Sina Lohrasb, Radzuan Junin

a Department of Petroleum Engineering, School of Chemical and Energy Engineering, Universiti Teknologi Malaysia, UTM, Skudai, 81310, Johor, Malaysia

b Institute for Oil and Gas, Universiti Teknologi Malaysia, 81310, Johor, Malaysia

Keywords: Analytical method Carbonate acidizing Pore volume to breakthrough Hydrochloric acid

ABSTRACT Acidizing treatment is considered as a significant process in the oil well stimulations to form wormholes in carbonate formation in order to enhance the reservoir fluid production.Obtaining the number of pore volumes to breakthrough is an important objective in matrix acidizing, for it contributes to determining the wormhole characteristics such as type, shape, and size.Finding this number in experimental works requires a considerable amount of time, energy and cost.Therefore, this study aimed to establish an analytical method in which a reasonable result is achieved for the number of pore volumes to breakthrough.This purpose is accomplished by solely implementing acid and formation properties without performing any experimental works.The process of wormhole creation is done through developing a numerical model by utilizing the conservation of mass law method in which the carbonate core is considered as a closed system and the overall mass in the system as constant during the acid injection process.Furthermore, a constant number is added to the mathematical part of the model in order to eliminate the dimensionless Damk?hler number which is supposed to be calculated experimentally.The results of the numerical procedure of the model are further compared to four other experimental works, which led to calculating the average accuracy of this model that is shown to be 95.98%.This study puts forward a comprehensive numerical model to estimate the number of pore volumes to breakthrough with an acceptable accuracy rate merely through implementing known acid and core properties.

1.Introduction

Hydrochloric acid is the most commonly used acid in carbonate acidizing.The procedure of modeling in the study comprises scaling the acid injection rate by the number of wormholes created per unit surface area through using conservation of mass method [1].This method preserved the same relative rates of transport and reaction within the wormholes and thus, provided the same wormhole structure.Built upon this method, the carbonate core is considered to be a closed system and the overall mass in the system is constant throughout the process of injection and wormhole creation.It is assumed the wormhole is cylindrical.Based on this method, the created wormhole volume is shown through the injected mass of acid at the surface of carbonate core minus output mass of injected acid at the end of wormhole after reaction time.Also, the Damk?hler number is used to establish the mathematical part of the model [2-7].The Damk?hler number is named after the German chemist Gerhard Damk?hler.It is a dimensionless number and is described as the ratio of flow time scale to the chemical time scale [8].Furthermore, the number of dissolution rate constant is added to the mathematical part of the model in order to eliminate the dimensionless Damk?hler number which is supposed to be calculated experimentally.This constant number has a direct relation with Damk?hler number.This number indicates the rate of wormhole growing by chemical reaction time [9].There is a particular constant number equation, which is linear, for each acid and rock [10,11].The rock properties, acid concentration, and the temperature determine the optimum injection rate.Among them, core and acid properties are the most influential factors.In slow reactions in carbonate cores, acid mass needs to increase significantly and on the other hand, rate must decrease meaningfully.Also, the higher the temperature is, the higher the reaction rate of dolomite will be.Many investigations have been carried out regarding the optimum injection rate in carbonate acidizing [12-19].

In addition, concentration changes of injected acid that is reacted with carbonate core are accounted to calculate the chemical part of the model.Chemical equation balance rule is used to calculate the changes.The equation consists of the chemical formulas of the reactants and the chemical formula of the products.Because injected acid amount is not as much as carbonate core volume and cannot dissolve the whole core, limiting reagent method is used for molar mass and weight of chemical reaction products.The limiting reagent or limiting reactant in a chemical reaction is the substance that is entirely consumed when the chemical reaction is done [20].To develop the model, an average of the constant equation for hydrochloric acid (HCl) has been utilized.Porosity constant is another parameter that is implemented in the mathematical part of this study.To calculate the porosity constant number, the average porosity of the core is used.This number indicated the porosity changes in the model throughout acidizing.Only known properties of injected acid and carbonate core are used in the final equation of the model, which calculates the number of pore volumes to breakthrough.The obtained number has a satisfactory accuracy rate with no need for any experimental works.

2.Development of the model

Consider a horizontal model of porous media, which has the length ofXand cross section ofA, and is in a stable condition with injected fluid rate ofqand concentration ofCinjected to one side of this porous media as a solvent fluid flow to create a wormhole.The wormhole volume is calculated assuming to have a capillary tube shape.The system is depicted in Fig.1.

