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# Open Engineering

### formerly Central European Journal of Engineering

Editor-in-Chief: Ritter, William

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Volume 7, Issue 1

# Numerical Validation of Chemical Compositional Model for Wettability Alteration Processes

Bakhbergen Bekbauov
/ Abdumauvlen Berdyshev
/ Zharasbek Baishemirov
/ Domenico Bau
Published Online: 2017-12-29 | DOI: https://doi.org/10.1515/eng-2017-0049

## Abstract

Chemical compositional simulation of enhanced oil recovery and surfactant enhanced aquifer remediation processes is a complex task that involves solving dozens of equations for all grid blocks representing a reservoir. In the present work, we perform a numerical validation of the newly developed mathematical formulation which satisfies the conservation laws of mass and energy and allows applying a sequential solution approach to solve the governing equations separately and implicitly. Through its application to the numerical experiment using a wettability alteration model and comparisons with existing chemical compositional model’s numerical results, the new model has proven to be practical, reliable and stable.

## 2 Formulation of mathematical model

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 i is defined as $∂∂t(ϕρic~+∇⋅[ϕ^ρi∑α=1np(Sαciαu→α)]−−∇∑α=1np[K¯¯iα⋅∇(ϕ^ρiSαciα)=Ri,i=1,…,nc,$(1)

where ϕ is the porosity, ρi is the density of pure component i and 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 Sα is dehned as the fraction of the reduced pore volume occupied by phase α. cia is the modified volume fraction of component i in phase α.

The phase flux u⃗α is the modified average pore velocity vector of phase α owing to convection and calculated from Darcy’s law $u→α=k¯¯kraϕ^Sαμα(∇pα−γα∇z),α=1,…,np,$(2)

where k̿ is the permeability tensor, k is the relative permeability of fluid phase α, μα is the dynamic viscosity of fluid phase α, pα is the pressure in fluid phase α, γα is the specihc weight of fluid phase α and z represents depth.

Two components of dispersion tensor k̿ia for a homogeneous isotropic permeable medium [24] are $(Kxx)iα=Diατ+αlαuxα2+αtα(uyα2+uzα2)|u→α|,(Kxy)iα=(αlα−αtα)uxαuyα|u→α|,i=1,…,nc;α=1,…,np,$(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 is the effective binary diffusion coefficient of component i in phase α [25], α and α are the longitudinal and transverse dispersivities and τ is the permeable medium tortuosity.

For biodegradation model $Ri=−kiϕρi[(1−∑j=1ncvc^j)∑α=1npSαciα+c^i]+Qi,i=1,…,nc.$(4)

where ki is the reaction rate coefficient in units of inverse time and Qi represents physical sources. The overall concentration i is defined as $c~i=[(1−∑j=1ncvc^j)∑α=1npSαciα+c^i,i=1,…,nc.$(5)

where ĉj is the absorbed concentration of components j, ncv is the total number of volume-occupying components and np is the number of phases.

The magnitudes of the vector flux for each phase, |u⃗α|, are computed as follows:

$|u→α|=(uxα)2+(uyα)2+(uzα)2.$

The porosity depends on pressure due to rock compressibility. Therefore, $ϕ=ϕR[1+cr(p1−pS)],$(6)

where ϕR is the porosity at a specific pressure ps, p1 is the water phase pressure and cr is the rock compressibility at pS. For a slightly compressible fluid, the component density can be written as: $ρi=ρiR[1+ci0(p1−pR)],i=1,…,nc,$(7)

where ρiR is the density of component i at the standard pressure pR.

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

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 ρiR 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: $ϕRct∂p1∂t−∇⋅(k¯¯λrTc∇p1)==∇⋅(k¯¯∑α=1npλrαc∇pcα1)−∇⋅(k¯¯∑α=1npλ(rαcγα)∇z)−−ΔtF(c~i+∑i=1ncQiρiR,$(8)

where ct is the total system compressibility, p1 is the pressure of phase 1, p1 is the capillary pressure, z is the depth, λrαc is the relative mobility and λrTc is the total relative mobility.

