Aqueous-Minerals Reactions: Chemical reactions involving aqueous species and mineral species are heterogeneous and slower compared to aqueous species reactions, thus modeled as rate dependent chemical reactions. This means that when mineral species (N_m ) are not is equilibrium with the aqueous species, the minerals can either dissolve or precipitate. As shown in equation R11 of an example of calcite dissolution/precipitation, a mineral species (Calcite) is involved in a reaction with only aqueous species (〖CO〗_2,H〖〖CO〗_3〗^-,H^+ ) and no other mineral species. The law of mass action governing the mineral dissolution/precipitation reaction dictates that the reaction rates be calculated as: r_β=k_β A ̇_β (1-Q_β/K_(eq,β) )
…show more content…
Governing equations: For this isothermal process, the number of moles, N_i, of each species are characterized as a function of the gradients in terms of advective-dispersion transfer, chemical reaction rate variables analogous to the various reactions given in previous section and external sources/sinks given by the rate of change in the number of moles of species added or subtracted. Hence, the rate of change in the number of moles of each species must satisfy the general conservation equation for all components, N_t, written as:
V_b (∂N_i)/∂t-V_b ∇ ⃗∙[∑_(j=o,g,aq)▒(ξ_j y_ij (u_j ) ⃗-y_ij (D_ij ) ⃗ ⃗ ) ]-V_b (∂σ_(i,aq))/∂t-V_b (∂σ_(i,m))/∂t-q_i=0 for i=1,…..,N_t (20)
Total flux according to Darcy’s law is represented by:
(u_j ) ⃗=k ⃗ ⃗λ_ij (∇P_j-γ_j ∇Z) for j=o, g, aq …show more content…
Detailed explanation can be found in Nghiem et al. (2004a) and Nghiem et al. (2011). The discretized equation for mineral components is:
V_b/∇t (〖N_i〗^(n+1)-〖N_i〗^n )-V_b 〖r_(i,m)〗^(n+1)=0 for i=1,…..,N_m (27)
It is crucial to ensure that the sum of phase volumes equal to the pore volumes by employing the volume constraint equation
∑_(j=o,g,aq)▒[〖N_j〗^(n+1)/〖ρ_j〗^(n+1) ] -ϕ^(n+1)=0 (28)
Numerical Solution Approach: Elimination of the equilibrium reactions term from equation 25 using ERA matrix reduced the discretized equation to only primary aqueous component. Then, the number of all conservation equations including hydrocarbon and aqueous components becomes N_ia+N_c. There are N_tot=N_ia+3N_(c )+5+ R_aq+N_m equations that can be used to find solution to the same number of unknown variables as summarized
Governing equations: For this isothermal process, the number of moles, N_i, of each species are characterized as a function of the gradients in terms of advective-dispersion transfer, chemical reaction rate variables analogous to the various reactions given in previous section and external sources/sinks given by the rate of change in the number of moles of species added or subtracted. Hence, the rate of change in the number of moles of each species must satisfy the general conservation equation for all components, N_t, written as:
V_b (∂N_i)/∂t-V_b ∇ ⃗∙[∑_(j=o,g,aq)▒(ξ_j y_ij (u_j ) ⃗-y_ij (D_ij ) ⃗ ⃗ ) ]-V_b (∂σ_(i,aq))/∂t-V_b (∂σ_(i,m))/∂t-q_i=0 for i=1,…..,N_t (20)
Total flux according to Darcy’s law is represented by:
(u_j ) ⃗=k ⃗ ⃗λ_ij (∇P_j-γ_j ∇Z) for j=o, g, aq …show more content…
Detailed explanation can be found in Nghiem et al. (2004a) and Nghiem et al. (2011). The discretized equation for mineral components is:
V_b/∇t (〖N_i〗^(n+1)-〖N_i〗^n )-V_b 〖r_(i,m)〗^(n+1)=0 for i=1,…..,N_m (27)
It is crucial to ensure that the sum of phase volumes equal to the pore volumes by employing the volume constraint equation
∑_(j=o,g,aq)▒[〖N_j〗^(n+1)/〖ρ_j〗^(n+1) ] -ϕ^(n+1)=0 (28)
Numerical Solution Approach: Elimination of the equilibrium reactions term from equation 25 using ERA matrix reduced the discretized equation to only primary aqueous component. Then, the number of all conservation equations including hydrocarbon and aqueous components becomes N_ia+N_c. There are N_tot=N_ia+3N_(c )+5+ R_aq+N_m equations that can be used to find solution to the same number of unknown variables as summarized