The starting point of the model is writing the mass conservation equation for mixture acid concentration rate of fluid injected.The overall equation for this model is:

Whereq1andq2are fluid injection rate and rate of fluid coming out of the core,C1andC2are injected fluid concentration and product fluid concentration,ais wormhole cross section area,lis length of wormhole andtis time.

Dividing by concentration changes, the conservation of mass or continuity equation defined as below:

Equation (1) is further expanded using the following equations in order to determine the number of pore volume to breakthrough.

Wheremais mass of injected acid,Viis the volume of injected fluid acid andqis fluid injection rate.Put forward by Fredd and Fogler [3], the acid injection rate and pore volume to breakthrough by definition are calculated by Equations (5) and (6).

Whereπis equal to 3.14,dis the wormhole diameter,lis the wormhole length,Kis overall dissolution rate constant of acid and rock,NDais Damk?hler number andPVBTis pore volume to breakthrough.The pore volumes space of rock that acid penetrate in it is calculated by Equation (7).

Fig.1.Horizontal model of wormhole created in porous media.

WhereVwis the volume of wormhole.The differential of time equals the acid reaction time with the rock.The wormhole is created in the formation at this time.So, the total processing time in this model is shown byT.Therefore Equation (1) is developed as below.

For this model the assumption is, the acid is injected in the core with the flow rate ofqand due to being a single shot injection with a certain amount of acid to create wormhole in the core and there is no flow rate at the other side of the core andq2is equals zero.Therefore Equation (9) is drive as

The concentration of acid before the injection is shown asC1.The acid concentration is inconsistent and it changes during reaction with the rock.In this model, the injected acid is assumed to be the only fluid that flows in the rock and the acid has a reaction with the rock to make a wormhole.Thus, the acid concentration changes during theTtime.The changes in fluid flow concentration are accounted asΔC.

According to definition of pore volumes to breakthrough, the number of pore volumes to break through (PVBT) as the ratio of the volume of fluid injected to achieve channel breakthrough to the volume of the pore space in the core as shown in Equation (6).TheXApart of this equation is the whole core volume that is presented byVc.Also, the injected acid concentration is defined as mass of acid (ma) divided by volume of fluid injected to the core (Vi).

The following changes are done to Equation (10) to create an equation for pore volume to breakthrough.

The termVc?is further multiplied by both side of Equation (13) and theqmapart is shifted to the other side of the equation.WhereVcis volume of the core and?is porosity of the core.

Therefore,

In Equation (5), the termKindicates the dissolution rate constant reaction between core and acid with dimension of [L.T-1] andπdlKin Equation (5) shows the rate of wormhole creation in core with dimension of [L3.T-1] that is always calculated by experimental work for every acid and rock type.The wormhole creation rate in the rock is as follows.

So, Equation (5) changes into

Table 2 Concentration of injected acid and product fluid.

Table 3 Calculated coefficient numbers by Equation 27.

Fig.2.Coefficient number and Damk?hler number relation for 0.5 M HCl.

Table 4 Overview of the others works.

And Equation (15) becomes

To make the analysis less complicated, the dissolution rate constant (DR) is shown as follows:

And Equation (19) is transformed into

So, the main equation to calculate the number of pore volume to breakthrough is presented in Equation 21

2.1.Dissolution rate equation

In order to calculate the concentration of acid before injection, the molarity of acid with the dimension of [mol.L-3] is multiplied by the molar mass of acid with the dimension of [m.mol-1].For example, the concentration of acid before injection is 0.5 MHCl(0.5 mol/lit) multiplied by the molar mass ofHCl(36.46094 g/mol) that is equal toC1(0.018 g/cm3).The chemical balance equation of the reaction between hydrochloric acid and limestone is:

To determine the mass of other compounds, the amount of acid mass is accounted as shown in Equation (22).Table 1 presents the example of primary mass amounts.Furthermore, the mass of limestone (CaCO3) core is 259 g and is calculated by size, probity and density of the core [3].The post-reaction injected fluid flow concentration is calculated by Equation (23) with mass and volume of chemical equation products.

Fig.3.Comparison of the model with Wang, Hill [21] results in dolomite core.

Fig.4.Comparison of the model with Buijse and Glasbergen [22] results.

Fig.5.Comparison of the model with Etten [24] results.

Table 5 Comparison of the model with Wang, Hill [21] results in limestone core.

