Simulation of In-situ Combustion in a Matrix-Fracture System at Laboratory Scale - Selected Topics of Computational and Experimental Fluid Mechanics - page 309

Ta b l e 1 Phases and
components

Component Phase

Wa t er Oil Gas Solid

Oxygen, O 2 X

Inert gas, IG X

Rock X

Coke X

Oil X

Wa t er, H 2 O X X

Kk r ʱ
μ ʱ

u

ʱ =−

( ∇ p

ʱ − ˁ ʱ

g ) ,ʱ = o , w , g , (2)

μ

with K being permeability, k r , and p are relative permeability, viscosity and

pressure, respectively, whilst g is the gravity acceleration.
The energy balance assuming local thermal equilibrium is

ˁ

c p eq ∂

t +∇· ˆ s o ˁ o c p , o u o +

s w ˁ w c p , w u w T

T
∂

s g ˁ g c p , g u g +

k eff ·∇

T

(3)

q o q g q w + q s =∇· + + + −

where

T ˁ o c p , o − ˁ g c p , g −

/ M g ∂

s o
∂

= ˆ c g R

t

(4)

ˁ w c p , w − ˁ g c p , g −

/ M g ∂

t + c p , g −

c g R s g ∂ˁ g
∂

s w
∂

c g R

t

ˁ

c p eq = ˆ s o ˁ o c p , o + s g ˁ g c p , g − / M g

c g R

) ˁ s c p , s

(5)

+ s w ˁ w c p , w + ( 1 /ˆ − 1

with k eff being the total thermal dispersion tensor, q T the thermal source/sink, T the
temperature, M g the average gas molecular weight, c p the heat capacity, c g the gas
compressibility, and R the ideal gas constant.
The oxygen balance is given by,
∂ ˆ

s g ˉ gO 2 ˁ g
∂

s g ˁ g D gO 2 ·∇ ˉ gO 2 +∇· ˆ s g ˉ gO 2 ˁ g u g =∇· ˆ + R O 2 (6)

t

And for coke we have that,

− ˆ) ∂ (ˁ s ˉ sCoke )
∂

( 1 = R Coke (7)

t