Transport of kinetically sorbing solutes in heterogeneous sediments with multimodal conductivity and hierarchical organization across scales

Abstract

Solute transport in subsurface environments is controlled by geological heterogeneity over multiple scales. In reactive transport characterized by a low Damköhler number, it is also controlled by the rate of kinetic mass transfer. A theory for addressing the impact of sedimentary texture on the transport of kinetically sorbing solutes in heterogeneous porous formations is derived using the Lagrangian-based stochastic methodology. The resulting model represents the hierarchical organization of sedimentary textures and associated modes of log conductivity (K) for sedimentary units through a hierarchical Markov Chain. The model characterizes kinetic sorption using a spatially uniform linear reversible rate expression. Our main interest is to investigate the effect of sorption kinetics relative to the effects of K heterogeneity on the dispersion of a reactive plume. We study the contribution of each scale of stratal architecture to the dispersion of kinetically sorbing solutes in the case of a low Damköhler number. Examples are used to demonstrate the time evolution and relative contributions of the auto- and cross-transition probability terms to dispersion. Our analysis is focused on the model sensitivity to the parameters defined at each hierarchical level (scale) including the integral scales of K spatial correlation, the anisotropy ratio, the indicator correlation scales, and the contrast in mean K between facies defined at different scales. The results show that the anisotropy ratio and integral scales of K have negligible effect upon the longitudinal dispersion of sorbing solutes. Furthermore, dispersion of sorbing solutes depends mostly on indicator correlation scales, and the contrast of the mean conductivity between units at different scales.

Appendices

Appendix 1: Derivation of spectral density of fluctuations in ln(K)

Considering an exponential covariance function of ln(K) (Dagan 1989; Rubin 2003):

$$ C_{Y} (h) = \sigma _{Y}^{2} {\mkern 1mu} e^{{\left( { - \left| {{\kern 1pt} {\kern 1pt} \frac{h}{\mu }{\kern 1pt} {\kern 1pt} } \right|} \right)}} = \sigma _{Y}^{2} {\mkern 1mu} e^{{\left( { - \sqrt {{\kern 1pt} (\frac{{h_{1} }}{{\mu _{1} }})^{2} + {\kern 1pt} (\frac{{h_{2} }}{{\mu _{2} }})^{2} + {\kern 1pt} (\frac{{h_{3} }}{{\mu _{3} }})^{2} } } \right)}} $$

(33)

where μ _i are the integral scales. The Fourier transform of ln(K) is found by:

$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{C}_{Y} (k) = \frac{1}{{(2\pi)^{3}}}\,\,\int {\int {\int\limits_{- \,\infty}^{\infty} {e^{- ik.h}}}} \,C_{Y} (h)\,dh $$

(34)

Substituting (33) into (34) yields:

$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{C}_{Y} (k)\, = \frac{1}{{(2\pi)^{3}}}\,\,\int {\int {\int\limits_{- \infty}^{\infty} {e^{- ik.h}}}} \,\sigma_{Y}^{2} \,e^{{\left(- \left|{\frac{h}{\mu}} \right|\right)}}dh $$

(35)

Integrating (35) leads to the following expression for the Fourier transform of ln(K):

$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{C}_{Y} (k) = \frac{{\sigma_{Y}^{2} \mu_{1} \mu_{2} \mu_{3}}}{{\pi^{2}}}\,\,\frac{1}{{(1 + (\mu k)^{2})^{2}}} $$

(36)

where $(\mu k)^{2} = (\mu_{1} k_{1})^{2} + (\mu_{2} k_{2})^{2} + (\mu_{3} k_{3})^{2}$. (36) is the Fourier transform of ln(K) for the unimodal porous media. For the multimodal porous media, the multimodal covariance function of ln(K) (Eq. (8) and/or Eq. (9)) is easily integrated to obtain the corresponding Fourier transform of ln(K).

Appendix 2: Derivation of the dispersivity of nonreactive solute

Here we use the integration method suggested by Dagan (1989) to derive the related expression for the dispersivity of a nonreactive solute. Equation (15) contains the following integral:

$$ \alpha_{pq}^{(nr)} (t) = \frac{{\int\nolimits_{0}^{t} {u_{p\,q}} \,(U\,t^{\prime})\,dt^{\prime}}}{U} $$