The amount of product fluid concentration will not change for each amount of acid mass after the reaction.The reason is the limiting reagent, which is the substance that is totally consumed when the chemical reaction is fully done.This reagent limits the amount of created product because the reaction is unable to continue without it.Injected acid is always the limiting reagent compound in carbonate acidizing.

Similar process is accomplished on the reaction of dolomite with hydrochloric acid (HCl) to calculateC2for each reaction.

Table 2 shows the concentration amounts of injected acid and product fluid for each acid and core that used in this study.

Table 6 Comparison of the model with Izgec, Zhu [23] results.

2.2.Coefficient number for model calibration

A dimensionless coefficient number (Co.) is further added in order to develop the model.The coefficient number is created by equalizing the number of pore volumes in the model with the actual number of pore volume to the breakthrough calculated by Fredd and Fogler [3].

Table 3 shows the coefficient number that is calculated through Equation (25) for0.5 M HClwhose Damk?hler number had been provided by Fredd and Fogler [3].

The relation between acid injection rates and Damk?hler numbers with the coefficient number is analyzed so that the dissolution rate (DR) as an experimental factor can be excluded in the model.The outcome indicates that these coefficient numbers have a direct relation with Damk?hler numbers as shown in Fig.2.

The following equations are calculated for hydrochloric acid (HCl) to represent the numerical relation between the coefficient number and Damk?hler number.

For

To find the final equation useable for hydrochloric acid with any concentration and carbonate cores, Equation (27) is the concluding equation in which the final coefficient number is discovered to be 1.3791.

2.3.Evaluation of the model

For the deviation and accuracy evaluation of the model, the final results are compared to the pore volume to breakthrough results provided by Wang, Hill [21], Buijse and Glasbergen [22], Izgec, Zhu [23] and, Etten [24] (as listed in Table 4) on different hydrochloric acids and carbonate cores.Figs.3-5 and Tables 5 and 6 compare the results.

2.4.Model evaluation by deviation and accuracy

The accuracy of the model is measures using the standard deviation formula as presented in Equations (28)-(30).Error, deviation and accuracy are shown in Table 7.

Table 7 Evaluation of the model.

WhereSDis the standard deviation,MAPEis the mean absolute percentage error,PVactualis the number of pore volumes to the breakthrough that was calculated experimentally;PVmodelis the number of pore volumes to the breakthrough that is calculated by this model andNis the number of observations.

3.Conclusions

By comparing the results with 96 samples from four experimental works, the average accuracy of the model has been measured as shown to be 95.98%.Comparing the results of the model to the first group of experimental results indicates that the accuracy of dolomite equals to 90.81% while standard deviation shows that the model have better results when temperature is increased.Furthermore, the comparison with the second group of experimental results shows that the accuracy of 1.5 MHClwith limestone core is very high (99.55%) and again increasing of temperature leads to better results.Also, the comparison with the third and fourth group of data reveals that core porosity plays a significant effect on the accuracy of the model.

Acknowledgments

The authors would like to thank the Department of Petroleum Engineering, School of Chemical and Energy Engineering, Univesiti Teknologi Malaysia (UTM) for providing support.

Nomenclature

ΔCFluid concentration changes (gr/cm3)

aCross section of the wormhole (cm2)

C1Acid concentration before injection (gr/cm3)

C2Fluid concentration after reaction (gr/cm3)

CaCO3Limestone (Calcium Carbonate)

CaMg(CO3)2Dolomite (Calcium Magnesium Carbonate)

CoCoefficient number

dWormhole diameter (cm)

DRDissolution rate constant (cm3/min)

HClHydrochloric acid (HCl)

KOverall dissolution rate (cm/min)

lWormhole length (cm)

MMolarity (mol/lit)

maMass of acid (gr)

MAPEMean absolute percentage error (%)

NNumber of samples

NDaDamk?hler number

?Porosity

?cPorosity constant

PV1Pore volume of rock before acid injection (cm3)

PV2Pore volume of rock after acid injection (cm3)

PVBTPore volume to breakthrough

qinjection rate (cm3/min)

SDStandard deviation

TProcessing time (min)

tTime (min)

VcCore volume (cm3)

ViVolume of injected fluid (cm3)

VwWormhole volume (cm3)

xlength of core (cm)

πPi number (3.14)

Appendix A.Supplementary data

Supplementary data to this article can be found online at https:// doi.org/10.1016/j.petlm.2019.08.001.

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