Random Walk Methods for Modeling Hydrodynamic Transport in Porous and Fractured Media from Pore to Reservoir Scale

Abstract

Random walk (RW) methods are recurring Monte Carlo methods used to model convective and diffusive transport in complex heterogeneous media. Many applications can be found, including fluid mechanic, hydrology and chemical reactors modeling. These methods are easy to implement, very versatile and flexible enough to become appealing for many applications because they generally overlook or deeply simplify the building of explicit complex meshes required by deterministic methods. RW provides a good physical understanding of the interactions between the space scales of heterogeneities and the transport phenomena under consideration. In addition, they can result in efficient upscaling methods, especially in the context of flow and transport in fractured media. In the present study, we review the applications of RW to several situations that cope with diverse spatial scales and different insights into upscaling problems. The advantages and downsides of RW are also discussed, thus providing a few avenues for further works and applications.

Appendices

Appendix 1: Langevin Equation and Fokker–Planck Equation

In this section, we show the equivalence between the Fokker–Planck Eq. (1) and the Langevin Eq. (2). To this end, we use a duality argument. Let $f({\mathbf{x}})$ be a twice differentiable function. We consider now the average $\langle f[{\mathbf{x}}(t)] \rangle$. By virtue of (3), this average may be written as

$$\begin{aligned} \langle f[{\mathbf{x}}(t)] \rangle = \int d{\mathbf{x}} P(\mathbf{x}, t) f({\mathbf{x}}). \end{aligned}$$

(57)

We consider now $\langle f[{\mathbf{x}}(t +\hbox {d}t)] \rangle$. To this end, we note that ${\mathbf {x}}(t + \hbox {d}t)$ is according to (2)

$$\begin{aligned} {\mathbf {x}}(t + \hbox {d}t) = {\mathbf {x}}(t) + {\mathbf {v}}[{\mathbf {x}}(t)] \hbox {d}t + \sqrt{2 {\mathbf {B}}[{\mathbf {x}}(t)]} \cdot \mathbf \eta (t) \equiv {\mathbf {x}}(t) + {\Delta } {\mathbf {x}}(t) \end{aligned}$$

(58)

where we use the Ito interpretation of the stochastic integral (Risken 1996); $\varvec{\eta }(t)$ is a Gaussian random variable with 0 mean and unit variance. Taylor expansion of $f[{\mathbf {x}}(t) +{\Delta } {\mathbf {x}}(t) ]$ consistently up to order $\hbox {d}t$ then gives

$$\begin{aligned} f[{\mathbf{x}}(t + \hbox {d}t)] - f[{\mathbf{x}}(t)] = \nabla f[\mathbf{x}(t)] \cdot {\mathbf {v}}[{\mathbf {x}}(t)] \hbox {d}t + \nabla \otimes \nabla f[{\mathbf {x}}(t)] : {\mathbf {B}}[{\mathbf {x}}(t)] \hbox {d}t. \end{aligned}$$

(59)

It is worthwhile noting that this equation is also called the Ito formula, or chain rule of stochastic calculus (Risken 1996). Taking the average of (59) and using (57) gives after integration by parts the Fokker–Planck Eq. (1).

Appendix 2: Dual-Porosity Models

Fractured porous media are characterized by a high property contrast between fractures and matrix. This leads to introducing a new class of models, starting from the steady-state double-porosity models (Barenblatt and Zheltov 1960; Warren and Root 1963) as derived by Arbogast et al. (1990) and Quintard and Whitaker (1993) that couple matrix and fracture by means of a linear exchange term:

$$\begin{aligned} \left\{ \begin{array}{rcl} \phi _f V_f {{\partial P_f({\mathbf{x}},t)}\over {\partial t}} &{} = &{} {D_f}{\nabla }^2 P_f({\mathbf{x}},t) + Q({\mathbf{x}},t)\\ \phi _m V_m{{\partial P_m({\mathbf{x}},t)}\over {\partial t}} &{} = &{} {D_m}{\nabla }^2 P_m({\mathbf{x}},t) - Q({\mathbf{x}},t). \end{array} \right. \end{aligned}$$

(60)

Here, $\phi _f V_f$ and $\phi _m V_m$ represent, respectively, the overall proportions of fracture and matrix volumes (weighted by the relevant porosity and compressibility). The model is closed once the interporosity flux $Q({\mathbf{x}},t)$ is expressed as a function of $P_f({\mathbf{x}},t)$ and $P_m({\mathbf{x}},t)$. In the steady-state case, $Q({\mathbf{x}},t)$ is given by:

$$\begin{aligned} Q({\mathbf{x}},t) = \lambda \left( P_m({\mathbf{x}},t) - P_f({\mathbf{x}},t) \right) . \end{aligned}$$

(61)

The transfer coefficient $\lambda$, reciprocal of a time depends mainly on the geometry of the matrix blocks. It is proportional to $D_m$. Its determination from the detailed DFN geometry is discussed in Sect. 3.4.1.

