R + 1)DrefPeh (cf. equation (11)). The reason for including the factor R+1 is clarified below. This yields:

display math(21)

where δL = 1–δT (Table 2). It is generally observed that βL > βT, which is due to the fact that longitudinal dispersion is mainly affected by channels in which advection dominates (as discussed above), while the velocity spread inside such channels is larger than in subdominant channels.

Table 2. The General Expression (Equation (21)) for Dispersion Coefficients as a Function of Péclet Number Written Out for Heat and Solute, and for Transverse and Longitudinal Dispersion
 Heat (R ≠ 0)Solute (R = 0)
Longitudinal math formula math formula
Transverse math formula math formula

[25] In the limit R → 0, this expression is equivalent to the expressions derived in van Milligen and Bons [2012] for solute dispersion, which is reasonable in view of the fact that at R = 0, heat cannot enter the solid matrix (due to the zero heat capacity of the solid), so that the equation describes the situation of solute dispersion. To obtain the equation for solute dispersion at R = 0, all subindices “h” (heat) are replaced by “p” (particles) (Table 2). Thus, we have obtained a single equation that describes the dispersion coefficient in both the longitudinal and transverse directions, for both particles (e.g., solutes) and heat. In general, Pehc ≠ Pepc.

[26] In the limit Peh,p/Peh,pc >> 1, dispersion is completely dominated by advection (assuming no other effects are present). This limit can be achieved either by increasing the flow velocity v or by reducing the diffusion coefficients to zero. Increasing the flow velocity is the strategy followed in many experimental setups, as diffusion coefficients are material properties, which cannot be changed [e.g., Chiogna et al., 2010]. However, the limit is easier understood by performing a thought experiment where the diffusion coefficient is reduced while maintaining a constant v. In this case, it is clear that this limit corresponds to pure mechanical dispersion, in which both particle and heat tracers are advected by the fluid flow. Evidently, dispersion should be equal for heat and particles in this case. Hence βhL,T = βpL,T (which also clarifies the reason for including the factor R + 1 in the definition of Dv). One may therefore drop the subindex (h or p) of βL,T. We note that the numerical values of βL reported in literature are systematically larger than the corresponding values of βT (in the same porous material sample) by about an order of magnitude or more [Bear and Verruijt, 1987; Bijeljic and Blunt, 2007; Delgado, 2007].

[27] Equation (21) for longitudinal dispersion has triple asymptotic behavior. For Peh,tr → 0, one obtains

display math(22)

[28] This simply means that the effective diffusion coefficient is a weighted mean between the solid and fluid diffusion coefficients in the case of heat dispersion, or equals the fluid diffusion coefficient in the case of particle dispersion (R = 0). As mentioned above, in the mechanical limit, Peh,p/Peh,pc >> 1, one recovers equation (11) where math formula. Interestingly, there is an intermediate limiting case for Dh,p;f /(βL(R + 1)DrefPeh,p) << Peh,p/Peh,pc << 1, which only occurs for longitudinal dispersion:

display math(23)

[29] This intermediate limiting regime is characterized by a quadratic dependence of the effective longitudinal dispersion coefficient on the Péclet number. This quadratic dependence can be understood as follows. In the intermediate regime, as Peh,p is raised, not only does the flow velocity, and hence flow variability, in the pore channels increase (which is characteristic of mechanical dispersion, and corresponds to the factor βLPeh,p), but simultaneously, the transport in an ever greater fraction of channels is dominated by advection (corresponding to the factor Peh,p/Peh,pc). These two effects combine to give the quadratic dependence of equation (23).

