MODELING OF LOW SALINITY EFFECTS IN SANDSTONE OIL ...

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10 OMEKEH, EVJE, AND FRIIScoefficient (longitudinal and transversal dispersion lengths are here taken to be equal), which issplit into molecular diffusion contribution and the so called mechanical/advective contribution. Awidely quoted formulation by Sahimi [32] is of the form(32)DD m= ϕ p s q + aP e + bP δ e + cP 2 e ,where D m is the free molecular diffusion coefficient, p, q, a, b, c, and δ are experimentally determinedconstants and P e is the Péclet number given by(33) P e = α|V|D m,where α is the characteristic dispersion length which varies from millimeters (in laboratory scale)to metres(field scale). Several authors [8, 32] have experimentally determined that the diffusionco-efficient is determined by only the first term of equation(32) when P e is less than 0.3, and atransition zone between 0.3 and 5 where D is not clearly defined. The other terms of the equationbecomes significant at higher Péclet numbers. For our application of the model, the Péclet numberfalls below 0.3, hence subsequently we use only the first term of equation (32). In (32) the coefficientp is referred to as the cementation exponent, q as the saturation exponent. The cementationexponent is often close to 2 whereas the saturation exponent is also often fixed at a value in thesame range, see for example [8, 10, 11]. Using (30) in (27) yields(34)∂ t (ϕs o C o ) + ∇ · (C o V o ) = 0,∂ t (ϕsC l ) + ∇ · (C l V l ) = 0,∂ t (ϕsC na ) + ∂ t (M c β na ) − ∇ · (D∇C na ) = −∇ · (C na V),∂ t (ϕsC cl ) − ∇ · (D∇C cl ) = −∇ · (C cl V),∂ t (ϕsC ca ) + ∂ t (M c β ca ) − ∇ · (D∇C ca ) = −∇ · (C ca V),∂ t (ϕsC so ) − ∇ · (D∇C so ) = −∇ · (C so V),∂ t (ϕsC mg ) + ∂ t (M c β mg ) − ∇ · (D∇C mg ) = −∇ · (C mg V).In particular, summing the equations corresponding to C na , C cl , C ca , C so , and C mg , we obtain anequation for C g in the form(35) ∂ t (ϕsC g ) + ∂ t (M c [β na + β ca + β mg ]) − ∇ · (D∇C g ) = −∇ · (C g V).In a similar manner, using C l V l = C l V − C g U g (obtained from (25), (26), and (24)) in the secondequation of (34), the following equation is obtained(36)∂ t (ϕsC l ) + ∇ · (C l V) = ∇ · (C g U g ).Summing (36) and (35), we get the following equation for the concentration of the water phasewith its different chemical components, represented by C = C g + C l ,(37) ∂ t (ϕsC) + ∂ t (M c [β na + β ca + β mg ]) + ∇ · (CV) = 0.To sum up, we have a model in the form(38)∂ t (ϕs o C o ) + ∇ · (C o V o ) = 0,∂ t (ϕsC) + ∂ t (M c [β na + β ca + β mg ]) + ∇ · (CV) = 0,where D = D(ϕ, s) as given by (32).∂ t (ϕsC na ) + ∂ t (M c β na ) + ∇ · (C na V) = ∇ · (D∇C na ),∂ t (ϕsC cl ) + ∇ · (C cl V) = ∇ · (D∇C cl ),∂ t (ϕsC ca ) + ∂ t (M c β ca ) + ∇ · (C ca V) = ∇ · (D∇C ca ),∂ t (ϕsC so ) + ∇ · (C so V) = ∇ · (D∇C so ),∂ t (ϕsC mg ) + ∂ t (M c β mg ) + ∇ · (C mg V) = ∇ · (D∇C mg ),
MODELING OF LOW SALINITY EFFECTS IN SANDSTONE OIL ROCKS 114.1. Simplifying assumptions. Before we proceed some simplifying assumptions are made:• The oil and water component densities C o and C are assumed to be constant, i.e., incompressiblefluids;• The effect from the water-rock chemistry in the water phase equation (second equation of(38)) is neglected which is reasonable since the concentration of the water phase C is muchlarger than the concentrations of the ion exchange involved in the chemical reactions;• Constant porosity ϕ, absolute permeability κ, viscosities µ, µ o ;• One dimensional flow in a horizontal domain.• Capillary pressure is currently neglected as discussed in the introduction (Section 1.2).This results in the following simplified model:(39)∂ t (ϕs o ) + ∂ x (V o ) = 0,∂ t (ϕs) + ∂ x (V ) = 0,∂ t (ϕsC na ) + ∂ t (M c β na ) + ∂ x (C na V ) = ∂ x (D(ϕ, s)∂ x C na ),∂ t (ϕsC cl ) + ∂ x (C cl V ) = ∂ x (D(ϕ, s)∂ x C cl ),∂ t (ϕsC ca ) + ∂ t (M c β ca ) + ∂ x (C ca V ) = ∂ x (D(ϕ, s)∂ x C ca ),∂ t (ϕsC so ) + ∂ x (C so V ) = ∂ x (D(ϕ, s)∂ x C so ),∂ t (ϕsC mg ) + ∂ t (M c β mg ) + ∂ x (C mg V ) = ∂ x (D(ϕ, s)∂ x C mg ).In view of (28) and (29) in a 1D domain, we get(40)(41)V = − κλp x , λ(β ca , β mg ) = k(β ca, β mg )µV o = − κλ o p o,x , λ o (β ca , β mg ) = k o(β ca , β mg ),µ oMoreover, capillary pressure P c is defined as the difference between oil and water pressure(42) P c = p o − p,and is assumed to be zero in the following. Total velocity v T is given by(43)where total mobility λ Tv T := V + V o = −κ(λp x + λ o p o,x ) = −κλ T p x ,(44) λ T = λ + λ o ,has been introduced. Summing the two first equations of (39) and using that 1 = s+s o , implies that(v T ) x = 0, i.e., v T =constant and is determined, for example, from the boundary conditions. Fromthe continuity equation for s given by the second equation of (39) it follows (since V = −κλp x )(45) (ϕs) t + (−κλp x ) x = 0,where, in view of (43),Thus,(46)−κp x = v Tλ T.(ϕs) t + (v Tλλ T) x = 0.The fractional flow functions f(β ca , β mg ) and f o (β ca , β mg ) are defined as follows(47) f(s, β ca , β mg ) :=def(48)f o (s, β ca , β mg ) :=defUsing this in (46) implies thatλ(s, β ca , β mg )λ(s, β ca , β mg ) + λ o (β ca , β mg ) ,λ o (s, β ca , β mg )λ(s, β ca , β mg ) + λ o (β ca , β mg ) = 1 − f(β ca, β mg ).(49) (ϕs) t + v T f(s, β ca , β mg ) x = 0.
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