Volume Averaged Equations for Mass Transport and Reaction for In-Situ Combustion

Carlos G. Aguilar-Madera; Octavio Cazarez-Candia; Francisco J. Valdés-Parada

doi:10.1515/ijcre-2017-0124

Article

Volume Averaged Equations for Mass Transport and Reaction for In-Situ Combustion

Carlos G. Aguilar-Madera , Octavio Cazarez-Candia and Francisco J. Valdés-Parada

Published/Copyright: August 31, 2017

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From the journal International Journal of Chemical Reactor Engineering Volume 15 Issue 5

Abstract

In-situ combustion (ISC) is an oil recovery technique where many phenomena can take place simultaneously such as: chemical reactions, phase change, heat transfer, mass transport, thermodynamic equilibrium, and so on. Each one of these phenomena may have important contributions over the ISC behavior at any scale of interest as lab-scale, inter-wells or reservoir-scale. In this work, a mass transport study is presented. Firstly, the appropriate phase and interface governing equations at pore-scale are set up. Later, the volume averaged equations valid at macroscale are rigorously derived using the volume averaging method (VAM). The theoretical analysis is general and applies for typical oil-water-gas-rock systems found in petroleum reservoirs, and for any number of chemical species distributed in the phases. The model also allows the existence of several heterogeneous and homogeneous chemical reactions. From this general point of view, the volume averaged equations governing species and phase mass transport at macroscale, along its closure scheme to predict the effective transport parameters, are presented. We have clearly identified the length scale constraints and assumptions that support our derivations. In future works, we shall expand the range of applicability of the model by relaxing some of these assumptions. To demonstrate the applicability of the average models, we numerically predicted the longitudinal mass dispersion of oxygen for passive and reactive mass transport problems at lab-scale. The general trends of theoretical results are in concordance with previous works.

Keywords: in-situ combustion; species mass transport; volume averaging; closure scheme

Acknowledgements

The author CGAM really thanks the postdoctoral fellowship from the Mexican Institute of Petroleum through the project Y.00101.

A Volume averaging theory

In order to carry out an up-scaling procedure to the mass transport problem, we need to define a Representative Elementary Volume (REV) where primary variables as mass fractions will be averaged. The size of REV must be large enough to smooth fluctuations from the microscale over the averaged variables, and in turn, small enough to avoid significant variations from changes in the medium structure (macroscale heterogeneities). In a qualitatively way, these requirements of REV are written as

(116)lα\lt\ltr0\lt\ltL,α=o,g,w,r

Here l is the characteristic length of phases, whilst L is the size of the multiphase system (see Figure 2) and r0 is the size of the REV. Here it must be clarified that for fractured reservoirs, L is the length of homogeneous zones as cores, or zones with weakly position-dependent porosity and permeability. This length -scale constraint allows neglecting terms that are larger than the first one when expanding volume-averaged quantities about the centroid of the averaging domain (Whitaker 1999). This is convenient, since it leads to local volume-averaged models, in which volume-averaged quantities can be treated as constants within the averaging domain.

Let us define the superficial average operator acting over the variable ψα defined in the α-phase as,

(117)〉ψα〉(x,t)=1|V|∫Vα(x,t)⁡ψα(x+yα,t)dV

Here x is a vector position locating the centroid of REV whilst yα is a vector position locating the α-phase inside the REV with respect to its centroid. In the above equation |V| is the REV volume and Vα is the portion of the REV occupied by the α-phase.

As seen, the notation used in eq. (117) may be complicated, and hereinafter we will use a simplified version. For instance, instead of eq. (117) we will just write

(118)〉ψα〉=1|V|∫Vα⁡ψαdV

Another useful operator is the intrinsic average given by

(119)〉ψα〉α=1|Vα|∫Vα⁡ψαdV

Both types of averaging operators are related as

(120)〉ψα〉=εα〉ψα〉α

Here εα is the volume fraction of α-phase inside the REV.

It is worth noting that the volume |V| is independent of time, but the same is not true for the phases-volumes inside the REV. Thus, when the integral operators are applied to governing equations, two important theorems will be necessary in order to interchange integration and derivation. These are the spatial averaging theorem (Howes and Whitaker 1985; Soria and De Lasa 1991 )

(121)〉∇ψα〉=∇〉ψα〉+∑ββ≠α⁡1|V|∫Aαβ⁡(ψαnαβ)dA,α=o,g,w,β=o,g,w,r

and the general transport theorem

(122)〉∂ψα∂t〉=∂〉ψα〉∂t−∑ββ≠α⁡1|V|∫Aαβ⁡(ψαwαβ)⋅nαβdA,α=o,g,w,β=o,g,w,r

From these theorems we can obtain the spatial and time variations of volume fraction as follows

(123)∇εα=−∑ββ≠α⁡1|V|∫Aαβ⁡nαβdA,α=o,g,w,β=o,g,w,r

(124)∂εα∂t=∑ββ≠α⁡1|V|∫Aαβ⁡(wαβ⋅nαβ)dA,α=o,g,w,β=o,g,w,r

Appendix

B Definitions

The definitions of source-like terms appearing in eq. (52) are:

