Similitude Analysis of Experiment and Modelling of Immiscible Displacement Effects with Scaling and Dimensional Approach

This paper presents the use of scaling and dimensional analysis to assess the viability of conventional modelling of immiscible displacement occurring when water is injected into the oil-saturated, porous rock—a conventional secondary oil-recovery method. A brief description of the laboratory tests of oil displacement with water performed on long core sets taken from wells operating on a Polish oil reservoir was presented. A dimensionless product generator based on dimensional analysis and Buckingham Π theorem was used to generate all possible combinatorial sets of dimensionless products for physical variables describing the phenomenon. The mathematical model of the phenomenon was transformed to its dimensionless form, using a selected set of the products. The results of the laboratory tests were analyzed as functions of the products. Statistically verified quantities describing both dependent and independent experiment variables were subject to a regression analysis to study dependencies of the experimental results upon selected dimensionless products. The degrees of the dependencies were determined and compared with the model coefficients. The conclusions are drawn for the purposes of model application to correctly describe the laboratory and, consequently, field scale processes of immiscible oil displacement by water.


Introduction
Immiscible displacement is a phenomenon occurring, e.g., during the process of water injection to an oil field as a secondary method of oil recovery [1,2]. Effectiveness of this phenomenon is estimated in laboratories by performing experiments on bore-hole cores [3,4]. The obtained results of laboratory tests are typically used to model this phenomenon with full-scale reservoir models [5,6]. However, to quantitatively characterize the phenomenon, it is necessary to apply appropriate models of laboratory experiments. To the authors' best knowledge, there are relatively few papers reporting immiscible displacement experimental results associated with their modelling and analyses of its correctness [7,8]. This paper presents a unique report on the subject with regard to the carbonate rocks and reservoir fluids found in Polish petroleum formations.
Both small-and large-scale modelling are conventionally performed by an approximate description of the real-world phenomena. To assess the viability of such modelling, a scaling and dimensional analysis is performed as applied to the immiscible flow data obtained from the laboratory experiments. Scaling laws are derived by dimensional analysis from the general standpoint according to the Buckingham Π theorem [9].

Laboratory Tests
This paper takes advantage of the results of five laboratory tests of oil displacement by water performed on various long core sets [16]. Each set consisted of four cores arranged according to diminishing permeability. The cores were of constant sizes: 2.5 cm in diameter and 5 cm in length. The cores in the first four tests featured similar permeability parameters, ranging between 30 and 60 mD, while in test No. 5, cores with a bigger permeability (up to 400 mD) were used. Prior to starting the displacement tests, all cores were saturated with water and then with oil, to take the irreducible water into account in tests and to estimate the effective porosity of the cores.
Displacement experiments differed between themselves in the rate and total volume of injected water. In tests No. 1 and No. 2, water was injected at the rate of 0.05 cm 3 /min, and altogether 1.06 of the cores pore volume (PV) was injected. In tests No. 3 and No. 4, water was injected at the same rate as in previous tests, while altogether 1.09-1.10 of the cores PV was injected. Test No. 5 differed from the others in the injection rate, which was 0.03 cm 3 /min, and, in a total 1.08 of the cores, PV was injected into it. The same reservoir fluids of known properties were used in all the tests. The tests were performed under constant initial and outflow pressure (a boundary condition) of P ini = P out = 424 bars and constant temperature of T = 119 • C. The other boundary condition referred to the constant injection rate at the inflow end of the core sets.
A list of physical variables describing displacement experiments in relation to the laboratory tests, together with the variable dimensions, is shown in Table 1.
As the pressure variation in the core sets was relatively low (below 1.2 bar, equivalent to approx. 0.003 of the initial pressure, P ini ) during the experiments, the above variables of viscosity, µ, density, ρ, and interfacial tension, σ, determined in Reference [17], were treated as constant values. The relative permeabilities for reservoir oil and water were determined from separate measurements on rock samples of the same formation [17]. It should be noted that the rock of the cores is water-wet [17]. As the injected water used in the tests is the original reservoir water, no changes of core wettability are expected.
Because the considered experiments are carried out on batteries of cores with a diameter much lower than their lengths and the boundary conditions (the injection rate at the inflow end and the pressure at the outflow end) were assumed to be transversely constant, the fluid flow can be modelled as a 1D phenomenon in first approximation. Consequently, only one parameter related to the position dimension was among parameters affecting this displacement phenomenon, i.e., the length of the core batteries-L.
The next parameters include the following: the final time of experiment performance-t; averaged properties of cores-their porosities, φ (not specified in Table 1, as they are dimensionless); absolute permeabilities-k; and phase permeabilities of oil and water defined at the residual saturations of reservoir fluids.
Other parameters, describing the process of oil displacement with water, apply to properties of reservoir fluids, such as the phase pressures, viscosities, densities, and interface tension. The last considered parameter, substantially affecting the performed experiments, was the water injection velocity-v w,inj , calculated directly from the injection rate divided by the area of the core cross-section.
The main results of the analyzed laboratory tests consisted in the obtained oil outflow and the displacement coefficient as functions of the injection time. The characteristic displacement coefficient at 1 PV of injected water amounted to 56.1%, 56.7%, 56.9%, 56.5%, and 52.8% in test Nos. 1, 2, 3, 4, and 5, respectively.

