Prediction of Oil Recovery Factor in Stratified Reservoirs after Immiscible Water-Alternating Gas Injection Based on PSO-, GSA-, GWO-, and GA-LSSVM

Abstract

In this study, we solve the challenge of predicting oil recovery factor ($\mathrm{RF}$) in layered heterogeneous reservoirs after 1.5 pore volumes of water-, gas- or water-alternating-gas (WAG) injection. A dataset of ~2500 reservoir simulations is analyzed based on a Black Oil 2D Model with different combinations of reservoir heterogeneity, WAG hysteresis, gravity influence, mobility ratios and WAG ratios. In the first model MOD1, RF is correlated with one input (an effective WAG mobility ratio $M^{*}$). Good correlation (Pearson coefficient −0.94), but with scatter, motivated a second model MOD2 using eight input parameters: water–oil and gas–oil mobility ratios, water–oil and gas–oil gravity numbers, a reservoir heterogeneity factor, two hysteresis parameters and water fraction. The two mobility ratios exhibited the strongest correlation with RF (Pearson coefficient −0.57 for gas-oil and −0.48 for water-oil). LSSVM was applied in MOD2 and trained using different optimizers: PSO, GA, GWO and GSA. A physics-based adaptation of the dataset was proposed to properly handle the single-phase injection. A total of 70% of the data was used for training, 15% for validation and 15% for testing. GWO and PSO optimized the model equally well (R² = 0.9965 on the validation set), slightly better than GA and GSA (R² = 0.9963). The performance metrics for MOD1 in the total dataset were: RMSE = 0.050 and R² = 0.889; MOD2: RMSE = 0.0080 and R² = 0.998. WAG outperformed single-phase injection, in some cases with 0.3 units higher RF. The benefits of WAG increased with stronger hysteresis. The LSSVM model could be trained to be less dependent on hysteresis and the non-injected phase during single-phase injection.

1. Introduction

Oil recovery through water or gas injection often lacks efficiency due to the unfavorable mobility ratio between the oil and the displacing phase. Viscous fingering, gravity segregation and heterogeneity can also lead to poor sweep. Gas features low viscosity and density and can achieve channeling and early breakthrough [1,2]. Water-alternating gas injection (WAG) is an enhanced oil recovery (EOR) technique in which water and gas are injected in cycles to displace the oil. This type of technique mitigates the exponential rate decline seen in most fields after peak production [3]. The mobility of each injected fluid is reduced by the presence of the other, producing a more favorable mobility ratio to oil. Gravity segregation becomes less detrimental, since gas sweeps the top of the reservoir, while water sweeps the bottom [2,4]. Gas usually results in lower residual oil saturation than water, but this can be further lowered by WAG. Thus, WAG utilizes the advantages of water and gas injection and minimizes their individual downsides. In several field implementations, it has been beneficial to use WAG; the oil recovery factor in 59 fields increased when WAG was introduced, by 5% to 10 % of the oil originally in place [2]. Sanchez [5] reported that 80% of US WAG projects were beneficial. Micromodel studies also demonstrate better oil recovery with WAG than with single-phase displacement [6].

WAG introduces more design and operational parameters compared to water or gas injection, such as the WAG ratio (volume water to volume gas injected), number of cycles, cycle volume, injection rates and pressures. This may further affect optimal well placements. A 1:1 WAG ratio is considered common or even optimal [4]. Variation in WAG ratio with project time (tapering) has been conducted, partly due to limited gas access, for limiting gas production or for optimization. [7] used ensemble-based optimization of injector and producer well controls at each WAG cycle to maximize the net present value for a channeled reservoir model. Whether the reservoir pressure is above the system’s minimum miscibility pressure (MMP) determines whether the gas is miscible or immiscible with the oil [8]. During miscible displacement, the oil and gas become practically the same phase and residual oil saturation can approach zero. Kulkarni [9] found that miscible gas (CO₂) core flooding (continuous or WAG) outperformed immiscible injection. The choice or modification of the injected fluids can also improve the outcome. Foam, surfactant, polymer, low salinity brine and CO₂ are some alternatives [4].

Reservoir geology or heterogeneity is important during any field development. During gas injection, heterogeneity in terms of thief zones, stratification or fractures can cause gas channeling and early breakthrough, which can be mitigated by WAG [4]. Favorable well placement relative to the dip angle can provide more stable frontal displacement of oil. In heterogeneous reservoirs, gravity and capillary forces can divert flow from highly permeable layers to less permeable layers. These effects are more important in naturally fractured reservoirs, where advective forces are unable to mobilize oil [10,11].

The simultaneous flow of oil, water and gas requires the detailed measurement, quantification and correlation of three-phase relative permeabilities [12,13,14]. Injecting water and gas alternately causes gas and water saturations to rise and fall, resulting in hysteresis [15,16,17,18,19]. During WAG, the relative permeability of gas is more affected by hysteresis than oil and water [4] and hysteresis tends to decrease gas mobility. This reduction delays gas breakthrough and reduces gravity segregation. The Land [12] and Carlson [15] models are widely used to model relative permeability hysteresis.

Machine learning (ML) has gained increased popularity in the petroleum industry in recent years. ML algorithms can be useful for understanding trends in complex datasets and provide multivariate nonlinear regression or classification. Their applications include lithology classification [20], selecting EOR methods [21], locating optimal drilling spots [22], correlating asphaltene precipitation [23] or predicting CO₂ viscosity [24]. Important steps in developing ML models include selecting appropriate input and output variables, acquiring sufficient quality data, applying a suitable ML algorithm and tuning its metaparameters to prevent over- or under-fitting, usually via optimization algorithms.

This study makes use of the least squares support vector machine (LSSVM) algorithm, based on the works [25,26,27], for nonlinear regression. This algorithm has been applied in many contexts, such as predicting drilling fluid density [28], gas solubility [29,30,31,32], water availability [33], energy consumption [34,35], shale gas adsorption [36], wind power [37,38] and even tourism flow [39]. LSSVM has been successfully combined with optimizers such as particle swarm optimization (PSO), genetic algorithm (GA) and grey wolf Optimization (GWO) and has, in many cases, outperformed regression algorithms such as artificial neural networks, radial basis function, gene expression programming and adaptive neuro-fuzzy interference system [29,30,40,41].

In recent studies, [42,43] simulated CO₂ WAG injection in a reservoir model and developed machine learning proxy models with different algorithms to predict current rates of oil, gas and water based on current time, gas and water injection rates, half cycle time and operational constraints. The recovery factor and cumulative production were calculated from the produced output. The calibrated proxy models were used to optimize the WAG process. [41] used LSSVM and other ML approaches to predict two-phase relative permeabilities and combined them via correlations in previous research to estimate three-phase relative permeabilities and the performance of a WAG core flood. [40] correlated the oil recovery performance of EOR carbonated water injection using LSSVM. [44] used ML to optimize well placement during WAG injection. [45,46] used ML to co-optimize CO₂ injection for oil recovery and storage during WAG under different operational constraints.

In this study, our main contribution is to predict the reservoir oil recovery factor (RF) in layered reservoirs during immiscible WAG and single-phase (gas or water) injection for different fluid, reservoir, geometrical and operational conditions. This is a relatively complex task given the number of parameters involved and their coupled nature. Based on a comprehensive simulation database generated in both [47] and this work, we present two predictive approaches. In the first (MOD1), a dimensionless number $M^{*}$, derived from Nygård and Andersen’s study [47], is applied as a single input parameter. The second approach (MOD2) applies eight physics-motivated dimensionless input parameters to improve predictive power compared to the first method: two mobility ratios, two gravity numbers, injected water fraction, reservoir heterogeneity factor and two hysteresis parameters. In both models, the input variables incorporate all the system information. The latter approach, MOD2, utilizes the ML regression algorithm, LSSVM, with metaparameters optimized by either PSO (Particle Swarm Optimization), GSA (Gravity Search Algorithm), GA (Genetic Algorithm) or GWO (Grey Wolf Optimization). LSSVM has been optimized successfully in other works using PSO [20,23,36], GSA [37,38], GA [28] and GWO [29,32,33]. We propose a methodology to ensure the physical behavior of the machine learning model MOD2. We then adapt the dataset to be independent of hysteresis and the non-injected phase when single-phase injection is performed. Some of the research questions we investigate are:

-

How well do the models predict WAG performance?