(125)Φiα,conv=−∇⋅(ρα〉ω~iαv~α〉),i=1,…,n−1,α=o,g,w

(126)Φiα,diffx=∇⋅[ραDiα(∑ββ≠α1|V|∫Aαβω~iαnαβdA)],i=1,…,n−1,α,β=o,g,w

(127)Φiα,diffT=∇⋅[Diα,T(∑ββ≠α1|V|∫AαβT~αnαβdA)],i=1,…,n−1,α,β=o,g,w

(128)Φiα,inter=Diα,T(∑ββ≠α1|V|∫Aαβ∇T~α⋅nαβdA)+Diαρα(∑ββ≠α1|V|∫Aαβ∇ω~iα⋅nαβdA)−(εα)−1〉vα〉⋅[ρα(∑ββ≠α1|V|∫Aαβω~iαnαβdA)]−ρα{∑ββ≠α1|V|∫Aαβω~iα(v~α−wαβ)⋅nαβdA}−ρα〉ωiα〉α(∑ββ≠α1|V|∫Aαβv~α⋅nαβdA),i=1,…,n−1,α,β=o,g,w

Here we have taken into account that integrals, containing spatial derivatives of average variables, are zero. In addition, integrals of products of spatial deviations have been kept because such quantities are generally non-zero. In addition, in the above equations we have considered that density and diffusivity coefficients are constants inside the REV.

Now, the Γ-coefficients of eq.(64) are defined as follows,

(129)Γα,I=−∇⋅(ραv~α)+ρα(∑ββ≠α1|V|∫Aαβv~α⋅nαβdA),α,β=o,g,w

(130)Γiα,II=∇(εα)−1εαραDiα−ραv~α,i=1,…,n−1,α=o,g,w

(131)Γiα,III=∇(εα)−1εαDiα,T,i=1,…,n−1,α=o,g,w

(132)Γiα,IV=(εα)−1∇⋅(ρα〉ω~iαv~α〉)−(εα)−1Diαρα∑ββ≠α1|V|∫Aαβ(∇ω~iα)⋅nαβdA+(εα)−2〉vα〉⋅(ρα∑ββ≠α1|V|∫Aαβω~iαnαβdA)+(εα)−1ρα∑ββ≠α1|V|∫Aαβω~iα(v~α−wαβ)⋅nαβdA−(εα)−1Diα,T∑ββ≠α1|V|∫Aαβ∇T~α⋅nαβdA−(εα)−1(Φiα,diffx+Φiα,diffT),i=1,…,n−1,α=o,g,w

The Ω parameters appearing in eqs (69) and (70) are expressed as,

(134)Ωiαr=∑k=1k≠in(ω~kα∂kiαr∂ωkα|…)+∑k=1n(ω~kαr∂kiαr∂ωkαr|…)+T~αr∂kiαr∂Tαr|…+p~αr∂kiαr∂pαr|…+…,i=1,…,n−1,α=o,g,w

The effective parameters of eq. (78) are defined in terms of closure variables as follows:

(135)Diα,α=Diα[I+∑ββ≠α(1|Vα|∫Aαβbiα,3nαβdA)]−1|Vα|∫Vαbiα,3v~αdV+…,i=1,…,n,α,β=o,g,w

(136)Diα,β=∑ββ≠α[Diα|Vα|∫Aαβbiα,f(β)nαβdA]−1|Vα|∫Vαbiα,f(β)v~αdV+…,i=1,…,n,α,β=o,g,w

(137)Diα,α,T=Diα,T{I+∑ββ≠α[1|Vα|∫Aαβ(sα,1+ραDiαDiα,Tbiα,5)nαβdA]}−ρα|Vα|∫Vαbiα,5v~αdV+…,i=1,…,n,α,β=o,g,w

(138)Diα,β,T=ρα[∑ββ≠α(Diα|Vα|∫Aαβbiα,f(β)nαβdA)−1|Vα|∫Vαbiα,f(β)v~αdV]+…,i=1,…,n,α,β=o,g,w

(139)aαβκiαβ=1|V|∫Aαβ⁡[Diαρα∇biα,f(β)−ραbiα,f(β)(vα−wαβ)]⋅nαβdA+…,i=1,…,n,α,β=o,g,w,β≠α

In these definitions we have just written the most relevant contributions as demonstrated elsewhere (Aguilar-Madera et al. 2011; Quintard, Kaviany, and Whitaker 1997; Valdés-Parada, Aguilar-Madera, and Alvarez-Ramirez 2011; Whitaker 1999). However, it is expected the inclusion of more terms related with other macroscale sources as written in eq. (77).

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Published Online: 2017-8-31

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https://doi.org/10.1515/ijcre-2017-0124

Keywords for this article

in-situ combustion; species mass transport; volume averaging; closure scheme