Microbial enhanced oil recovery: interfacial tension and biosurfactant-bacteria growth

Abstract

Microbial enhanced oil recovery (MEOR) is a method that utilises bacteria or bioproducts to increase oil recovery at the tertiary stage. Clostridium sp. produces biosurfactant that alters rock–fluid properties and increases oil detachment. The interaction between bacteria and surfactant is interesting relation to study. We revisit and develop models for biosurfactant-producing bacteria’s growth and the interfacial tension (IFT) response. The biosurfactant-producing bacteria growth model (BBG model) mimics the predator–prey interaction and the IFT response model derived from analogy. Both models form an integrated model called coupled-simultaneous model. We deliver the suitability of these models to experimental datasets by conducting parameter estimation. The decreased number of parameter in BBG model is with the help of rate estimation model. It estimates the bacteria growth rate and biosurfactant production rate. This research introduces a graphical method to narrow parameters initial guess in the IFT model. The method comes with a proposed index to compare surfactant performance called as surfactant performance index (SPI). The paper exposes the logic of each parameter, physics behind the models, and addresses the mathematical artefacts. The significant findings are valuable to anticipate bacterial performance for MEOR.

Introduction

MEOR mechanisms and aims

Microbial enhanced oil recovery (MEOR) is a method that utilises bacteria or bioproducts to alter rock–fluid properties. It is suggested in 1926 (Beckmann 1926) and systematically investigated in 1947 (ZoBell 1947a, b). A detailed literature study has summarised its recent developments and worldwide applications (Patel et al. 2015). Table 1 reports the bacteria bioproducts associated with possible MEOR mechanisms. MEOR mechanism adapts the chemical-based EOR (Hartono et al. 2017), however, it corresponds to a combined distinct-EOR because microbes could produce biosurfactant, biopolymer, and gas simultaneously. There are other mechanisms such as the consumption of hydrocarbon and degradation of heavy long-carbon-chains into light and short (Sen 2008; Gudina et al. 2012; Cheng et al. 2014; Patel et al. 2015; Cai et al. 2015; Varjani and Upasani 2016; Zhao et al. 2018; Iraji and Ayatollahi 2018).

Table 1 Role of microbial bioproducts in EOR

There are several possible schemes in MEOR: (i) injecting exogenous bacteria, (ii) injecting active indigenous bacteria, (iii) supplying the nutrients only, (iv) combination of scheme i–iii. The common approach is to conduct huff-and-puff. Microbial huff-and-puff involves nutrient and microbes injection into production wells, soak them for a period of time, and then continue to produce the oil. Therefore, understanding how bacteria behave becomes essential, from the growth, bioproducts formation, and the effect on rock–fluid properties.

Time constraint is critical when the bacteria or bioproducts make an alteration in the reservoir because it guides the huff-and-puff shutting and injection duration scheme. Interfacial tension (IFT) is also important to represent the rock–fluid and oil-brine properties. A lower IFT contributes to wettability alteration, to higher oil-detachment from rock surface, and to influence recovery (Jadhunandan and Morrow 1995; Hakiki et al. 2015). Unfortunately, a wettability alteration by biosurfactant could be significant at freshwater but not in brine (Veshareh et al. 2018). A contributing lower IFT to increase capillary number is proven to obtain a less residual-oil saturation (Hakiki et al. 2017).

This research aims to develop an integrated model of biosurfactant-producing bacteria growth and its IFT response. The model focuses on Clostridium sp. as the biosurfactant producer. The results can provide a better understanding of bacteria growth and the IFT with respect to time. This information can help optimising MEOR operation.

Biosurfactant-producing bacteria growth model

Growth model research has been going for a long time even from 190 years ago. The subject may be animate such as human population, animals, microorganisms, and trees in the forest. It also includes an abstract object such as geologic, business, and economic growth.

The model may describe microbial growth under constant temperature or it may be temperature dependent. The basic form of model which describes microbial growth under constant temperature is called primary model (Swinnen et al. 2004; Corradini et al. 2005; Juneja et al. 2007, 2009; Gospavic et al. 2008; Theys et al. 2008; Huang 2013). The two most widely used growth models are the Baranyi and Gompertz models (Buchanan et al. 1997). Most of the growth model is empirical, usually sigmoid model, but some are more mechanistic such as the application of kinetic modelling (Altiok et al. 2006; Yang et al. 2011; Gahlawat and Srivastava 2013). Table 2 summarises the existing growth models.

Table 2 Summary of growth models

It is common to include a temperature-dependent model complemented with the growth model for temperature-dependent growth model. There are many studies on microbial growth with the temperature-dependent model (Zwietering et al. 1994; Daughtry et al. 1997; Bovill et al. 2000; Fujikawa et al. 2004; Corradini et al. 2005; Dominguez and Schaffner 2007; Juneja et al. 2007, 2009; Gospavic et al. 2008; Salih et al. 2012). Other alternative may include salinity, pH, temperature, and nutrient concentration into the model (Hosseininoosheri et al. 2016).

