A lattice-Boltzmann study of permeability-porosity relationships and mineral precipitation patterns in fractured porous media


Mineral precipitation can drastically alter a reservoir’s ability to transmit mass and energy during various engineering/natural subsurface processes, such as geothermal energy extraction and geological carbon dioxide sequestration. However, it is still challenging to explain the relationships among permeability, porosity, and precipitation patterns in reservoirs, particularly in fracture-dominated reservoirs. Here, we investigate the pore-scale behavior of single-species mineral precipitation reactions in a fractured porous medium, using a phase field lattice-Boltzmann method. Parallel to the main flow direction, the medium is divided into two halves, one with a low-permeability matrix and one with a high-permeability matrix. Each matrix contains one flow-through and one dead-end fracture. A wide range of species diffusivity and reaction rates is explored to cover regimes from advection- to diffusion-dominated, and from transport- to reaction-limited. By employing the ratio of the Damköhler (Da) and the Peclet (Pe) number, four distinct precipitation patterns can be identified, namely (1) no precipitation (Da/Pe < 1), (2) near-inlet clogging (Da/Pe > 100), (3) fracture isolation (1 < Da/Pe < 100 and Pe > 1), and (4) diffusive precipitation (1 < Da/Pe < 100 and Pe < 0.1). Using moment analyses, we discuss in detail the development of the species (i.e., reactant) concentration and mineral precipitation fields for various species transport regimes. Finally, we establish a general relationship among mineral precipitation pattern, porosity, and permeability. Our study provides insights into the feedback loop of fluid flow, species transport, mineral precipitation, pore space geometry changes, and permeability in fractured porous media.

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This work was supported by ETH Grant ETH-12 15-2. The Werner Siemens Foundation (Werner Siemens-Stiftung) is further thanked by Martin Saar for its support of the Geothermal Energy and Geofluids (GEG.ethz.ch) Group at ETH Zurich. We thank the two anonymous reviewers for their helpful comments and suggestions that improved this paper.

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Appendix A: fluid dynamics solver

In a D2Q9 lattice-Boltzmann method (LBM), the particle velocity vector, vi, in Eq. 1 is defined as

$$ \boldsymbol{v}_{i} = \frac{\Delta x}{\Delta t}\times\left\{ \begin{array}{lllll} &(0,0), & & i=0 \\ & \left( \cos{\frac{(i-1)\pi}{2}}, \sin{\frac{(i-1)\pi}{2}} \right), & & i=1-4 \\ & \left( \cos{\frac{(2i-1)\pi}{4}}, \sin{\frac{(2i-1)\pi}{4}} \right), & & i=5-8, \end{array} \right. $$

where i = 0 indicates the rest velocity, i = 1 − 4 indicates velocities pointing to North-South-East-West, respectively, and i = 5 − 8 indicates the diagonal velocities.

The collision operator, \(\boldsymbol {\Omega }_{i}^{f}(x,t)\), is reduced here to a single relaxation time (SRT or BGK) operator that redistributes momentum and mass among particles [41, 43],

$$ \boldsymbol{\Omega}_{i}^{f}(\boldsymbol{x},t)=\frac{1}{\tau_{f}}\Big({f_{i}^{0}}(\boldsymbol{x},t)-f_{i}(\boldsymbol{x},t)\Big), $$

where \({f_{i}^{0}}\) is the local equilibrium distribution,

$$ {f_{i}^{0}}(\boldsymbol{x},t)={w_{i}^{f}}\rho(\boldsymbol{x},t)\left( 1 + \frac{\boldsymbol{u}\cdot \boldsymbol{v}_{i}}{{c_{s}^{2}}} + \frac{1}{2}\frac{(\boldsymbol{u}\cdot \boldsymbol{v}_{i})^{2}}{{c_{s}^{4}}} - \frac{1}{2}\frac{\boldsymbol{u}\cdot \boldsymbol{u}}{{c_{s}^{2}}}\right). $$

The lattice weights, \({w_{i}^{f}}\), for the D2Q9 scheme are

$$ {w_{i}^{f}} = \left\{ \begin{array}{lllll} &4/9, & & i=0 \\ &1/9, & & i=1-4 \\ &1/36, & & i=5-8. \end{array} \right. $$

The fluid density, ρ(x, t), is calculated as

$$ \rho(\boldsymbol{x},t)=\sum\limits_{i=0}^{8} f_{i}(\boldsymbol{x},t). $$

Accordingly, the fluid pressure is given as \(p=\rho {c_{s}^{2}}\). Following the scheme proposed by [16], the macroscopic fluid velocity, u, is calculated from the momentum of the probability distribution function, f(x, t), and the drag force, fdrag,

$$ \boldsymbol{u}(\boldsymbol{x},t)= \frac{1}{\rho(\boldsymbol{x},t)}\left( \sum\limits_{i=0}^{8} f_{i}(\boldsymbol{x},t)\boldsymbol{v}_{i}+\frac{\Delta t}{2}\boldsymbol{f}_{drag}\right). $$

The Fdrag, i in Eq. 1 is calculated by

$$ F_{drag,i}= {w_{i}^{f}}\left( 1-\frac{1}{2\tau_{f}}\right)\left( \frac{\boldsymbol{v}_{i}-\boldsymbol{u}}{{c_{s}^{2}}}+\frac{\boldsymbol{u}\cdot \boldsymbol{v}_{i}}{{c_{s}^{4}}}\boldsymbol{v}_{i}\right)\cdot \boldsymbol{f}_{drag}. $$