Dimensionless Π Products for Immiscible Displacement
A universal generator of Π products was developed based on the Buckingham Π theorem. The algorithm implemented in the generator was adopted from the literature [18]. Figure 1 presents the block diagram of this algorithm. Moreover, Π products are generated from all possible combinations (without repetition) of dimensional variables, a i , of different dimensions by writing out k-element string from the n-element a i set. Every k-element string is a base of the whole Π product set. Every such set is generated by complementing k-element string with one of the n − k remaining elements of the a i set. Thus, there are n − k quantities equal to the products of k + 1 dimensional variables a i , each raised to an exponent that is determined from the condition of Π being dimensionless with respect to each of containing − dimensionless  products; however, the effective number of the sets is smaller than , as some of the original sets are identical. In the analyzed case of immiscible displacement, input variables of the algorithm included the list of n = 13 physical variables from Table 1, and basic k = 3 dimensions from which the dimensions of these variables are derived, i.e., L-length, M-mass, and t-time. Therefore, according to the Buckingham theorem, ten (n − k) i parameters, where i = 1, 2, …,10 can be defined to describe the immiscible displacement experiments.
Altogether there are 173 possible sets of  products for immiscible displacement that were generated. Table 2 presents examples of six sets which were used in the further analysis where the basic model equations applied to describe the phenomenon were transformed to a dimensionless form. In the analyzed case of immiscible displacement, input variables of the algorithm included the list of n = 13 physical variables from Table 1, and basic k = 3 dimensions from which the dimensions of these variables are derived, i.e., L-length, M-mass, and t-time. Therefore, according to the Buckingham theorem, ten (n − k) Π i parameters, where i = 1, 2, . . . ,10 can be defined to describe the immiscible displacement experiments.
Altogether there are 173 possible sets of Π products for immiscible displacement that were generated. Table 2 presents examples of six sets which were used in the further analysis where the basic model equations applied to describe the phenomenon were transformed to a dimensionless form.