Growth model of biosurfactant-producing bacteria may couple logistic-growth and modify predator–prey model (see Eq. 1, Halim et al. 2009).

$$\frac{{{\text{d}}P}}{{{\text{d}}t}}={\mu _1}P - \frac{{{\mu _1}}}{{{K_0}}}{P^2} - \gamma PS$$

(1a)

$$\frac{{{\text{d}}S}}{{{\text{d}}t}}={\mu _2}P - \delta PS$$

(1b)

$P$ is the number of bacteria in base-10 logarithm for each cell/mL, $S$ is biosurfactant concentration (g/L), ${\mu _1}$ is bacteria growth rate (h⁻¹), ${\mu _2}$ is biosurfactant production rate (mg/cell/h), ${K_0}$ is carrying capacity (cell/mL), $\gamma$ is toxicity constant (L/g/h), and $\delta$ is predation factor (1/h). Equation (1) assumes that there is a significant reduction of bacteria due to environmental adjustment at the initial stage (Halim et al. 2009). The initial condition is at the next time step of the data i.e. after reduction. The reduction is normal when bacteria is exposed to a high pressure condition (Pruitt and Kamau 1993). There are mismatches and confusing physical units in Eq. 1 that this paper will address.

Interfacial tension response model

Biosurfactant produced by bacteria decreases the tension at the oil–water interface, which eases oil to detach from the rock. The model for IFT response due to biosurfactant concentration is proposed as in Eq. (2) (Hakiki 2014).

$$- \frac{{{\text{d}}\sigma }}{{{\text{d}}S}}=\frac{{{\sigma _0} - \sigma }}{S} - \eta ({\sigma _0} - \sigma ).$$

(2)

$\sigma$ is IFT (mN/m), ${\sigma _0}$ is initial IFT (mN/m), $S$ is biosurfactant concentration (g/L), and $\eta$ is a damping constant (L/g). The minus sign represents IFT reduction with respect to biosurfactant concentration. The reduction rate decreases as the concentration gets higher. Solution of Eq. (2) through variable separation is Eq. (3)

$$\sigma ={\sigma _0}[1 - CS\exp ( - \;\eta S)],$$

(3)

where $C$ and $\eta$ are constants (L/g). IFT response model is useful in determining critical micelle concentration (CMC).

Methods

Experimental data

Table 3 lists the detailed experimental data, usage in this research, bacteria name or type, medium of cultivation, and the supplied nutrient. Detailed composition through GCMS analysis of Handil Field crude oil is available in the literature (Fulazzaky et al. 2015). All data have incubation temperature of 50–55 °C. The experimental data used stone mineral salt solution (SMSS) as the growth substrate in a batch system under anaerobic condition.

Table 3 List of the experimental data

DATA 1: Three experimental datasets of bacteria growth and biosurfactant concentration are from indigenous bacteria in Handil Field, East Kalimantan, Indonesia (Primeia 2008). The isolated cultures of bacteria from this field are Flavimonas oryzihabitants, Amphibacillus xylanus, Bacillus polymyxa, Bacillus macerans, and Clostridium sp. (Purwasena 2006; Primeia 2008). We only use the data of Clostridium sp. growth and biosurfactant concentration to validate biosurfactant-producing bacteria growth model.

DATA 2: The literature provides six experimental datasets of biosurfactant and commercial surfactant, and IFT (Bitticaca 2009). The data are also from Handil Field in East Kalimantan, Indonesia. Four of them are biosurfactant concentration and IFT response data of Bacillus sp. and Bacillus polymyxa consortium. Two others are the surfactant and IFT response of commercial synthetic surfactant coded S7A and S13A*. Those data benefit for the validation of IFT response model. A latter study reveals a similar trend of producing biosurfactant from Bacillus sp. at early 30 h cultivation (Heryani and Putra 2017).

DATA 3: The data consists of one experimental dataset of Clostridium sp. growth, biosurfactant concentration, and IFT (Primeia 2008). The dataset are useful to evaluate whether or not the proposed models (biosurfactant-producing bacteria growth model and IFT response model) can be coupled and solved simultaneously.

Parameter estimation

The main problem in solving the proposed model is to fit the simultaneous differential equation solution with the data. We deploy the sum of residual sum of squares (SRSS) as the objective of optimisation. Equation (4) shows the objective equation

$$\mathop {\hbox{min} }\limits_{p} {\varepsilon _{{\text{tot}}}}=\mathop {\hbox{min} }\limits_{p} \left[ {\mathop \sum \limits_{{i=1}}^{m} {{\left( {\mathop \sum \limits_{{j=1}}^{n} \varepsilon _{j}^{2}} \right)}_i}} \right],$$

(4)

where $p$ are parameters to be estimated, ${\varepsilon _{{\text{tot}}}}$ is the SRSS, which sums up each $i$-th residual sum of squares (RSS) out of $m$ number of equation, and $\varepsilon _{j}^{2}$ is the $j$-th squared error out of $n$ number of data. The parameter estimation and optimisation uses explicit Runge–Kutta algorithm (Dormand and Prince 1980) to solve the simultaneous differential equation numerically, then Nelder–Mead simplex algorithm optimises it (Lagarias et al. 1998). We develop the code in MATLAB as a platform for parameter estimation and optimisation.

