Towards prediction of oil recovery by spontaneous imbibition of modified salinity brine into limestone rocks: A scaling study

Abstract

Spontaneous imbibition is a key mechanism of oil recovery in naturally fractured reservoirs. Many enhanced oil recovery techniques, such as modified salinity brine injection, have been suggested to improve spontaneous imbibition efficiency. To predict oil recovery by spontaneous imbibition process, scaling equations have been developed in the literature where almost none of them include the effect of two critical aspects. One aspect is the different ionic composition of injecting brine from connate brine. Another aspect is the effect of combination/interaction of a lower salinity imbibing (injecting) brine with connate brine. This research takes into account these two aspects to propose a new empirical scaling equation to scale oil recovery by modified salinity imbibing brines in limestone rocks. For this purpose, the results of available 59 tests from 14 references performed on various limestone rock samples collected from different formations and regions were used. The tests had been performed at high temperatures and on aged cores, which makes the proposed scaling equation more realistic and applicable to reservoir conditions. For the first time, the imbibing and connate brines ionic strengths are included in the equation due to the mechanism of the modified salinity brine injection method. In addition, the scaled spontaneous imbibition recovery data by the new equation was matched using two mathematical expressions based on the Aronofsky model and Fries and Dreyer model which can be used to derive transfer functions for simulation of spontaneous imbibition oil recovery by modified salinity brine injection in fractured limestone reservoirs.

Introduction

More than half of the world’s oil reserves are in carbonate reservoirs, where reservoir rocks are classified into three main categories: limestones, chalks and dolomites (Levorsen and Berry, 1967). Due to carbonate rocks natural brittleness, most of them host natural fractures with varying degrees of fracture properties (Saidi, 1987). A fractured system comprises two distinct media: fracture network and matrix blocks that are responsible for conductance and storage of the fluids in the system, respectively (Behbahani et al. 2006; Gilman and Kazemi, 1983; Warren and Root, 1963).

At the early stages of oil production, fractures are depleted and water can invade the oil zone through the fractures. Consequently, a considerable portion of the oil zone turns into a water invaded zone (van Golf-Racht, 1982). In this situation, water spontaneous imbibition into the matrix block will become the primary mechanism of oil recovery. Each block spontaneous imbibition rate and ultimate recovery inside the water invaded zone depend on many variables such as rock, oil, connate and imbibing brines and their common properties. Moreover, size and shape of matrix block and system’s boundary condition are other important factors controlling oil recovery by spontaneous imbibition (Hatiboglu and Babadagli, 2004; Hatiboglu and Babadagli, 2006; Kazemi et al. 1992).

Based on the boundary condition, there are two kinds of spontaneous imbibition: co-current (COCSI) and counter-current (COUCSI). In the COCSI, the displacing and displaced phases flow directions are the same, whereas, in the COUCSI, they have opposite directions (Bourbiaux and Kalaydjian, 1990; Mirzaei-Paiaman et al. 2017; Mirzaei‐Paiaman and Masihi, 2014; Pooladi-Darvish and Firoozabadi, 2000a; Pooladi-Darvish and Firoozabadi, 2000b). Most of the spontaneous imbibition tests in the laboratory have been performed on cylindrical cores saturated with a non-wetting phase, i.e., oil, and in contact with a wetting phase, i.e., pure water/brine, in one or more than one outer surfaces (Anderson, 1986; Fischer and Morrow, 2006; Fischer et al. 2008). For the representation of the COCSI and COUCSI, usually four typical boundary conditions have been applied which are named as all faces open (AFO), one end open (OEO), two ends open (TEO) and two ends closed (TEC). In AFO in small cores, COUCSI with a combination of linear and radial imbibition occurs, while OEO, TEO and TEC cases can represent linear COUCSI, linear COCSI and radial COUCSI, respectively (Fischer and Morrow, 2006; Fischer et al., 2008; Hatiboglu and Babadagli, 2004; Hatiboglu and Babadagli, 2006). This research will evaluate spontaneous imbibition tests with AFO boundary conditions (considering COUCSI only) representing a matrix block fully immersed in water.

To predict spontaneous imbibition oil recovery in an oil/brine/rock system, scaling equations in the form of dimensionless time have been proposed in the literature. These equations are developed based on the effective parameters that determine the oil recovery by spontaneous imbibition. The main purpose of scaling of scattered tests data is finding a universal curve in a semi-log plot of recoverable oil recovery versus a dimensionless time. The recoverable oil is defined as the amount of oil produced as a fraction of a reference quantity. The reference quantity could be the final oil production (${\text{R}}_{\infty }$), pore volume, or initial oil in place (IOIP). In this work, the reference quantity for expressing the recovery data is the final oil production.