-

Which parameters affect RF the most?

-

Do the parameters have a positive or negative effect on RF?

-

Will the models properly account for WAG injection and single phase injection?

The paper is structured as follows. The model serving as the basis for the simulation results is outlined in Section 2.1. The dimensionless number $M^{*}$ is outlined in Section 2.2. This number is used in the single-input parameter model. The machine learning approach and dataset follow in Section 2.3. The eight input parameters of the second approach are also presented in those latter sections. The results from analyzing the data are shown in Section 3 and the paper is concluded in Section 4.

2. Theory

2.1. Mathematical Model

We consider the same modeling approach for immiscible WAG injection as [47]: A 2D reservoir layered in a horizontal direction, with one injector and one producer, both vertical and perforated along the full reservoir height. See Figure 1 for an illustration. A black oil model is assumed with an incompressible and immiscible three-phase flow of oil, water and gas and negligible capillary pressure. WAG is applied from the start, rather than as a tertiary method. Relevant equations are presented below:

$${\partial }_t\left(\varphi s_o\right)={\partial }_x\left(f_o{u}_{Tx}\right)+{\partial }_z\left(f_o{u}_{Tz}\right)+{\partial }_z\left(K_{z}g{\lambda }_w{f}_o\mathsf{\Delta }{\rho }_{wo}\right)-{\partial }_z\left(K_{z}g{\lambda }_g{f}_o\mathsf{\Delta }{\rho }_{og}\right)$$

(1)

$${\partial }_t\left(\varphi s_w\right)={\partial }_x\left(f_w{u}_{Tx}\right)+{\partial }_z\left(f_w{u}_{Tz}\right)-{\partial }_z\left(K_{z}g{\lambda }_o{f}_w\mathsf{\Delta }{\rho }_{wo}\right)-{\partial }_z\left(K_{z}g{\lambda }_g{f}_w\mathsf{\Delta }{\rho }_{wg}\right)$$

(2)

$${\partial }_x{u}_{Tx}+{\partial }_z{u}_{Tz}=0,$$

(3)

$$u_{Tx}=-K_x{\lambda }_T{\partial }_{x}p,\hspace{1em}{u}_{Tz}=-K_z{\lambda }_T{\partial }_{z}p+K_{z}g\left({\lambda }_o{\rho }_o+{\lambda }_w{\rho }_w+{\lambda }_g{\rho }_g\right)$$

(4)

where $\varphi$ denotes porosity, $s_i$ saturation of phase $i$, $f_i$ fractional flow, $K$ permeability, ${\lambda }_i$ mobility, ${\rho }_i$ density, $u_T$ total Darcy flux and $p$ pressure. Corey correlation was applied for relative permeabilities, while gas relative permeability hysteresis was incorporated using Land’s trapping model [12] (which reduces the mobile gas saturation interval based on the parameter $C$) and Carlson’s hysteresis model [15] with parameter $\alpha$ (which reduces gas relative permeability).

Nygård and Andersen [47] ran simulations systematically to investigate the role of gravity segregation, the mobility ratios between the three phases, heterogeneity, hysteresis and WAG ratio and how they affected RF after 1.5 pore volumes of fluid were injected. The simulations were scaled using a combined dimensionless mobility ratio $M^{*}$ stating how effectively the injected fluids displaced oil under the given conditions, summarized as follows. In the design of this number, the mechanisms were incorporated one at a time. We refer to

Table A1. Rock/grid properties and operational parameters. $N$ denotes number of cells in each direction, $L$ the respective lengths, $Q$ the volumetric rate.

${\mathit{N}}_{\mathit{x}}$	100	${\mathit{L}}_{\mathit{x}}$	1000 m	${\mathbf{\varphi }}_{\mathit{j}}$	0.30	${\mathit{Q}}_{\mathit{w}}$	1014.6 m3/d	Half cycle duration	45 d
${\mathit{N}}_{\mathit{y}}$	1	${\mathit{L}}_{\mathit{y}}$	100 m	${\mathit{h}}_{\mathit{j}}$	3 m	${\mathit{Q}}_{\mathit{g}}$	1014.6 m3/d	Total injection volume, PVs	1.5 PVs
${\mathit{N}}_{\mathit{z}}$	81	${\mathit{L}}_{\mathit{z}}$	81 m

,

Table A2. Reservoir flow properties in terms of relative permeability end points, Corey exponents, initial and residual saturations.

${\mathit{k}}_{\mathit{r}\mathit{o}\mathit{w}}^{\mathit{m}\mathit{a}\mathit{x}}$	0.25	${\mathit{n}}_{\mathit{o}\mathit{w}}$	2	${\mathit{S}}_{\mathit{o}\mathit{i}}$	0.842
${\mathit{k}}_{\mathit{r}\mathit{o}\mathit{g}}^{\mathit{m}\mathit{a}\mathit{x}}$	0.25	${\mathit{n}}_{\mathit{o}\mathit{g}}$	2	${\mathit{S}}_{\mathit{w}\mathit{i}}={\mathit{S}}_{\mathit{w}\mathit{r}}$	0.158
${\mathit{k}}_{\mathit{r}\mathit{w}}^{\mathit{m}\mathit{a}\mathit{x}}$	0.05	${\mathit{n}}_{\mathit{w}}$	2	${\mathit{S}}_{\mathit{g}\mathit{i}}={\mathit{S}}_{\mathit{g}\mathit{r}}$	0.00
${\mathit{k}}_{\mathit{r}\mathit{g}}^{\mathit{m}\mathit{a}\mathit{x}}$	0.005	${\mathit{n}}_{\mathit{g}}$	2	${\mathit{S}}_{\mathit{o}\mathit{r}\mathit{w}}$	0.20
				${\mathit{S}}_{\mathit{o}\mathit{r}\mathit{g}}$	0.10

and

Table A3. Specification of model heterogeneities. Each model had 9 layers with permeability distributed as specified. It was assumed that vertical and horizontal permeabilities were equal in each layer: $K_{z,j}=K_{x,j}$.

	${\mathit{K}}_{\mathit{x}} \left[\mathbf{mD}\right]$
Layer 1 (top)	300	300	500	1000
2	300	100	50	20
3	300	900	500	1000
4	300	300	50	20
5	300	100	500	1000
6	300	900	50	20
7	300	300	500	1000
8	300	100	50	20
9 (bottom)	300	900	500	1000
FH	1.0	2.1	3.0	12.9

in Appendix A for several important simulation input parameters or model configurations that were constant in the simulations. More details can be found in the original paper. The fact that these parameters remained constant was mainly due to prioritization. However, these input parameters were incorporated in the dimensionless numbers presented in the following sections.

Figure 1. The geometrical configuration of the model (modified from [47]). is the distance from the injector, while is the distance from the top of the reservoir.

Figure 1. The geometrical configuration of the model (modified from [47]). is the distance from the injector, while is the distance from the top of the reservoir.

Oil recovery factor ($\mathrm{RF}$) is the output parameter of interest, defined as:

$$\mathrm{RF}=\frac{volume oil produced}{volume oil initially in place}=1-\frac{{{\displaystyle \sum }}_{j=1}^{N_z}{{\displaystyle \sum }}_{k=1}^{N_x}\varphi \left(z_j\right)s_o\left(x_k,z_j\right)}{{{\displaystyle \sum }}_{j=1}^{N_z}{{\displaystyle \sum }}_{k=1}^{N_x}\varphi \left(z_j\right)s_{oi}}$$

(5)

Every grid block features same dimension $\mathsf{\Delta }x\mathsf{\Delta }z$. $\mathrm{RF}$ is reported after 1.5 pore volumes are injected.