Results

Proposed models

Biosurfactant-producing bacteria growth (BBG Model)

This research proposes similar model such as Eq. (1). We found a mismatch between Eq. (1) and the predator model; also known as Lotka–Volterra model in Eq. (5).

$$\frac{{{\text{d}}x}}{{{\text{d}}t}}=\alpha x - \beta xy$$

(5a)

$$- \frac{{{\text{d}}y}}{{{\text{d}}t}}=\gamma y - \delta xy.$$

(5b)

The biosurfactant-producing bacteria growth model uses the predator–prey model as in Eq. (6).

$$\frac{{{\text{d}}P}}{{{\text{d}}t}}={\mu _1}P - \gamma PS$$

(6a)

$$- \frac{{{\text{d}}S}}{{{\text{d}}t}}={\mu _2}S - \delta PS.$$

(6b)

The logistic growth model transforms Eq. (6) into (7).

$$\frac{{{\text{d}}P}}{{{\text{d}}t}}={\mu _1}P - \frac{{{\mu _1}}}{{{K_0}}}{P^2} - \gamma PS$$

(7a)

$$- \frac{{{\text{d}}S}}{{{\text{d}}t}}={\mu _2}S - \delta PS.$$

(7b)

The idea of predator–prey is that the predator and prey are interacting towards each other. This phenomenon supports the modification of Eq. (1) into (7) because $P$ and $S$ must interact. Predation and interaction factor appears as $\gamma$ and $\delta$, respectively. This research explores which model provides better approximation for the experimental data. This paper also proposes that the designation of global growth rates ${\mu _1}$ and ${\mu _2}$ should be consistent, where ${\mu _1}$ is used for $P$ and ${\mu _2}$ is used for $S$ so that the parameters’ dimension is consistent. We recommend that $P$ and $S$ must be normalised by 1 cell/mL and 1 g/L, respectively, so that both variables are finally dimensionless. The term “quasi-dimension” appears to overcome this confusion.

This research proposes another model that is modified from the existing model (Eq. 1, Halim et al, 2009), the consistent model, as shown by a set formula in Eq. (8). The term “consistent” means the growth constant is ${\mu _1}$ for controlling $P$.

$$\frac{{{\text{d}}P}}{{{\text{d}}t}}={\mu _1}P - \frac{{{\mu _1}}}{{{K_0}}}{P^2} - \gamma PS$$

(8a)

$$\frac{{{\text{d}}S}}{{{\text{d}}t}}={\mu _1}P - \delta PS.$$

(8b)

The large carrying capacity ${K_0}$ can diminish the second term in RHS of Eq. (8). It leads to another alternative model as shown by Eq. (9).

$$\frac{{{\text{d}}P}}{{{\text{d}}t}}={\mu _1}P - \gamma PS$$

(9a)

$$\frac{{{\text{d}}S}}{{{\text{d}}t}}={\mu _1}P - \delta PS.$$

(9b)

The last model aims to check the effect of parameter consistency towards the model. The second inconsistent model is shown as Eq. (10). “Inconsistent” term signifies the inconsistent quasi-dimension that ${\mu _1}$ for the growth of $P$ and $S$ depends on their own concentration. Variable $S$ should have its own constant, i.e. ${\mu _2}$.

$$ \frac{{{\text{d}}P}}{{{\text{d}}t}}={\mu _1}P - \frac{{{\mu _1}}}{{{K_0}}}{P^2} - \gamma PS$$

(10a)

$$- \frac{{{\text{d}}S}}{{{\text{d}}t}}={\mu _1}S - \delta PS.$$

(10b)

In summary, the inconsistent models are Eqs. (1) and (10); and the consistent models emerge in Eqs. (7)–(9).

Rate estimation model (REM)

The rate parameters in Eq. (1): growth rate ${\mu _1}$ and biosurfactant production rate ${\mu _2}$ can be estimated with parameter estimation method such as multivariate regression or differential equation solution fitting. The rate parameters should vary through the time instead of using global growth rates. Equation (11) covers the proposed rate estimation model (REM). The estimated rate will be discrete because it follows the use of discrete experimental data point. We suggest using polynomial regression to capture the trend of the estimated rate. These results in a continuous time-dependent rate model $\mu$ = $f(t)$.

$${\mu _M}=\frac{{\ln \left( {\frac{M}{{{M_i}}}} \right)}}{{t - {t_i}}}$$

(11)

$M$ is either bacteria population $P$ or biosurfactant concentration $S$, $t$ is time h, and ${M_i}$ is $M$ at initial condition ${t_i}$.

The initial condition sets the time step of the data as assumed in “Biosurfactant-producing bacteria growth model”. Indeterminate form emerges at the application of initial condition to Eq. (11), hence, we exclude it from the rate estimation. Important to note that bacteria may grow differently as they are supplied with different nutrients. We draw this thought because different nitrogen sources can influence bacteria to degrade n-alkane in unique manner (Purwasena et al. 2014).

