Dilution of Reactive Plumes: Evolution of Concentration Statistics Under Diffusion and Nonlinear Reaction

Abstract

Concentration fields of solutes in porous media often exhibit large fluctuations, driven by physical and chemical heterogeneity from the pore to the Darcy scale. For many applications, ranging from reactive transport modeling to toxicology, the knowledge of mean concentrations is not sufficient, and quantifying concentration variability is necessary. The probability density function (PDF) of concentration quantifies the frequency of occurrence of concentration values throughout a spatial domain. While evolution equations and analytical solutions for the concentration PDF exist for conservative solutes, less is known about its evolution under the joint action of transport and reaction. In this work, we investigate how dilution of a reactive plume by diffusion affects the statistics of concentrations. While mixing has no effect on first-order reactions, its coupling with nonlinear reactions leads to non-trivial effective kinetics relevant for a broad range of reactive transport problems. We study the evolution of the concentration PDF under diffusion and nonlinear reaction in one spatial dimension, which represents a critical step toward further coupling with heterogeneous advection. We show that the dependence of the scalar dissipation rate on concentration encodes the impact of diffusive transport on the concentration PDF and derive a dynamical equation for its time evolution. Using a weak-coupling approximation for the reaction and diffusion dynamics, we derive analytical predictions for the concentration PDF and its moments. Our results provide new insights into how diffusion and reaction control concentration variability and open new opportunities for coupling mixing models with chemical reactions.

Article Highlights

• We introduce a general framework quantifying the link between concentration PDFs and spatial concentration profiles.

• We derive a dynamical equation for the evolution of the concentration PDF under diffusion and nonlinear reaction in 1D.

• We derive analytical predictions for the concentration PDF and its moments using a weak-coupling approximation.

Appendices

A Discretization

When computing the concentration PDF numerically, discretizations are typically employed both spatially and for the concentration values. As a simple example that highlights the central concepts, consider a regular spatial discretization into a grid with constant cell volume $V_g$, with each grid cell associated with the average concentration within it. According to the argument for constant concentration regions developed in the previous section, for each fixed time t, the concentration PDF is then estimated as

$$\begin{aligned} p(c;t;N_g)=\frac{1}{N_g}\sum _{i=1}^{N_g}\delta [c-c_i(t)], \end{aligned}$$

(60)

where $N_g=|\varOmega |/V_g$ is the number of grid points and $c_i$ is the average concentration in cell i. Note that the $c_i$ associated with different cells are not necessarily all different.

Further discretizing concentration into bins leads to a probability mass function with values for the probability of concentration in each bin $B_k=[c^{(k)},c^{(k+1)}[$, $k\geqslant 0$:

$$\begin{aligned} p_k(t;N_g)=\frac{1}{N_g}\sum _{i=1}^{N_g}H[c_i(t) -c^{(k)}]H[c^{(k+1)}-c_i(t)], \end{aligned}$$

(61)

where $H(\cdot )$ is the Heaviside step function. This means that the probability of finding a concentration value in bin k is the fraction of cells where the concentration falls within bin k. If we take concentration bin widths $\varDelta c$ to be constant, $c^{(k)}=k\varDelta c$ for $k\geqslant 0$, and approximate the PDF of concentration by dividing probabilities by $\varDelta c$ (as in Fig. 2), we obtain

$$\begin{aligned} p(c;t;N_g,\varDelta c)=\frac{1}{N_g\varDelta c}\sum _{i=1}^{N_g} H[c_i(t)-k\varDelta c]H[(k+1)\varDelta c-c_i(t)]. \end{aligned}$$

(62)

These discretization procedures generalize directly to a multispecies system. In that case, concentration bins refer to the simultaneous attainment of concentration values of each species. The associated probabilities are computed as above by counting spatial cells where these values occur simultaneously. The procedure also generalizes directly to non-uniform and/or time-dependent spatial cell sizes and/or concentration bins. More involved techniques employing kernel reconstructions of the concentration field may be formalized in a similar manner (Morariu et al. 2008; Fernàndez-Garcia and Sánchez-Vila 2011; Sole-Mari et al. 2019).