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

$λrαc=λrα∑i=1ncρ¯iciα,λrα=krαμα,α=1,…,np,$(9)

$λrTc=∑α=1npλrαc,$(10)

$ct=cr+[1+cr(2p1−pS−pR)]∑i=1nc(ci0c~i).$(11)

The term $ΔtF(c~i)=ϕ(p1−pR)∑i=1nc(ci0∂c~i∂t)$(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 $∂∂t[ϕ^∑α=1npραSαCVα+(1−ϕ^)ρsCs]T++∇⋅(ϕ^∑α=1npραSαCpαu→αT−KT∇T)=qc−qL,$(13)

where T is the temperature; C and C are the heat capacities of phase α at constant volume and pressure, respectively; Cs is the heat capacity of the solid phase; kT is the thermal conductivity; qc is the heat source term; and qL 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].

## 3 Wettability alteration model

In this section we present a brief description of the wettability alteration model. Relative permeabilities are calculated based on the Corey model as follows: $krl=krl0S¯lel,l=1,2,3,$(14)

where subscript l indicates water, oil or microemulsion phase,

$\begin{array}{}{k}_{rl}^{0}\end{array}$

is the relative permeability endpoint for phase l, el is the Corey exponent of phase l and Sl is the normalized saturation of phase l. The normalized saturations are calculated as $S¯l=Sl−Slr1−∑l=13Slr,l=1,2,3,$(15)

where Sl is the saturation of phase l and Slr is the residual saturation of phase l. Residual phase saturations are modeled as follows $Slr=Slrhigh+Slrlow−Slrhigh1+TlNTl,l=1,2,3,$(16)

where

$\begin{array}{}{S}_{lr}^{high}\end{array}$

and $\begin{array}{}{S}_{lr}^{low}\end{array}$

are residual saturations of phase l at high and low trapping numbers, respectively (given as input parameters), Tl is a positive input trapping parameter of phase l and NTl is the trapping number of phase l.

$\begin{array}{}{S}_{lr}^{high}\end{array}$

are typically zero.

The trapping number for phase l displaced by phase ĺ is obtained by a force balance on the displaced blob of phase l and is defined as follows: $NTl=|−k¯¯∇Φl′−k¯¯[g(ρl′−ρl)∇h]|σll′l=1,2,3,$(17)

where l and ĺ are the displaced and displacing fluids, respectively, Φ is the flow potential, g is the gravitational acceleration, k̿ is the permeability tensor, h is the height to a reference datum, ρl and ρpĺ are densities of the displaced and displacing fluids, respectively, and σ is the IFT.

The endpoint relative permeability enhancements caused by residual-saturation reduction of the conjugate phase as a function of the trapping number are modeled using the following correlation validated against experimental data: $krl0=krl0low+Sl′rlow−Sl′rSl′rlow−Sl′rhigh××(krl0high−krl0low),l=1,2,3,$(18)

where $\begin{array}{}{k}_{rl}^{0{\phantom{\rule{thinmathspace}{0ex}}}^{low}}\end{array}$

and $\begin{array}{}{k}_{rl}^{0{\phantom{\rule{thinmathspace}{0ex}}}^{high}}\end{array}$

represent the endpoint relative permeability of phase l at low and high trapping numbers, respectively, $\begin{array}{}{S}_{{l}^{\prime }r}^{low}\end{array}$

and $\begin{array}{}{S}_{{l}^{\prime }r}^{high}\end{array}$ are residual saturations for phase l at low and high trapping numbers, respectively, and Sĺr is the residual saturation of the conjugate phase (e.g. oil is the conjugate phase for microemulsion phase).

The following equation gives the relative permeability exponents as a function of the trapping number: $el=ellow+Sl′rlow−Sl′rSl′rlow−Sl′rhigh××(el′rhigh−ellow),l=1,2,3,$(19)

where $\begin{array}{}{e}_{l}^{low}\end{array}$

and $\begin{array}{}{e}_{l}^{high}\end{array}$ represent the Corey exponents for low and high trapping numbers, respectively, specified as input parameters.