(37)

where u _pq is the flow velocity covariance. The Fourier transform of flow velocity, Eq. (14), is used to evaluate (37). By using (36) we get the Fourier transform of flow velocity in the longitudinal direction as:

$$ \begin{aligned} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{u}_{11} (k) = U^{2} \,\left(1 - \frac{{k^{2}_{1}}}{{k^{2}}}\right)^{2} \,\,\,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{C}_{Y} (k)\,\,\, \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\, = U^{2} \left(1 - \frac{{k^{2}_{1}}}{{k^{2}}}\right)^{2} \,\,\, \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{C}_{Y} (k)\,\,\, \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{U^{2} \sigma_{Y}^{2} \mu_{1} \mu_{2} \mu_{3}}}{{\pi^{2}}}\left(1 - \left(\frac{{k_{1}}}{k}\right)^{2}\right)^{2} \frac{1}{{(1 + (\mu k)^{2})^{2}}} \hfill \\ \end{aligned} $$

Therefore,

$$ u_{11} (h) = \frac{{U^{2} \,\sigma_{Y}^{2} \,\mu_{1} \,\mu_{2} \,\mu_{3}}}{{\pi^{2}}}\int\limits_{- \infty}^{\infty} {\int\limits_{- \infty}^{\infty} {\int\limits_{- \infty}^{\infty} {\left(1 - \left(\frac{{k_{1}}}{k}\right)^{2}\right)^{2} \,\,\,\frac{1}{{(1 + (\mu k)^{2})^{2}}}\cos (k{\cdot}h)\,dk}}} $$

Hence,

$$ \begin{aligned} \frac{{u_{11} (U\,s,0,0)}}{U} = \frac{{U\,\sigma_{Y}^{2} \,\mu_{1} \,\mu_{2} \,\mu_{3}}}{{\pi^{2}}}\int\limits_{- \infty}^{\infty} {\int\limits_{- \infty}^{\infty} {\int\limits_{- \infty}^{\infty} {\left(1 - \left(\frac{{k_{1}}}{k}\right)^{2}\right)^{2} \,\,\,\frac{1}{{(1 + (\mu k)^{2})^{2}}}\cos \left(k_{1} U\,s \right)\,dk}}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathop = \limits_{{k^{\prime}_{i} = \mu_{i} k_{i}}} \,\frac{{U\,\sigma_{Y}^{2} \,}}{{\pi^{2}}}\int\limits_{- \infty}^{\infty} {\int\limits_{- \infty}^{\infty} {\int\limits_{- \infty}^{\infty} {\left(\frac{{\mu_{2}^{- 2} \,k_{2}^{2} \, + \mu_{3}^{- 2} \,k_{3}^{2} \,}}{{\,\mu_{1}^{- 2} \,k_{1}^{2} \, + \mu_{2}^{- 2} \,k_{2}^{2} \, + \mu_{3}^{- 2} \,k_{3}^{2}}}\right)^{2} \,\,\frac{1}{{(1 + k^{2})^{2}}}\cos \left(k_{1} \frac{U\,}{{\mu_{1}}}s\right)\,dk}}} \hfill \\ \end{aligned} $$

Changing variables $s^{\prime} = \mu^{- 1}_{1} \,\,U\,s$ and τ = Ut/μ ₁ gives

$$ \alpha_{11}^{(nr)} (t) = \frac{{\,\sigma_{Y}^{2} \,\mu_{1} \,}}{{\pi^{2}}}\int\limits_{0}^{\tau} {\int\limits_{- \infty}^{\infty} {\int\limits_{- \infty}^{\infty} {\int\limits_{- \infty}^{\infty} {\left(\frac{{\mu_{2}^{- 2} \,k_{2}^{2} \, + \mu_{3}^{- 2} \,k_{3}^{2} \,}}{{\,\mu_{1}^{- 2} \,k_{1}^{2} \, + \mu_{2}^{- 2} \,k_{2}^{2} \, + \mu_{3}^{- 2} \,k_{3}^{2}}}\right)^{2} \,\,\,\frac{1}{{(1 + k^{2})^{2}}}\cos (k_{1} \,s^{\prime})\,dk}}}} \,\,ds^{\prime} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, $$

Now we use $\mu_{1} \, = \mu_{2} = \lambda$ and μ ₃ = ɛλ in which ɛ is the anisotropy ratio. Thus,

$$ \alpha_{11}^{(nr)} (t) = \frac{{\sigma_{Y}^{2} \,\lambda}}{{\pi^{2}}}\int\limits_{0}^{\tau} {\,\int\limits_{- \infty}^{\infty} {\int\limits_{- \infty}^{\infty} {\int\limits_{- \infty}^{\infty} {\left(1 - \frac{{k^{2}_{1} \,\,}}{{\,k^{2}_{1} \, + \,k^{2}_{2} \, + \varepsilon^{- 2} \,k^{2}_{3}}}\right)^{2} \,\frac{1}{{(1 + k^{2})^{2}}}\cos (k_{1} \,s)\,dkds}}}} $$