Mattax and Kyte (Mattax and Kyte, 1962) used Rapoport’s equation (Rapoport, 1955) to scale four tests on cylindrical alundum (aluminum oxide) cores with OEO boundary condition and two tests on cylindrical sandstone cores with AFO boundary condition, separately, and offered the following equation:

$$t_{D - MK} = t\sqrt {\frac{k}{\varphi }} \frac{\sigma }{{L^{2} }}\frac{1}{{\mu_{w} }}$$

(1)

where MK denotes Mattax and Kyte and t, k, φ, σ, L and µ_w are time, rock permeability, rock porosity, interfacial tension (IFT) between crude oil and imbibing brine, core length and the imbibing brine viscosity, respectively.

Ma et al. (Ma et al. 1997) improved Mattax and Kyte’s scaling equation by two modifications. They believed that two reasons for failing simultaneous scaling of Mattax and Kyte’s tests on alundum and sandstone cores were their difference in the viscosity ratios and various boundary conditions. The modified scaling equation by Ma et al. is as follows:

$$t_{D - MMZ} = t\sqrt {\frac{k}{\varphi }} \frac{\sigma }{{L_{c}^{2} }}\frac{1}{{\sqrt {\mu_{w} \mu_{o} } }}$$

(2)

where MMZ denotes Ma, Morrow and Zhang. In addition to Mattax and Kyte’s tests, Ma et al.’s scaling equation could scale Hamon and Vidal (Hamon and Vidal, 1986) and Zhang et al. (Zhang et al. 1996) tests. In Eq. (2), the length of the core and the imbibing brine viscosity in Mattax and Kyte’s equation are replaced by characteristic length (${\text{L}}_{{\text{C}}}$) and the geometric mean of imbibing brine viscosity and crude oil viscosity (${\upmu }_{{\text{o}}}$), respectively. The characteristic length includes the effect of different boundary conditions and is defined by Ma et al. as:

$$L_{c} = \sqrt {\frac{{V_{b} }}{{\mathop \sum \nolimits_{i = 1}^{n} \frac{{A_{i} }}{{l_{{A_{i} }} }}}}}$$

(3)

where ${\text{V}}_{\text{b}}$ is the matrix block bulk volume, ${\text{A}}_{\text{i}}$ is the i-th surface area that is open to flow and ${\text{l}}_{{\text{A}}_{\text{i}}}$ is the distance from the i-th imbibition face to the corresponding no-flow boundary. For a cylindrical core with AFO boundary condition, the characteristic length is:

$$L_{c} = \frac{LD}{{2\sqrt {D^{2} + 2L^{2} } }}$$

(4)

where D is the core diameter. In this case, the no-flow boundary will be at the center of the matrix block.

Mason et al. (Mason et al. 2010) proposed a more comprehensive scaling equation version to scale tests with a wide range of viscosity ratios (0.008 < $\frac{{{\upmu }_{{\text{o}}} }}{{{\upmu }_{{\text{w}}} }}$<173) such as Fischer and Morrow (Fischer and Morrow, 2006) and Fischer et al. (Fischer et al. 2008) experiments. Mason et al.’s scaling equation is:

$$t_{D - MFMR} = t\sqrt {\frac{k}{\varphi }} \frac{\sigma }{{L_{c}^{2} }}\frac{2}{{\mu_{w} + \sqrt {\mu_{w} \mu_{o} } }}$$

(5)

where MFMR denotes Mason, Fischer, Morrow and Ruth. Table 1 summarizes all important parameters of crude oil/brine/rock systems of the tests scaled by each scaling equation.

Table 1 Tests conditions scaled by various scaling equations

There are other scaling equations proposed in the literature that cannot be used in this evaluation; because some of them were defined for the COCSI conditions that are not the focus of this paper (Cai et al. 2012, 2010; Mirzaei-Paiaman et al. 2017; Mirzaei‐Paiaman and Masihi, 2014), and some others needed special core analysis data (e.g., capillary pressure and relative permeability data) that is not available for our collected tests (Behbahani and Blunt, 2005; Li and Horne, 2005, 2006; Mirzaei-Paiaman et al. 2017; Schmid and Geiger, 2012, 2013; Standnes and Andersen, 2017; Zhou et al. 2002).

Due to the difference in the surface charge of different lithologies, carbonate rocks usually tend to be oil-wet/mixed-wet whereas sandstone rocks remain water-wet in reservoir conditions. This discrepancy is originated from the difference in their points of zero charge (PZC) (Jaafar et al. 2014). The PZC is a pH level where the rock surface charge is zero. At higher pH levels, the rock surface charge tends to be negative and at the lower pH values, it is positively charged. The PZC of sandstone rocks lies between 2.2 and 3.3 while in carbonate rocks, this value ranges from 8 to 9.6 (Farooq et al. 2011; Jaafar et al. 2014).

The presence of acidic components in the crude oil makes the pH level of oil reservoirs usually in the acidic range (i.e., lower than 7). Consequently, the rock surface charge in sandstone reservoirs typically becomes negative and in carbonate reservoirs is positive (Madsen and Ida, 1998; Mwangi et al. 2018; Skauge et al. 1999). During the aging of a limestone rock sample in the lab, adsorption of organic acids (especially carboxylic acids) from their negatively charged head (RCOO⁻) on the positively charged calcite sites of the rock surface becomes the major mechanism of wettability alteration of the limestones from water-wet to oil-wet/mixed-wet state. (Fathi et al. 2011a, 2010b).

