Governing equations for Multi-phasic reservoir modeling
CMG-STARS is a finite difference numerical tool that describes
mathematical equations for fluid flow in a petroleum reservoir. In
multiphase flow equations, the simulation model is governed by the
conservation of mass, energy and momentum. These functions relate
conservation equations with an elementary volume or specified region of
interest; wherein each component of volumetric change is related to the
fluids entering or leaving the system [38,39]. This includes the
material balance equations, Darcy’s law, relative permeability
correlations, capillary pressure equations, and phase equilibrium
equations in two- and three-phase porous media [39-41]. The
conservation law states that the conserved quantity within a volume or
at a point depends on the net rate of fluids that flow in and out of the
volume (or region). With the depiction of an appropriate set of
initial/boundary conditions, the governing models are applied to develop
an understanding of simultaneous flow of two or more fluid phases. The
conservation equation for mass is presented for a flowing and/or
adsorbed component ‘i ’ within the system as Eq. (1):
\(\frac{\partial}{\partial t}\left[V_{f}\left(\rho_{w}S_{w}w_{i}+\rho_{o}S_{o}x_{i}+\rho_{g}S_{g}y_{i}\right)+V_{v}\text{Ad}_{i}\right]=\sum_{k=1}^{n_{f}}\left[T_{w}\rho_{w}w_{i}\text{ΔΦ}_{w}+T_{o}\rho_{o}x_{i}\text{ΔΦ}_{o}+T_{g}\rho_{g}y_{i}\text{ΔΦ}_{g}\right]+V\sum_{k=1}^{n_{r}}{\left(s_{\text{ki}}^{{}^{\prime}}-s_{\text{ki}}\right)r_{k}}+\sum_{k=1}^{n_{f}}\left[\phi D_{\text{wi}}\rho_{w}{\Delta w}_{i}+\phi D_{\text{oi}}\rho_{o}\text{Δx}_{i}+\phi D_{\text{gi}}\rho_{g}{\Delta y}_{i}\right]+\rho_{w}q_{\text{wk}}w_{i}+\rho_{o}q_{\text{ok}}x_{i}+\rho_{g}q_{\text{gk}}y_{i}+\delta_{\text{iw}}\sum_{k=1}^{n_{f}}\rho_{w}\text{qaq}_{\text{wk}}\)(1)
where,\(\frac{\partial}{\partial t}\left[V_{f}\left(\rho_{w}S_{w}w_{i}+\rho_{o}S_{o}x_{i}+\rho_{g}S_{g}y_{i}\right)+V_{v}\text{Ad}_{i}\right]\)is the time-derivative for material accumulation. The total fluid volume
and void volume are represented by terms, Vf andVv respectively. In above relation,wi , xi , andyi refer to mole fraction of component ‘i’in water, oil and gas respectively, whereas ρ and S stand
for density and saturation of different phases.\([T_{w}\rho_{w}w_{i}\text{ΔΦ}_{w}+T_{o}\rho_{o}x_{i}\text{ΔΦ}_{o}+T_{g}\rho_{g}y_{i}\text{ΔΦ}_{g}+\phi D_{\text{wi}}\rho_{w}{\Delta w}_{i}+\phi D_{\text{oi}}\rho_{o}\text{Δx}_{i}+\phi D_{\text{gi}}\rho_{g}{\Delta y}_{i}]\)is the flow term for component ‘i’ .
[\(\rho_{w}q_{\text{wk}}w_{i}+\rho_{o}q_{\text{ok}}x_{i}+\rho_{g}q_{\text{gk}}y_{i}]\)is well source/sink term, and\(V\sum_{k=1}^{n_{r}}{\left(s_{\text{ki}}^{{}^{\prime}}-s_{\text{ki}}\right)r_{k}}\)stands for the reaction source/sink term for component ‘i’ . For
water component, \(\sum_{k=1}^{n_{f}}\rho_{w}\text{qaq}_{\text{wk}}\)is aquifer source/sink term wherein qaqwkrepresents the volumetric flow rate through the block face kto/from adjacent aquifer.