The maximum resolution of the discretized concentration PDF described above is given by $1/(N_g\varDelta c)$, and the maximum PDF value is $1/\varDelta c$. We now discuss the impact of local spatial extrema, which can be associated with divergences of the continuous PDF as discussed in Sect. 2.3, on the discretized computation. It is also important to note that sources of error typically come into play in the determination of the spatial concentration field. For example, if the latter is computed based on a particle tracking simulation of some transport and reaction dynamics, fluctuations arise due to the finite number of particles, whereas in a standard Eulerian simulation the spatial discretization impacts the determination of the true concentration field. We focus here on the error resulting from concentration discretization $\varDelta c$, which is dominant given a sufficiently resolved spatial concentration field.

Consider first a region of constant concentration. A delta peak $\delta (c-c_0)|\varOmega _0|/|\varOmega |$ in the concentration PDF, associated as discussed above to a subdomain $\varOmega _0$ of volume $|\varOmega _0|$ where $C(\varvec{x};t)=c_0$, corresponds under sufficiently fine spatial discretization to a discretized contribution $|\varOmega _0|/(|\varOmega |\varDelta c)$. Thus, halving the concentration discretization $\varDelta c$ leads to a doubling of the numerically computed peak. More generally, refining the discretization as $\varDelta c \rightarrow a \varDelta c$, $a<1$, yields $p(c_0;t;N_g,a\varDelta c)=a p(c_0;t;\varDelta c)$. On the other hand, smooth extrema correspond to divergences only for spatial dimension $d=1$, for which they lead to an inverse-square-root divergence, as shown in Sect. 2.3. In this case, averaging the concentration over a range $\varDelta c$ near the spatial extreme value $c_0$ yields $p(c_0;t;\varDelta c)\propto 1/\sqrt{\varDelta c}$, so that $p(c_0;t;a\varDelta c)\approx \sqrt{a} p(c_0;t;\varDelta c)$. Thus, observing these behaviors in a numerical computation is a signature of the presence of a spatial extremum, and the scaling behavior with concentration discretization refinement indicates its type.

B Dynamical Equation for the Concentration PDF

In this appendix, we provide a detailed derivation of the dynamical equation for the concentration PDF discussed in the main text, Eq. (34), by explicitly determining the diffusive transport and reaction contributions in Eq. (29). We first consider the effect of diffusion, $\varDelta p_D(c;t)$. For one-dimensional diffusion and nonlinear concentration decay, assuming a symmetric initial condition about the origin and monotonically decreasing with distance from the latter, the spatial concentration profile retains these properties for all times. Therefore, we have $|\varLambda (c;t)|=2$ for all concentrations within the range observed at time t, corresponding to the two points $x=\pm x_c(c;t)$ where $C(x;t)=c$. Furthermore, the concentration gradient magnitude is the same at $\pm x_c$, so that its harmonic average is simply $g_h(c;t)=\nabla C[-x_c(c;t),t]=-\nabla C[x_c(c;t),t]$. Using Eq. (8) and the fact that $|\varLambda (c;t)|$ is time-independent within the concentration range observed, we can write the change in the concentration PDF as

$$\begin{aligned} \frac{\partial p(c;t)}{\partial t}=-p(c;t) \left[ \frac{\partial \ln |\varOmega (t)|}{\partial t} +\frac{\partial \ln g_h(c;t)}{\partial t}\right] . \end{aligned}$$

(63)

If a fixed reference volume is considered, the term corresponding to the change of $|\varOmega (t)|$ in time is zero. As discussed in the main text, we focus here on the case of a minimum detection limit $c_m$ and a time-varying domain $\varOmega (t)$ where $c>c_m$.

