Derivation of the model equations presented in this paper is given in our previous publication [7] where we theoretically proved that the newly developed model satisfies the conservation laws of mass and energy precisely, as opposed to the existing chemical compositional model.

The mass conservation equation for the overall concentration *c̃*_{i} is defined as
$$\begin{array}{c}{\displaystyle \frac{\mathrm{\partial}}{\mathrm{\partial}t}(\varphi {\rho}_{i}\stackrel{~}{c}+\mathrm{\nabla}\cdot [\hat{\varphi}{\rho}_{i}\sum _{\alpha =1}^{{n}_{p}}({S}_{\alpha}{c}_{i\alpha}\overrightarrow{u}\alpha )]-}\\ {\displaystyle -\mathrm{\nabla}\sum _{\alpha =1}^{{n}_{p}}[{\overline{\overline{K}}}_{i\alpha}\cdot \mathrm{\nabla}(\hat{\varphi}{\rho}_{i}{S}_{\alpha}{c}_{i\alpha})={R}_{i},i=1,\dots ,{n}_{c},}\end{array}$$(1)

where *ϕ* is the porosity, *ρ*_{i} is the density of pure component *i* and *c̃*_{i} is the overall concentration of component *i*. The modfied porosity *ϕˆ* is defined as the fraction of the bulk permeable medium occupied by pore space remaining after adsorption. The modified phase saturation *S*_{α} is dehned as the fraction of the reduced pore volume occupied by phase *α*. *c*_{ia} is the modified volume fraction of component *i* in phase *α*.

The phase flux *u⃗*_{α} is the modified average pore velocity vector of phase *α* owing to convection and calculated from Darcy’s law
$$\begin{array}{c}{\displaystyle {\overrightarrow{u}}_{\alpha}=\frac{\overline{\overline{k}}{k}_{ra}}{\hat{\varphi}{S}_{\alpha}{\mu}_{\alpha}}(\mathrm{\nabla}{p}_{\alpha}-{\gamma}_{\alpha}{\mathrm{\nabla}}_{z}),\alpha =1,\dots ,{n}_{p},}\end{array}$$(2)

where *k̿* is the permeability tensor, *k*_{rα} is the relative permeability of fluid phase *α*, *μ*_{α} is the dynamic viscosity of fluid phase *α*, *p*_{α} is the pressure in fluid phase *α*, *γ*_{α} is the specihc weight of fluid phase *α* and *z* represents depth.

Two components of dispersion tensor *k̿*_{ia} for a homogeneous isotropic permeable medium [24] are
$$\begin{array}{}{\displaystyle ({K}_{xx}{)}_{i\alpha}=\frac{{D}_{i\alpha}}{\tau}+\frac{{\alpha}_{l\alpha}{u}_{x\alpha}^{2}+{\alpha}_{t\alpha}({u}_{y\alpha}^{2}+{u}_{z\alpha}^{2})}{|{\overrightarrow{u}}_{\alpha}|},}\\ {\displaystyle ({K}_{xy}{)}_{i\alpha}=\frac{({\alpha}_{l\alpha}-{\alpha}_{t\alpha}){u}_{x\alpha}{u}_{y\alpha}}{|{\overrightarrow{u}}_{\alpha}|},}\\ i=1,\dots ,{n}_{c};\alpha =1,\dots ,{n}_{p},\end{array}$$(3)

where the subscript *l* refers to the spatial coordinate in the direction parallel or longitudinal to bulk flow and *t* is any direction perpendicular or transverse to *l*. *D*_{iα} is the effective binary diffusion coefficient of component *i* in phase *α* [25], *α*_{lα} and *α*_{tα} are the longitudinal and transverse dispersivities and *τ* is the permeable medium tortuosity.

For biodegradation model
$$\begin{array}{}{\displaystyle {R}_{i}=-{k}_{i}\varphi {\rho}_{i}[(1-\sum _{j=1}^{{n}_{cv}}{\hat{c}}_{j})\sum _{\alpha =1}^{{n}_{p}}{S}_{\alpha}{c}_{i\alpha}+{\hat{c}}_{i}]+{Q}_{i},i=1,\dots ,{n}_{c}.}\end{array}$$(4)

where *k*_{i} is the reaction rate coefficient in units of inverse time and *Q*_{i} represents physical sources. The overall concentration *c̃*_{i} is defined as
$$\begin{array}{}{\displaystyle {\stackrel{~}{c}}_{i}=[(1-\sum _{j=1}^{{n}_{cv}}{\hat{c}}_{j})\sum _{\alpha =1}^{{n}_{p}}{S}_{\alpha}{c}_{i\alpha}+{\hat{c}}_{i},i=1,\dots ,{n}_{c}.}\end{array}$$(5)

where *ĉ*_{j} is the absorbed concentration of components *j*, *n*_{cv} is the total number of volume-occupying components and *n*_{p} is the number of phases.