[30] The logarithmic growth rate of the complete dispersion curve equation (21) (i.e., its local power-law exponent) is defined as

display math(24)

[31] Typically, it does not reach the maximum value of 2 corresponding to the limiting case of equation (23), as the two conditions for this regime are usually only marginally met. In a typical practical case, a maximum value of nh,pL = 1.2 was obtained by van Milligen and Bons [2012]. This result is quite significant, in view of the fact that such behavior (equation (2)), including max(nh,pL) = 1.2, has indeed been observed in the intermediate regime by many authors [Saffman, 1959; De Arcangelis et al., 1986; Bijeljic et al., 2004, and references therein]. Anomalous diffusion or fractality have been used to model dispersion behavior in porous media [e.g., Sahimi, 1993; Berkowitz and Scher, 1997; Makse et al., 2000; Meerschaert et al., 2001; Levy and Berkowitz, 2003]. Some authors explained the emergence of nonlinearity (n ≠ 1) with such fractality or non-Fickian diffusion [Sahimi, 1993; Bjieljic and Blunt, 2006]. Other authors argue that nonlinearity in the transition regime is caused by the heterogeneity of the porous system [e.g., Sorbie and Clifford, 1991; Yao et al., 1997; Bruderer and Barnabé, 2001]. However, we show here that the emergency of nonlinearity is to be expected, even in homogeneous systems, without the need to invoke heterogeneity, fractality, or anomalous diffusion.

[32] The validity of the approach described in this section can be tested in multiple ways. Successful microscopic simulations of dispersion in porous media have been performed in recent years [Ovaysi and Piri, 2011; Mostaghimi et al., 2012; and others]. By analyzing the statistical behavior of tracers in such detailed simulations, it should be possible to verify the validity of the assumptions made here. In the next section, we present such an analysis of solute dispersion in a simplified numerical model, confirming the basic assumption underlying equation (15). In section 7, we apply equation (21) to published heat and solute dispersion coefficients covering an ample range of Péclet numbers.

6. Simplified Numerical Model

[33] In order to check the basic assumptions underlying the general expression for the dispersion coefficients, equation (21), we have performed a simulation in simplified geometry. The numerical model consists of a two-dimensional network of straight (pore) channels. These channels connect nodes, and transport along each channel is described by the one-dimensional transport equation

display math(25)

where z is a local coordinate along the channel. At each node, a continuity equation applies, so that the solute concentration p is conserved globally. The diffusion coefficient Dm = 1 mm2/s is taken constant everywhere, while v, constant within each channel, is determined by solving a Laplace equation for the pressure, imposed at the boundaries of the network. Thus, a fairly standard model is obtained for solute (p) transport through a channel network with imposed pressure drop (taken to occur in the x-direction) [Bijeljic, 2004, 2007].

[34] The network considered here is a foam structure obtained from the numerical simulation platform “Elle”, with mean grain size G = 1 mm [Jessell et al., 2001; Roessiger et al., 2011]. Figure 2 shows the grid used in the simulation, with 31,776 nodes. The network is irregular on the smallest scale, but homogeneous and isotropic on larger scales. We initialize simulations with a delta function distribution, p0(x,y) = δ(0,0) at t = t0= 0. Then, we evolve the system in time according to the above system of equations. When the stop criterion θ = maxedge(p)/max(p) exceeds a threshold value, θmax = 0.001, the integration is stopped (at time t = t1). Finally, we determine the longitudinal and transverse dispersion coefficients from the second moment of the spatial distribution of p:

display math(26)

where the angular brackets refer to means taken over the distribution p(x,y) at time t1. Figure 3 shows the evolution of DL,T with the Péclet number. These results resemble the experimental results in various respects: the molecular diffusion and mechanical dispersion limits are reproduced, and the there is a transition region where nL > 1 (while nT < 1).

Figure 2.

Two-dimensional network used for the numerical simulations. The grid has 3972 grains with a mean size of 1 mm2 and 31,776 nodes.

Figure 3.

Results of dispersion simulations with the network shown in Figure 2. (a) DL,T versus Pep. (b) nL,T versus Pep, showing the intermediate regime where nL > 1 (while nT < 1). (c) Nv/Nf versus Pep. The dashed line is Nv/Nf math formula Pep. The simulations are performed for log10(Pep) = −3.0, −2.8, …, 3.8, and 4.0.