The equations above are solved once for the initial reservoir wettability condition $\begin{array}{}\left({k}_{rl}^{initial}\right)\end{array}$ and once for the altered condition of strongly water-wet $\begin{array}{}\left({k}_{rl}^{final}\right).\end{array}$

Two sets of relative permeabilities $\begin{array}{}\left({k}_{rl}^{0},\phantom{\rule{thinmathspace}{0ex}}{S}_{rl},\phantom{\rule{thinmathspace}{0ex}}{e}_{l}\right)\end{array}$ and trapping parameters (Tl) corresponding to each wettability state are required as model inputs. The relative permeability in each gridblock (krl) is then obtained by linear interpolation between the relative permeabilities corresponding to the two different wettability conditions, provided the concentration of surfactant in the gridblock is greater than the critical micelle concentration. Interpolation is based on a scaling factor ω

$krl=ωkrlfinal+(1−ω)krlinitial,l=1,2,3,$(20)

where ω is the interpolation scaling factor and $\begin{array}{}{k}_{rl}^{final}\end{array}$

and $\begin{array}{}{k}_{rl}^{initial}\end{array}$ represent the relative permeabilities corresponding to the two extreme wetting states (i.e., final and initial wettability states, respectively).

The scaling factor is either a constant user input parameter or related to the adsorbed surfactant concentration in each gridblock as follows:

$ω=C^surfC^surf+Csurf,$(21)

where ĉsurf and Csurf represent the adsorbed and total concentration of surfactant, respectively.

The capillary pressure Pc is scaled with the oil/microemulsion and oil/water IFT (σom and σow) as follows: $Pc=σomσowCpc(1−S¯l)Epc,$(22)

where Cpc and Epc are endpoint and exponent for capillary pressure, respectively. Cpc takes into account the effect of permeability and porosity using the Leverett J-function.

The capillary pressure as a function of wettability is also modeled using linear interpolation between the initial and final wetting state capillary pressures, as follows: $Pc=ωPcfinal+(1−ω)Pninitial,l=1,2,3,$(23)

## 4 Numerical simulation of wettability alteration processes

While we developed the mathematical formulation for chemical flooding simulation, we did not provide enough numerical evidence to support our theoretical arguments made in our previous publication [7].

For validation of the enhanced simulation model, two laboratory experiments on wettability alteration are modeled using the modified code.

The first experiment is a static imbibition cell test conducted by Hirasaki and Zhang (2004) and the second is a dynamic fracture block experiment conducted at UT Austin (Fathi et al., 2008) [13], [14]. Fathi Najafabadi (2009) gives full description of the matching procedure and obtained results for each experiment [15].

In this work, an attempt was made to match the simulation results on wettability alteration modeling with those that have been published in the literature as a validation of the newly developed chemical compositional model formulation.

## 4.1 Numerical simulation of wettability alteration processes in a static imbibition cell test

To validate the implemented wettability alteration model described in section 3, the laboratory alkaline/surfactant imbibition experiments reported by Hirasaki and Zhang (2004) were used [13]. For numerical study of wettability effects on oil/water relative permeability and oil capillary-desaturation curve, we used the data measured by Mohanty (1983) and Morrow et al. (1973) [26], [27]. The parameters of capillary pressure and relative permeabilities used in our simulation are listed in Table 1.

Table 1

Relative Permeability and Capillary Pressure Parameters (Low Trapping Number in Matrix)

Residual saturations, endpoint relative permeabilities and relative permeability exponents are given in Figs. 1 through 3 as functions of trapping number for different wetting conditions. Relative permeabilities calculated using Eqs. 14 through 20 with a constant wettability scaling factor of 0.5 for trapping number of 10−7 are given in Fig. 4. The capillary pressure curves calculated for water-wet and oil-wet conditions using Eq. 22 and a mixed-wet curve using the scaling factor of 0.5 in Eq. 23 are shown in Fig. 5.

Figure 1

Capillary desaturation curves used in simulations

Figure 2

Endpoint relative permeabilities varying with trapping number

Figure 3

Relative permeability exponents varying with trapping number

Figure 4

Relative permeability curves for different wettability conditions at low trapping number of 10−7

The properties of the core and liquid are given in Table 2. Surfactant solution is a mixture of 0.025 weight% CS-330 (C12-3EO-sulfate) and 0.025% by weight of TDA-4PO-sulfate (C13-4PO-sulfate) added with 0.3 mole of sodium carbonate to reduce the adsorption of surfactants. The alkalinity of sodium carbonate is also a reason for the formation of surfactant. Description of the entire experimental and modeling procedure can be found in Fathi Najafabadi (2009) [15].