Immiscible Displacement Equations
Representation of a mathematical model in a dimensionless form enables determination of coefficients, on which this model depends. Equations, conventionally referred to in the case of immiscible displacement in the water-oil system, are the two following differential equations resulting from the equation of continuity and from the Darcy's law [19]: supplemented with the saturation confining relationship: oil/water potential definitions: and capillary pressure definition: where φ-porosity; S w,o -water/oil saturation; t-time; µ w/o -water/oil viscosity; x, y, z-coordinates; k w/o -water/oil phase permeability; Φ w/o -water/oil potential; P o/w -water/oil phase pressure; P c -capillary pressure; ρ o/w -water/oil density; ∆ρ-difference of fluids density; and g-acceleration of gravity.
In the 1D horizontal case, the gravity term in the formulae for potentials is neglected, and the above equations take the following form: Here, the porosity, φ, and permeability, k, are assumed constant and equal to their average values. Moreover, the pressure dependence of viscosities, µ o/w , is assumed negligibly small, and corresponding terms in (6) and (7) are omitted.
Transformation of the discussed mathematical model to a dimensionless form was performed by using various sets of dimensionless variables' definitions. The most convenient and natural one turned out to be the following: where v w,inj -injection velocity, L-cores length, and S wr -residual water saturation; − position: While the definitions of dimensionless position, time, and saturations were of natural and conventional type, those of the pressures were more arbitrary and related to the sets of Π products of Table 2.
Using the above definitions, the Equations (6)-(9) are transformed to the dimensionless form of (10)- (13): Here, the interface tension, σ, is assumed to be a negligible function of the water saturations, S w , in the observed range of S wr < S w < 1 − S or .
As a result of the above transformation, three dimensionless coefficients of Equations (10), (11), , are identified as Π products of Set No.
17, i.e., Π 6 ,Π 7 , and Π 10 . We assume that two systems (the real one and mathematical model) are similar, and the model is scalable when the following are present: − Dimensionless initial and boundary conditions in the model and in the real system are identical; − Relative permeabilities k rw , k ro and the function J(S * w ) are the same functions in the model and in the real system, where k rw = k w k , k ro = k o k ; − The assumed dimensionless parameters are the same function of the reduced water saturation, S * w , in both systems, from which it results that P * o , P * w , S * o , and S * w are the same functions of t * and x * .

Model Parameters
Characteristic parameters that describe the model used to simulate laboratory tests are listed as M i , i = 1 to 7, on the right part of Table 3. M 1 and M 2 are independent variables that characterize spatial and temporal results of the tests. As no measurements were done for intermediate positions (0 < x* < 1), M 1 is not used for further analysis. M 2 describes test results as a function of time and is used to analyze time-dependent measurements, such as total reservoir fluid outflow. Parameters M 3 and M 4 refer to initial conditions of the experiments that were fixed for all the experiments. Parameters M 5 , M 6 , and M 7 are essential coefficients of the model Equations (9), (10), and (12), respectively. Table 3. Dimensionless parameters of the 1D immiscible displacement process. On the left is the generated set of products, and on the right are the parameters on which model results depend.

Set No. 17
Variability in Experiments Model Parameters Comment The complete set of dimensionless parameters generated above as Set No. 17 and selected for the comparison with model parameters is listed as Π i , i = 1 to 10, on the left side of Table 3, together with each one's variability in the experiments. As show in the table, parameters Π 2, Π 6, Π 7, Π 8, Π 9, and Π 10 take different values in the analyzed experiments, and four of them are equal to the corresponding model parameters: Π 2 = M 2 , Π 6 = M 5 , Π 7 = M 6 , and Π 10 = M 7 . Meanwhile, the other two (Π 8 and Π 9 ) are expected not to influence the experimental results.
It is worth nothing that definite physical meanings can be ascribed to some of the above Π products. They follow from the below relations, (14) and (15): where Re w/o is the Reynolds number for water/oil, according to Formula (16): and We w/o is the Weber number for water/oil, according to Formula (17): As a consequence of Reynolds and Weber number meanings given above, and of the constant value of Π 1 , Π 6/7 corresponds to the ratio of interface tension to viscous forces, while Π 8/9 is a measure of the relative importance of the fluid inertia to their interface tension. It should be noted that values of the parameters Π 6 and Π 7 , as well as Π 8 and Π 9 , are strongly correlated (co-dependent) in the analyzed experiments. As a consequence, only two of them (Π 6 and Π 8 ) are taken into account in the dependency analysis below.
Two types of experimental results are used for the quantitative analysis below: − The relative oil outflow velocity, v r = (q o /A)/v w,inj , where q o is the oil outflow rate, and A is the cross section area of the cores; − The relative total oil outflow, N = N p /N p,max , where N p is the current total oil outflow, and N p,max is the maximum total oil outflow.

