Advective Transport Through Three-Dimensional Anisotropic Formations of Bimodal Hydraulic Conductivity

Abstract

Transport of a non-reactive solute in anisotropic bimodal porous formations is investigated through the evaluation of the breakthrough curve. The hydraulic conductivity $K$ structure is modelled as an ensemble of anisotropic and non-overlapping inclusions, with given hydraulic conductivity $K_{i}$, randomly placed in an homogeneous matrix with hydraulic conductivity $K_{0}$. The conductivity contrast $\kappa =K_{i}/K_{0}$ between the inclusions and the matrix represents the main source of heterogeneity. The resulting $K$ structure is a theoretical representation of natural porous formations where inclusions and matrix play the role of two distinct hydrofacies. Semi-analytical solutions for flow and transport are carried out in both 3D and 2D domains following a Lagrangian approach by adopting the dilute limit approximation. The procedure, formally valid for media with low inclusions–matrix relative volume fraction, is extended to dense media by a simplified formulation of the self-consistent approach. Natural porous formations are intrinsically anisotropic due to the depositional process; hence, particular emphasis is addressed to the quantification of the effects of the anisotropy on solute transport. The solution of our physically based model depends on three structural parameters: the hydraulic conductivity contrast $\kappa$, the relative volume fraction $n$, and the statistical anisotropy ratio $e$. The procedure allows for the physical representation of high-velocity channels and stagnation zones, which can be encountered in bimodal media, and it is particular advantageous for a quick estimation of the solute breakthrough curve. Our analysis shows different transport dynamics, as function of the conductivity contrast $\kappa =K_{i}/K_{0}$, leading to fast preferential flow ($\kappa >1$) or mass retention ($\kappa <1$), as a consequence of the lack of symmetry of the flow and transport parameters (velocity and travel time residuals). High dispersion is typically found when $\kappa <1$, i.e. for low conductive inclusions, where the latter typically trap the solute particles, determining large travel times. Conversely, when $\kappa >1$ flow in the inclusions is constrained by the outer flow, limiting the solute velocity inside the inclusions, which results is a rather limited dispersion. Anisotropy typically enhances the solute spreading for $\kappa <1,$ as a consequence of the slower velocity predicted for more anisotropic inclusions, while its impact on formations with high $\kappa$ is rather limited.

Appendices

Appendix 1: Potential and Velocity for an Oblate Spheroidal Inclusion

In this section, we reproduce the analytical solution of the flow potential through an unbounded domain made by one oblate spheroid $\omega$ with hydraulic conductivity $K$ embedded in a homogeneous matrix with hydraulic conductivity $K=K_{0}$; for a more detailed discussion, the reader can refer to Carslaw and Jaeger (1959). The spheroid is aligned with the mean flow direction $\mathbf {U}$; the Cartesian reference systems $(x,y,z)$ are placed in the inclusion centroid $\mathbf {x}=0$ with the $x$ axis is parallel with $\mathbf {U}$ so that the spheroid semi-axes are $(R,R,eR)$. The flow problem is governed by the following equations

$$\begin{aligned} \phi _{j}(\mathbf {x})&= -KH_{j};\quad \nabla ^{2}\phi _{j}(\mathbf {x})=0 \end{aligned}$$

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$$\begin{aligned} \nabla \phi _{j}&= \mathbf {U};\quad |x|=\pm L/2 \\ \frac{\phi _{j}^{\mathrm{ex}}}{K_{0}}&= \frac{\phi _{j}^{\mathrm{in}}}{K_{i}};\quad \frac{\partial \phi _{j}^{\mathrm{ex}}}{\partial \nu }=\frac{\partial \phi _{j}^{\mathrm{in}}}{\partial \nu };\quad x\in \partial \omega _{i} \nonumber \end{aligned}$$