In order to compute the terms in square brackets, we consider the time evolution of quantities on a given concentration surface (in one dimension, at the points $\pm x_c(c;t)$). By definition, the change in time of concentration over such a surface is zero, so that

$$\begin{aligned} \frac{\partial C[x_c(c;t);t]}{\partial t} =\left[ \frac{\partial x_c(c;t)}{\partial t} \frac{\partial C(x;t)}{\partial x} +\frac{\partial C(x;t)}{\partial t}\right] _{x=x_c(c;t)} = 0 \end{aligned}$$

(64)

Taking into account that, at $x=x_c$, $\partial /\partial x=-g_h(c;t)\partial /\partial c$, the changes associated with transport lead to

$$\begin{aligned} \frac{\partial x_c(c;t)}{\partial t} =\frac{1}{2g_h(c;t)}\frac{\partial \chi (c;t)}{\partial c}, \end{aligned}$$

(65)

where $\chi (c;t) = \chi _x[\pm x_c(c;t);t] = D g_h^2(c;t)$ is the concentration-dependent scalar dissipation rate. The same approach for the variation of the concentration gradient leads to

$$\begin{aligned} \frac{\partial g_h(c;t)}{\partial t} =-\left[ \frac{\partial x_c(c;t)}{\partial t} \frac{\partial ^2 C(x;t)}{\partial x^2} +\frac{\partial }{\partial t}\frac{\partial C(x;t)}{\partial x} \right] _{x=x_c(c;t)}, \end{aligned}$$

(66)

and we find

$$\begin{aligned} \frac{\partial \ln g_h(c;t)}{\partial t} =\frac{1}{2}\frac{\partial ^2\chi (c;t)}{\partial c^2} -\frac{1}{4}\frac{\partial \ln \chi (c;t)}{\partial c} \frac{\partial \chi (c;t)}{\partial c}. \end{aligned}$$

(67)

The change in domain volume $\varOmega (t)=2x_c(c_m;t)$ due to dilution of concentration below the detection limit $c_m$ obeys

$$\begin{aligned} \frac{\partial |\varOmega (t)|}{\partial t} = 2\frac{\partial x_c (c_m;t)}{\partial t}, \end{aligned}$$

(68)

Using Eq. (65), this leads to

$$\begin{aligned} \frac{\partial |\varOmega (t)|}{\partial t} = \frac{1}{g_h(c_m;t)} \frac{\partial \chi (c;t)}{\partial c}\Bigg |_{c=c_m}, \end{aligned}$$

(69)

so that, dividing through by $|\varOmega (t)|$ and using Eq. (8),

$$\begin{aligned} \frac{\partial \ln |\varOmega (t)|}{\partial t} =\frac{p(c_m;t)}{2}\frac{\partial \chi (c;t)}{\partial c}\Bigg |_{c=c_m}. \end{aligned}$$

(70)

Using Eq. (63) for the transport contribution, these results lead to

$$\begin{aligned} \varDelta p_D(c;t)=p(c;t)\left[ \frac{1}{4}\frac{\partial \ln \chi (c;t)}{\partial c}\frac{\partial \chi (c;t)}{\partial c} -\frac{1}{2}\frac{\partial ^2\chi (c;t)}{\partial c^2} -\frac{p(c_m;t)}{2}\frac{\partial \chi (c;t)}{\partial c} \Bigg |_{c=c_m}\right] . \end{aligned}$$

(71)

We now turn to the reaction term, $\varDelta p_R$. Consider a small change dc in the concentrations due to reaction only, over a small time interval dt. The probability $p(c;t+dt)\,dc$ of finding concentrations in the infinitesimal vicinity dc of c decreases due to reaction at rate r(c) away from c, and increases due to decrease in nearby concentrations toward c. Thus,

$$\begin{aligned} {[}p(c;t+dt)-p(c;t)]\,dc = \sum _{j=1}^{n_s}[r(c+dc) p(c+dc;t)-r(c) p(c;t)]\,dt. \end{aligned}$$

(72)

Expanding the first term on each side in Taylor series, and dividing through by dt and dc, we obtain

$$\begin{aligned} \frac{\partial p(c;t)}{\partial t} =\frac{\partial r(c) p(c;t)}{\partial c}. \end{aligned}$$

