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.
  1. Results and Discussion
  2. Experimental investigations
  3. 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.