Dependence Analysis
The conventional regression analysis is used to study dependencies of experimental results (v r and N) upon the dimensionless parameters (Π 2, Π 6, Π 8, and Π 10 ). As regression diagnostic tests, the distributions of both dependent (v r and N) and independent quantities (Π 2, Π 6, Π 8, and Π 10 ) are determined as their histograms (shown in Figures 2-7) and checked against their normal-like form. The appropriate sets of experimental data points are restricted by rejecting of outliers. In the case of N and Π 2 , the residuals of their distributions were shown after subtracting the fitted trends of these quantities.
Energies 2020, 13, 5224 9 of 15 and N) upon the dimensionless parameters (П2, П6, П8, and П10). As regression diagnostic tests, the distributions of both dependent (vr and N) and independent quantities (П2, П6, П8, and П10) are determined as their histograms (shown in Figures 2-7) and checked against their normal-like form. The appropriate sets of experimental data points are restricted by rejecting of outliers. In the case of N and П2, the residuals of their distributions were shown after subtracting the fitted trends of these quantities.

Relative frequency [-]
П 6 [-] It is concluded that all the analyzed quantities except П10 satisfy the requirement of their distribution, being appropriate for the conventional linear regression analysis applied to vr versus П6 and П8, as well as N versus П2.
The analyses of adjusted R square and residual distributions for various models of the regression fitting result in the selection of a bilinear model of vr vs. П8 and П6 and a quadratic model of N vs. 1/П2 (the reversal of П2). Detailed results of the regression analysis are presented in Tables 5 and 6 for vr and N, respectively. Note that the format and entries of these tables follow the generally accepted convention of the regression results. The quality of regression fittings is shown as vr vs. П8 and П6, and N vs. П2 in Figures 8, 9, and 10, respectively.  It is concluded that all the analyzed quantities except Π 10 satisfy the requirement of their distribution, being appropriate for the conventional linear regression analysis applied to v r versus Π 6 and Π 8 , as well as N versus Π 2 .
The analyses of adjusted R square and residual distributions for various models of the regression fitting result in the selection of a bilinear model of v r vs. Π 8 and Π 6 and a quadratic model of N vs. 1/Π 2 (the reversal of Π 2 ). Detailed results of the regression analysis are presented in Tables 5 and 6 for v r and N, respectively. Note that the format and entries of these tables follow the generally accepted convention of the regression results. The quality of regression fittings is shown as v r vs. Π 8 and Π 6 , and N vs. Π 2 in Figure 8, Figure 9, and Figure 10, respectively.  It is concluded that all the analyzed quantities except П10 satisfy the requirement of their distribution, being appropriate for the conventional linear regression analysis applied to vr versus П6 and П8, as well as N versus П2.
The analyses of adjusted R square and residual distributions for various models of the regression fitting result in the selection of a bilinear model of vr vs. П8 and П6 and a quadratic model of N vs. 1/П2 (the reversal of П2). Detailed results of the regression analysis are presented in Tables 5 and 6 for vr and N, respectively. Note that the format and entries of these tables follow the generally accepted convention of the regression results. The quality of regression fittings is shown as vr vs. П8 and П6, and N vs. П2 in Figures 8, 9, and 10, respectively.        The main results of the regression analysis imply that the experimental results for the oil output velocity, v r , depend in the significantly higher degree on Π 6 (p-value = 0.0125) than on Π 8 (p-value = 0.0378). According to the physical meanings of the products given in Section 5, the experimental results for the oil output velocity, v r , are determined mostly by the relationship between interfacial tension and viscous forces, while they are weakly dependent upon the ratio of inertial forces to the interfacial tension. The latter dependence results from both the low flow velocity of the reservoir fluids observed in the displacement experiments ( Π 8 ∼ v w,inj 2 ) and a large value of Π 1 ( Π 8 ∼ 1 Π 1 , Π 1 ≈ 1 × 10 6 ). The analysis of N vs. Π 2 shows the significant dependence of the total oil production upon the quadratic function of 1/Π 2 (with p-value = 0.0210 for linear term and p-value = 0.0002 for quadratic term). Because Π 2 has a direct meaning of the relative range of displacing fluid (injected water), the above quadratic dependence of N vs. Π 2 indicates dispersive effects of the displacement process.
The above results lead to the following conclusions: -The experimental results are consistent with the model predictions, i.e., explicit dependence upon the following: (1) Model coefficients Π 6 (and Π 7 ); (2) Model independent variables (experiment duration) Π 2 -linear dependence on Π 2 .
-The dependence of experimental results upon other parameters (such as Π 8 and Π 9 ) that do not enter the model description are much weaker and may be explained by small effects from the inertial forces under the conditions of small fluid velocities; in principle, including inertial effects goes beyond the Darcy law of fluid flow in the porous media.
Energies 2020, 13, 5224 13 of 15 -Non-linear dependence of the total oil outflow upon the displacement time cannot be taken into account in a simple 1D flow model with no dispersion effects; a typical smoothing-out of the displacement front obtained from such models results from a numerical dispersion defect of the standard numerical solvers of the flow equations; the correct modelling of the physical dispersion effect, responsible for the above mentioned non-linear dependence of the total oil outflow upon the displacement time, can be achieved by applying 3D model of non-uniform transport properties of the porous media and by explicit modelling of the physical dispersion effects. - The last two points indicate deficiency of the modelling approach analyzed in the paper.