The magnitudes of the vector flux for each phase, |*u⃗*_{α}|, are computed as follows:

$$\begin{array}{}{\displaystyle |{\overrightarrow{u}}_{\alpha}|=\sqrt{({u}_{x\alpha}{)}^{2}+({u}_{y\alpha}{)}^{2}+({u}_{z\alpha}{)}^{2}}.}\end{array}$$The porosity depends on pressure due to rock compressibility. Therefore,
$$\begin{array}{}{\displaystyle \varphi ={\varphi}_{R}[1+{c}_{r}({p}_{1}-{p}_{S})],}\end{array}$$(6)

where *ϕ*_{R} is the porosity at a specific pressure *p*_{s}, *p*_{1} is the water phase pressure and *c*_{r} is the rock compressibility at *p*_{S}. For a slightly compressible fluid, the component density can be written as:
$$\begin{array}{}{\displaystyle {\rho}_{i}={\rho}_{iR}[1+{c}_{i}^{0}({p}_{1}-{p}_{R})]\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thinmathspace}{0ex}}i=1,\dots ,{n}_{c},}\end{array}$$(7)

where *ρ*_{iR} is the density of component *i* at the standard pressure *p*_{R}.

$\begin{array}{}{c}_{i}^{0}\end{array}$

is the compressibility of component *i*.

The pressure equation is formed by summing up the mass balances over all volume-occupying components after dividing both sides by *ρ*_{iR} and substituting Darcy’s law in each of the phase flux terms. By using the capillary pressure definition, the pressure equation in terms of the reference phase pressure (phase 1) will be:
$$\begin{array}{c}{\displaystyle \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\varphi}_{R}{c}_{t}\frac{\mathrm{\partial}{p}_{1}}{\mathrm{\partial}t}-\mathrm{\nabla}\cdot (\overline{\overline{k}}{\lambda}_{rTc}\mathrm{\nabla}{p}_{1})=}\\ {\displaystyle =\mathrm{\nabla}\cdot (\overline{\overline{k}}\sum _{\alpha =1}^{{n}_{p}}{\lambda}_{r\alpha c}\mathrm{\nabla}{p}_{c\alpha 1})-\mathrm{\nabla}\cdot (\overline{\overline{k}}\sum _{\alpha =1}^{{n}_{p}}{\lambda}_{(r\alpha c\gamma \alpha )}\mathrm{\nabla}z)-}\\ {\displaystyle \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}-{\mathit{\Delta}}_{t}F({\stackrel{~}{c}}_{i}+\sum _{i=1}^{{n}_{c}}\frac{{Q}_{i}}{{\rho}_{iR}},}\end{array}$$(8)

where *c*_{t} is the total system compressibility, *p*_{1} is the pressure of phase 1, *p*_{cα}_{1} is the capillary pressure, *z* is the depth, *λ*_{rαc} is the relative mobility and *λ*_{rTc} is the total relative mobility.

The relative mobilities and total compressibility in (8) are calculated using the following equations:

$$\begin{array}{}{\displaystyle {\lambda}_{r\alpha c}={\lambda}_{r\alpha}\sum _{i=1}^{{n}_{c}}{\overline{\rho}}_{i}{c}_{i\alpha},\phantom{\rule{1em}{0ex}}{\lambda}_{r\alpha}=\frac{{k}_{r\alpha}}{{\mu}_{\alpha}},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\alpha =1,\dots ,{n}_{p},}\end{array}$$(9)

$$\begin{array}{}{\displaystyle {\lambda}_{rTc}=\sum _{\alpha =1}^{{n}_{p}}{\lambda}_{r\alpha c},}\end{array}$$(10)

$$\begin{array}{}{\displaystyle {c}_{t}={c}_{r}+[1+{c}_{r}(2{p}_{1}-{p}_{S}-{p}_{R})]\sum _{i=1}^{{n}_{c}}({c}_{i}^{0}{\stackrel{~}{c}}_{i}).}\end{array}$$(11)

The term
$$\begin{array}{}{\displaystyle {\mathit{\Delta}}_{t}F({\stackrel{~}{c}}_{i})=\varphi ({p}_{1}-{p}_{R})\sum _{i=1}^{{n}_{c}}({c}_{i}^{0}\frac{\mathrm{\partial}{\stackrel{~}{c}}_{i}}{\mathrm{\partial}t})}\end{array}$$(12)
can be treated as a source type function.

To increase the stability and robustness in numerical modeling, on the contrary to what is commonly used in the literature, we do not neglect higher order terms in the conservation equations.

The energy conservation equation reads
$$\begin{array}{}{\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{\mathrm{\partial}}{\mathrm{\partial}t}[\hat{\varphi}\sum _{\alpha =1}^{{n}_{p}}{\rho}_{\alpha}{S}_{\alpha}{C}_{V\alpha}+(1-\hat{\varphi}){\rho}_{s}{C}_{s}]T+}\\ {\displaystyle +\mathrm{\nabla}\cdot (\hat{\varphi}\sum _{\alpha =1}^{{n}_{p}}{\rho}_{\alpha}{S}_{\alpha}{C}_{p\alpha}{\overrightarrow{u}}_{\alpha}T-{K}_{T}\mathrm{\nabla}T)={q}_{c}-{q}_{L},}\end{array}$$(13)

where *T* is the temperature; *C*_{Vα} and *C*_{pα} are the heat capacities of phase *α* at constant volume and pressure, respectively; *C*_{s} is the heat capacity of the solid phase; *k*_{T} is the thermal conductivity; *q*_{c} is the heat source term; and *q*_{L} is the heat loss to overburden and underburden formations or soil.

All equations other than the basic equations (1), (2), (8) and (13) and auxiliary relations in the recently developed model formulation, including the adsorption, chemical reaction, phase behavior and well modeling approaches, remain unchanged from the currently used chemical compositional model, details of which can be found in [4] and [7].

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.