[35] In accordance with the discussion of section 4, we define a local Péclet number Peloc = vlocG/D for each channel. Thus, for each value of the externally imposed flow, all individual channels are classified as being either “diffusive” (Peloc ≤ 1) or “advective” (Peloc > 1). Then, at time t1, we determine how much solute (p) resides in “diffusive” channels (yielding Nf) versus “advective” channels (Nv). Figure 3c shows Nv/Nf versus Pep. The figure shows that the statistical quantifier Nv/Nf indeed grows monotonically with Pep, as stated in section 4, with a growth exponent that is close to the exponent of 1 that was assumed in section 4 for simplicity. Future work may clarify if and how this exponent depends on the geometrical details of the network. Thus, the simulation confirms one of the basic assumptions underlying the description elaborated in the preceding section: namely, the variation of the importance of advective versus diffusive channels with Péclet number. Note that the equilibration point (Nv/Nf = 1) immediately precedes the transition region where nL > 1, and marks the value of the critical Péclet number.

[36] It should be noted that the statistical quantifier Nv/Nf is affected by the finite size of the grid: due to the finite number of channels, at very low values of Pe, all channels are diffusive (Nv = 0), while at very high values of Pe, all channels are advective (Nf = 0). This situation means that the quantifier is only valid over a limited range of Péclet numbers (here: 6 < Pe < 1000). However, this range will expand if larger simulation grids are used.

7. Comparison With Experimental Data

[37] To test our approach, we have performed fits of the function (equation (21)) to data from laboratory dispersion experiments on homogeneous porous media available in literature (Table 3). Figure 4 shows a fit to a large data set of experimental measurements of DL and DT for solute dispersion (R = 0) in porous granular aggregates, including gas and liquid flowing through systems of packed beads of regular shape or beds of sand with a wide range of bead or grain sizes [Delgado, 2007]. The scatter is partly due to variations in material properties, such as grain shape and size distribution. Here and in the following, fit quality is quantified using the coefficient of determination

display math(27)

where i enumerates the data points, angular brackets indicate a mean over i, and y is the logarithm of the relevant dispersion coefficient. We draw particular attention to the correct behavior of the model in the intermediate regime (ii + iii), where the simple addition model (equation (1)) fails.

Table 3. Experimental Data, Type, and Best-Fit Parameters for the Proposed Model
Data SourceExperimental DataTypePecβRfit2
Figure 4: Figures 1 and 2 in Delgado [2007] and references thereinGranular porous aggregates (regular shaped beads and beds of sand)Solute, DpL,T3.4 ± 0.2βpL = 1.4 ± 0.10.9776
βpT = 0.064 ± 0.006
Figure 6: Guedes de Carvalho and Delgado [2003, 2005]Packed glass beadsSolute, DpL13.5 ± 1.02.13 ± 0.050.9998
Figure 7: Figures 11 and 13 of Levec et al. [1985b]Urea formaldehyde microspheres (diameters of 0.55 cm and 0.25 cm);data only for Peh < 300Heat, DhL,T23 ± 8βhL = 1.8 ± 0.30.885
βhT = 0.052 ± 0.006
Figure 8: Figure 10 of Rau et al. [2012]Well-sorted quartz sand (d50 ∼ 2 mm, uniformity U =d60/d10 ∼ 1.68)Solute (KCl) and heat, Dp,hL,TPehc = 12 ± 2βhL = βpL =1.33 ± 0.070.9977
Pepc = 37 ± 8
Figure 9: Figure 10 of Delgado [2006]Granular porous aggregatessolute, DpLPepcSc0.14β LSc0.22≥0.9771
Figure 4.

Longitudinal (circles) and transverse (crosses) solute dispersion data normalized to the molecular diffusion coefficient (Dm), after Figure 2 in van Milligen and Bons [2012]. Data from experiments in a wide range of granular porous media from Delgado [2007], and references therein. The fit of the model to the longitudinal dispersion data set yielded D0/Dm = 0.8 ± 0.2, Pepc = 3.4 ± 0.2. The full longitudinal and transverse dispersion data set was fitted keeping Pepc fixed, yielding βL = 1.4 ± 0.1 and βT = 0.064 ± 0.006. Rfit2 = 0.9776. The vertical dashed line indicates Pepc.

[38] Using equation (21), one can now predict heat dispersion (Figure 5a, assuming R = 10). Only the slightly deviating DhL is shown, as there is no discernable difference between DhT and DpT. Figure 5b shows the local effective power-law exponent, cf. equation (24). In the intermediate regime, the transverse dispersion exponent is clearly below 1, while the longitudinal exponent exceeds 1, in accordance with numerous studies [see e.g., reviews of Delgado, 2006; Delgado, 2007].