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The flow potential can be written as the sum of the mean flow potential $Ux$ and the disturbance induced by the inclusion $\varphi$

$$\begin{aligned} \phi (\mathbf {x})=Ux+\varphi (\mathbf {x}) \end{aligned}$$

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The analytical expression for $\varphi$ is split into two terms: $\varphi ^{(\mathrm in)}$ for $\mathbf {x}\in \omega$ and $\varphi ^{(\mathrm{ex})}$ for $\mathbf {x} \notin \omega$. Superscripts in and ex indicate zones inside and outside the inclusion.

$$\begin{aligned} \frac{\varphi ^{\left( \mathrm{in}\right) }}{\hbox {UR}}&= -\frac{(\kappa -1)F_{0}\left( e\right) }{1+\left( \kappa -1\right) F_{0}\left( e\right) }\frac{r_{x}}{R} \quad \text { }\,\left( \mathbf {r\in }\omega \right) \\ \frac{\varphi ^{\left( \mathrm{ex}\right) }}{\hbox {UR}}&= -\frac{(\kappa -1)}{1+\left( \kappa -1\right) F_{0}\left( e\right) }F_{\lambda }\left( e,\mathbf {r} \right) \frac{r_{x}}{R}\;\ \;\;\left( \mathbf {r}\notin \omega \right) \; \nonumber \end{aligned}$$

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where

$$\begin{aligned} F_{\lambda }&= \frac{\sqrt{1-e^{2}}}{2e^{3}}\left( \cot ^{-1}\zeta \left( \mathbf {r}\right) -\frac{\zeta \left( \mathbf {r}\right) }{\zeta ^{2}\left( \mathbf {r}\right) +1}\right) ;\quad F_{0}=\frac{\sqrt{1-e^{2}}}{2e^{3}} \left( \cot ^{-1}\zeta _{0}-\frac{\zeta _{0}}{\zeta _{0}^{2}+1}\right) \nonumber \\ \zeta \left( \mathbf {r}\right)&= \sqrt{\frac{\left( Re\right) ^{2}+\lambda \left( \mathbf {x}\right) }{R^{2}\left( 1-e^{2}\right) }};\quad \zeta _{0}=\frac{e}{\sqrt{1-e^{2}}} \end{aligned}$$

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with $\mathbf {r}=\mathbf {x-0}$, $r=\left| \mathbf {r}\right|$, $\kappa =K_{i}/K_{0}$ and

$$\begin{aligned} \lambda \left( \!\mathbf {r}\right) =2^{-1}\left( -R^{2}\left( 1\!+e^{2}\right) +r^{2}\!+\sqrt{R^{2}\left( e^{2}-1\right) \left[ \left( e^{2}-1\right) R^{2}+2\left( r_{x}^{2}+r_{y}^{2}-r_{z}^{2}\right) \right] +r^{4}}\right) \end{aligned}$$

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Closed-form expressions for the velocity $V_{x}$ along the streamline $\mathbf {X_{t}}=(x,0,0)$ can be obtained through derivation of Eq. (25). Velocity disturbances $u^{\mathrm{in}};u^{\mathrm{ex}}$, which are not null only in the longitudinal direction, are given in the following expressions with the aid of the dimensionless variable $x_{1}=x/R;\kappa =K_{i}/K_{0}$

$$\begin{aligned}&\frac{u_{x}^{\mathrm{in}}}{U} =\frac{2\left( e^{2}-1\right) ^{2}\kappa }{\left( e^{2}-1\right) \left( e^{2}(\kappa +1)-2\right) +e\sqrt{1-e^{2}}(\kappa -1)\cot ^{-1}\left( \frac{e}{\sqrt{1-e^{2}}}\right) } \nonumber \\&\frac{u_{x}^{\mathrm{ex}}(x_{1},0,0)}{U}\\&\quad =\frac{e\left( e^{2}-1\right) (\kappa -1)\left( x_{1}^{2}\sqrt{-\frac{e^{2}+x_{1}^{2}-1}{e^{2}-1}}\cot ^{-1}\left( \sqrt{\frac{e^{2}+x_{1}^{2}-1}{1-e^{2}}}\right) +e^{2}-x_{1}^{2}-1\right) }{x_{1}^{2}\sqrt{e^{2}+x_{1}^{2}-1}\left( \left( e^{2}-1\right) \left( e^{2}(\kappa +1)-2\right) +e\sqrt{1-e^{2}}(\kappa -1)\cot ^{-1}\left( \frac{e}{\sqrt{1-e^{2}}}\right) \right) }\nonumber \end{aligned}$$