T is the component transmissibility between two regions/points,
which accounts for cross-sectional area, distance between the elementary
volumes, and fluid permeability. Eq. (2) depicts the relation between
volumetric flow rate, v and transmissibility, T , as:
\(v_{j}=T\left(\frac{k_{\text{rj}}}{\mu_{j}r_{j}}\right){\Phi}_{j}\)(2)
In this equation, the term ‘j ’ can be applied to different
phases, i.e. water (w ), oil (o ) and gas (g ).ΔΦj represents potential difference for phase‘j’, and it may be either positive or negative, depending on the
inflow/outflow of fluid component. rj is phase
resistance factor, krj is relative permeability
and µj is viscosity of phase j . Component
dispersibility in water, oil, and gas phases are shown byDwi , Doi , andDgi . The well rate (qjk )
of any phase ‘j ’ in the layer ‘k ’ is shown in Eq. (3):
\(q_{\text{jk}}=I_{\text{jk}}\left(p_{\text{wfk}}-p_{k}\right)\)(3)
where, pwfk is wellbore pressure,pk is pressure existing within volume andIjk refers to phase index. The phase index
property of the system depends on various factors such as geometry,
permeability, layer thickness, and skin factor. Therefore, individual
changes in each component contribute toward conservation studies for
mass. Total energy of the material volume is an important field of
research analysis, which has significant repercussion in defining fluid
flow behavior. During CMG simulation, the rock volume does not change,
and the internal energy of the rock remains constant. The energy
conservation equation is shown in Eq. (4) as:
\(\frac{\partial}{\partial t}\left[V_{f}\left(\rho_{w}S_{w}U_{w}+\rho_{o}S_{o}U_{o}+\rho_{g}S_{g}U_{g}\right)+V_{v}{c_{s}U}_{s}+V_{r}U_{r}\right]=\sum_{k=1}^{n_{f}}\left[T_{w}\rho_{w}H_{w}\text{ΔΦ}_{w}+T_{o}\rho_{o}H_{o}\text{ΔΦ}_{o}+T_{g}\rho_{g}H_{g}\text{ΔΦ}_{g}\right]+\sum_{k=1}^{n_{f}}{KT}+\rho_{w}q_{\text{wk}}H_{w}+\rho_{o}q_{\text{ok}}H_{o}+\rho_{g}q_{\text{gk}}H_{g}+V\sum_{k=1}^{n_{r}}{H_{\text{rk}}r_{k}}+HL_{o}+HL_{v}+HL_{c}+\sum_{k=1}^{n_{f}}\left(HA_{\text{CV}}+HA_{\text{CD}}\right)_{k}\)(4)
In the above equation,\(\frac{\partial}{\partial t}\left[V_{f}\left(\rho_{w}S_{w}U_{w}+\rho_{o}S_{o}U_{o}+\rho_{g}S_{g}U_{g}\right)+V_{v}{c_{s}U}_{s}+V_{r}U_{r}\right]\)is the time-derivative for energy accumulation, andUj is the internal energy of the rock system. It
is primarily influenced by two factors, namely, temperature and phase
composition. Hj denotes the enthalpy of the
respective phases. Reaction source/sink term for energy is represented
by \(V\sum_{k=1}^{n_{r}}{H_{\text{rk}}r_{k}}\), whereinHrk and rk stand for
enthalpy and volumetric rate of reaction in layer ‘k’respectively. HLo , HLv andHLc represents the total heat transfer rate, heat
transfer rate for convection model and constant heat transfer model
respectively.\(\sum_{k=1}^{n_{f}}\left(HA_{\text{CV}}+HA_{\text{CD}}\right)_{k}\)describes the aquifer source/sink term for energy, whereinHACV and HACD represent
respective rates of heat transfer via convection; and conduction to/from
adjacent aquifer.
[\(T_{w}\rho_{w}H_{w}\text{ΔΦ}_{w}+T_{o}\rho_{o}H_{o}\text{ΔΦ}_{o}+T_{g}\rho_{g}H_{g}\text{ΔΦ}_{g}+KT]\)represents the energy term for flow between two regions, and
[\(\rho_{w}q_{\text{wk}}H_{w}+\rho_{o}q_{\text{ok}}H_{o}+\rho_{g}q_{\text{gk}}H_{g}]\)is the well source/sink term for energy.