2.2. WAG Efficiency Characterization Using Dimensionless Number

The characteristic mobility ratio $M^{*}$ defined by Nygård and Andersen [47] features the following functional relation:

$$M^{*}={\left(\frac{r_w}{M_{w/o}^{*}{F}_H{F}_G^{w/o}}+\frac{1-r_w}{M_{g/o}^{*}{F}_H{F}_G^{g/o}}\right)}^{-1}$$

(6)

$r_w$ is the volume fraction of water in each cycle. Larger $M^{*}$ is associated with lower recovery factor. Characteristic two-phase mobilities ${\lambda }_i^{*}$ for each phase $i$ were found by averaging their mobility over their mobile saturation interval and used to define two-phase mobility ratios $M_{w/o}^{*}$, $M_{g/o}^{*}$:

$$M_{w/o}^{*}=\frac{{\lambda }_w^{*}}{{\lambda }_{ow}^{*}},\hspace{1em}{\lambda }_w^{*}=\frac{k_{rw}^{max}}{{\mu }_w}\frac{\left(1-\frac{s_{wr}}{s_{w,max}}\right)}{\left(n_w+1\right)},\hspace{1em}{\lambda }_{ow}^{*}=\frac{k_{row}^{max}}{{\mu }_o}\frac{\left(1-\frac{s_{orw}}{s_{ow,max}}\right)}{\left(n_{ow}+1\right)}$$

(7)

$$M_{g/o}^{*}=\frac{{\lambda }_g^{*}}{{\lambda }_{og}^{*}},\hspace{1em}{\lambda }_g^{*}=\frac{k_{rg}^{max}}{{\mu }_g}\frac{\left(1-\frac{s_{gr}}{s_{g,max}}\right)}{\left(n_g+1\right)},\hspace{1em}{\lambda }_{og}^{*}=\frac{k_{rog}^{max}}{{\mu }_o}\frac{\left(1-\frac{s_{org}}{s_{og,max}}\right)}{\left(n_{og}+1\right)}$$

(8)

$s_{i,max}$ denotes the saturation of phase $i$ where end-point relative permeability $k_{ri}^{max}$ is obtained, $n_i$ is the Corey exponent and ${\mu }_i$ is viscosity. A heterogeneity factor $F_H$ was derived from the horizontal permeability $K_{xj}$ and layer height $h_j$ distribution over layers $j=1:N_L$:

$$F_H=\frac{{\overline{K}}_x^{ar}}{{\overline{K}}_x^{ha}},\hspace{1em}{\overline{K}}_x^{ar}={\left({\displaystyle \sum }_{j=1}^{N_L}{h}_j\right)}^{-1}{\displaystyle \sum }_{j=1}^{N_L}{h}_j{K}_{x,j},{\overline{K}}_x^{ha}=\left({\displaystyle \sum }_{j=1}^{N_L}{h}_j\right){\left({\displaystyle \sum }_{j=1}^{N_L}\frac{h_j}{K_{x,j}}\right)}^{-1}$$

(9)

Two-phase gravity numbers were defined using the ratio of two-phase segregation time $t_{seg}$ and the residence time $t_{res}$ of the injected phase:

$$N_G^{w/o}=\frac{t_{res}^w}{t_{seg}^{w/o}},\hspace{1em}{t}_{res}^w=\frac{L_x{L}_y{{\displaystyle \sum }}_{j=1}^{N_L}{\varphi }_j{h}_j}{Q_w},\hspace{1em}{t}_{seg}^{w/o}=\frac{H\varphi }{K_z^{ha}{\mathsf{\Delta }\rho }_{wo}g}\left(\frac{1}{{\lambda }_w^{*}}+\frac{1}{{\lambda }_{ow}^{*}}\right)$$

(10)

$$N_G^{g/o}=\frac{t_{res}^g}{t_{seg}^{g/o}},\hspace{1em}{t}_{res}^g=\frac{L_x{L}_y{{\displaystyle \sum }}_{j=1}^{N_L}{\varphi }_j{h}_j}{Q_g},\hspace{1em}{t}_{seg}^{g/o}=\frac{H\varphi }{K_z^{ha}{\mathsf{\Delta }\rho }_{go}g}\left(\frac{1}{{\lambda }_g^{*}}+\frac{1}{{\lambda }_{og}^{*}}\right),$$

(11)

It was found that the role of gravity depended on heterogeneity and two-phase gravity factors $F_G^{w/o},F_G^{w/o}$ accounting for this coupling were introduced:

$$F_G^{w/o}=\frac{1+a_1{\left(N_G^{w/o}\right)}^{a_2}}{1+a_1\left(F_H-1\right){\left(N_G^{w/o}\right)}^{a_2}},\hspace{1em}{F}_G^{g/o}=\frac{1+a_1{\left(N_G^{g/o}\right)}^{a_2}}{1+a_1\left(F_H-1\right){\left(N_G^{g/o}\right)}^{a_2}}$$

(12)

Note the unitless tuning parameters $a_1=3$ and $a_2=0.5$. Finally, hysteresis was incorporated into the gas characteristic relative permeability. Land’s parameter $C$ defines a hysteresis residual gas saturation $s_{gr}^{hyst}$:

$$s_{gr}^{hyst}=s_{gr}+\frac{s_{g,max}-s_{gr}}{1+C\left(s_{g,max}-s_{gr}\right)}$$

(13)

A further modification according to $r_w$ was made:

$$s_{gr}^{wag}=s_{gr}\left(1-r_w\right)+r_w{s}_{gr}^{hyst}$$

(14)

Additionally, the gas relative permeability end point $k_{rg}^{max}$ in ${\lambda }_g^{*}$, (see Equation (8)), was reduced due to hysteresis. The reductions were performed individually for the gas–oil mobility ratio and the gas–oil gravity number first based on the parameter $\alpha$ and heterogeneity factor $F_H$ using unitless tuning parameters $b_1=1,b_2=0.5$ and $b_3=10,b_4=2$:

$$k_{rg,M}^{max,hyst}=\frac{k_{rg}^{max}}{1+b_1{F}_H^{b_2}\alpha },\hspace{1em}{k}_{rg,N_G}^{max,hyst}=\frac{k_{rg}^{max}}{1+b_3{F}_H^{b_4}\alpha }$$

(15)

Next, the fraction $r_w$ was incorporated according to:

$$k_{rg,M}^{wag}={\left(\frac{1-r_w}{k_{rg}^{max}}+\frac{r_w}{k_{rg,M}^{max,hyst}}\right)}^{-1},\hspace{1em}{k}_{rg,N_G}^{wag}={\left(\frac{1-r_w}{k_{rg}^{max}}+\frac{r_w}{k_{rg,N}^{max,hyst}}\right)}^{-1}$$

(16)

We then obtained the hysteresis-corrected characteristic gas mobilities ${\lambda }_{g,M}^{*}$ and ${\lambda }_{g,N_G}^{*}$ by replacing $s_{gr}$ with $s_{gr}^{wag}$ in (8), while the end-point relative permeability $k_{rg}^{max}$ in (8) was replaced by $k_{rg,M}^{wag}$ in the gas-oil mobility ratio $M_{g/o}^{*}$ in (8) and by $k_{rg,N_G}^{wag}$ in the gas–oil gravity number $N_G^{g/o}$ in (11):

$${\lambda }_{g,M}^{*}=\frac{1}{{\mu }_g}\left(1-\frac{s_{gr}^{wag}}{s_{g,max}}\right)\frac{k_{rg,M}^{wag}}{n_g+1},\hspace{1em}{\lambda }_{g,N_G}^{*}=\frac{1}{{\mu }_g}\left(1-\frac{s_{gr}^{wag}}{s_{g,max}}\right)\frac{k_{rg,N_G}^{wag}}{n_g+1}$$

(17)

Every input parameter is incorporated in the dimensionless number $M^{*}$. Note that during single-phase injection $\left(r_w=0 \mathrm{or} 1\right),$ the two-phase parameters involving the phase not injected do not affect $M^{*}$. Similarly, hysteresis does not affect $M^{*}$ during single-phase injection.