Now changing to spherical coordinate system and defining:

$$ \begin{aligned} k_{1} \, = r\,\cos \,\beta,\,k_{2} \, = r\,\sin \,\beta,\,\,\,k_{3} \, = k_{3} \hfill \\ dk_{1} \,dk_{2} \,dk_{3} = r\,dr\,d\theta \,dk_{3} \hfill \\ \end{aligned} $$

Therefore,

$$ \alpha_{11}^{(nr)} (t) = \frac{{\,\sigma_{Y}^{2} \,\lambda}}{{\pi^{2}}}\int\limits_{0}^{\tau} {ds\,\int\limits_{0}^{\infty} {r\,dr\,\,\int\limits_{0}^{2\,\pi} {d\theta \,\int\limits_{- \infty}^{\infty} {\left(1 - \frac{{r^{2} \cos^{2} \theta}}{{r^{2} + \varepsilon^{- 2} \,k^{2}_{3}}}\right)^{2} \,\frac{\,\cos (sr\,\cos \,\theta)}{{(1 + r^{2} + \,k^{2}_{3})^{2}}}dk_{3}}}}} $$

(38)

In order to derive the above integral the following general integrals are used;

$$ \int\limits_{- \infty}^{\infty} {\frac{1}{{(a^{2} + x^{2})^{2}}}dx = \frac{\pi}{{2a^{3}}}} $$

(39a)

$$ \int\limits_{- \infty}^{\infty} {\frac{1}{{(a^{2} + x^{2})^{2} (b^{2} + x^{2})}}dx = \frac{\pi}{{2(b^{2} - a^{2})a^{3}}}} \, - \frac{\pi}{{(b^{2} - a^{2})^{2}}}\left(\frac{1}{a} - \frac{1}{b}\right) $$

(39b)

$$ \begin{aligned} \int\limits_{- \infty}^{\infty} {\frac{1}{{(a^{2} + x^{2})^{2} (b^{2} + x^{2})^{2}}}dx = \frac{\pi}{{2(b^{2} - a^{2})^{2}}}\left(\frac{1}{{a^{3}}} + \frac{1}{{b^{3}}}\right)} \hfill \\ - \frac{\pi}{{(b^{2} - a^{2})^{3}}}\left(\frac{1}{a} - \frac{1}{b}\right)\, \hfill \\ \end{aligned} $$

(39c)

To evaluate the integral in (38) the following parameters are used in (39a)–(c):

$$ \begin{aligned} a = 1 + r^{2} = u \hfill \\ b = \varepsilon \,r \hfill \\ a{}^{2} - b^{2} = 1 + r^{2} - \varepsilon^{2} r^{2} = v \hfill \\ \end{aligned} $$

Thus, one can derive (38) as follows

$$ \alpha _{{11}}^{{(nr)}} (t) = \sigma _{Y}^{2} {\mkern 1mu} \lambda {\mkern 1mu} {\mkern 1mu} \int\limits_{0}^{\tau } {{\mkern 1mu} ds{\mkern 1mu} \int\limits_{0}^{\infty } {\left\{ {\frac{r}{{2u^{{\frac{3}{2}}} }}A(sr){\mkern 1mu} dr + 2\varepsilon ^{2} r^{3} \left[ {\frac{1}{{2vu^{{\frac{3}{2}}} }} + \frac{1}{{v^{2} u^{{\frac{3}{2}}} }} - \frac{1}{{v^{2} {\mkern 1mu} \varepsilon r}}} \right]{\mkern 1mu} B(sr) + \varepsilon ^{4} r^{5} \left[ {\frac{1}{{2v^{2} u^{{\frac{3}{2}}} }} + \frac{1}{{2v^{2} (\varepsilon r)^{3} }} + \frac{2}{{v^{3} u^{{\frac{1}{2}}} }} - \frac{2}{{v^{3} \varepsilon r}}} \right]C(sr)} \right\}} } {\mkern 1mu} dr{\text{ }} $$

(40)

where

$$ A(sr) = 2J_{o} (sr) $$

(41a)

$$ B(sr) = \frac{{2J_{o} (sr)sr - 2J_{1} (sr)}}{sr} $$

(41b)

$$ C(sr) = \frac{{2(sr)^{3} J_{o} (sr) - 6J_{o} (sr)(sr) + 12J_{1} (sr) - 4J_{1} (sr)(sr)^{2}}}{{(sr)^{3}}} $$

(41c)