CMG-STARS employs various equations to generate relative permeability
curves, which convey rock-fluid interactions in porous media flow
studies. Corey’s correlation is an important alternative to calculate
relative permeability curves, particularly in situations wherein
displacing/displaced fluid properties are not available in detail
[42-44]. This behavior is described in Eqs. (5), (6), (7), and (8)
as:
\(K_{\text{rw}}=K_{\text{rwiro}}\left(\frac{S_{w}-S_{\text{wcrit}}}{1.0-S_{\text{wcrit}}-S_{\text{oirw}}}\right)^{N_{w}}\)(5)
\(K_{\text{row}}=K_{\text{rocw}}\left(\frac{S_{o}-S_{\text{orw}}}{1.0-S_{\text{wcon}}-S_{\text{orw}}}\right)^{N_{\text{ow}}}\)(6)
\(K_{\text{rog}}=K_{\text{rogcg}}\left(\frac{S_{l}-S_{\text{org}}-S_{\text{wcon}}}{1.0-S_{\text{gcon}}-S_{\text{org}}-S_{\text{wcon}}}\right)^{N_{\text{og}}}\)(7)
\(K_{\text{rg}}=K_{\text{rgcl}}\left(\frac{S_{g}-S_{\text{gcrit}}}{1.0-S_{\text{gcrit}}-S_{\text{oirg}}-S_{\text{wcon}}}\right)^{N_{g}}\)(8)
Krw , Krow ,Krog and Krg refer to
respective values of water phase relative permeability for water-oil
table, oil phase relative permeability for water-oil table, liquid phase
relative permeability for liquid-gas table, and gas phase relative
permeability for liquid-gas table. Different saturation terms were
employed to characterize wetting characteristics of reservoir rock
[43,44]. It is to be noted that Krocw isKro at connate water saturation,Krgcl is Krg at connate
liquid saturation, Krwiro isKrw at irreducible oil saturation,andKrogcg is Krog at connate
gas saturation. Connate water saturation, critical water saturation,
irreducible oil saturation for water-oil table, and residual oil
saturation for water-oil table are represented bySwcon , Swcrit ,Soirw and Sorwrespectively. Irreducible oil saturation, residual oil saturation,
connate gas saturation and critical gas saturation are depicted by
respective terms Soirg ,Sorg , Sgcon andSgcrit for liquid-gas table.Nw , Now ,Nog and Ng are exponent
terms determined from relative permeability curves. Equations (5) and
(6) describe the water-oil permeability table, whereas equations (7),
(8) are used to generate liquid-gas relative permeability data.
Alterations occurring within physicochemical properties of reservoir
fluids due to presence of chemicals i.e. surfactant, polymer and/or
nanoparticle lead to varying fluid flow profiles during EOR.
Interpolation of relative permeability curve is performed by
corresponding to relative permeability datasets in-between high and
ultralow IFT conditions [45,46]. The interpolated relative
permeability data is described as function of dimensionless parameters,
as shown in Eqs. (9), (10) and 11):
\(k_{\text{rw}}=k_{\text{rwA}}.\left(1-\text{ratn}^{\text{WCRV}}\right)+k_{\text{rwB}}.\text{ratn}^{\text{WCRV}}\)(9)
\(k_{\text{ro}}=k_{\text{roA}}.\left(1-\text{ratn}^{\text{OCRV}}\right)+k_{\text{roB}}.\text{ratn}^{\text{OCRV}}\)(10)
\(k_{\text{rg}}=k_{\text{rgA}}.\left(1-\text{ratn}^{\text{GVRV}}\right)+k_{\text{rgB}}.\text{ratn}^{\text{GCRV}}\)(11)
In the above equations, ratw and ratn refer to
interpolation parameters with values ranging between zero and unity. The
curvature interpolation parameters are represented by WRCV ,OCRV , and GCRV , with the default value of one.
Furthermore, the interpolation parameters are related to the capillary
number as shown in Eq. (12) and Eq. (13) as follows:
\(ratw=\frac{\operatorname{}{\left(N_{c}\right)-DTRAPWA}}{DTRAPWB-DTRAPWA}\)(12)
\(ratn=\frac{\operatorname{}{\left(N_{c}\right)-DTRAPNA}}{DTRAPNB-DTRAPNA}\)(13)
where, Nc is the capillary number, whereasDTRAPWA and DTRAPNA describe interpolation parameters for
high IFT value (low Nc condition) and ultralow IFT value
(favorably high Nc condition) respectively, for wetting
phase. On the contrary, DTRAPNA and DTRAPNB are similar
interpolation terms for the non-wetting phase. Relative permeability
plot analysis is important during CMG-STARS modelling studies for
accurate investigation of fluid-rock interactions and flooding
performance of injection fluids.
- Results and Discussion
- Experimental investigations
- Rock-wetting characteristics of 14-6-14 GS
Wettability alteration characteristics was investigated by sessile drop
analyses of aqueous surfactant fluid onto crude oil-saturated sandstone
rock. Figs. 1(a) and 1(b) show the variation of contact angle with time.
At initial time (t = 0), contact angle was measured as 102.8°,
confirming the intermediate wet nature of rock surface. For sandstone
rock, contact angle progressively reduced to 72.6°, 61.9°, 50.9°, 32.1°
and 14.2°; at the end of 30 s, 60 s, 120 s, 240 s and 480 s
respectively. This trend is indicative of “spreading” of aqueous
chemical fluid and “detachment” of crude oil molecules from rock
surface [47]. Rock-wetting process is mainly dependent on rock
morphology, inter-ionic electrostatic interactions and attractive and
attractive/hydrophobic interactions among 14-6-14 GS molecules and crude
oil components. As time elapses, surfactant molecules gradually
destabilize the ordered arrangement of previously adsorbed oil molecules
and spread onto the rock substrate. Hence, 14-6-14 GS possess the
capability to favorably “wet” oil-saturated rock and mobilize
entrapped crude oil within reservoir formations effectively.