Interfacial tension response (IFT model)

The authors propose an IFT model as shown by Eq. (12). We assume a general IFT for the whole ranges of biosurfactant concentrations from $S<{S_{{\text{CMC}}}}$ to $S>{S_{{\text{CMC}}}}$. The IFT at near CMC is considerably the interface of oil–water only. Otherwise, there exists microemulsion at $\sigma>{\sigma _{{\text{CMC}}}}$, hence there are three interfaces being considered: oil–water, microemulsion–water, and microemulsion–oil.

$$\frac{{\sigma - {\sigma _{{\text{CMC}}}}}}{{{\sigma _0} - {\sigma _{{\text{CMC}}}}}}={\text{erfc}}(KS).$$

(12)

Equation (13) results from Eq. (12) rearranged into a more convenient form.

$$\sigma ={\sigma _{{\text{CMC}}}}+({\sigma _0} - {\sigma _{{\text{CMC}}}})~{\text{erfc}}(KS).$$

(13)

The subscripts assert the surfactant concentration. Therefore, ${\sigma _0}$ means the IFT at zero-surfactant concentration and ${\sigma _{{\text{CMC}}}}$ means the IFT at CMC (mN/m). $S$ is surfactant concentration (g/L) and $K$ is surfactant performance constant (L/g). However, we present the $S$ and $K$ both are dimensionless because they are normalised by 1 g/L and 1 L/g, respectively, to avoid confusions due to the existence of growth constants. The function ${\text{erfc}}$ is a complementary error function. Note that the model is applicable to either biosurfactant or surfactant. Equation (13) can estimate the parameters and the ability to compare surfactant performance quantitatively by means of surfactant performance index $\psi$.

Coupled-simultaneous model (CSM)

The coupling of proposed models (Eqs. 7 and 13) into first three simultaneous model shown as Eq. (14). This provides an integrated model of bacteria growth and IFT-response. Figure 1 displays the workflow of how to use the coupled model. The model assumes that the external varying factors such as temperature, pH, salinity, and surfactant affinity are constant. The surfactant affinity itself depends on temperature, pH, salinity, surfactant compositions (hydrophilic lipophilic balance HLB), and also oil composition (Salager et al. 2000). A MEOR bacteria, Pseudoxanthomonas sp. G3, for instance, can withstand with emulsion level up to 50% (v/v), wide pH range (2–12), and salinity up to 10% of NaCl (w/v) (Astuti et al. 2019). A latter comprehensive study has modelled IFT and phase behaviour relations for a wide range of surfactant affinity conditions (see Eq. 14d; Torrealba and Johns 2017). The model covers three phases due to the presence of microemulsion. We adapt it in a simplified case for the least IFT, i.e. no hydrophilic–lipophilic deviation (HLD = 0). This implies that the IFT would only be at oil–water interface (Torrealba and Johns 2017). Our model focus is to link it to the bacteria–surfactant interactions.

Fig. 1

Workflow of the integrated model

$$\frac{{{\text{d}}P}}{{{\text{d}}t}}={\mu _1}P - \frac{{{\mu _1}}}{{{K_0}}}{P^2} - \gamma PS$$

(14a)

$$- \frac{{{\text{d}}S}}{{{\text{d}}t}}={\mu _2}S - \delta PS$$

(14b)

$$\sigma ={\sigma _{{\text{CMC}}}}+({\sigma _0} - {\sigma _{{\text{CMC}}}})~{\text{erfc}}(KS)$$

(14c)

$$\sigma _{\text{CMC}}={c_\gamma }HL{D^2}+\sigma _{{\text{ow}}}^{{\text{min}}} $$

(14d)

Parameters ${\mu _1}$ and ${\mu _2}$ are directly estimated with REM (Eq. 11). Variable ${c_\gamma }$ is a constant and derived in the literature (Torrealba and Johns 2017). The graphical method estimates the parameters ${\sigma _{{\text{CMC}}}}$, ${\sigma _0}$ and $K$ (“Surfactant Performance Index $\psi$”), and together with $\gamma$ and $\delta$ will be tuned to achieve minimum SRSS with Nelder–Mead simplex algorithm. The SRSS is calculated from history data (time-based only) includes the IFT model because IFT can be a function of $S$ or $t$.

Surfactant performance Index $\varvec{\psi}$

Nelder–Mead simplex algorithm needs an initial guess for each parameter. The graphical approach can narrow the guess ranges for the proposed IFT-response model. A consideration of how good surfactant performance is the ability to obtain IFT as low as possible with the least CMC. Thus, we define surfactant performance index $\psi$ (mN/m) as Eq. (15). Typical surfactant by bacteria cannot reduce IFT up to smaller order like a synthetic surfactant, for example, by an anionic ethoxy carboxylate at 4 × 10⁻⁴ mN/m (Hakiki et al. 2015). Hence, a synthetic surfactant with an oil–water IFT of 10⁻¹ mN/m has a similar performance index $\psi$ than a surfactant with an oil–water IFT of 10⁻³ mN/m, in fact, their oil-mobilisation behaviour is markedly different. Thus, surfactant performance index $\psi$ in linear fashion such as Eq. (15) is suitable for biosurfactant case. Further study for more options of synthetic surfactants should extend performance index $\psi$ in logarithm trend instead of linear.

$$\psi =({\sigma _0} - {\sigma _{{\text{CMC}}}})K.$$

(15)

This index can work to compare several surfactants and estimate $K$ for an initial guess. The initial gradient of the IFT-response is calculated with the first derivative of Eq. (13) with respect to the surfactant concentration. This is $2\psi /\sqrt \pi$ (see “Appendix”). The initial gradient is, thus, equal to ${\sigma _0}/{S^*}~$ (Eq. 16) where ${S^*}$ is an extrapolated surfactant concentration (see Fig. 2a).