2.3. Workflow

2.3.1. Model Input Parameters

In the first model, MOD1, we take

$$x_0={\log }_{10}{M}^{*}$$

(18)

as the only input parameter to predict RF. We also consider a machine learning model (MOD2) in which the following eight dimensionless numbers are used as input parameters:

$$\begin{gathered} x_1=r_w,\hspace{1em}{x}_2={\log }_{10}{F}_H,\hspace{1em}{x}_3=\alpha ,\hspace{1em}{x}_4={\log }_{10}C, \\ {x}_5={\log }_{10}{M}_{g/o}^{*},\hspace{1em}{x}_6={\log }_{10}{M}_{w/o}^{*},\hspace{1em}{x}_7={\log }_{10}{N}_G^{w/o},\hspace{1em}{x}_8={\log }_{10}{N}_G^{g/o} \end{gathered}$$

(19)

These numbers reflect injected fluid fractions $x_1$, heterogeneity $x_2$, hysteresis $x_3,x_4$, relative magnitude of fluid mobilities $x_5,x_6$ and gravity vs. advective forces $x_7,x_8$. They incorporate all the input parameters used in the Eclipse model and the number $M^{*}$. The overall workflow is demonstrated in Figure 2, where the two modeling approaches after the data collection step are indicated. In MOD1, a polynomial regression is performed, while machine learning is used for MOD2. The data and the detailed steps for developing MOD2 are explained below.

2.3.2. Reservoir Simulation Dataset and Model Approaches

In addition to the 1648 WAG and 96 single phase injection simulations generated by [47], 824 new WAG simulations were performed with new $C$ and $\alpha$ values combined with existing combinations of heterogeneity, density and mobility. In the previous study, $C$ and $\alpha$ were selected primarily to cover no or significant hysteresis. The values $\alpha =0$ and $C=1000$ were assigned to points without hysteresis influence from the respective parameters. For MOD1, each simulation allows the calculation of $M^{*}$, which is input to the corresponding output $\mathrm{RF}$. The $1648+96+824=2568$ data points were analyzed with MOD1 and correlated using a polynomial expression between RF and $x_0$.

For the single-phase injection of water $\left(r_w=1\right)$, values for $N_G^{g/o}$ and $M_{g/o}^{*}$ are not well defined since gas is not injected. The case is similar regarding $N_G^{w/o}$ and $M_{w/o}^{*}$ at $r_w=0$ (gas injection). Further, hysteresis parameters $C,\alpha$ should not matter. The insensitivity of $M^{*}$ to the mentioned parameters under single-phase injection was ensured during its derivation. To properly define the input values under single-phase injection for MOD2 the following approach was taken:

Each single-phase data point was duplicated to 16 data points, in which all combinations of high and low values of the four missing parameters were assigned. Specifically, for gas injection (points with $r_w=0$, indexed ‘g’), the following values were set for $x_3,x_4,x_6,x_7$:

$$\begin{gathered} x_{3,g}={\overline{x}}_{3,WAG}\pm X{\sigma }_{3,WAG},\hspace{1em}{x}_{4,g}={\overline{x}}_{4,WAG}\pm X{\sigma }_{4,WAG}, \\ {x}_{6,g}={\overline{x}}_{6,WAG}\pm X{\sigma }_{6,WAG},\hspace{1em}{x}_{7,g}={\overline{x}}_{7,WAG}\pm X{\sigma }_{7,WAG}, \end{gathered}$$

(20)

where ${\overline{x}}_{i,WAG} \left(i=3,4,6,7\right)$ indicate the average of the data point values applied in the WAG cases and ${\sigma }_{i,WAG} \left(i=3,4,6,7\right)$ the corresponding standard deviations. $X$ is a multiplier. Similarly, for water injection (indexed ‘w’) the following values were set for $x_3,x_4,x_5,x_8$:

$$\begin{gathered} x_{3,w}={\overline{x}}_{3,WAG}\pm X{\sigma }_{3,WAG},\hspace{1em}{x}_{4,w}={\overline{x}}_{4,WAG}\pm X{\sigma }_{4,WAG}, \\ {x}_{5,w}={\overline{x}}_{5,WAG}\pm X{\sigma }_{5,WAG},\hspace{1em}{x}_{8,w}={\overline{x}}_{8,WAG}\pm X{\sigma }_{8,WAG}, \end{gathered}$$

(21)

The 96 single-phase simulations resulted in $16\cdot 96=1536$ points to the ML model. In total, $1648+824+16\cdot 96=4008$ data points were then applied in MOD2.

2.3.3. Machine Learning Dataset Preparation

The 2472 WAG cases and 96 single-phase injection cases were both divided randomly between three sets: 70% in the training set, 15% in the validation set and the remaining 15% in the testing set [48]. Single-phase data points within each set were further split into 16 points, as described previously. See

Table 1. Number and type of points in different datasets and models.

MOD1	Single Phase Cases	WAG Cases	Total
	96	2472	2568
MOD2	Single phase cases	WAG cases	Total
Training (70%)	68 × 16 = 1088	1730	2818
Validation (15%)	14 × 16 = 224	371	595
Testing (15%)	14 × 16 = 224	371	595
Total	96 × 16 = 1536	2472	4008

for an overview of points in the models and datasets.

2.3.4. Machine Learning Workflow

We apply LSSVM with radial basis kernel (RBK) function and either PSO, GWO, GA or GSA as optimizers. Each of the optimizers features its own strengths and disadvantages in finding global optima efficiently, as well as depending on its individual tuning parameters. They are swarm-based algorithms, making use of many potential solutions simultaneously and improving these solutions according to those performing best at a given iteration. The result is the existence of differently optimized LSSVM models, as indicated in Figure 2. Detailed explanations of the algorithms are provided in Appendix B and Appendix C, respectively. Each input parameter $x_i$ was normalized (denoted $x_N$) to a range between −1 and +1 based on the maximum and minimum values of the total dataset.

$$x_N=2\frac{x-x_{min}}{x_{max}-x_{min}}-1$$

(22)

Assuming predefined values of the metaparameters ($\sigma ,\gamma$), LSSVM provides the function $y\left(x\right)$ and its coefficients ${\alpha }_k,b$ that minimize the error between the model predictions and observations of a given dataset (usually the training dataset) for those parameter choices [27]:

$$y\left(x\right)={\displaystyle \sum }_{i=1}^n{\alpha }_i\exp \left(-\frac{\left|\right|x_{Ni}-x_N{}^2\left|\right|}{{\sigma }^2}\right)+b$$

(23)

For given metaparameters, the LSSVM algorithm is calibrated on the training set to provide choices of ${\alpha }_k,b$. The optimizer algorithm is used to search for the metaparameters $\sigma ,\gamma$ that minimize the model prediction error on the validation dataset (i.e., many LSSVM models are calibrated on the training set and the one giving best prediction on the validation set is taken as the best). This systematically determines the best choice of metaparameters to avoid over- or under-fitting. The optimized LSSVM models were finally used to predict the data in the testing set. The model performing best overall was selected for further sensitivity analysis. For proper comparison, the optimizers were implemented with the same random initial solutions guesses (and velocities if applicable), search space and number of iterations. The optimizer parameters can be found in

Table A4. Optimization algorithm parameters. No algorithm specific parameters were required for GWO.

Common			PSO			GA
# particles /chromosomes/wolves	$N_p$	$20$	Acceleration constants	${\varphi }_1,$ ${\varphi }_2$	$1.5,$ $1.5$	Mutation rate	${\mu }_r$	$0.15$
# variables/genes	$N_{var}$	$2$	Damping factor	$\omega$	$0.8$	Mutation factor	${\mu }_f$	$0.1$
# iterations	$N_{it}$	$30$	GSA			# elite chromosomes	$N_{elit}$	$2$
Search range variable 1	${\beta }_1^{min},$ ${\beta }_1^{max}$	$-2,$ $+8$	Initial gravity	$G_0$	$2$	GWO
Search range variable 2	${\beta }_2^{min},$${\beta }_2^{max}$	$-3,$ $+3$	Gravity reduction factor	$\alpha$	$5$	-
Initial velocity range			Small constant	$\epsilon$	${10}^{-4}$

.

An advantage of LSSVM is its few (two) metaparameters and the automated optimization of its internal tuning parameters [27]. In comparison, artificial neural networks often need subjective selection of the number of nodes and layers and then comprehensive tuning of a vast number of weights and biases to train the network [49].