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Appendix 2: Potential and Velocity for an Elliptical Inclusion

In this section, we provide the solution of the flow through an elliptical inclusion; a more comprehensive study can be found in Dagan and Lessoff (2001). The ellipse $\omega$ is placed in a unbounded domain, the hydraulic conductivity ratio is $\kappa =K_{i}/K_{0}$ with $K_{i}$ and $K_{0}$ pertaining, respectively, to the inclusion and to the matrix; the velocity $\mathbf {U}$ applied on the boundary is aligned with the ellipse major axis. The Cartesian coordinate system is placed on the ellipse centroid so that its semi-axes are $(R;eR)$. Because of the approximation in Eq. 12, we are interested only in the flow potential $\phi$ and in the longitudinal velocity $V_{x}=\partial \phi /\partial x$ along the particular streamline $\mathbf {X}=0$. In the following equations, two terms indicated with the superscript in and ex represent the longitudinal velocity $V_{x}$ in the domain exterior and interior to the inclusion

$$\begin{aligned} \frac{V_{x}^{\mathrm{in}}}{U}&= \frac{(e+1)\kappa \sqrt{e^{2}\left( \kappa ^{2}+1\right) -\left( e^{2}-1\right) \left( \kappa ^{2}-1\right) +4e\kappa +\kappa ^{2}+1}}{\sqrt{2}(e+\kappa )(e\kappa +1)} \\ \frac{V_{x}^{\mathrm{ex}}(x,0,0)}{U}&= \frac{1}{2}\left( \frac{x_{1}}{\sqrt{e^{2}+x_{1}^{2}-1}}+1\right) \left( \frac{\left( e^{2}+1\right) (e\kappa -1) }{(e\kappa +1)\left( \sqrt{e^{2}+x_{1}^{2}-1}+x_{1}\right) ^{2}}+1\right) \nonumber \end{aligned}$$

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with $x_{1}=x/R$. The flow potential is given by

$$\begin{aligned}&\displaystyle \frac{\phi ^{\mathrm{in}}(x,0,0)}{\hbox {UR}} =V_{x}x \end{aligned}$$

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$$\begin{aligned}&\displaystyle \frac{\phi ^{\mathrm{ex}}(x,0,0)}{\hbox {UR}} =\frac{\left( \sqrt{\left( e^{2}-1\right) R^{2}+x^{2}}+x\right) ^{2}+4\mu }{2\left( \sqrt{\left( e^{2}-1\right) R^{2}+x^{2}}+x\right) } \end{aligned}$$

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with

$$\begin{aligned} \mu =\pm \frac{\left( e^{2}+1\right) R^{2}\sqrt{\left( e^{2}k+e\left( k^{2}-1\right) -k\right) ^{2}}}{4(e+k)(ek+1)} \end{aligned}$$

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where the positive and negative signs hold, respectively, for $\kappa <1$ and $\kappa >1$.

Appendix 3: Effective Hydraulic Conductivity

In this section, we summarize the results in terms of effective hydraulic conductivity $\mathbf {K}_\mathrm{ef}$ obtained through the application of the self-consistent approach (SCA). The procedure was introduced in the study of porous formations by Dagan (1979); it was thoroughly described and tested in both isotropic and anisotropic domains (e.g. Janković et al. 2003b; Suribhatla et al. 2011). The SCA is recognized to be more suitable for the study of bimodal formation (Pozdniakov and Tsang 2004); moreover, in this particular media, the discontinuity of hydraulic conductivity could leave theoretical issues on the application of alternatives models.