Fig. 2

Surfactant performance constants. a The initial gradient is $2\psi /\sqrt \pi$. The gradient can also be calculated by dividing the opposite and adjacent side, which is ${\sigma _0}/{S^*}$. Equating both gradient results in Eq. (16). b${\sigma _0}$ is estimated with the initial IFT, ${\sigma _{{\text{CMC}}}}$ is estimated by horizontal line of the lower boundary of the trend, and $K$ is estimated by with Eq. (17). c Example on using graphical method to obtain approximate value of parameters in Eq. (13). Calculate $K$ with Eq. (17) results in $K=0.56$. Hence, plotting Eq. (13) with ${\sigma _0}=15~\;{\text{mN/m}}$, ${\sigma _{{\text{CMC}}}}=6.2\;{\text{~mN/m}}$ and $K=0.56$ results in frame d. d Sensitivity of $K$ to IFT with a dummy data

$$\frac{{2\psi }}{{\sqrt \pi }} \approx 1.13\psi =\frac{{{\sigma _0}}}{{{S^*}}}$$

(16)

${S^*}$ is an extrapolated surfactant concentration. Equation (17) appears to compute surfactant performance constant $K$.

$$K=\frac{{\sqrt \pi }}{{2{S^*}}}\frac{{{\sigma _0}}}{{({\sigma _0} - {\sigma _{{\text{CMC}}}})}}~$$

(17)

Figure 2b, c depicts the procedure to determine initial guesses for IFT-response model parameters $K$, $~{\sigma _0}$, and ${\sigma _{{\text{CMC}}}}$. Figure 2d provides exemplary data from perturbed Eq. (13) to withdraw the value of the parameters. Figure 2d illustrates the sensitivity of surfactant performance constant $K$.

Discussion

This research delivers an exploration on the suitability of dynamic models. These include bacteria and the selected metabolite, i.e. biosurfactant. The paper addresses to question whether the global growth rates ${\mu _1}$ and ${\mu _2}~$ should exist to multiply for each $P$ and $S$, respectively; or there should only be ${\mu _1}$. Once we select ${\mu _1}$ only, there will be next questions whether that the biosurfactant production depends on the population ${\text{d}}S/{\text{d}}t \propto {\mu _1}P$ (see Eq. 9); or depends on the surfactant concentration ${\text{d}}S/{\text{d}}t \propto {\mu _1}S$ (see Eq. 10).

Equations (1), (8), and (9) emphasise that the growth of population and biosurfactant relies only on population regardless of the growth constant either ${\mu _1}$ or ${\mu _2}$. Reverse case applies for Eqs. (7) and (10) to describe that growth of population and biosurfactant depends on their own.

This paper introduces terms “inconsistent” and “consistent” to differentiate the quasi-dimension consistency. The term “inconsistent” is applicable for Eqs. (1) and (10); and the “consistent” is for in Eqs. (7)–(9). “Consistent” signifies that ${\mu _1}$ multiplies $P$ and ${\mu _2}$ works for $S$. The ultimate model is the Eq. (14). This equation set designates coupled-simultaneous computation and results from Eqs. (7) and (13) and a multiphase IFT model (Torrealba and Johns 2017).

BBG model

Without REM: Table 4 presents the parameter estimation results and SRSS. The proposed “consistent” models (Eqs. 7–9) have relatively similar SRSS with “inconsistent” model (Eq. 1), except for Eq. (7) with DATA 1C. Figure 3 describes the modelling results in detail.

Table 4 Parameter estimation and optimisation of BBG and IFT model

Fig. 3

BBG modelling without REM. BBG biosurfactant-producing bacteria growth, REM rate estimation model. The bacteria growth model (blue line) is overlayed with the experimental data (blue circle) and the biosurfactant concentration (red line) is overlayed with the experimental data (red square). Note that the bacteria population is in base-10 logarithmic value. The five rows represent different equations of each: 1, 7, 8, 9 and 10. The three columns use the DATA 1A, 1B, and 1C, respectively

The parameter values of Eqs. (1) and (7) are insignificantly different except ${\mu _2}$. The values of ${\mu _2}$ in Eq. (7) are higher than in Eq. (1) up to two orders of magnitude. This informs that ${\mu _2}$ salient controls $S$ than $P$.