To resolve the unfavorable reservoir rock wettability, many enhanced oil recovery (EOR) techniques have been proposed for the wettability alteration such as using surfactants (Alagic and Skauge, 2010; Bassir and Shadizadeh, 2020; Das et al. 2021; Hou et al. 2021; Johannessen and Spildo, 2014; Khayati et al. 2020; Sekerbayeva et al. 2020; Shi et al. 2021), nanoparticles (Abhishek et al. 2018a, 2018b; Hamouda and Abhishek, 2019; Olayiwola and Dejam, 2020; Shirazi et al. 2019), ionic liquids (Bera et al. 2020; Nandwani et al. 2019; Pillai and Mandal, 2019; Velusamy et al. 2017) and modified salinity brine injection (Hao et al. 2019).

Modified salinity brine injection, also known as smart water, ion-controlled water, ion-engineered water or engineered water injection, is a relatively inexpensive method that has attracted more attention in the last decade. In this technique, a brine with lower salinity than connate brine, such as seawater, with/without manipulation of some ions concentrations is injected into the reservoir (Zhang and Austad, 2006; Zhang et al. 2006). In this study, our focus is on the spontaneous imbibition tests in the literature that have used modified salinity brine and are performed on limestone rock samples.

The primary mechanism of improving oil recovery by the modified salinity brine is wettability alteration by multi-ion exchange with the help of electrical double layer expansion (Ligthelm et al. 2009). Two layers exist around any charged particle; Stern and diffuse layers filled with a high- and low-density concentration of ions, respectively (Shirazi et al. 2020). The electrical double layer of rock pores surface will be expanded and the multi-ion exchange will be facilitated due to reduced imbibing brine salinity. Multivalent ions such as calcium (${\text{Ca}}^{2+}$), magnesium (${\text{Mg}}^{2+}$) and sulfate (${\text{SO}}_{4}^{2-}$) ions are considered as potential determining ions (PDIs) in the modified salinity brine research area. These ions are the main contributors to multi-ion exchange in such a way that ${\text{Ca}}^{2+}$ and ${\text{Mg}}^{2+}$ have operative roles while ${\text{SO}}_{4}^{2-}$ is a catalyst (Zhang and Austad, 2006).

${\text{Ca}}^{2+}$ and ${\text{Mg}}^{2+}$ can adsorb negatively charged carboxylic acids and detach them from the rock surface while a positive rock surface can behave as a barrier to this desired process. Mixing a lower salinity injected brine with high salinity connate brine will expand the electrical double layer and prepare a suitable condition for the detachment of the acidic components of crude oil from rock surface and thus wettability alteration. Meanwhile ${\text{SO}}_{{4}}^{{2 - }}$ can lower the positive charge of the rock surface and facilitate the wettability alteration process by ${\text{Ca}}^{{2 + }}$ and ${\text{Mg}}^{{2 + }}$. These chemical reactions can be explained by (Purswani and Karpyn, 2019):

$$RCOO^{ - } - CaCO_{3} (s) + Ca^{2 + } (aq) + SO_{4}^{2 - } (aq) \leftrightarrow \left( {RCOO - Ca} \right)^{ + } (aq) + SO_{4}^{2 - } - CaCO_{3} (s)$$

(6)

$$RCOO^{ - } - CaCO_{3} (s) + Mg^{2 + } (aq) + SO_{4}^{2 - } (aq) \leftrightarrow \left( {RCOO - Mg} \right)^{ + } (aq) + SO_{4}^{2 - } - CaCO_{3} (s)$$

(7)

Equations (6) and (7) are the sum products of three intermediate reactions illustrated for ${\text{Ca}}^{{2 + }}$ in Eqs. (8) to (10). For ${\text{Mg}}^{{2 + }}$, similar reactions can be considered (Purswani and Karpyn, 2019).

$$RCOO^{ - } - CaCO_{3} \left( s \right) + SO_{4}^{2 - } \left( {aq} \right) \leftrightarrow RCOO^{ - } - - - CaCO_{3} \left( s \right) - - - SO_{4}^{2 - }$$

(8)

$$RCOO^{ - } - - - CaCO_{3} \left( s \right) - - - SO_{4}^{2 - } + Ca^{2 + } \left( {aq} \right) \leftrightarrow Ca^{2 + } - - - RCOO^{ - } - - - CaCO_{3} \left( s \right) - - - SO_{4}^{2 - }$$

(9)

$$Ca^{2 + } - - - RCOO^{ - } - - - CaCO_{3} \left( s \right) - - - SO_{4}^{2 - } \leftrightarrow RCOO - Ca^{2 + } \left( {aq} \right) + SO_{4}^{2 - } - CaCO_{3} \left( s \right)$$

(10)

In addition to the imbibing brine salinity effect, some studies have concentrated on the connate brine salinity effect on oil recovery. The mechanisms of the connate brine impact on oil recovery are similar to those of the imbibing brine, i.e., lowering the salinity, as well as the participation of PDIs in the connate brine had a positive effect on the oil recovery in a modified salinity brine scheme (Mohammadkhani et al. 2018; Sharma and Filoco, 2000; Shehata and Nasr-El-Din, 2017).

