FRACTIONAL DIFFUSION MODEL FOR TRANSPORT THROUGH POROUS MEDIA

2016

A nonlinear transport model for single-phase gas through tight rocks, is combined with a fractional calculus method, to produce a new time-fractional advection-diffusion transport model for the pressure field, p = p(x, t) in the flow of gas through tight porous reservoirs. Solutions for different fractional order, 0 <∝< 1, and for different nonlinear models with different apparent diffusivity K and apparent velocity U are compared. These systems could represent the gas transport in porous media where the representative control volumes are small but not infinitesimal. Applications are possible in many areas, such to shale gas recovery, and also aquifers.

The prospects of meeting the future’s high energy demands lie in the exploration of unconventional hydrocarbon reservoirs, of which the shale gas and the tight gas are two important resources. The deep understanding of such reservoirs is crucial to the economical recovery of such energy resources. With the advancement in the technological sides, such as, hydraulic fracturing and horizontal drilling, new mathematical models are needed that can precisely capture the complexity of the physical phenomena and can describe the flow of gas through the natural and induced fractures. The performance and future behavior of such reservoirs can be enhanced through careful modeling. We develop a new mathematical model based on time fractional derivative combined with the consideration of various flow regimes and a nonlinear treatment of reservoir parameters. The model describes the transport of gas in tight porous media (such as shale formations). The derivation of the model is done by using the mass balance equation and momentum conservation equation (basically modified time-fractional form of Darcy’s law) which incorporates the properties of tight porous media and accounts on the previous behavior. We find the pressure equation by considering that the rock properties, such as, permeability, viscosity, porosity, are pressure dependent. The pressure equation can be used to study the pressure distribution in the reservoir.

Energies

Due to the complexity imposed by all the attributes of the fracture network of many naturally fractured reservoirs, it has been observed that fluid flow does not necessarily represent a normal diffusion, i.e., Darcy’s law. Thus, to capture the sub-diffusion process, various tools have been implemented, from fractal geometry to characterize the structure of the porous medium to fractional calculus to include the memory effect in the fluid flow. Considering infinite naturally fractured reservoirs (Type I system of Nelson), a spatial fractional Darcy’s law is proposed, where the spatial derivative is replaced by the Weyl fractional derivative, and the resulting flow model also considers Caputo’s fractional derivative in time. The proposed model maintains its dimensional balance and is solved numerically. The results of analyzing the effect of the spatial fractional Darcy’s law on the pressure drop and its Bourdet derivative are shown, proving that two definitions of fractional derivati...

2014

Unconventional hydrocarbon reservoirs, such as, shale gas deposits, offer a new source of energy resources. These reservoirs consist of tight porous rocks which are characterized by nano-scale size porous networks with ultra-low permeability. The mathematical modeling of phenomena through such tight porous media provides its own challenges. In this study, we consider a relatively new approach by modeling the transport of gas by the time-fractional advection-diffusion equation, , p p p K U t x x x α α ∂ ∂ ∂ ∂     = −     ∂ ∂ ∂ ∂     0 t > , 0 1 x ≤ ≤ ; (with suitable initial and boundary conditions) in order to study the pressure distribution ( , ) p x t in unconventional reservoirs. In our model, ( ) K K p = is the diffusivity (which is related to the rock permeability) and ( , ) x U U p p = is a convection velocity; and both K and U are highly non-linear. This model is derived from mass balance and momentum balance equations. An approximate series solution is found b...

In reservoir engineering, an oil reservoir is commonly modeled using Darcy’s diffusion equation for a porous medium. In this work we propose a fractional diffusion equation to model the pressure distribution, p(x, t), of fluid in a horizontal one-dimensional homogeneous porous reservoir of finite length, L, and uniform thickness. A chief concern in this work is to examine the sensitivity of the pressure distribution, p(x, t), to different forms of pseudo-diffusivity, K, including cases when it depends upon the order of the fractional derivative (), 0 ≤  < 1 (e.g., K  (1 – )), which may be more realistic for some types of rock formations. In all cases the systems show a near-linear increase in the pressure difference P(x, t) = (p(x, t) – pi)/pi in the reservoir for large times, where pi = p(x, t = 0). For x/L < 0.4, the rate of increase of P with time increases with , but there is a crossover at x/L = 0.4 and this trend reverses for x/L > 0.4. When K = 10k (k is the conventional permeability when  = 0), the solutions are almost independent of , and when K = 0.1k the rate of increase in P depends upon . This effect is enhanced when K = (1 – )k; furthermore, in this case towards the closed end of the reservoir the pressure distribution remains practically undisturbed as  ¡1. These results show that the pressure distribution in a porous reservoir is very sensitive to the dependence of the pseudo-diffusivity on the order of the fractional derivative, alpha.

5th International Conference on Porous Media and Their Applications in Science, Engineering and Industry, 2014

Unconventional hydrocarbon reservoirs, such as, shale gas deposits, offer a new source of energy resources. These reservoirs consist of tight porous rocks which are characterized by nano-scale size porous networks with ultra-low permeability. The mathematical modeling of phenomena through such tight porous media provides its own challenges. In this study, we consider a relatively new approach by modeling the transport of gas by the time-fractional advection-diffusion equation,

Applied Mathematical Modelling, 2017

Nonlinearity, 2009

In this paper we study the heat transfer with a general fractional diffusion term of an incompressible fluid in a porous medium governed by Darcy's law. We show formation of singularities with infinite energy and for finite energy we obtain existence and uniqueness results of strong solutions for the sub-critical and critical cases. We prove global existence of weak solutions for different cases. Moreover, we obtain the decay of the solution in $L^p$, for any $p\geq2$, and the asymptotic behavior is shown. Finally, we prove the existence of an attractor in a weak sense and, for the sub-critical dissipative case with $\alpha\in (1,2]$, we obtain the existence of the global attractor for the solutions in the space $H^s$ for any $s > (N/2)+1-\alpha$.

The analysis of the low-frequency conductivity spectra of the clay-water mixtures is presented. The conductivity spectra for samples at different water content values are shown to collapse to a single master curve when appropriately rescaled. The frequency dependence of the conductivity is shown to follow the power-law with the exponent n=0,67 before reaching the frequency-independent part. It is argued that the observed conductivity dispersion is a consequence of the anomalously diffusing ions in the clay-water system. The fractional Langevin equation is then used to describe the stochastic dynamics of the single ion.

Journal of Mathematical Sciences: Advances and Applications, 2020

This article presents a novel system of flow equations that models the pressure deficit of a reservoir considered as a triple continuous medium formed by the rock matrix, vugular medium and fracture. In non- conventional reservoirs, the velocity of the fluid particles is altered due to physical and chemical phenomena caused by the interaction of the fluid with the medium, this behaviour is defined as anomalous. A more exact model can be obtained with the inclusion of the memory formalism concept that can be expressed through the use of fractional derivatives. Using Laplace transform of the Caputo fractional derivative and Bessel functions, a semi-analytical solution is reached in the Laplace space.