(73)

When a fixed minimum concentration detection limit is considered, as discussed above, it is necessary to take the change in volume (in one dimension, length) $|\varOmega (t)|$ due to reactive decay of the minimum concentration into account. Using the same techniques as before, we obtain for the change in the domain size due to reaction:

$$\begin{aligned} \frac{\partial \ln |\varOmega (t)|}{\partial t} = -p(c_m;t)r(c_m). \end{aligned}$$

(74)

Thus, the complete effect of reaction is

$$\begin{aligned} \varDelta p_R(c;t)=\frac{\partial r(c) p(c;t)}{\partial c}+p(c;t) p(c_m;t)r(c_m). \end{aligned}$$

(75)

Note that integration of the right-hand side from $c=c_m$ to $c=\infty$ yields zero, which ensures the reactive contribution conserves probability for arbitrary r(c). Substituting the effects of transport and reaction, Eqs. (33) and (33), in Eq. (29) leads to the dynamical Eq. (34) for the evolution of the concentration PDF under one-dimensional diffusion and nonlinear decay.

C Numerical Methods

In this appendix, we provide details on the numerical methods used to integrate the weak-coupling equations (45) and the reaction–diffusion equation (35). Regarding Eqs. (45), which are ordinary differential equations, we implemented a standard fourth-order Runge–Kutta method in the C++ language. This method was chosen for its simplicity of implementation and high accuracy, and also because, as an explicit method, it provides a convenient approach to integrate these nonlinear equations without requiring numerical root-finding methods. We employed a time step $\varDelta t=10^{-2}\min \{\mathrm{Da},1/\mathrm{Da}\}$ for the temporal discretization, which we verified led to consistently converged results.

Table 1 Discretization parameters used in computing solutions of Eq. (35) to determine the mass and concentration peak, mean, and variance. For these computations, we employed a temporal discretization $\varDelta t = a\min \{\mathrm{Da},1/\mathrm{Da}\}$ and a spatial discretization $\varDelta x=b\sqrt{\varDelta t/\mathrm{Da}}$. The corresponding values of (a, b) for different reaction orders $\beta$ and Damköhler numbers $\mathrm{Da}$ are given in table

Table 2 Discretization parameters used in computing solutions of Eq. (35) to determine the concentration PDF and scalar dissipation rate

For the fully coupled reaction–diffusion problem, Eq. (35), we employed the py-pde open-source Python package for solving partial differential equations (Zwicker 2020). We used a regular finite difference discretization of a one-dimensional domain of half-width L and second-order centered differences for the spatial derivative approximations. For the time integration, we employed an explicit Forward Euler scheme. We set reflecting boundary conditions at the edges of the computational domain, but we verified that the latter was sufficiently large that no appreciable mass reached the edges, rendering the choice of boundary conditions irrelevant. Since the late-time variance growth is approximately diffusive, $L=10\sqrt{t_m/\mathrm{Da}}$, where $t_m$ is the maximum simulation time, may be used as a simple estimate of necessary domain size. However, because of the lower detection limit $c_m=10^{-6}$ used in the computation of the quantities of interest, we found that in practice it was never necessary to use $L>1500$ for the simulations conducted here. We note that, for $\beta <1$, where complete depletion of concentrations can happen in finite time, the increase of reaction rates with decreasing concentration values can lead to numerical issues, because very low concentrations can drop below zero within a time step. We avoid this issue by setting negative concentrations to zero before computing reaction rates. We chose the spatial and temporal discretizations so as to ensure good accuracy while maintaining reasonable simulation times. The discretization parameters for different system parameters $\beta$ and $\mathrm{Da}$ are summarized in Table 1 for the mass and concentration peak, mean, and variance calculations, and in Table 2 for the concentration PDF and scalar dissipation rate.