There are other proposed mechanisms for improving the oil recovery due to the modified salinity brine technique such as mineral dissolution and formation of microdispersions but none of them is the main mechanism of recovery improvement by spontaneous imbibition of modified salinity brine in carbonates (Mahani et al. 2015; Mahzari et al. 2019). Therefore, the multi-ion exchange is considered as the principal mechanism of enhancement of the oil recovery during modified brine salinity imbibition into the limestones.

Most of the scaling equations presented in the literature are for a water-wet condition where the wettability is almost unchanged during the test. During the injection of modified salinity brine, there is a wettability alteration. In other words, wettability is not water-wet and is changing over time of the process. All the gathered tests in this work are long enough until no oil is produced from the core sample to demonstrate a perfect wettability alteration process. This research aims to find a scaling equation that can scale imbibition recovery during such a complex condition.

In this paper, 59 tests of modified salinity brine performed on the cylindrical limestone cores available in the literature were gathered and used. 47 tests (about 80% of the collected tests) were used to find a new scaling equation, and 12 tests were used to validate the scaling equation. Finally, two approaches based on the Aronofsky model and Fries and Dreyer model were utilized to fit the scaled data with a mathematical relationship. As part of this study, we will show that current scaling equations cannot scale the spontaneous imbibition oil recovery by modified salinity brine.

Data acquisition and methodology

Spontaneous imbibition tests data acquisition

For this study, 59 tests from 14 references published between 2010 and 2020 were gathered (Al-Harrasi et al. 2012; Chandrasekhar and Mohanty, 2013; Feldmann et al. 2020; Nasralla et al. 2018; Rashid et al. 2015; Ravari, 2011; Romanuka et al. 2012; Roostaei, 2014; Shariatpanahi et al. 2010; Shirazi et al. 2020; Song et al. 2020; Torrijos, 2017; Zaeri et al. 2018; Zaeri et al. 2019). All the tests were spontaneous imbibition of a modified salinity brine into a limestone rock sample hosting a higher salinity connate brine.

Some required parameters of the scaling equations were not reported in the references. Some non-reported parameters such as crude oil and brine viscosities and the crude oil-imbibing brine IFT were estimated using generally accepted correlations which were developed based on experimental data (Abooali et al. 2019; Glaso, 1980; Numbere et al. 1977) published in the literature with acceptable minor errors (correlations are explained in Appendix 1). However, the other non-reported data such as core dimensions and crude oil total acid number (TAN) cannot be estimated. This omitted data was obtained through communication with the authors of the papers. In Table 2, the parameters of the gathered tests applied in our new scaling equation are listed.

Table 2 Properties of the collected tests

In Table 3, the minimum, arithmetic mean and maximum of the other important parameters of the collected tests are tabulated (Details are given in Appendix 2). It should be reminded that all the tests were performed at the ambient pressure and the crude oils used in the tests were “dead oil”. For a high block with altered wettability in a fractured reservoir that is immersed in water, in addition to capillary force, gravity force can also affect oil production. In our study, we concentrate on oil recovery by spontaneous imbibition due to capillary force changes as a result of wettability alteration toward a more water-wet state made by injected modified salinity brine and ignore the oil recovery due to gravity. The experimental data used in the study is in line with this purpose as the core samples' heights (2.54 to 8.30 cm, as shown in Table 2) are small enough to neglect the effect of gravitational forces.

Table 3 Range of properties of the collected tests

Methodology

A stepwise methodology of this research is as follows:

• 1. At first, the scaling capability of the three aforementioned scaling equations (Mattax and Kyte, Ma et al. and Mason et al.) is evaluated for 59 collected tests. The arithmetic mean of data scatter is utilized for the comparison.

• 2. The most accurate scaling equation with the lowest arithmetic mean of data scatter (Mason et al.) is selected for the development of the new scaling equation.

• 3. All 59 gathered tests are categorized into two datasets: training (47 tests) and testing (12 tests) with about 80% and 20% of all the tests, respectively. The training dataset is used for the development of the new scaling equation and the testing dataset is applied to evaluate the scaling ability of the new scaling equation for unseen data.

• 4. After a detailed discussion about the mechanisms behind the oil recovery improvement by the modified salinity in limestone rocks, an empirical term comprising the ionic strengths of both imbibing and connate brines is added to Mason et al.’s scaling equation.

• 5. The empirical exponent of the added term is determined using a trial-and-error method. The arithmetic mean of data scatter is used as criteria for the selection of the best answer for the empirical exponent that has the lowest value of arithmetic mean of data scatter.