Figure 3 illustrates that the results from Eqs. (8) and (9) are indistinguishable. The errors in Table 4 also claim that they are almost exact because the value of carrying capacity in Eq. (8) is huge. If ${K_0}$ → $\infty$, there is no interaction between bacteria ${\mu _1}/{K_0}$ → 0. The factor ${\mu _1}/{K_0}$ actually points the inter-bacteria interactions, e.g. competition in nutrient consumption. In mammalian creatures, the ${\mu _1}/{K_0}$ may mean an intercourse factor to promote the growth and a competition factor in case of demoting the growth.

The parameter values except ${\mu _2}$, SRSS and the numerical results from Eq. (10) are similar to Eq. (7). DATA 1A and 1B have a concentration “peak” around 120 h and Eq. (1) cannot capture this peak even it has the least error. Equations (7) and (10) capture the peaks with DATA 1A. Comparison of parameter values in Eqs. (7) and (10) concludes that the term ${\mu _2}P$ in Eq. (1) is negligible. Obvious control in the surfactant growth ${\text{d}}S/{\text{d}}t$ is the predation or interaction activity between $P$ and $S$.

With REM: This paper develops REM to span growth duration from initial time to maximum simulated time, 170 h. A growth model may make a dichotomy of “growth” and “death” to address the cultivating and declining population (Jones and Walker 1993). This REM works for a general change in the number of population includes the declining population. We do not segregate the growth and death REM like an earlier study on Petrotoga sp. (Daryasafar et al. 2014). That study limits the dynamics of populations because it is possible to have a second growth due to nutrient injection in the middle of decaying process. It means the growth constant developed by that literature needs the latest data to update the model.

The study uses Eq. (11) to establish the rate, named REM. Figure 4 represents the computed REM. Those REMs account the nature of time-dependent growth rate and biosurfactant production rate. REMs have a significant decrease in early 50 h. It suggests supplying more nutrient in necessity of boosting the bacteria population and the consequent biosurfactant production. The parameters and SRSS for BBG model with REM may refer to Table 4. It shows that Eq. (10) is not compatible with the REM because it provides negative predation factor $\delta$ in all DATA 1.

Fig. 4

Rate estimation model (REM). The first and second columns are for bacteria and biosurfactant, consecutively. The dotted data are calculated with the proposed REM (Eq. 11) from DATA 1, which are then captured by polynomial regression (third degree) to provide continuous time-dependent growth rate and biosurfactant production rate

Figure 5 portrays the numerical results of BBG modelling with REM. Equation (1) does not show an agreement with all data because the model reaches the concentration peak at earlier time, i.e. 30 h. Equation (10) does not either confirm a trend because it exhibits a linear to exponential alike for surfactant concentration. Figure 5 confirms that BBG model with REM is in a good agreement with Eqs. (7)–(9). The SRSS error supports this statement as depicted in Table 4.

Fig. 5

BBG modelling with REM. The REM refers to Fig. 4. The bacteria growth model (blue line) is overlayed with the experimental data (blue circle) and the biosurfactant concentration (red line) is overlayed with the experimental data (red square). Note that the bacteria population is in base-10 logarithmic value and the rate is now time-dependent following the estimated rate in Fig. 4. The five rows represent different equations of each: 1, 7, 8, 9 and 10. The three columns use the DATA 1A, 1B, and 1C, respectively

All numerical simulations using DATA 1C provide very high carrying capacity ${K_0}$. Only results from Eqs. (8) and (9) are similar for all data. This proves that no inter-bacteria relation ${\mu _1}/{K_0}$ should appeal even though the ${K_0}$ spans 13 orders of magnitude different. The least errors and the best curvature following the experimental data withdraws Eq. (7) as the best.

IFT model

Table 4 also specifies the parameters of IFT model. Figure 6 emerges the detailed results and compares between Eq. (3) and the proposed model, Eq. (13). Both models agree with the experimental data but Eq. (3) is limited to small concentration. Term $[1 - CS\exp ( - \eta S)]$→1 as $S$ → $\infty$, hence Eq. (3) will reach back initial IFT ${\sigma _0}$. Equation (13) derives more physical-sense because IFT will be at stationary for a high concentration. The erfc function gives a minute error of $\sigma ({S_{{\text{CMC}}}})$ with respect to ${\sigma _{{\text{CMC}}}}$ and cannot hit an exact equality. The magnitude of ${\sigma _0}$ is in the order of 10–100 mN/m, ${\sigma _{{\text{CMC}}}}$ is in the order of 1–10 mN/m, and $K$ is in the order of 0.1–10 L/g. The deviations of IFT at around of CMC are not observable (Fig. 6). Thus, the use of erfc is still reasonable in this range.

Fig. 6

IFT model. Both Eqs. (3) and (13) show good agreement towards DATA 2 even though Eq. (3) results in higher SRSS in most cases (see Table 4)

All data of the least IFT and the CMC are more or less same for all surfactants. Figure 7 summarises the surfactants in the sense of surfactant performance index $\psi$. The surfactant performance index (SPI) recommends the scheme applied in DATA 2A because it displays the highest $\psi$. The scheme is the 1:1 consortium of Bacillus sp. and Bacillus polymyxa with 0.01% (m/v) NPK and 20% (v/v) crude oil. This recommendation concurs with the conclusion in the literature (Bitticaca 2009).