Table 2

The properties of the core used for the absorption experiment

A 3D numerical model was created to simulate the experiment and validate wettability alteration model. A homogeneous Cartesian grid with 7×7×7 grid blocks was created to simulate the rock and fluids in the cell that surround the core. The middle 5×5×5 portion of the grids was given petrophysical properties representing the rock (Table 3) and the remaining gridblocks were given properties representing the imbibition cell.

Table 3

The parameters of relative permeability and capillary pressure

At first, the simulation model was run to determine the oil recovery based on the assumption that the wettability is not altered from the original oil-wet conditions. It only models the effect of surfactant on interfacial tension reduction and oil mobilization. Surfactant concentration in gridblocks comprising the oil-wet core initially increases primarily due to an effective molecular diffusion and influences the onset of oil being produced from the core, but with very little impact on final oil recovery. An effective molecular diffusion/dispersion coefficient of 6.5 × 10−5 ft2/day was used in the simulation, which is the same as that used in the previous simulation studies. Based on published data of Lam and Schechter (1987) and others, the expected surfactant molecular diffusion would be several times smaller than that was simulated [28]. Therefore, the value presented here can be described as a pseudo diffusion/dispersion coefficient. The vertical cross section through the center of the model is shown in Fig. 6.

Figure 5

Capillary pressure curves for different wettability conditions

Figure 6

Initial surfactant concentration (volume fraction) for the imbibition test model. The vertical XZ cross section through the center (at Y-slice=4) of the model.

Next, the enhanced simulator was used to model combined interfacial tension reduction and wettability alteration effects of surfactant/alkali solution on oil recovery. It was assumed that the altered wettability state was water-wet with relative permeability and capillary pressure parameters as shown in Table 3. The interpolation scaling factors for this simulation were assumed to be a constant value of 0.5. The distribution of surfactant concentration and oil saturation after 10 days of imbibition are given in Figs. 7 and 8.

Figure 7

Surfactant concentration (volume fraction) after 10 days of imbibition. The vertical XZ cross section through the center (at Y-slice=4) of the model.

Figure 8

Oil saturation distribution after 10 days of imbibition. The vertical XZ cross section through the center (at Y-slice=4) of the model.

There is a good agreement between the surfactant concentration and oil saturation as shown in Figs. 7 and 8. The rock gridblocks with higher surfactant concentration have lower oil saturation. These two figures also indicate the role of gravity in the oil recovery from the core plug. The rock gridblocks in the lower parts of the core have smaller oil saturations due to gravity drainage of the oil. The result with wettability alteration gives a much better agreement with the laboratory data than the case without wettability alteration. The simulation with wettability alteration has a faster response to oil production and a higher cumulative oil recovery. This is due to the increase in oil relative permeability and initial change in capillary pressure from negative to positive during the wettability alteration process. Therefore, more surfactant solution is imbibed into the rock gridblocks displacing more oil before the interfacial tension reduction decreases the capillary pressure to zero. Alteration of the wettability towards more water-wet conditions increases oil mobility, recovery rate and its final recovery. Successful modeling of this experiment validates implementation of wettability alteration model in the modified simulation tool for the spontaneous imbibition test.

## 4.2 Numerical simulation of wettability alteration processes in a fractured block

The test problem for the numerical study of the wettability change due to surfactants in the fractured reservoirs in the computational grid with 31×11×3 number of nodes was examined. The Fractured Block experiment was carried out by Dr. Q. P. Nguyen and J. Zhang. The details and steps taken in modeling the fractured block experiment using the UTCHEM simulator with the wettability alteration model were described in [15].

A numerical experiment was designed to get a better understanding of the mechanisms occurring in the flow of fluids such as alkali and surfactant in naturally fractured formations. Alkali agent was used as a wettability modifier and surfactant solution was used to lower the interfacial tension and enhance oil recovery by oil emulsification (Fathi et al., 2008) [14]. A 3D discrete fracture model consisting of 6 fractures was used (Fig. 9).