• 6. The testing dataset is properly scaled by the resulted new scaling equation.

• 7. Two mathematical models (Aronofsky model and Fries and Dreyer model) are fitted to the scaled tests by the new scaling equation. The root of mean square error (RMSE) is used for determining the matching parameters of these two models.

The details of the aforementioned steps are described in the next section.

Results and discussion

Checking current scaling equations

The collected tests were classified into two categories: training and testing datasets with 47 and 12 tests, each accounting for about 80% and 20% of the collected tests, respectively. The training dataset was used for finding a new scaling equation, whereas the testing dataset was used for the validation of the new equation. Using the random function of Microsoft Excel, tests with the IDs 4, 7, 17, 20, 23, 32, 36, 38, 40, 53, 56 and 57 were selected as testing data and the rest of the tests were allocated to the training set. Three scaling equations (Mattax and Kyte’s, Ma et al.’s and Mason et al.’s) were used to scale oil recovery data.

As shown in Table B-1, each test has different ${\text{R}}_{\infty }$ and t_end. Hence, the data of each experiment is normalized by ${\text{R}}_{\infty }$ and the normalized data of each test lies between zero and one. The results of scaling are plotted on a semi-log plot where the vertical axis is oil recovery as the fraction of ${\text{R}}_{\infty }$ and the horizontal axis is dimensionless time. For a fair comparison, non-scaled and scaled oil recovery curves are drawn on a similar log scale from 0.001 to 1000. The non-scaled oil recovery and scaled data by Mattax and Kyte’s, Ma et al.’s and Mason et al.’s scaling equations for all 59 tests are compared in Fig. 1.

Fig. 1

a Non-scaled oil recovery and scaled data by b Mattax and Kyte’s, c Ma et al.’s and d Mason et al.’s scaling equations for all 59 tests

As shown in Fig. 1, the scaling of tests data by all three scaling equations is insufficient because they could not collapse data points on a universal curve. As the data points become more collapsed, the scaling equation has more accuracy. Although Ma et al.’s and Mason et al.’s scaling equations show a little better result than Mattax and Kyte’s equation, the scattered scaling results need to be analyzed to find the discrepancy’s possible reason(s). Due to differences in the tests conditions and the data used to develop former scaling equations, five reasons could be considered for the scattered results:

• 1. The first difference may be high temperature of the gathered tests. However, test temperature is implicitly considered in the scaling equations because crude oil and brine viscosities and oil–water IFT are all functions of temperature (as discussed in Appendix 1). Therefore, it does not seem that the high temperature of the tests is responsible for their weak scaling.

• 2. Another affecting parameter could be the wetting state of the rock samples. In the collected tests, each core is initially aged at a high temperature. Zhang and Austad (2005) stated that rather than the aging temperature, the TAN of the crude oil used during aging has a crucial role in carbonate rock wettability alteration by the aging process (Zhang and Austad, 2005). The oil TAN affects oil–water IFT because TAN is used to calculate IFT by Abooali et al.’s correlation (Appendix 1). Therefore, we can assume the aging process effect is also implicitly included in the scaling equations.

• 3. It seems that the main reason for the poor scaling ability of the scaling equations could be ignoring the variation of the injected and connate brines’ ionic composition.

• 4. Moreover, there is no parameter in the current scaling equations related to the connate brine whereas connate brine properties are effective in oil recovery by the modified salinity brine technique.

• 5. Furthermore, the rock samples of all the collected tests are limestone cores. It seems porosity and absolute permeability are not adequate parameters to represent a complex carbonate rock and other properties such as tortuosity, mineral composition, pore/grain size distribution and grain sorting and packing may be needed to include in the scaling equation.

The three latest reasons will be considered in our investigations of the new scaling equation. To better compare the scaling quality of different scaling equations, the arithmetic mean of data scatter is employed. For calculation of this parameter, the data scatter at distinct y-axis values (i.e., $\frac{R}{{R_{\infty } }} = 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9\; {\text{and}}\; 1$) is determined (in log cycles) and then arithmetically averaged. The calculated arithmetic mean of data scatter for Mattax and Kyte’s, Ma et al.’s and Mason et al.’s scaling equations are 1.874, 1.795 and 1.781 log cycles, respectively. Therefore, we selected Mason et al.’s scaling equation for our modification study because of having the lowest value of the arithmetic mean of data scatter compared to the other scaling equations.