The correlation between input variables $x$ and output $y$ (RF) is quantified using Pearson correlation $r_{xy}^P$, Spearman rank $r_{xy}^{Sp}$ and distance correlation $r_{xy}^D$ coefficients. These indices, respectively, indicate linear correlation, nonlinear monotonic correlation and nonlinear nonmonotonic correlation. The former two range between −1 and 1, while the latter ranges from 0 to 1. For all of them, 0 indicates no correlation. The goodness-of-fit between the calculated RF from MOD1 or MOD2 and the data values of RF were quantified using the coefficient of determination $R^2$ and the root mean square error $\mathrm{RMSE}$. The definitions of the mentioned quantities are in Appendix D.

3. Results and Discussion

3.1. Preliminary Dataset Analysis

The range and mean of the data points used in MOD1 and MOD2 (using $X=0.5$) are listed in

Table 2. Range of values for the total datasets used in MOD1 and MOD2 (using $X=0.5$).

MOD1	Train			Val			Test			Tot
	Min	Mean	Max	Min	Mean	Max	Min	Mean	Max	Min	Mean	Max
$M^{*}$	−0.2	1.4	3.4	−0.1	1.5	3.2	−0.1	1.4	3.2	−0.2	1.4	3.4
$y$	0.14	0.49	0.88	0.20	0.49	0.84	0.19	0.50	0.85	0.14	0.49	0.88
MOD2	Train			Val			Test			Tot
	Min	Mean	Max	Min	Mean	Max	Min	Mean	Max	Min	Mean	Max
$x_1$	0	0.5	1	0	0.5	1	0	0.4	1	0	0.5	1
$x_2$	0	0.5	1.1	0	0.5	1.1	0	0.4	1.1	0	0.5	1.1
$x_3$	0	1.2	2.5	0	1.2	2.5	0	1.1	2.5	0	1.2	2.5
$x_4$	0	1.3	3	0	1.3	3	0	1.4	3	0	1.3	3
$x_5$	0.1	1.4	2.4	0.1	1.4	2.4	0.1	1.4	2.4	0.1	1.4	2.4
$x_6$	0.0	1.4	2.3	0.0	1.4	2.3	0.0	1.3	2.3	0.0	1.4	2.3
$x_7$	−4.6	−2.6	−0.9	−4.6	−2.7	−0.9	−4.6	−2.7	−0.9	−4.6	−2.6	−0.9
$x_8$	−7.9	−3.0	−0.8	−7.9	−3.1	−0.8	−7.9	−3.0	−0.8	−7.9	−3.0	−0.8
$y$	0.14	0.45	0.88	0.20	0.44	0.84	0.19	0.48	0.85	0.14	0.45	0.88

. The use of logarithms made the range of the different variables span a few units rather than orders of magnitude.

The Pearson, Spearman and distance correlation coefficients evaluated between $\mathrm{RF}$ and the input parameters were calculated for the two model datasets and are listed in

Table 3. Pearson, Spearman and Distance correlation coefficients $r_{xy}$ evaluated for the total dataset between $\mathrm{RF}$ and the involved input parameters for MOD1 and MOD2.

	MOD1	${\mathit{r}}_{\mathit{x}\mathit{y}}^{\mathit{P}}$	${\mathit{r}}_{\mathit{x}\mathit{y}}^{\mathit{S}\mathit{p}}$	${\mathit{r}}_{\mathit{x}\mathit{y}}^{\mathit{D}}$
	$x_0$	$-0.94$	$-0.95$	$0.93$
	MOD2	$x_1$	$x_2$	$x_3$	$x_4$	$x_5$	$x_6$	$x_7$	$x_8$
	WAG cases	$-0.053$	$-0.34$	$0.16$	$-0.15$	$-0.62$	$-0.49$	$0.078$	$0.0045$
$r_{xy}^P$	$X=0.25$	$-0.055$	$-0.27$	$0.11$	$-0.10$	$-0.58$	$-0.49$	$0.059$	$0.0087$
	$X=0.5$	$-0.055$	$-0.27$	$0.10$	$-0.099$	$-0.57$	$-0.48$	$0.057$	$0.0085$
	$X=1$	$-0.055$	$-0.27$	$0.087$	$-0.083$	$-0.53$	$-0.45$	$0.053$	$0.0079$
$r_{xy}^{Sp}$	$X=0.5$	$-0.045$	$-0.25$	$0.095$	$-0.095$	$-0.53$	$-0.47$	$0.056$	$0.017$
$r_{xy}^D$	$X=0.5$	$0.19$	$0.25$	$0.15$	$0.15$	$0.54$	$0.47$	$0.092$	$0.077$

. For MOD1, the two former coefficients were $\approx -0.94$ and the latter $0.93$. Their magnitude being close to 1 indicates a strong linear correlation between $\mathrm{RF}$ and $x_0={\log }_{10}{M}^{*}$ and the negative sign indicates that a larger $M^{*}$ reduces RF.

For MOD2, Pearson correlation coefficients were reported for WAG cases only and for all cases when $X=0.25$, $0.5$ and $1$ in Table 3. Spearman rank and distance correlation were calculated only for $X=0.5$. Considering the WAG cases, several variables correlate with $\mathrm{RF}$, especially $x_5, x_6$ which are the log of gas/oil and water/oil mobility ratios. They feature Pearson coefficients $r_{xy}^P~-0.5$ to $-0.6$ indicating that when they increase (less favorable mobility ratio towards the oil), RF is reduced. Heterogeneity, represented by $x_2$, also correlates with $\mathrm{RF}$ with a lower $r_{xy}^P~-0.27$, indicating that RF generally reduces when the heterogeneity factor increases. The hysteresis parameters $x_3,x_4$ correlate with RF in opposite ways to each other, with $r_{xy}^P~0.15$ for $x_3$ and $-0.15$ for $x_4$. When $x_3$ (i.e., $\alpha$) increases, gas relative permeability is reduced and should improve RF. Higher $x_4$ (i.e., $\log C$) leads to less gas trapping and RF therefore decreases. The gravity numbers feature relatively poor Pearson correlation with $\mathrm{RF}$, with $r_{xy}^P~0.08$ for water-oil and $0.0045$ for gas-oil. Similarly, the water volume fraction correlates little with $\mathrm{RF}$, and $r_{xy}^P~-0.05$ is slightly negative. These results could be related to their coupled nature, as is discussed below.

When considering the MOD2 datasets with single-phase data included for different $X$, we note that the magnitude and sign of the different Pearson coefficients are similar to when only the WAG cases were considered. The main difference is that the correlation is somewhat lower, especially the parameters with unspecified information during single-phase injection. This was expected, since we added points where $\mathrm{RF}$ does not vary with changes in these parameters.

When evaluating the dataset with Pearson rank correlation for $X=0.5,$ we observe similar, but slightly lower values as for the Pearson coefficient, except for $x_8$, where the correlation doubles, but remains very low. When considering the distance correlation coefficient, however, several input parameters correlate more strongly with recovery, indicating that their relation is nonmonotonic. In particular, the water fraction, $x_1$, features a higher distance correlation coefficient, of $0.19$. WAG was expected to perform better than single-phase injection, with $\mathrm{RF}$ not changing linearly with $x_1$, but peaking. Gravity can be a cause of both low and improved sweep and the gravity numbers $x_7,x_8$ feature distance correlation coefficients around 0.08, where more impact is attributed the gas–oil gravity number in particular. Furthermore, the hysteresis parameters $x_3,x_4$ now seem to correlate more strongly. The three correlation coefficients are similar for $x_2$, indicating a relatively linear and monotic relation, so if all other parameters are constant, increased heterogeneity should reduce recovery.

Note that all the variables in MOD2 feature less correlation than the variable $x_0$ in MOD1, since they individually do not contain all the involved system parameters. The aim is for them to provide better predictions when combined.