Figure 5.

(a) Normalized longitudinal and transverse solute dispersion coefficient curves derived from Figure 4, compared to model prediction for heat dispersion at R = 10. Only longitudinal heat dispersion is shown, as there is no visible difference between transverse heat and solute dispersion at the given R. (b) Apparent power law exponent of fit lines of Figure 4a, computed as math formula.

Figure 6.

Longitudinal dispersion coefficients in a packed bed of glass beads (after Figure 3 in van Milligen and Bons [2012]). Diamonds: data from Guedes de Carvalho and Delgado [2005] at Sc = 754. Circles: data from Guedes de Carvalho and Delgado [2003] at Sc = 424. Closed symbols: DL/Dm. Open symbols: the same data represented as PeLPepDm/DL. The continuous curve corresponds to the fit of the data to equation (21), yielding D0/Dm = 0.86 ± 0.05, Pepc = 13.5 ± 1, and βL = 2.13 ± 0.05. The long dashed line is the same curve, represented as PeL. The vertical dashed line indicates Pepc. Note the very close agreement between the data points and the model curve, reflected in the high value of Rfit2 = 0.9998. The dotted curve is PeL computed from a simplified model that is often used, according to which the effective dispersion is the sum of diffusion and mechanical dispersion, DL/Dm = D0/Dm + βLPep, cf. equation (1). It is seen that this simplified model fails to fit the data in the intermediate regime.

Figure 7.

Normalized heat dispersion coefficients for urea formaldehyde spheres (grain size 5.5 mm) and water from Figures 11 and 13 in Levec et al. [1985b]. Fit results: Pehc = 23 ± 8, βhL = 1.8 ± 0.3, and βhT = 0.052 ± 0.006. Rfit2 = 0.885. Note that Peh in this graph is recalculated cf. our equation (7) as Levec et al. [1985b] used a slightly different definition.

Figure 8.

Fit to normalized longitudinal heat and solute dispersion coefficients obtained in the same well-sorted sand (grain size 2 mm) aggregate (R = 1.33) from Figure 10 in Rau et al. [2012]: Pehc = 12 ± 2, Pepc = 37 ± 8, βhL = βpL = 1.33 ± 0.07, with a coefficient of determination Rfit2 = 0.9977. Note that Peh in this graph is recalculated cf. our equation (7) as Rau et al. [2012] used a slightly different definition.

Figure 9.

(a) A collection of longitudinal solute dispersion coefficients at different values of the Schmidt number (viscosity/diffusivity), presented in Delgado [2006], and fits to the current expression. Data points at very high values of Pep (to the right of the corresponding fit lines) are ignored, as this is the region where inertia and turbulence become important. Rfit2(Sc = 0.72) = 0.9771. Rfit2(Sc = 57) = 0.9997. Rfit2(Sc = 102) = 0.9999. Rfit2(Sc = 200) = 0.9993. Rfit2(Sc = 424) = 0.9993. Rfit2(Sc = 754) = 0.9997. Rfit2(Sc = 1930) = 0.9998. (b) Values of Pepc and βL, derived from the fits, versus Sc. The dashed lines are power laws: PepcSc0.14and βLSc0.22.

[39] The precision of the analytical curve becomes even more apparent when considering longitudinal and transverse dispersion data obtained under very similar experimental conditions and covering a significant range of Péclet numbers in the important intermediate regime between the diffusive and mechanical dispersion limits [Guedes de Carvalho and Delgado, 2003; Guedes de Carvalho and Delgado, 2005]. The result shows close agreement between the data points and the model curve, as reflected in the very high value of Rfit2 = 0.9998 (Figure 6). A fit to the data using the popular expression of equation (1), giving the total dispersion as a sum of diffusion and mechanical dispersion, clarifies the degree of failure of the latter in the intermediate regime of Péclet numbers.