New scaling equation

As explained in the introduction section, in addition to the lower salinity of imbibing brine, PDIs also play an influential role in improving oil recovery by the modified salinity brine method. On the other hand, the imbibing brine viscosity which already appears in the scaling equations represents the brine’s total salinity instead of the brine’s ionic composition (see Appendix 1) and does not include the effect of individual ions. To include the effect of ionic composition in the scaling equation, we look for a parameter that can make a difference between different (monovalent and multivalent) ions i.e., PDIs of the brines. A property that could cover this circumstance is ionic strength (I). This brine parameter is introduced by Lewis and Randall (1921) as:

$${\text{I = }}\frac{{1}}{{2}}\mathop \sum \limits_{{\text{i = 1}}}^{{\text{n}}} {\text{c}}_{{\text{i}}} {\text{z}}_{{\text{i}}}^{{2}}$$

(11)

where c and z are molar concentration (in moles per liter (mol/L)) and charge number of each ion, respectively (Hückel and Debye, 1923). Based on Eq. (11), due to the exponent of the charge number, the effect of the divalent ions is four times that of monovalent ions. The ionic strength has the advantage of being independent of the temperature. To include the impact of imbibing and connate brines’ PDIs, we used both of their ionic strengths to add a parameter to Mason et al.’s scaling equation. The simplest dimensionless form of imbibing and connate brines ionic strengths is their ratio. The ionic strength ratio becomes unity when the imbibing and connate brines have identical ionic compositions.

A challenge that may be encountered is the selection of either imbibing to connate or connate to imbibing brines ionic strengths ratio. To this end, an investigation should be carried out on the impact of these ratios on spontaneous imbibition oil recovery. The effects of variation of brines ionic compositions on the imbibing to connate brines ionic strengths ratio and oil recovery are discussed based on the implications explained in the introduction at macroscale and microscale:

• In large variations of ionic composition of imbibing and connate brines, lower salinity of each brine (decrease in the ionic strength) yields an enhancement in oil recovery. Decreasing imbibing or connate brines ionic strengths can decrease or increase the imbibing to connate brines ionic strengths ratio, respectively.

• In small variations of ionic composition of imbibing and connate brines (approximately constant salinity) an increase in the PDIs concentration (increase in the ionic strength) of each brine results in higher oil recovery. Increasing imbibing or connate brines ionic strengths can increase or decrease the imbibing to connate brines ionic strengths ratio, respectively.

All in all, the imbibing to connate brines ionic strengths ratio and the oil recovery do not have a specified direct or inverse relationship. For the clarification of this complicated issue, an empirical exponent is assigned for this dimensionless ratio. Because the collected tests were performed on limestones, this exponent may be assumed to be specific to limestone lithology. To this end, a sufficient range of negative to positive values will be evaluated for collapsing the gathered data as possible. The form of our new scaling equation is as follows:

$$t_{D - New} = t\sqrt {\frac{k}{\varphi }} \frac{\sigma }{{L_{c}^{2} }}\frac{2}{{\mu_{w} + \sqrt {\mu_{w} \mu_{o} } }}\left(\frac{{I_{w} }}{{I_{wi} }}\right)^{N}$$

(12)

where N is the proposed empirical exponent.

The ionic strength assigns an equal weight for the ions with the same charge number (e.g., Ca²⁺, Mg²⁺ and SO₄²⁻). After an investigation of the literature (especially references used for our data collection), it can be concluded that these three determinant ions (PDIs) have an approximately similar weight in the improvement of the oil recovery due to wettability alteration (Chandrasekhar and Mohanty, 2013; Rashid et al. 2015; Shirazi et al. 2020; Zaeri et al. 2019).

We used a trial-and-error method to scale the training dataset using the proposed scaling equation to find a proper empirical exponent for the new term. Arbitrary values between -2 to 2 are tested and this interval is narrowed step by step until the answer for the exponent is gained. As shown in Fig. 2, an empirical exponent of 0.38 with an arithmetic mean of data scatter of 1.279 log cycles, 28.2% lower than that of Mason et al.’s scaling Eq. (1.781 log cycles), is achieved. This positive value (0.38) shows a direct relationship between the imbibing to connate brines ionic strengths ratio and spontaneous imbibition oil recovery in limestones. Due to the increasing trend of the arithmetic mean of data scatter for the exponent values above one and below zero, the achieved power seems unique. Our newly proposed scaling equation is as follows:

$$t_{D - New} = t\sqrt {\frac{k}{\varphi }} \frac{\sigma }{{L_{c}^{2} }}\frac{2}{{\mu_{w} + \sqrt {\mu_{w} \mu_{o} } }}\left(\frac{{I_{w} }}{{I_{wi} }}\right)^{0.38}$$

(13)

Fig. 2

Gaining the answer for the empirical exponent

Validity test of new scaling equation

In this section, the testing dataset (12 tests) is applied for checking the validity of our new scaling equation for unseen data. Our new scaling equation can scale these new data, too. The scaled oil recovery of all 59 tests data is shown in Fig. 3.