3.2. Development of MOD1

RF is plotted against $x_0$ in Figure 3 and demonstrates a clear correlation where higher $x_0$ gives lower RF. There is also significant scatter, meaning a given $x_0$ can be associated with a range of RF values. The data were fitted to a third-order polynomial function, given by the blue curve in Figure 3 and Equation (24). A higher order polynomial did not further reduce the RMSE, which means the remaining error was associated primarily with the scatter in the data.

$$\begin{gathered} \mathrm{RF}={\displaystyle \sum }_{i=1:4}{p}_i{\left(x_0\right)}^{4-i}, \left(R^2=0.889; \mathrm{RMSE}=0.0498\right) \\ {p}_1=0.01645,\hspace{1em}{p}_2=-0.06302,\hspace{1em}{p}_3=-0.1393,\hspace{1em}{p}_4=0.7676 \end{gathered}$$

(24)

The performance of the model is also shown by comparing the estimated $\mathrm{RF}$ and the data $\mathrm{RF}$ in Figure 4a together with a histogram, Figure 4b, of the residual errors (the difference between the estimated RF and the data point RF). The R² = 0.889 is relatively high. As seen in the histogram, the residuals are symmetrically distributed around zero and roughly 95% of the points estimate RF correctly within $\pm 0.1$. The RMSE, which can be considered a more typical error, is 0.050.

3.3. Development of LSSVM Model MOD2

The ML model MOD2 was developed using LSSVM and a dataset assuming $X=0.5$. The best LSSVM model was determined using different approaches. First, a random choice of metaparameters $\left(\sigma ,\gamma \right)=\left(1,1\right)$ was used. Next, LSSVM was applied with the different optimization algorithms, PSO, GA, GWO and GSA, to systematically find the best metaparameters. As previously mentioned, for any combination of the metaparameters, LSSVM models were fitted to the training set and used to forecast the validation set. The metaparameters that resulted in the best performance in the validation set, after using a given optimizer algorithm, determined the best model. The test set was then forecasted.

In Figure 5, the performance of the different algorithms is illustrated as a function of the iterations performed. R² and RMSE for the validation set are plotted for the best solution at the given iteration, together with the corresponding values of $\log \gamma$ and $\log \sigma$ in plots a to d, respectively. The same initial solutions and number of iterations were applied in all the algorithms (different colors). Two different initializations were applied for robustness (dashed and full lines).

In all cases, a high $R^2\approx 0.996$ and low $\mathrm{RMSE}\approx 0.009$ were obtained after 30 iterations for all the algorithms and both starting points, although GSA deviated from initially good solutions and converged slowly or to inferior solutions. Furthermore, GA seemed to not produce as good results as PSO and GWO. The two algorithms, PSO and GWO, exhibited very similar values of $\log \gamma \approx 5.5$ and $\log \sigma \approx 0.3$ and the lowest error indicating that they were better able to find the global optima. Notably, the values of $\log \gamma$ and $\log \sigma$ varied significantly during the iterations, but mostly exhibited very good performance. This may have been due to the ability of LSSVM to tune its internal parameters ${\alpha }_i$ and $b$ for any given $\gamma ,\sigma$.

The best metaparameters obtained during the 30 iterations are listed in

Table 4. Optimized LSSVM metaparameters using different optimizers and different initializations (marked 1 and 2) and corresponding performance metrics on the training, validation and testing datasets. The parameters used in the final model, MOD2, are indicated.

				RMSE			R2
	Seed	$\log \left(\gamma \right)$	$\log \left(\sigma \right)$	Train	Val	Test	Train	Val	Test
LSSVM (preset)		$0$	$0$	$0.0220$	$0.0202$	$0.0279$	$0.9821$	$0.9817$	$0.9691$
PSO-LSSVM	$1$	$5.6106$	$0.32535$	$0.0056$	$0.0088$	$0.0142$	$0.9988$	$0.9965$	$0.9920$
	$2$	$5.4335$	$0.30305$	$0.0055$	$0.0088$	$0.0142$	$0.9989$	$0.9965$	$0.9920$
GSA-LSSVM	$1$	$6.6812$	$0.42982$	$0.0058$	$0.0089$	$0.0150$	$0.9988$	$0.9964$	$0.9911$
	$2$	$7.0506$	$0.49871$	$0.0064$	$0.0090$	$0.0156$	$0.9985$	$0.9963$	$0.9904$
GWO-LSSVM	$1$	$5.6564$	$0.32883$	$0.0056$	$0.0088$	$0.0142$	$0.9988$	$0.9965$	$0.9919$
	$2$	$5.6698$	$0.32230$	$0.0055$	$0.0088$	$0.0144$	$0.9989$	$0.9965$	$0.9918$
GA-LSSVM	$1$	$7.3404$	$0.49280$	$0.0059$	$0.0090$	$0.0155$	$0.9987$	$0.9963$	$0.9904$
	$2$	$4.9708$	$0.25298$	$0.0054$	$0.0089$	$0.0140$	$0.9989$	$0.9964$	$0.9922$
Range (opt.)		$~2.4$	$~0.25$	$0.0010$	$0.0002$	$0.0016$	$0.0004$	$0.0002$	$0.0018$
Final (MOD2)		$5.6$	$0.32$	$0.0056$	$0.0088$	$0.0143$	$0.9988$	$0.9965$	$0.9919$

, considering all four algorithms and both initializations. The corresponding metrics (R² and RMSE) were calculated on the training, validation and testing sets. All the optimized models performed better than the algorithm with the arbitrarily preset metaparameters, although this choice also performed well, with $R^2$ greater than 0.969 on all three sets. The optimized models exhibited very consistent performance in the three datasets, with $R^2\approx 0.999$ in the training set, $\approx 0.996$ in the validation set and $\approx 0.992$ on the testing set. The difference in $R^2$ was in the fourth digit for the first two sets and the third digit in the latter set. The RMSE was around $0.006$ on the training set, $0.009$ on the validation set and $0.015$ on the test set, with GSA standing out with the highest RMSE. As final metaparameter values in the optimized model, we took an average of the four similar results from the PSO and GWO runs with two significant digits. Calculating the RMSE and R² metrics on the datasets confirmed that the performance with these values was still optimal (see Table 4). The LSSVM model with these parameters is referred to as MOD2 in what follows. Note especially that MOD2 is capable of predicting unseen single-phase data and thus accounts for the physics introduced during the modification of the dataset.

The RMSE and R² were calculated with MOD2 for the total dataset as 0.0080 and 0.9976, respectively. These metrics are greatly improved compared to MOD1, which featured a corresponding RMSE of 0.0498 and an R² of 0.889. The calculated (with MOD2) and observed RF data are plotted against each other for the three datasets in Figure 6. For all three datasets, there is little scatter around the perfect match line. The residual errors were calculated for each datapoint in the full dataset and the results are plotted as a histogram in Figure 7. Approximately 90% of the data feature errors in the estimated RF of less than $0.01$, and 95% of the data feature errors less than 0.02.

Partial derivatives with respect to each normalized variable, $\frac{\partial y}{\partial x_{Ni}}$, were calculated for each data point using MOD2 and histograms were created for each variable, as shown in Figure 8. The derivatives were calculated numerically and two choices of $\mathsf{\Delta }{x}_N=5\cdot {10}^{-2}$ and ${10}^{-3}$ were used. The two choices produced practically identical histograms, suggesting that the optimized LSSVM function did not suffer from oscillations (a sign of over-fitting). For each variable, a large fraction of the points featured positive and negative derivatives. Hence, changing the variable can affect RF positively or negatively, indicating coupling and room for finding optimal conditions.

For $\frac{\partial y}{\partial x_{N1}}$ we see positive and negative values, which is reasonable, since RF should be higher for WAG than single-phase injection. $\frac{\partial y}{\partial x_{N5}}$ and $\frac{\partial y}{\partial x_{N6}}$ are both dominated by negative values, since increasing the mobility ratio between gas and oil or between water and oil, respectively, should reduce RF. Higher water–oil gravity segregation is considered negative for RF, with $\frac{\partial y}{\partial x_{N7}}$ mainly negative. On the other hand, gas–oil gravity segregation is considered mainly positive for RF with a majority of points having $\frac{\partial y}{\partial x_{N8}}$ positive. This could be attributed to the better sweep of low-permeable layers in heterogeneous cases. Hysteresis appears to benefit recovery, as seen by a majority of positive $\frac{\partial y}{\partial x_{N3}}$, although the effect of $\frac{\partial y}{\partial x_{N4}}$ seems to be equally negative and positive.