Summary and Conclusions
Correctness of using numerical modelling to quantitatively characterize the immiscible displacement phenomenon occurring in the water-oil system was discussed in the paper by studying the results of experimental tests on core sets with scaling and dimensional analysis. To this end, a complete procedure including generation of dimensionless Π products as of the Buckingham Π theorem, identifying the dimensionless parameters of the models, and regression analyses of the experimental results dependence upon the dimensionless Π products were applied.
The following conclusions were drawn from the obtained results: -Using conventional mathematical flow description and 1D approximation, it is reasonable to model laboratory tests of immiscible displacement in the water-oil system of bore-hole cores. - The experimental results are consistent with the model predictions, i.e., they significantly depend upon the following: Explicit model coefficients (Π 6 and Π 7 ) related with the ratio of the Reynolds number to the Weber number that is a measure of the relationship between interfacial tension and viscous forces; Model independent variables (experiment duration-Π 2 ).
-The dependence of experimental results upon other parameters ((Π 8 and Π 9 -corresponding to the ratio of inertial forces to the interfacial tension-the Weber number) that do not explicitly enter the model description is much weaker and results from both the low flow velocity of the reservoir fluids observed in the displacement experiments ( Π 8/9 ∼ v w,inj 2 ) and a typically large value of Π 1 parameter ( Π 8/9 ∼ 1 Π 1 , Π 1 ≈ 1 × 10 6 ). -Non-linear dependence of the total oil outflow upon the displacement time (Π 2 ) cannot be taken into account in a simple 1D flow model with no dispersion effects. - The last two observations show the imperfection of the standard modelling approach used to analyze the immiscible displacement of oil by water in porous media. - The potential way of model improvements consists in including the following: Inertial effects beyond the Darcy law of fluid flow in the porous media; Physical dispersion effects by applying 3D model of non-uniform transport properties of the porous media and by explicit modelling of the dispersion phenomena.

Conflicts of Interest:
The authors declare no conflict of interest.

Nomenclature
A cross-section area of the cores g acceleration of gravity J(S * w ) J-Leverette function k absolute permeability k w/o water/oil phase permeability k orw oil permeability at irreducible water saturation, S wr k wro water permeability at residual oil saturation, S or L total length of the core assembly length dimension M mass dimension M 1 , M 2 , M 3 , M 4 , M 5 , M 6 , M 7 dimensionless model parameters N number of observations relative total oil outflow n number of physical variables N p current total oil outflow N p,max maximum total oil outflow P c capillary pressure P o/w oil/water phase pressure P o/w,ini initial oil/water pressure P o/w,out outflow oil/water pressure P * o/w dimensionless oil/water phase pressure q o oil outflow rate S o/w oil/water saturation S * o/w reduced oil/water saturation S o/w,r residual oil/water saturation t final time of experiment performance time independent variable t * dimensionless time independent variable T temperature v r relative oil outflow velocity v w,inj water injection velocity x, y, z coordinates independent variables x* dimensionless x-coordinate variables Symbols φ porosity Φ o/w oil/water potential ∆ρ difference of fluid densities µ o/w oil/water viscosity Π 1, Π 2, Π 3, Π 4, Π 5, Π 6, Π 7, Π 8, Π 9, Π 10 dimensionless Π products ρ o/w oil/water density σ oil-water interface tension