More general models using memory functions accounting for more details of the diffusion inside the matrix can be introduced (Odeh 1965; Swaan 1976; Carslaw and Jaeger 1986; Daviau 1986; Chen 1989; Swann and Ramirez-Villa 1993). These models belong to the general class of multiple-rate mass transfer (MRMT) models or multiple interacting continua (MINC) (Narasimhan et al. 1988; Haggerty and Gorelick 1995; de Dreuzy et al. 2013). These models correspond to quite different formulations of the same physics differ through the formulation of the exchange term. The latter appears as a time convolution expressed by:

$$\begin{aligned} Q({\mathbf{x}},t)= & {} \int _0^t G(t-\tau )\left( {{d(P_m({\mathbf{x}},\tau )- P_f({\mathbf{x}}, \tau ))}}\over {d\tau }\right) d\tau . \end{aligned}$$

(62)

In all cases, the exchange kernel G(t) is scaled by a parameter $\lambda$ which depends only on the geometry of the matrix blocks. It was shown in Landereau et al. (2001) and Babey et al. (2015) that multiple porosity models, MRMT models and transient models are equivalent and correspond to different formulations of the same idea.

In most cases, the term ${D_m}{\nabla }^2 P_m({\mathbf{x}},t)$ may be neglected in the double-porosity Eq. (60), so $P_m({\mathbf{x}},t)$ may be eliminated from the equations to provide the following generic form:

$$\begin{aligned} \int _0^t d\tau (\phi _f V_f \delta (t-\tau )+\phi _m V_m f(t-\tau ))\frac{\partial {P}_{f} ({\mathbf{x}}, \tau )}{\partial \tau } = \nabla .( D_{f} \nabla {P}_f ({\mathbf{x}}, t)). \end{aligned}$$

(63)

The quantity f(t) is the time-dependent exchange function. Introducing the average pressure in the fractures ${<}{{\hat{P}}}_f{>}(t)$ solution of the following initial value problem:

$$\begin{aligned}&\phi ({\mathbf{x}})\frac{\partial P_f({\mathbf{x}}, t)}{\partial t} = \nabla . (D({\mathbf{x}})\nabla P_f({\mathbf{x}}, t)) \end{aligned}$$

(64)

$$\begin{aligned}&\forall {\mathbf{x}} \in {\varOmega }_f P_f({\mathbf{x}}, t=0)=1\end{aligned}$$

(65)

$$\begin{aligned}&\forall {\mathbf{x}} \in {\varOmega }_m P_f({\mathbf{x}}, t=0)=0\end{aligned}$$

(66)

$$\begin{aligned}&{<}P_f{>}(t) = \frac{1}{\vert {\varOmega }_f \vert } \int _{{\varOmega }_f } d{\mathbf{x}} P_f({\mathbf{x}}, t). \end{aligned}$$

(67)

It is possible to show the following relation in the Laplace domain:

$$\begin{aligned} {<}P_f{>}(s) = \frac{\phi _f V_f}{s(\phi _f V_f + \phi _m V_m f(s)) }. \end{aligned}$$

(68)

The practical interest of introducing the f(s) function is that it can be shown that the general solution of a double-porosity system (60) can be directly related to a solution of a single-porosity equation replacing the argument s of the single-porosity solution by the argument $s(\phi _f V_f +s\phi _m V_m f(s))$. The large amount of analytical single-porosity solutions that are well known is sufficient for most practical situations. This means that all the double-porosity behavior is captured by f(s), which appears to be a renormalized apparent storativity. The initial value problem (67) defining ${<}{{\hat{P}}}_f{>}(t)$ has in turn a simple RW interpretation. The quantity ${<}P_f{>}(t)$ corresponds to the average proportion of particle undergoing RW (with diffusivity $D_m$) that belongs to the fractures at time t given they was released at a random location in the fractures at time $t=0$. In the steady-state case, the function f(s) is given by (Noetinger and Estebenet 2000; Noetinger et al. 2001a):

$$\begin{aligned} f(s) = \frac{\lambda }{\phi _m V_m s + \lambda }. \end{aligned}$$

(69)

It appears that $\lambda$ is a characteristic diffusion time in the matrix. Explicit expressions may be given for f(s) for either a layered medium, or for spherical blocks:

• for the layered case

  $$\begin{aligned} f(s) =\sqrt{ \frac{\lambda }{3s V_m} } \tanh \sqrt{{\frac{3 V_m s}{\lambda }}}, \end{aligned}$$

  (70)

• for the spherical case

  $$\begin{aligned} f(s) = \frac{\lambda }{5 sV_m}\left( \sqrt{\frac{15 V_m s}{\lambda }}\mathrm cotanh\sqrt{\frac{15V_m s}{\lambda }}- 1 \right) . \end{aligned}$$

  (71)

These generic forms, or others can be used for large-scale applications, solving (63) using any numerical approach. It remains to be able to evaluate the transfer coefficient $\lambda$ or the full f(t) function. This is the objective of Sect. 3.4.1.