3.4. Sensitivity Analyses with Optimized LSSVM Model MOD2

The calibrated model, MOD2, was much better at predicting RF than MOD1 and was therefore pursued in the sensitivity analysis. Below, we present contour plots showing RF as a function of different input variables, while keeping the others constant. The parameters are kept within the total dataset range (see Table 2) in order to ensure model validity.

3.4.1. Variation of Oil Viscosity

Oil viscosity can vary greatly from one reservoir to another. It proportionally impacts mobility ratios $M_{w/o}^{*}$ and $M_{g/o}^{*}$ in Equations (7) and (8), represented by $x_6$ and $x_5$. For low oil mobilities, the gravity numbers $N_G^{w/o}$ and $N_G^{g/o}$ (represented by $x_7$ and $x_8$) increase proportionally with oil viscosity but are less dependent if water or gas feature mobility that is similar to or lower than that of oil (see Equations (10) and (11)). For simplicity, we assume they are proportional. We vary the oil viscosity by 2.0 orders of magnitude, which is less than the smallest range of the four dimensionless numbers (2.3 for $x_6$), as seen in Table 2.

Four cases are defined in

Table 5. Parameter selections for MOD2, where oil viscosity is varied and influences mobility ratios and gravity numbers. Four cases are considered according to heterogeneity and hysteresis.

	Low Het, Low Hyst	High Het, Low Hyst	Low Het, High Hyst	High Het, High Hyst
x1	0:1
x2	0.25	1	0.25	1
x3	0	0	2.5	2.5
x4	3	3	0.5	0.5
x5	0.2:2.2
x6	0.2:2.2
x7	−3:−1
x8	−3:−1

with low or high heterogeneity (low or high $x_2$), and a low or high degree of hysteresis (low $x_3$ and high $x_4$ and opposite, respectively). For each of these cases, $\mathrm{RF}$ is plotted as a function of $x_1$ (the water fraction) and $x_5$ (representing the gas–oil mobility ratio) representing different viscosities (see Figure 9).From the figure, we observe that:

-

Optimal RF values were mainly obtained at an intermediate water fraction $0<x_1$ (consider any line parallel with the x-axis), suggesting that WAG gives higher RF than single-phase injection. Cases with low hysteresis and favorable mobility ratios seem to give similar RF for water injection and WAG (although WAG with a low water fraction seems optimal) (see Figure 9a,b (low and high heterogeneity)).

-

The advantage of WAG over single-phase injection was most clear when hysteresis was significant (see Figure 9c,d). The best water fraction produced RF up to 0.3 units higher than the worst fraction. This strong impact was mainly at low oil viscosity (low $x_5$) with optimal water fraction around 0.5–0.6. For higher oil viscosity or lower heterogeneity cases, WAG was in many cases only marginally better (~0.05 units) than the best single-phase injection.

-

Increased oil viscosity reduced RF for a given water fraction (follow any line parallel with the y-axis). This was dominant over the WAG fraction at high viscosities, except for the highly heterogeneous cases with high hysteresis (Figure 9d). This demonstrates the benefit of WAG in heterogeneous formations and that hysteresis is an important contributor.

-

For a given heterogeneity (low or high), increased hysteresis improved RF (compare Figure 9c,d (high hyst) with Figure 9a,b (low hyst)). This was related to the improved gas–oil mobility ratio and reduced gravity segregation, which improves volumetric sweep. The optimal water fraction shifted to more central values, since both phases are needed for hysteresis.

-

For a given hysteresis state, increased heterogeneity reduced RF, especially for cases with less viscous oil (compare Figure 9b,d (high het) with Figure 9a,c (low het)). For high hysteresis cases, increased heterogeneity increased RF in cases with more viscous oil.

To better understand the relation between viscosity, heterogeneity and hysteresis, we plotted RF as a function of $x_2=log{F}_H$ and $x_5=log\left(M_{go}^{*}\right)$ for $x_1=0.5$ (WAG injection with equal volume fractions of gas and water) for the two hysteresis cases in Figure 10. Each value of $x_5$ represents fixed oil viscosity and the curves cover the same viscosity range as before. We observed that that:

-

For low hysteresis (Figure 10a), RF was very sensitive to heterogeneity for low oil viscosities and increased heterogeneity reduced RF. For high viscosity, RF changed little with heterogeneity.

-

With significant hysteresis (Figure 10b), low-viscosity cases produced reduced RF at higher heterogeneity, while high-viscosity cases produced increased RF.

3.4.2. Variation of Well Distance, Injection Rate or Density Difference

The distance between wells can vary from a dense pattern of a few hundred meters onshore to ~1000 m offshore. For fixed injection rates, a longer well distance $L_x$ proportionally increases the residence time and, hence, the gravity numbers (see (10)), represented by $x_7$ and $x_8$. Similarly, increasing the injection rates of water $Q_w$ and gas $Q_g$ equally reduces the residence times and the gravity numbers. Increased density differences reduce the segregation time and increase the gravity numbers. If the height is varied but the injection rate is the same, we note that both segregation time and residence time change equally and there is no net change in the gravity numbers. Varying the aforementioned parameters does not affect the variables $x_1$ to $x_6$; we can thus investigate cases in which they are constant and only the gravity numbers change.

We plotted RF as a function of injected water fraction $x_1$ and log gravity number (equal values of $x_7$ and $x_8$). We investigated the role of mobility ratio, heterogeneity and hysteresis one by one. The different cases are listed in

Table 6. Parameter selections for MOD2 with cases demonstrating influence of gravity numbers according to heterogeneity, mobility ratio and hysteresis.

	Lo Het	Hi Het	Fav	Unfav	Lo Hyst	Hyst
x1	0:1
x2	0	1	0.8		0.3
x3	1		0		0	2.5
x4	3		3		3	0
x5	2		0.5	2	1.5
x6	2		0.5	2	1.5
x7	−4:−1.5
x8	−4:−1.5

. The gravity numbers varied equally by 2.5 orders of magnitude.

When low heterogeneity $x_2=0$ is considered (Figure 11a), $\mathrm{RF}$ stays fairly constant at low $N_g$ (when the impact from gravity is negligible) and decreases when $N_g$ is large due to gravity segregation and reduced vertical sweep. At high heterogeneity ($x_2=1$) in Figure 11b, RF is generally lower, but increases significantly with increases in the gravity number. Gravity therefore exerts a positive effect as more of the low-permeable layers are swept by gravity drainage into the highly permeable layers [47].

A relatively heterogeneous case ($x_2=0.8$ ) is considered where either mobility ratio is favorable, Figure 12a, or unfavorable, Figure 12b. In both cases, increased gravity number improves RF, but the effect is more pronounced in the favorable mobility ratio case. In the unfavorable mobility case, gravity exerts little impact until the gravity number exceeds −3. RF is generally higher in favorable mobility ratio cases compared to corresponding unfavorable mobility ratio cases.

In a relatively uniform case ($x_2=0.3$) with intermediate mobility ratios $x_5=x_6=1.5$), hysteresis is varied. At low hysteresis, Figure 13a, increased gravity numbers increase RF moderately towards an optimal gravity number. At high hysteresis, Figure 13b, the optimal gravity number occurs at a lower value (for a given injected fraction). The peak can be related to improved sweep in low mobility layers, which becomes dominated by gravity segregation at the highest gravity numbers.

3.4.3. Handling Single Phase Data

The model MOD2 was trained to provide the same $RF$ during single-phase injection when varying input variables related to hysteresis and the phase not injected (for example, gas during water injection). This was performed by generating points with different input values for parameters that should not exert an influence, but with the same output. To check how effectively this was captured by the calibrated model, we ran cases in which the hysteresis parameters $x_3,x_4$ and mobility ratio parameters $x_5,x_6$ were varied individually. RF was plotted against the relevant variable and $r_w$ ranging from gas to water injection. The input parameters are listed in

Table 7. Parameter selections for MOD2 cases to check response in going from multiphase to single-phase scenarios.