The concentration PDF was obtained by counting discretized spatial locations where the concentration value fell within prescribed bins (see also Section A). In order to accurately resolve both low and high concentrations, we employed $n_\ell$ logarithmically spaced concentration bins for concentrations between the lower detection limit $c_m$ and $2c_M(t)/3$, where the time-dependent peak value $c_M(t)$ was determined from the numerical concentration profiles, and $n_h$ linearly space bins for the remaining concentrations between $2c_M(t)/3$ and $c_M(t)$. For the lower, intermediate, and higher time examined in each case, we employed $(n_\ell ,n_h)=(20,10)$, (15, 8), and (10, 6), respectively. The scalar dissipation rate was calculated according to Eq. (77b) by numerically computing the spatial derivative at each discretized spatial location (in the rising limb of the symmetric concentration profile), using second-order central differences. The corresponding concentration values at each spatial location were recorded and used to obtain the scalar dissipation rate as a function of concentration.

D Problem Setup and Non-dimensionalization

This appendix provides additional details on the non-dimensionalization used in Sect. 4. Denoting non-dimensionalized quantities by an asterisk, we have

$$\begin{aligned} C_*(x_*;t_*) = \frac{C(s_0 x_*,\tau _R t_*)}{c_0}, \quad r_*(c_*) = \frac{\tau _R}{c_0}r(c_0 c_*)=c_*^\beta , \end{aligned}$$

(76)

where the characteristic reaction time $\tau _R = \kappa ^{-1} c_0^{1-\beta }$. The minimum and maximum concentrations $c_M(t)$ and $c_m(t)$ are normalized in the same manner. Similarly, the non-dimensional spatial variance $\sigma ^2_*(t_*)=\sigma ^2(\tau _R t_*)/s_0^2$. The concentration PDF and scalar dissipation are then non-dimensionalized accordingly as

$$\begin{aligned} p_*(c_*;t_*)&= c_0p(c_0 c_*; \tau _R t_*), \end{aligned}$$

(77a)

$$\begin{aligned} \chi _*(c_*;t_*)&= \frac{\tau _R}{c_0^2} \chi (c_0 c_*; \tau _R t_*) = \frac{1}{2\mathrm{Da}} \left( \frac{\partial C_*(x_*,t_*)}{\partial x_*} \right) ^2_{x_*=x_c(c)/s_0}, \end{aligned}$$

(77b)

with the Damköhler number $\mathrm{Da}=\tau _D/\tau _R$.

Note that, in non-dimensional units, the pulse initial condition is given by a unit-width rectangle centered at the origin,

$$\begin{aligned} C_*(x_*;0) = H\left( 1/2-x_*\right) H\left( 1/2+x_*\right) , \end{aligned}$$

(78)

which implies $p_*(c_*;0)=\delta (c_*-1)$. The batch concentration $c_B(t)$ for the well-mixed problem is non-dimensionalized as above, and $c_{B*}(0)=1$.

E Batch Dynamics

Here, we provide some details on the equations governing the well-mixed batch problem discussed in Sect. 4 and its relation to the concentration PDF. Noting that $p(c;t)=\delta [c-c_B(t)]$ for the batch problem, where $c_B(t)=C(x;t)$ is the homogeneous concentration over the domain, and assuming $c_B(t)>c_m$, Eq. (34) becomes

$$\begin{aligned} \frac{\partial p(c;t)}{\partial t}=\frac{\partial r(c) p(c;t)}{\partial c}. \end{aligned}$$

(79)

Multiplying through by c and integrating over c (using integration by parts on the right hand side), we recover the standard well-mixed rate law for the batch concentration as a function of time,

$$\begin{aligned} \frac{dc_B(t)}{dt} = -r[c_B(t)]. \end{aligned}$$

(80)

Substituting Eq. (76) for the rate yields Eq. (37).

Once $c_B(t)$ drops below $c_m$, at some time $t_m$, the domain $\varOmega (t)$ where concentrations are above this detection limit becomes empty, and the concentration PDF becomes ill-defined. By convention, we can set $c_B(t>t_m)=0$ and $p(c;t>t_m)=\delta (c)$, which conveys the meaning that concentrations are zero everywhere (below the detection limit).