Fig. 3

Verification of the new scaling equation for testing dataset

Matching data

Two approaches were employed for matching the scaled data with a mathematical relationship. The first method is an exponential recovery with time similar to the Aronofsky model (Aronofsky et al. 1958):

$$\frac{R}{{R_{\infty } }} = 1 - e^{{ - \lambda t_{D} }}$$

(14)

where λ is the oil production decline constant used as the tuning parameter (Behbahani and Blunt, 2005; Ma et al. 1997). The second model was proposed by Fries and Dreyer (2008) with the aid of the Lambert W function (Mirzaei-Paiaman, 2015; Mirzaei-Paiaman et al. 2011; Standnes, 2010):

$$\frac{R}{{R_{\infty } }} = 1 + W( - e^{{ - 1{ - }at_{D} }} )$$

(15)

where a is a matching parameter. The following equation can calculate the Lambert W function for ${\text{ - e}}^{{ - 1}} { } \le {\text{ x }} \le { 0}$ with a maximum relative error of 0.1% (Fries and Dreyer, 2008):

$$W\left( x \right) \approx - 1 + \frac{{\sqrt {2 + 2ex} }}{{1 + \frac{{4.13501\sqrt {2 + 2ex} }}{{12.7036 + \sqrt {2 + 2ex} }}}}$$

(16)

where e is the Euler’s number (2.71828…). A trial-and-error method with the help of the RMSE was applied for matching these two models to our scaled data. Stages of gaining the tuning parameters are similar to those of Sect. 3.2 with a difference that here, achievement of the lowest RMSE is favored. RMSE can be computed by:

$${\text{RMSE}} = \sqrt {\frac{{\mathop \sum \nolimits_{i = 1}^{n} (x_{{{\text{predicted}}_{i} }} - x_{{{\text{actual}}_{i} }} )^{2} }}{n}}$$

(17)

where $x_{{{\text{predicted}}_{i} }} {\text{ and }}x_{{{\text{actual}}_{i} }}$ are the predicted values by model and actual value for each data point, respectively. The best matches to all 59 scaled tests data using t_D-New from Eq. (13) into Eqs. (14) and (15) are as follows:

$$\frac{R}{{R_{\infty } }} = 1 - e^{{ - 0.24t_{D - New} }}$$

(18)

$$\frac{R}{{R_{\infty } }} = 1 + W( - e^{{ - 1{ - }0.083t_{D - New} }} )$$

(19)

The calculated RMSE values for the Aronofsky model and Fries and Dreyer model are 0.122 and 0.133, respectively. Therefore, the Aronofsky model has higher precision than Fries and Dreyer model for predicting spontaneous imbibition oil recovery by modified salinity brine in limestone rocks, as shown in Fig. 4. Equation (18) could be used to derive a transfer function for field-scale dual-porosity simulation of the modified salinity brine flow in fractured limestone reservoirs (Mirzaei-Paiaman et al. 2011; Tavassoli et al. 2005a, b).

Fig. 4

Matched models to the scaled data

Recommendations for future research

This study reveals that there is a long road to improve scaling equations to be applied at different reservoir conditions, i.e., for each lithology and various EOR schemes. Many tests, especially with AFO boundary conditions, have been published in the literature. Several spontaneous imbibition experiments were published for water-based EOR methods, such as surfactants (Hou et al. 2015; Santanna et al. 2014; Standnes, 2004; Standnes and Austad, 2000; Standnes et al. 2002) and hybrid techniques such as modified salinity brine and surfactants (Das et al. 2021; Eslahati et al. 2020; Khaleel et al. 2019; Mohammadi et al. 2019; Shi et al. 2021; Udoh and Vinogradov, 2019), modified salinity brine and nanoparticles (Nowrouzi et al. 2019a; Shirazi et al. 2019) and carbonated modified salinity brine (Amarasinghe et al. 2019; Ghandi et al. 2019; Nowrouzi et al. 2019b, 2020a, b). However, few scaling equations have been released to scale these abundant tests (Keykhosravi and Simjoo, 2020).

This research tried to scale experiments data of limestone rocks for the modified salinity brine method. We recommend future works to find new scaling equations to scale the tests of the modified salinity brine into chalk (Fathi et al. 2010a, 2011a, b; Fathi et al. 2010b; Shariatpanahi et al. 2010, 2011; Zhang and Austad, 2006; Zhang et al. 2006, 2007), dolomite (Eslahati et al. 2020; Torrijos, 2017) and sandstone rocks (Al-Saedi et al. 2020; Al-Saedi et al. 2019; Suijkerbuijk et al. 2012; Wickramathilaka et al. 2011). Our proposed scaling equation can be used as a base equation to suggest more scaling equations for scaling the spontaneous imbibition of modified salinity brine into lithologies other than limestones. Moreover, tests with boundary conditions other than the AFO for any water-based EOR technique are needed. However, tests with the AFO boundary condition for scaling are still required for hybrid EOR methods such as modified salinity brine and surfactants, modified salinity brine and nanoparticles, carbonated modified salinity brine, etc. This long road’s ultimate goal may be to reach a comprehensive scaling equation with a specific term defined for each EOR technique and lithology.

Conclusions

The following concluding remarks can be drawn from this paper:

• This work proposed an empirical scaling equation to scale spontaneous imbibition tests (collected from the literature) of the modified salinity brine into limestone rock samples from different reservoirs and regions.