Figure 9

The computational domain and pressure distribution resolved with 31×11×3 (X×Y×Z) grid points

Two of these fractures are parallel and four are perpendicular to the flow direction. The end caps were modeled as fractures and the injection and production wells were placed in these openings to model the same conditions as in the experiment. The matrix gridblock size was 1/3″ × l/3″ ×l/3″. Fracture aperture was 1 mm (0.039370"). Fracture gridblocks were assumed to have a porosity of 1 as opposed to 0.298 for matrix blocks (based on laboratory material balance). A uniform and isotropic matrix permeability of 3.36 × 10−14m2 (34 md) was used based on laboratory measurements. A uniform initial saturation was assumed based on the measured values. Table 4 summarizes the base case simulation parameters.

Table 4

Simulation input parameters for base case model of fracture block experiment

Oil recovery was measured for the three different fluid injection steps. Injection of 4.8 wt% NaCl solution at 5 ml/hr resulted in an ultimate recovery of about 15% OOIP. During the first few hours of waterflood, no oil was produced from the setup. This delay was due to the volume of the tube connecting the fracture block setup to the sample collection unit. This was not properly communicated to the modeling group at the time of modeling of the experiment. Next, water containing 1 wt% sodium metaborate (NaBO2) and 3.8 wt% NaCl injected at the same rate mobilized additional 15% OOIP. The producing oil cut was reduced to zero at the end of the alkaline flood. A mixture of alkali/surfactant was then injected at the same flow rate, rendering incremental recovery of 6% OOIP. The surfactant solution contained 1.5 wt% PetroStep® S-1 and 0.5 wt% PetroStep® S-2, 2 wt% secondary butanol as co-solvent, 1 wt% sodium metaborate and 3.8 wt% NaCl. This surfactant formulation formed a microemulsion with the crude oil which exhibits an IFT of 0.0008 mN/m. It is important to note that the pressure gradient was around 0.8 psi/ft throughout the experiment. This pressure gradient corresponds to the pressure difference at the inlet and outlet of the setup.

The result of this model for the range of 0 - 4 PV for average pressure in the form of integral curve is shown in Figure 10.

Figure 10

Average pressure vs. injected pore volume

Model results were compared with validated and published chemical compositional model results to demonstrate that the chemical compositional model equations were solved accurately. New simulator successfully provided well-correlated data when compared with the validated and published results (see Fig. 11).

Figure 11

Comparison of oil recovery curves simulated using the currently used and the newly developed chemical compositional models for the waterflood and alkali flood part of the fractured block experiment

As it can be seen from the above numerical analysis, the new and currently used formulation results are very much the same. In this work, through its application to the above-mentioned numerical experiments and comparisons with existing chemical compositional model’s numerical results, the newly developed formulation has proven to be practical and reliable. The main mechanism for oil recovery from fractured carbonate reservoirs is a combination of viscous, gravity and capillary forces. Capillary pressure and relative permeability curves are the main petrophysical properties affected by the wettability alteration. The capillary pressure shift towards positive values (more water-wet conditions) and the shift of relative permeability curves towards more ideal straight line water-wet conditions are a good indication of an efficient chemical EOR process. These processes can be modeled in the newly developed simulator for modeling chemical EOR processes.

Residual oil saturation does not decrease significantly due to small trapping number in gridblocks during chemical process. This process can be simulated by the alkali option in the new simulator with no effect on IFT and residual oil saturation.

## 5 Conclusion

The new chemical compositional model formulation was validated against existing chemical compositional simulator for imbibition test data using matrix carbonate rocks and coreflood cell tests. Numerical oil recovery findings for imbibitions and coreflood cell tests are in good agreement with observed results in UTCHEM.

Through its application to the numerical experiment using a wettability alteration model and comparisons with existing chemical compositional model’s numerical results, the new formulation has proven to be practical, reliable and more stable.

Test cases’ results of water and alkaline floods in a 3D fractured block were successfully compared with corresponding results from UTCHEM simulator. Very good match between the currently used chemical compositional model and the new model formulations’ results was obtained.

## Acknowledgement

This research work was financially supported by the Ministry of Education and Science of the Republic of Kazakhstan under grants No. 1735/GF4 and No. 0128/GF4.

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Accepted: 2017-11-28

Published Online: 2017-12-29

Citation Information: Open Engineering, Volume 7, Issue 1, Pages 416–427, ISSN (Online) 2391-5439,

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