	$\mathbf{Vary} {\mathit{x}}_{\mathbf{3}}\mathbf{=}\mathit{\alpha }$	Vary ${\mathit{x}}_{\mathbf{4}}\mathbf{=}\mathbf{log}\mathit{C}$	Vary ${\mathit{x}}_{\mathbf{5}}\mathbf{=}\mathbf{log}{\mathit{M}}_{\mathit{g}\mathbf{/}\mathit{o}}^{\mathbf{*}}$	Vary ${\mathit{x}}_{\mathbf{6}}\mathbf{=}\mathbf{log}{\mathit{M}}_{\mathit{w}\mathbf{/}\mathit{o}}^{\mathbf{*}}$
$x_1$	0:1
$x_2$	0.8
$x_3$	0:2.5	1	1	1
$x_4$	1	0:3	1	1
$x_5$	1.5	1.5	0.1:2.4	1.5
$x_6$	1.5	1.5	1.5	0.0:2.3
$x_7$	−2
$x_8$	−2

and the results are shown in Figure 14.

The variation of the hysteresis parameters $x_3$ and $x_4$, in Figure 14a,b, respectively, produces relatively constant RF with gas injection $r_w=0$ (RF~0.45) and water injection $r_w=1$ (RF~0.30), although the variation of $x_3$ during gas injection produces a wider range (RF = 0.30–0.45). The levels of RF differ, as is expected, since gas and water injection perform differently. Varying the gas-oil mobility ratio $M_{g/o}^{*}$ (via $x_5$ in Figure 14c) produces much less change in RF with water injection (RF~0.3) than gas or WAG injection. Similarly, varying the water–oil mobility ratio $M_{w/o}^{*}$ (via $x_6$ in Figure 14d) produces much less change in RF with gas injection (RF~0.45) than with water or WAG injection.

3.5. Application to a 3D Model

The dataset used to train MOD1 and MOD2 was based on a 2D layered model. By obtaining effective parameters from 3D reservoir models, we can predict RF from MOD1 and MOD2. An artificial 3D heterogeneous model was considered with curvature, faults and three layers (see Figure 15). The average permeability, average porosity and layer thickness are listed in

Table 8. Input and calculated parameters for the 3D model. ‘Low/hi’ indicates degree of hysteresis in parameters x₃, x₄, while for parameters x₅ to x₈, values are calculated from two oil viscosities.

Layer	${\mathit{K}}_{\mathit{x}} \mathbf{\left[}\mathbf{m}\mathbf{D}\mathbf{\right]}$	$\mathit{\varphi } \mathbf{[}\mathbf{-}\mathbf{]}$	$\mathit{h} \mathbf{\left[}\mathbf{m}\mathbf{\right]}$	${\mathit{x}}_{\mathbf{1}}$	0, 0.33, 0.5, 0.67, 1
1	2170	0.324	42	$x_2$	0.73
2	65.9	0.297	29	$x_3\left(low,hi\right)$	0, 2.5
3	589	0.323	33	$x_4\left(low,hi\right)$	3, 0
				$x_5\left(30,110cP\right)$	1.83, 2.40
$\mathsf{\Delta }{\rho }_{wo}$	250 kg/m3	$V_p$	$3.70\times {10}^7$ m3	$x_6\left(30,110cP\right)$	1.50, 2.06
$\mathsf{\Delta }{\rho }_{go}$	450 kg/m3	$T$	20 years	$x_7\left(30,110cP\right)$	−1.28, −1.84
$L$	1500 m	$s_{oi}$	0.84	$x_8\left(30,110cP\right)$	−0.96, −1.52
$W$	750 m	$Q$	7600 m3/d

. The vertical permeability was half of the horizontal permeability. An injector and producer were placed at a distance of 1500 m and the pore volume was based on a width of 750 m. The RF was calculated after the injection of 1.5 PV, assuming five injected fractions ($r_w$ equal 0, 0.33, 0.5, 0.67 and 1), two oil viscosities (30 and 110 cP) and low and high hysteresis (see values of $x_3$ and $x_4$).

In Figure 16, we plotted RF as a function of $x_1=r_w$ for the four cases, calculated with the 3D model, MOD1 and MOD2. In these examples, RF with water injection (0.13 and 0.35 for high and low oil viscosity) is higher than with gas injection (0.09 and 0.22) and RF is lower with more viscous oil. When hysteresis is high, WAG exhibits the best performance out of all the models (the RF peaks at $0<r_w<1$). When hysteresis is low, the 3D model features similar RF for a high WAG fraction to water injection, while MOD1 indicates water injection as optimal and MOD2 still clearly supports WAG. MOD1 predicts a level of RF and change in RF that are more similar to those seen in the 3D model than MOD2. MOD2 predicts the level of RF in low hysteresis relatively well, but appears very sensitive to adding hysteresis. This is also seen through a difference in the single-phase injection RF for the same oil viscosity: about 0.2 units for water injection and for gas injection with low amounts of viscous oil, but, more reasonably, 0.03 units for gas injection with high amounts of viscous oil. This indicates that this region of the model could be better calibrated. The water injection points in this case could be insufficiently near the single-phase points in the dataset. Furthermore, we do not expect MOD1 and MOD2 to predict the 3D model behavior identically as the geometries are not the same.

4. Conclusions

In this study, we interpreted a dataset of ~2500 points generated from single-phase and WAG injection reservoir model simulations to predict the recovery factor, RF. Two modeling approaches were selected. In MOD1, a universal dimensionless number $M^{*}$ derived from Nygård and Andersen [47] was selected as the single input variable and correlated by a polynomial expression. In MOD2, eight dimensionless numbers were used as input variables. Both choices included all the relevant input parameters to run the reservoir simulations. MOD2 was developed using LSSVM and optimized based on the best results from PSO, GA, GWO and GSA. The overall conclusions to this work can be summarized as follows:

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We demonstrated that it is possible to predict the recovery factor during single-phase and WAG injection.

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The LSSVM model optimized by GWO or PSO performed better than when optimized by GA or GSA.

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MOD2 with eight input variables clearly performed better than MOD1 with one input. Based on the total dataset, the RMSE and R² were 0.0080 and 0.998 for MOD2 and 0.050 and 0.889 for MOD1, respectively.

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The physics-based training of MOD2 was applied successfully. Single-phase injection data points were duplicated using different values in the input variables that should not affect RF, while keeping the same values for the relevant input variables and the output. The model correctly displayed little response to the irrelevant variables, but not for all conditions. Improvements could be made by adding more of these points or by training the model to include such constraints via an added penalty term in the objective function. MOD1 was analytically independent of these variables during single-phase injection.

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Plotting histograms of partial derivatives of RF showed that for most input variables, increasing them would increase RF for some conditions, but reduce RF under others, demonstrating coupling in the data.

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The best model (MOD2) predicted that under identical conditions, an optimal injected WAG fraction existed that outperformed single-phase injection (water or gas). The benefit of WAG was much clearer when gas relative permeability hysteresis was significant.

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The mobility ratios were important input variables. Increased values tended to reduce RF.

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The roles of gravity numbers, heterogeneity and hysteresis were coupled. Strong gravity effects reduced RF in low-heterogeneity cases, but improved RF in heterogeneous cases.

Finally, some limitations of the study and recommendations should be mentioned. Some parameters were not varied in the dataset and their role can therefore not be predicted by the model. This includes the reservoir dip angle, capillary forces, starting WAG in tertiary mode (with some period of gas or water injection first), gas miscibility and tapering (changing WAG ratio with time). Furthermore, the heterogeneity of the model was mainly described by one parameter although porosity and vertical permeability appear in other dimensionless numbers. It could also matter how the heterogeneity appears, i.e., permeability increasing up or down. We note that several parameters that were not varied are included in a physically meaningful manner into dimensionless numbers that were varied. Thus, considering new values of Corey parameters, the vertical-to-horizontal permeability ratio and reservoir layer configurations are accounted for. The proposed methodology can be applied to predict the performance of other EOR techniques as well, but requires a similar development of representative dimensionless numbers and parameters capturing the EOR effect.

It is recommended to explore the potential of physics-based machine learning [50,51] in combination with dimensionless numbers describing complex systems, as was considered in this study. The methodology of modifying the dataset as described offers the advantage of applying ML algorithms in their standard form. On the downside, the dataset is enlarged and the physics are added around the specific datapoints, not as inherent part of the model.