Fig. 7

Comparison of surfactant performance index $\varvec{\psi}$. The definition of better is the higher SPI. Higher SPI contributes to obtain the greatest IFT reduction

The graphical method of $\psi$ determination is influenced by ${S^*}$ because all cases work on the same oil sample (same ${\sigma _0}$; see Eq. 16). This extrapolated surfactant concentration ${S^*}$ is an indirect indication of CMC. The least CMC must have a smaller ${S^*}$. The advantage of $\psi$ is that it does cover how much the reduction of IFT (${\sigma _0} - {\sigma _{{\text{CMC}}}}$) and how fast $\sigma$ to decay and reach CMC. The aim of surfactant selection is to obtain the least CMC and the lowest IFT. However, some surfactants may experience an IFT rise as the concentration greater than CMC (Hakiki et al. 2015).

Coupled-simultaneous model (CSM)

Coupled-simultaneous model (CSM) use Eq. (14) and REM model, Eq. (11) because we want to capture the nature of time-dependent rate. Figure 8 depicts the CSM results and all parameters: ${K_0}$, $\gamma$, $\delta$, ${\sigma _0}$, ${\sigma _{{\text{CMC}}}}$, and $K$. Figure 8a–e is in time domain. All model extends the computation up to ~ 200 h. Figure 8b shows that the highest concentration is at normalised 2 g/L, hence Fig. 8f follows this maximum value. The growth rate also notifies negative values, which means a reduction (see Fig. 8d, e).

Fig. 8

Coupled-simultaneous model (CSM). The computation uses Eq. (14) and DATA 3

The concern addresses the extended time, $t \geq$150 h because there is a possibility that biosurfactant concentration may not decrease but remains constant. The decrease of surfactant is because the computation of population undergoes a decrement. The bacteria may vanish because of nutrition deprivation and live in a dormancy stage. There is no good physical justification of why biosurfactant decrease. It is a mathematical artefact. The proper condition is that the IFT is more or less stagnant ~ 9.7 mN/m because the biosurfactant concentration is constant. The mathematical artefact is due to the way to select growth rate, i.e. Figure 8d, e. The construction of bacteria and biosurfactant rate deploys a polynomial fitting and results in negative value.

Assumptions to elaborate the decrement of surfactant is that the bacteria “consumes” the surfactant and it is toxic for them. The term of $P$ and $S$ interactions have, thus, suitable name, i.e. “toxicity constant” $\gamma$ and “predation factor” $\delta$. Some studies prove that bacteria can degrade surfactants and polymers (Scott and Jones 2000; Lucas et al. 2008). Some bacteria can use their own metabolites. These includes oxygen (Ettwig et al. 2010), methanol and acetate (Raghoebarsing et al. 2006). It requires further study whether or not bacteria consume their own metabolites, e.g. biosurfactant and understand the toxicity level to the bacteria colony or consortium. A knowledge on how the particular nutrient supply also takes a critical role in the bacteria growth (Heryani and Putra 2017). Recent study reveals that denitrification is a multistage reduction process (Song and Liu 2015). The interaction of bacteria-to-bacteria in consortium contributes an important role as well (Zheng and Li 2016; Al-Bahry et al. 2016). A further study of nutrients and metabolites consumption should be assessed together to see each contribution. An earlier study shows that organic matter follows an exponential decay degradation due to soil bacteria consumptions (Manzoni et al. 2012). An evolution in the bacteria itself is not observable in our data. A study to consider a mutation, natural assortment, or adaptive evolution may consider fractional differential equation (Ibrahim et al. 2016). However, it is unlikely to happen for MEOR bacteria. The more possible to exist is the periodicity if the nutrient and/or bacteria is reinjected. This yields a periodic pattern of solutions that may adapt an earlier study (Mohammed et al. 2017).

Figure 9 elaborates the parameter sensitivity from the fitting values to be halved and doubled. The fitting values follow the results exactly in Fig. 8. The logic in understanding the Fig. 9 results are as follow:

Fig. 9

Parameter sensitivity in coupled-simultaneous model. The sensitivity study uses Eq. (14) and the parameters include: carrying capacity ${K_0}$, toxicity constant $\gamma$ (1/h), and predation factor $\delta$ (1/h). The continuous-blue line deploys the value in Fig. 8, the halved value in dashed-green line, and the doubled value appears as dot-dashed-magenta line. There are three effects under observations: population, biosurfactant concentration, and IFT

• Higher carrying capacity ${K_0}$ indicates a greater ability to carry more population. High population leads to more competition to consume nutrition. Hence, with the current amount of nutrition, it is not sufficient to induce more biosurfactant production. Another assumptions may interpret more bacteria bring more species to consume their biosurfactant. This eventually results in less biosurfactant concentration and higher IFT.

• Greater toxicity constant $\gamma$ guides more population reduction because it amplifies poisonous effects of biosurfactant. There is no need for in-depth physical discussion of why it causes higher biosurfactant concentration. It is because $\gamma$ belongs to population growth model ${\text{d}}P/{\text{d}}t$. The increased concentration is numerical artefact.