[40] Partly due to the different (larger) value of the heat diffusion coefficient, as compared to the particle diffusion coefficient, systematic heat dispersion data covering a wide range of Peh values are rather scarce in literature. In Figure 7, we show the DhL,T data of Levec et al. [1985b]. Here we consider only data for Peh < 300, as the data for higher Péclet numbers suffered from alterations in viscosity, as mentioned by the authors, even though they attempted to correct for these effects. Our proposed expression again describes the data reasonably well with only three fit parameters. Only at very low Peh does the measured DhL deviate from the expression, which may be due to the occurrence of convection and the use of finer-grained materials in this range, according to Levec et al. [1985b].

[41] A rather interesting challenge for the present approach is offered by the high-quality heat and solute dispersion data obtained in a single saturated sand bed, with a grain size of 2 mm, by Rau et al. [2012]. Figure 8 shows a simultaneous fit to these data. Note that the two lines are fit to the heat and solute dispersion data using only three free parameters. Given this very limited number of free parameters, the analytical curves provide a surprisingly good fit (Rfit2 = 0.998). One observes a slight difference between Pehc and Pepc. The critical Péclet number arises from the competition between advection and diffusion and indicates the Péclet number at which advection starts to dominate over diffusion [van Milligen and Bons, 2012]. Thus, it is reasonable that PehcPepc, as diffusion acts differently for the two transport problems.

[42] It is of interest to note that Rau et al. [2012] fit both the heat and solute dispersion data using a simple heuristic law, Dh,pL = Dh,p;ref + αh,pvn, where n = 2 for heat dispersion and n = 1 for solute dispersion. While such heuristic laws function well over a reduced range of Péclet numbers, they have no physical basis and cannot be used for extrapolation. In contrast, not only is our proposed description applicable to the full range of Péclet numbers, but it also provides a link between heat and particle dispersion, missing in such heuristic models.

[43] Evidently, many factors may modify the first-order behavior captured by the analytic description to some degree. For example, specific pore topologies or mobile and immobile regions in pore space [e.g., Coats and Smith, 1964; Schumer et al., 2003] may modify the generic behavior described here. Furthermore, several additional physical effects have not been considered here, such as viscosity, quantified via the Schmidt number (Sc), as shown, for example, by Delgado [2007]. Figure 9a shows solute dispersion data, for various values of Sc, obtained using gas and fluid flow through packed beds, from Delgado [2006]. Our proposed model provides an excellent fit to each of these data sets, showing that variation in Sc does not affect the generic dispersion behavior (as given by equation (21)), but modifies the values of Pec and β: both these parameters exhibit a slight dependence on Sc (Figure 9b).

8. Discussion

[44] In the current study, we propose a unified description for tracer (both solute and heat) dispersion in homogeneous and isotropic porous media. The unified description is an extension of an analytic description of solute dispersion presented in a previous study by van Milligen and Bons [2012]. Apart from a few standard assumptions, the main novel ingredient of the model is the competition between diffusive and advective transport inside the pore channels. Here we used intuitive or heuristic arguments to support this description, and checked with a simplified numerical model that the assumed behavior does indeed arise. It is, however, clear that the matter is far from trivial and further justification is therefore needed, for example based on microscopic arguments. This approach leads to an analytic expression that is capable of successfully describing dispersion coefficients in homogenous porous media over a large range of Péclet numbers, both in the longitudinal and transverse directions, and for both solute and heat.

[45] The present approach solves an old mystery related to the unexpected appearance of nonlinear growth of the dispersion coefficient as the flow rate is increased, as discussed in the introduction, and which has led to the use of equation (2) with various values of n and equation (3) with γ ≠ 1. Although the latter may provide excellent fits to experimental data over a limited range of Pe numbers [e.g., Chiogna et al., 2010; Rolle et al., 2012], equations (2) and (3) do not appropriately reflect the underlying physical process and are therefore of limited use. By contrast, the description proposed here has the advantage of covering the full range of Péclet numbers in the laminar flow regime, including the intermediate regime (ii + iii).