The J _o and J ₁ are the zero and first order Bessel functions, respectively. By changing the variable β = rτ the integration in (40) yields

$$ \begin{aligned} \alpha_{11}^{(nr)} (t) = \,\sigma_{Y}^{2} \,\,\lambda \,\int\limits_{0}^{\tau} {\,ds\,\int\limits_{0}^{\infty} {\frac{r}{{2u^{\frac{3}{2}}}}A(sr)\,dr\,\,}} \hfill \\ \quad + \sigma_{Y}^{2} \,\,\lambda \int\limits_{0}^{\infty} {4\varepsilon^{2} r^{3} \left[\frac{1}{{2vu^{\frac{3}{2}}}} + \frac{1}{{v^{2} u^{\frac{3}{2}}}} - \frac{1}{{v^{2} \,\varepsilon r}}\right]\,J_{1} (\beta)} \,dr \hfill \\ \quad + \sigma_{Y}^{2} \,\,\lambda \int\limits_{0}^{\infty} {\varepsilon^{4} r^{5} \left[\frac{1}{{2v^{2} u^{\frac{3}{2}}}} + \frac{1}{{2v^{2} (\varepsilon r)^{3}}} + \frac{2}{{v^{3} u^{\frac{1}{2}}}} - \frac{2}{{v^{3} \varepsilon r}}\right]\left[\,\frac{{2J_{1} (\beta)\beta^{2} \, - 4J_{1} (\beta) + 2J_{0} (\beta)\beta}}{{\beta^{2}}}\right]dr} \hfill \\ \end{aligned} $$

In the above integral one can use the following general integral

$$ \int\limits_{0}^{\tau} {\,ds\,\int\limits_{0}^{\infty} {\frac{r}{{2u^{\frac{3}{2}}}}A(s\,r)\,d\,r\,\,}} = \int\limits_{0}^{\infty} {\frac{{\tau \,J_{o} (\beta)}}{{1 + r^{2} + r\sqrt {1 + r^{2}}}}dr = 1 - e^{- s}} $$

(42)

Finally, α ^(nr)₁₁ (t) is found as follows

$$ \alpha_{11}^{(nr)} (t)\,=\,\sigma_{Y}^{2} \,\,\lambda \{\,1 - e^{- \tau} - \varepsilon \int\limits_{0}^{\infty} {[2\,r\,J_{1} (\beta)} \frac{{2u^{\frac{3}{2}} \, - \varepsilon \,r(v + 2u)}}{{v^{2} u^{\frac{3}{2}}}} + F(r)]dr $$

(43)

where

$$ \begin{aligned} F(r) = \left[\,\frac{{(2 - \beta^{2})J_{1} (\beta)\, - \beta J_{0} (\beta)}}{{r\,\tau^{2}}}\right]\left[\frac{{\varepsilon^{3} \,r^{3} (v + 4u) + u^{\frac{3}{2}} (5v - 4u)}}{{v^{3} \,u^{\frac{3}{2}}}}\right] \hfill \\ u = 1 + r^{2} \hfill \\ \beta = r\,\tau \hfill \\ v = 1 + r^{2} - \varepsilon^{2} r^{2} \hfill \\ \end{aligned} $$

Appendix 3: Statistical moments of t _m

The first two temporal moments of t _m are given by the following expressions (Quinodoz and Valocchi 1993; Massabó et al. 2008):

$$ \mu_{{t_{m}}} (t)\, = \int\limits_{0}^{\infty} {t_{m}} \,p(t_{m})\,dt_{m} = \frac{{t + K^{2}_{d} \,e^{- \Upsilon} t + 2K_{d} [1 - e^{- \Upsilon}]\,}}{{R_{d} [1 + K_{d} e^{- \Upsilon}][k_{r} R_{d}]}}\, $$

(44)

$$ \begin{aligned} \sigma_{{t_{m}}}^{2} (t)\, = \int\limits_{0}^{\infty} {(t_{m}} - \mu_{{t_{m}}})^{2} \,p(t_{m})\,dt_{m} \hfill \\ \quad = [R_{d}^{2} (1 + K_{d} \,e^{- \Upsilon})]^{- 1} \{t^{2} + \frac{{6K_{d}}}{{k_{r} R_{d}}}[t - K_{d} e^{- \Upsilon} t]\, \hfill \\ \quad + K_{d}^{3} e^{- \Upsilon} t^{2} + 6K_{d} (K_{d} - 1)[(k_{r} R_{d})^{- 2} - (k_{r} R_{d})^{- 2} e^{- \Upsilon}]\} - \,\mu_{{t_{m}}}^{2} (t) \hfill \\ \end{aligned} $$

(45)

where p(t _m ) is the pdf of t _m given by the expression (22), $R_{d} = 1 + K_{d}$, and $\Upsilon = k_{r} \,R_{d} \,t$.