For 2D domains, the $\mathbf {K}_\mathrm{ef}$ tensor can be obtained by solving the following systems.

$$\begin{aligned} \left\{ \begin{aligned} \left( e_0 \sqrt{\frac{K_{\mathrm{ef},h}}{K_{\mathrm{ef},v}}}+1 \right) \left[ FV_X (K_0) (1-n)+ FV_X (K_i) n \right] =1 \\ \left( e_0 \sqrt{\frac{K_{\mathrm{ef},v}}{K_{\mathrm{ef},h}}}+1 \right) \left[ FV_Z (K_0) (1-n)+ FV_Z (K_i) n \right] =1 \end{aligned}\right. \end{aligned}$$

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with

$$\begin{aligned} FV_{X}(K)&= K\frac{\sqrt{\frac{e^{2}K_{\mathrm{ef},h}^{3}+2eK\sqrt{K_{\mathrm{ef},h}^{3}K_{\mathrm{ef},v}}+K^{2}K_{\mathrm{ef},v}}{K_{\mathrm{ef},v}}}}{\left( eK_{\mathrm{ef},h}\sqrt{\frac{K_{\mathrm{ef},h}}{K_{\mathrm{ef},v}}}+K\right) \left( K_{\mathrm{ef},h}+eK\sqrt{\frac{K_{\mathrm{ef},h}}{K_{\mathrm{ef},v}}}\right) } \end{aligned}$$

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$$\begin{aligned} FV_{Z}(K)&= K\frac{\sqrt{\frac{K_{\mathrm{ef},v}\left( e^{2}K^{2}+2eK\sqrt{K_{\mathrm{ef},h}K_{\mathrm{ef},v}}+K_{\mathrm{ef},h}K_{\mathrm{ef},v}\right) }{K_{\mathrm{ef},h}}}}{\left( eK_{\mathrm{ef},v} \sqrt{\frac{K_{\mathrm{ef},v}}{K_{\mathrm{ef,}h}}}+K\right) \left( K_{\mathrm{ef},v}+eK\sqrt{\frac{K_{\mathrm{ef},v}}{K_{\mathrm{ef},h}}}\right) } \end{aligned}$$

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For 3D domains, the determination of the two independent components of the $\mathbf {K}_\mathrm{ef}$ tensor result from the solution of the following system

$$\begin{aligned} \left\{ \begin{aligned} (1-n) \frac{ K_0/K_{\mathrm{ef},h}-1}{F_0 \left( K_0/K_{\mathrm{ef},h}-1 \right) +1}+ n \frac{ K_i/K_{\mathrm{ef},h}-1}{F_0 \left( K_i/K_{\mathrm{ef},h}-1 \right) +1}=0 \\ (1-n) \frac{ K_0/K_{\mathrm{ef},v}-1}{F_0 \left( K_0/K_{\mathrm{ef},v}-1 \right) +1}+ n \frac{ K_i/K_{\mathrm{ef},v}-1}{F_0 \left( K_i/K_{\mathrm{ef},v}-1 \right) +1}=0 \end{aligned}\right. \end{aligned}$$

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with

$$\begin{aligned} F_{0}=\frac{1}{2}\left( \frac{e_\mathrm{s}}{\sqrt{(1-e_\mathrm{s}^{2})^{3}}}\cot ^{-1}\left( \frac{e_\mathrm{s}}{\sqrt{1-e_\mathrm{s}^{2}}}\right) -\frac{e_\mathrm{s}^{2}}{1-e_\mathrm{s}^{2}}\right) ;\quad e_\mathrm{s}=\sqrt{K_{\mathrm{ef},h}/K_{\mathrm{ef},v}} \end{aligned}$$

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