Appendix B: heterogeneous reaction transport solver

In the D2Q5 scheme, the lattice direction ei in Eq. 4 is given by

$$ \boldsymbol{e}_{i} = \frac{\Delta x}{\Delta t}\times\left\{ \begin{array}{lllll} &(0,0), & & i=0 \\ &\left( \cos{\frac{(i-1)\pi}{2}}, \sin{\frac{(i-1)\pi}{2}} \right), & & i=1-4 \end{array} \right. $$

where i = 0 indicates the rest velocity, and i = 1 − 4 indicates velocities pointing to North-South-East-West, respectively. The collision term, Ωg(x, t), is calculated using a so-called single relaxation time approach (SRT),

$$ \boldsymbol{\Omega}_{i}^{g}(\boldsymbol{x},t)=\frac{1}{\tau_{g}}\Big({g_{i}^{0}}(\boldsymbol{x},t)-g_{i}(\boldsymbol{x},t)\Big), $$

where \({g_{i}^{0}}\) is the local equilibrium distribution,

$$ {g_{i}^{0}}(\boldsymbol{x},t)={w_{i}^{g}} C(\boldsymbol{x},t)\left( 1~+~\frac{\boldsymbol{u}\cdot \boldsymbol{e}_{i}}{{c_{s}^{2}}}\right), $$

where \(C(\boldsymbol {x},t)={\sum }_{i=0}^{4} g_{i}(\boldsymbol {x},t)\) is the concentration of the transported species.

Appendix C: reactive model coupling

The coupling between flow and reactive transport in this study is illustrated by Fig. 11. Our procedure is slightly different from Huber et al. [19]. In the present study, the convergence of the velocity field is required to proceed to the transport and reaction step.

Fig. 11

Flow chart of the current lattice-Boltzmann reactive transport model

Appendix D: verification of the current LBM code

The current LBM code is validated with simulations of (i) flow and (ii) transport to their corresponding analytical solutions in a Hele-Shaw model with two parallel plates. Taking advantage of our drag force model (Eq. 1), the 3D Hele-Shaw model can be simulated using a 2D domain. Here, the width and length of the 2D domain are set to 600 lattices/nodes. Boundaries parallel to the flow/transport direction are set to no-slip boundaries. A constant pressure gradient is set from the inlet to the outlet boundaries (perpendicular to the main transport direction).

For the flow validation, we calculate the equivalent permeability of the Hele-Shaw model with various openings (i.e., the aperture of the Hele-Shaw model), ranging from 4 to 30 lattices/nodes ([4,5,6,8,10,14,17,20,25,30]). For these apertures, the Re number is calculated as ([0.006, 0.011, 0.018, 0.043, 0.084, 0.227, 0.405, 0.657, 1.275, 2.189]), respectively. In these validations, the physical dimension of one lattice is 50 μ m. The simulated permeability is compared with the permeability calculated using the cubic law [60],

$$ k=h^{2}/12 , $$

where k is the permeability and h is the aperture. The comparison of permeabilities, estimated by the LBM simulations and the cubic law is shown in Fig. 12. The results indicate that our current LBM code successfully reproduce fluid flow.

Fig. 12

Comparison of permeability values, calculated by the current LBM code and the cubic law [60]

For the transport validation, we simulate a non-reactive solute transport (front propagation) in a 2D Hele-Shaw model of the same size as the flow validation but with an aperture of 6 lattices/nodes. Each lattice has a physical dimension of 50 μ m. This aperture is the same as the one in the mineral precipitation simulations. Similar to the flow validation, no-slip boundaries are set to the boundaries parallel to the main transport direction and a constant pressure gradient is set across the inlet and outlet boundaries (perpendicular to the main flow/transport direction). For the given pressure boundary, the Reynolds (Re) number is 0.018. Initial solute concentration in the domain is set to zero and a constant solute concentration of 1 is set at the inlet boundary. The Peclet (Pe) number for the solute transport is Pe = 0.1. We compare the solute transport (front propagation) from the LBM simulations, with the following analytical solution [28],

$$ \begin{array}{lllll} \frac{\left( C(x,t)-C_{0}\right)}{\left( C_{inj}-C_{0}\right)} = \frac{1}{2}\times\left( \text{erfc}\left( \frac{x-vt}{\sqrt{4Dt}}\right)+\exp\left( \frac{xv}{D}\right)\text{erfc}\left( \frac{x+vt}{\sqrt{4Dt}}\right)\right) s , \end{array} $$

where C0 = C(x, 0) is the initial concentration, Cinj = C(0,t) is the inlet concentration, x is the coordinate along the center line of the Hele-Shaw domain in the flow/transport direction, v is the fluid velocity along the center line at steady-state conditions, D is the solute diffusion coefficient, erfc(y) is the complementary error function,

$$ \text{erfc}\left( y\right)= \frac{2}{\sqrt{\pi}}{\int}^{\infty}_{y} \exp(-t^{2}) dt, $$

and exp is the natural exponential function. The comparison at different lattice times shows a close-to perfect agreement between our LBM simulation and the analytical solution (Fig. 13).

Fig. 13

Comparison of solute front propagation between the LBM simulations and the analytical solutions at different lattice times

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Ahkami, M., Parmigiani, A., Di Palma, P. et al. A lattice-Boltzmann study of permeability-porosity relationships and mineral precipitation patterns in fractured porous media. Comput Geosci 24, 1865–1882 (2020). https://doi.org/10.1007/s10596-019-09926-4

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  • Lattice-Boltzmann method
  • Fractured porous media
  • Mineral precipitation patterns
  • Permeability-porosity relationships