• The suggested equation included the ionic strengths of both the imbibing and connate brines. As far as we are concerned, this is the first time that a property of connate brine is included in a scaling equation. The scaled tests by the new scaling equation were performed on aged cores and at high temperatures (near reservoir conditions).

• The proposed scaling equation collapses the scaled data by 28.2% more than the Mason et al.’s equation that had exhibited the best performance in comparison to other former scaling equations.

• In limestones, the oil recovery of spontaneous imbibition of modified brine salinity has a direct relationship with the ratio of imbibing to connate brines ionic strengths.

• The scaled data was matched by two single-parameter correlations (Aronofsky model and Fries and Dreyer model) that can be used to derive proper transfer functions for fractured reservoir simulation.

Appendices

Appendix 1

Some of the necessary not-reported data in the experiments references, such as crude oil and brine viscosities and crude oil-imbibing brine IFT, were estimated with the appropriate correlations suitable for their conditions. To calculate crude oil viscosity, Glaso’s correlation was used (Glaso, 1980):

$$\mu_{od} = \left[ {3.141\left( {10^{10} } \right)} \right]T^{ - 3.444} [\log (^\circ API)]^{{10.313\left[ {\log \left( T \right)} \right] - 36.447}}$$

(20)

where temperature (T) and dead oil viscosity (µ_od) are in degrees Fahrenheit (℉) and centipoises (cp), respectively (Glaso, 1980). Numbere et al.’s correlation is used to estimate brines’ viscosities (Numbere et al. 1977):

$$\frac{{\mu_{w} }}{{\mu_{pw} }} - 1 = - 1.87\left( {10^{ - 3} } \right)C_{S}^{0.5} + 2.18\left( {10^{ - 4} } \right)C_{S}^{2.5} + (T^{0.5} - 1.35\left( {10^{ - 2} } \right)T)(2.76\left( {10^{ - 3} } \right)C_{S} - 3.44(10^{ - 4} )C_{S}^{1.5}$$

(21)

where pure water viscosity (µ_pw) is in centipoises, brine salt concentration (${\text{C}}_{{\text{S}}}$) is in weight percent (wt%) and the temperature (T) is in degrees Fahrenheit (℉). The µ_pw at any temperature and pressure (P) can be calculated by:

$$\frac{{\mu_{pw} (T,P)}}{{\mu_{pw} (T,P_{s} )}} = 1 + \frac{{P{ - }P_{s} }}{{10^{6} }}(1.046T{ - }319.24)$$

(22)

where ${\text{P}}_{{\text{s}}}$ is saturation pressure in pounds per square inch (psia). Also, µ_pw at the saturation pressure can be computed by:

$$\mu_{pw} \left( {T,P_{s} } \right) = 241.4*10^{{\left[ {\frac{247.8}{{T - 140}}} \right]}}$$

(23)

And finally, the saturation pressure can be estimated by (Numbere et al. 1977):

$$P_{s} = 0.468\left(\frac{T}{100}\right)^{4.52}$$

(24)

For the prediction of IFT between the crude oil and brine, Abooali et al.’s correlation obtained by the genetic programming approach is applied (Abooali et al. 2019):

$$\begin{aligned} \sigma = 32.11T - 135.9\gamma _{o} - 3.26\tanh \left( {T.P.TAN} \right) \\ & - 259.2\gamma _{o}^{2} .T.\tanh \left( {\gamma _{o} } \right) - 135.9\tanh \left( {\gamma _{o}^{2} } \right) \\ + T.\gamma _{o} \left( {339.1\gamma _{o} - 173.8} \right) - 9.526\tanh \left( {T.\gamma _{o} .TAN^{2} } \right) \\ + 37.86\tanh \left( {\gamma _{o} + C_{S} } \right) + 0.00001111T.P \\ & + \tanh \left( {\tanh \left( {C_{S} } \right)} \right)\left( {14.52\tanh \left( P \right) - 30.05} \right) \\ - 0.1\left( {1 + 0.1TAN.\gamma _{o} .pH^{{0.5}} } \right)\left( {pH - 7} \right)^{2} \\ - 0.004048TAN\left( {1 - TAN} \right)T + 229.73 \\ \end{aligned}$$

(25)

where σ, T, P, C_S and TAN are in mN/m, Kelvin (K), atmosphere (atm), ppm and milligrams of potassium hydroxide (KOH) per gram of crude oil (mgKOH/g) and ${\upgamma }_{{\text{o}}}$ is crude oil specific gravity (Abooali et al. 2019).

Appendix 2

The other important parameters of the collected tests that were not used directly in the new scaling equation are shown in Table 4.

Table 4 Properties of the tests

It should be mentioned that in the references in which the TAN value is not reported, IFT is reported instead.

The imbibing and connate brines ionic compositions of all the collected tests are provided in Table 5.

Table 5 Ionic compositions of used brines in the tests

These five mentioned ions comprise the majority of salts dissolved in the brines and the total concentration of the other ions (such as Barium, Strontium, Potassium, Bicarbonate, etc.) is less than one percent of the total salt concentration of each brine.