• Stronger predation factor $\delta$ reduces biosurfactant concentration due to notable effects of biosurfactant consumption by bacteria. The $\delta$ engages surfactant growth model ${\text{d}}S/{\text{d}}t$.

Concluding remarks

This paper has modelled the biosurfactant-producing bacteria growth and the interfacial tension-response for ex-situ microbial enhanced oil recovery study. The model has been tested to indigenous Clostridium sp. from Handil Field, East Kalimantan, Indonesia. The significant observations from the modelling and implications follow:

• There are five models to elucidate the bacteria population $P$ and biosurfactant concentration $S$ dynamics. The models include variation in the growth constants ${\mu _1}$, ${\mu _2}$, $\mu (t)$, inter-bacteria relation ${\mu _1}/{K_0}$, and the interaction of bacteria–biosurfactant $\gamma$, $\delta$.

• There are two knowledges of growth to appeal: global growth constants ${\mu _1}$, ${\mu _2}$ and time-dependent constants $\mu$ = $f(t)$. The global growth constants derive a consideration to be consistent or inconsistent with respect to the quasi-dimension and physical aspects of the growth. Consistent defines the attachment of ${\mu _1}$ and ${\mu _2}$ should be each to multiply $P$ and $S$, consecutively. This studies the suitability of growth constants to $P$ or $S$. The option for time-dependent is introduced as the rate estimation model.

• Consistent models are in favourable numerical results with respect to the experimental data.

• Study on global growth constants proves: (i) ${\mu _2}$ controls $S$ more than $P$ in case of modelling ${\text{d}}S/{\text{d}}t$. (ii) Key domination in the biosurfactant production ${\text{d}}S/{\text{d}}t$ is the interaction activity between $P$ and $S$ through $\delta$.

• An integrated model couples models simultaneously: consistent bacteria–biosurfactant model and interfacial tension-response. The models account the nature of time dependent with the use of rate estimation model.

• The choice to develop rate estimation model is critical because it can contribute to numerical artefacts at the extrapolated time, $t \geq$150 h. However, the data emerge a biosurfactant concentration peak at ~ 120 h hence, there is no necessity to extent the experimental to more than 150 h. An engineering aspect suggests soaking time is around the peak production.

• The proposed interfacial tension model and the graphical method provide a surfactant performance index $\psi$ determination which benefits to compare the performance of either biosurfactant or synthetic surfactant, quantitatively. Surfactant performance index $\psi$ assists to a surfactant selection in conjunction with the least amount of critical-micellar-concentration and the lowest interfacial tension attainment.

• The composition of bacteria consortium, medium, and nutrient influences the bacteria growth, surfactant production, surfactant performance index, and the critical-micellar-concentration. These data are important to decide soaking-time or well shut-off period.

An advance experimental and simulation study can engage the proposed models to see some suggested studies:

• A study whether or not to include the bacteria and/or the surfactant transport.

• Fluid transport, porosity, and permeability distribution involvement will influence the bacteria distribution.

• The distribution takes account of the initial population and the growth.

• Porosity and permeability reduction due to bacteria entrapments.

• Study on the supplied nutrition. How it would affects the equations and the significances.

• Bacteria-to-bacteria interactions.

Appendix

The first derivative of Eq. (13) towards biosurfactant concentration is shown as follows:

$$\frac{{{\text{d}}\sigma }}{{{\text{d}}S}}=({\sigma _0} - {\sigma _{{\text{CMC}}}})\frac{{\text{d}}}{{{\text{d}}S}}[{\text{erfc}}(KS)].$$

(18)

The derivative of the error function comes up as follows:

$$\frac{{\text{d}}}{{{\text{d}}x}}{\text{erfc}}(x)= - \frac{2}{{\sqrt \pi }}\exp ( - {x^2})$$

(19)

We can obtain Eq. (20) by means of chain rule.

$$\frac{{{\text{d}}\sigma }}{{{\text{d}}S}}= - ({\sigma _0} - {\sigma _{{\text{CMC}}}})K\left\{ {\frac{2}{{\sqrt \pi }}\exp [ - {{(KS)}^2}]} \right\}$$

(20)

The first derivative evaluated at $S=0$ is

$${\left. {\frac{{{\text{d}}\sigma }}{{{\text{d}}S}}} \right|_{S=0}}= - \frac{2}{{\sqrt \pi }}({\sigma _0} - {\sigma _{{\text{CMC}}}})K$$

(21)

Figure 2a provides that an evaluation of ${\sigma _0}/{S^*}$ is the same as evaluating the initial gradient of Eq. (13). Note that the angle is in the second quadrant, while ${\sigma _0}/{S^*}$ is the same as initial gradient, so it is negative of the first derivative of Eq. (13) evaluated at $S=0$.

$$\frac{{{\sigma _0}}}{{{S^*}}}= - {\left. {\frac{{{\text{d}}\sigma }}{{{\text{d}}S}}} \right|_{S=0}}$$

(22)

Thus, from the definition of surfactant performance index (Eq. 15), Eq. (16) is proven as Eq. (23).

$$\frac{{{\sigma _0}}}{{{S^*}}}=\frac{{2\psi }}{{\sqrt \pi }}$$

(23)