[46] The model we propose is neither microscopic nor fully macroscopic. Microscopic modeling at the pore scale has progressed considerably due to the advent of large-scale computing. In recent studies, dispersional behavior has been reproduced successfully with detailed numerical models [Maier et al., 2000; Mostaghimi et al., 2012; Ovaysi and Piri, 2011]. Even so, microscopic models only provide limited insight into the resulting global behavior and do not immediately provide useful closed-form expressions to formulate predictions of material behavior under specific conditions. Our approach includes one specific novel ingredient to capture the competition between diffusion and advection in the pore channels at a statistical level. This is the critical Péclet number, which can be understood to reflect the point at which the distribution of solute or heat tracers over dominantly advective and dominantly diffusive pore channels is balanced (either in steady state or a sufficiently long time after injection into the flow). Hence, this critical Péclet number is a mesoscale property of the material, as a statistically large number of pore channels is involved in its definition, although it is unrelated to the total sample size (assumed much larger than the relevant mesoscale). However, the model does not predict the value of this quantity. From the analysis of a wide range of experimental data in the present work, we find that Pec is typically larger than one, although not more than by an order of magnitude or so (Table 3).

[47] We treated heat the same way as solute or particles, which allows applying the same description to heat and solute dispersion. The only difference is that contrary to solutes, heat “tracers” can diffuse through both pores and solid matrix. The practical advantage is that solute dispersion data can be used to predict heat dispersion, or vice versa, especially if the relationship between the critical Péclet numbers Pepc and Pehc is clarified. It should be noted that we have assumed local thermal equilibrium between fluid and solid (determining R in equation (18)). The other end member would be the case where there is no thermal interaction between fluid and solid, in which case R reduces to zero and thermal dispersion behavior is the same as for solute dispersion. A more fundamental advantage of this approach is that it opens up the possibility to treat more complex systems, such as the two-domain problem, double porosity materials and polyphase fluids (such as oil and gas).

[48] Equations derived here are approximate and only apply to homogeneous porous systems that are large enough (much larger than the pore dimensions) to reach asymptotic behavior [Bijeljic and Blunt, 2007]. It is implicitly assumed that the sample is large enough that a statistical approach to the flow through pores is meaningful and that pore geometry is appropriately random (so that the mechanical dispersion limit at high Péclet number is reached) and/or possesses full two or three-dimensional connectivity (so that hold-up dispersion is not important). In spite of these restrictions, the present approach appears to capture the generic first-order behavior of such materials well.

[49] More detailed confirmation of the model presented here requires new data for both longitudinal and transverse dispersion, simultaneously for heat and solute, similar to the study by Rau et al. [2012]. To determine the parameters of the model with precision (and in particular Pec), measurements must be made over a sufficiently wide range both above and below Pec. Such additional laboratory or numerical studies may also establish how the relevant model parameters scale with other fundamental parameters such as grain size, Schmidt number, and tortuosity. In addition, statistical analyses applied to tracer data taken from microscopic simulations might provide more detailed confirmation of the central hypothesis of the proposed model, i.e., the assumption of competition between transport mechanisms in pore channels.

9. Conclusions

[50] A unified expression for both heat and solute dispersion coefficients is proposed that is based on the concept that heat dispersion is largely analogous to particle dispersion of solutes (or tracers). The model is quite general in two aspects: it describes (1) both longitudinal and transversal dispersion coefficients for (2) both solute and heat. It is found that heat and solute dispersion coefficients are related via the retardation factor R. This opens up the possibility to predict heat transport properties from solute transport data and vice versa, which may be a considerable advantage for laboratory and field studies. Comparison with available data reveals that the model fits the data over a large range of Péclet numbers, spanning at least eight orders of magnitude (up to Pe > 106). Thus, the generic behavior of the dispersion of solute and heat in homogeneous porous media is captured by a single unified expression. The latter solves an old mystery related to the appearance of nonlinear growth of the dispersion coefficient as the flow rate is increased.


[51] Research sponsored in part by Ministerio de Economía y Competitividad of Spain under project Nr. ENE2012–30832. This study was carried out within the framework of DGMK (German Society for Petroleum and Coal Science and Technology) research project 718 “Mineral Vein Dynamics Modelling,” which is funded by the companies ExxonMobil Production Deutschland GmbH, GDF SUEZ E&P Deutschland GmbH, RWE Dea AG and Wintershall Holding GmbH, within the basic research program of the WEG Wirtschaftsverband Erdöl- und Erdgasgewinnung e.V. We thank the companies for their financial support and their permission to publish these results. Comments by the Editor and reviewers helped to significantly improve the manuscript.