DPR polymer gel treatment in oil reservoirs: A workflow for treatment optimization using static proxy models

Author’s Accepted Manuscript DPR Polymer Gel Treatment in Oil Reservoirs: A Workflow for Treatment Optimization Using Static Proxy Models Mohammad Norouzi, Hamed Panjalizadeh, Fariborz Rashidi, Mohammad Reza Mahdiani www.elsevier.com/locate/petrol

PII: DOI: Reference:

S0920-4105(16)30632-5 http://dx.doi.org/10.1016/j.petrol.2017.03.018 PETROL3906

To appear in: Journal of Petroleum Science and Engineering Received date: 9 October 2016 Revised date: 9 January 2017 Accepted date: 8 March 2017 Cite this article as: Mohammad Norouzi, Hamed Panjalizadeh, Fariborz Rashidi and Mohammad Reza Mahdiani, DPR Polymer Gel Treatment in Oil Reservoirs: A Workflow for Treatment Optimization Using Static Proxy Models, Journal of Petroleum Science and Engineering, http://dx.doi.org/10.1016/j.petrol.2017.03.018 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

DPR Polymer Gel Treatment in Oil Reservoirs: A Workflow for Treatment Optimization Using Static Proxy Models

Mohammad Norouzi*, Hamed Panjalizadeh, Fariborz Rashidi, Mohammad Reza Mahdiani Faculty of Petroleum Engineering, Amirkabir University of Technology, Hafez Ave, Tehran, Iran.

[email protected]

Abstract Excessive water production from oil producing reservoirs has become one of the operators’ major concerns. Various control methods have been used in recent years; however, they all have high degrees of uncertainty. Disproportionate permeability reduction (DPR) polymer gel treatment is classified as one of the chemical methods of water reduction and the risk which is associated with this type of treatment urges us to analyze it in a more detailed and exact way.

There is a need to have an accurate analysis of gel treatment behavior before applying it in a reservoir. There are various parameters that affect the gel treatment operation and even these parameters have interactions between themselves. If they can be found, in a simulator their effect should be measured and a good combination of them should be represented to maximize the profit of the process. Most of the literature works have not represented a comprehensive work that covers all above issues. Rarely a detailed analysis for screening the effective parameters and their interaction and their effect on final profit has been represented so far, and of course using simulator for testing distinctive values on the model and Net Present Value (NPV) is an expensive and time-consuming method. In addition, it should be mentioned that most previous studies were focused just on simple models (not threedimensionalthree-dimensional, three-phase) for simplification.

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Here for filling this gap, a comprehensive study about the gel treatment operation is represented. First, different possible candidate parameters that seem to have an effect on process were selected. Then, using Pareto charts the most effective ones were selected and their corresponding interaction based on their t-values was determined. Afterward, the need to a fast and accurate model was felt, and thus, an artificial neural network was created. Finally, the effect of these parameters on NPV is discussed and using the genetic algorithm, the best combination of parameters for maximizing the profit is represented. All these parts of study confirmed the good performance and applicability of the gel treatment operation.

Keywords: screening; effective parameters; interaction; response surface; artificial neural network, optimization, NPV

1. Introduction 1.1.

Gel Treatment

Excessive water production turns into a major problem for oilfield industry when hydrocarbon fields become mature. Oil and gas wells suffer from increasingly high water influx in their mature life (Kuzmichonok, et al., 2007).

Controlling water becomes a main objective as produced water significantly impacts economic profitability of the reservoir. To reduce water influx and maintaining oil flow, the wells could be treated with high molecular weight polymers. Cross-linkers are often added to the polymers to form gels with varying strength (Zitha & Vermolen, 1999). Applications of near-wellbore gel treatments in production wells are intended to reduce excess water production without sacrificing oil production (Liang, et al., 1992). In order to achieve this, polymer gels have been developed which, when injected into production wells, reduce formation permeability to water much more so than to oil; this phenomenon is known as

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disproportionate permeability reduction (DPR). In these treatments, a polymer gel is often formed through a reaction between a polymer (usually HPAM) solution and a cross-linker agent (Figure 1), creating a semi-solid material that is capable of modifying the permeability (Al-Sharji, et al., 1999).

Relative permeability modification (RPM) is a property that is exploited during certain oilfield water shut-off (WSO) treatments, and a property whereby many water-soluble polymers and aqueous polymer gels reduce the permeability to water flow to a greater extent than to oil or gas flow (Figure 2). The kinetics for gel reaction and the representation for equilibrium constant (k) are as follows (Lockhart, 1992): [

] [

[

[

]

]

] [

[

[ ] [

] [ ]

]

]

where the possible values for exponents (Lockhart, 1992) are: X4: 2.6 X14:0.6 X16:1.0 RPM is also referred to as disproportionate permeability reduction. Some practitioners reserve the term “DPR” for relatively strong polymer gels that impart a large degree of disproportionate permeability reduction and a large reduction in water permeability. These practitioners reserve the term “RPM” for systems such as solutions of water-soluble polymers or relatively “weak” gels that impart subtler disproportionate permeability reductions and 3

subtler reductions in water permeability. However, in this study, the terms RPM and DPR will be considered synonyms. At times in the literature, DPR and RPM have also been referred to as “selective-permeability reduction” and “selective-permeability blocking” (Sydansk & Seright, 2007).

DPR treatment may be successfully applied where some degree of excessive water production problems are appearing either in oil or gas production wells. Because of physical or economic constraints, remedial chemical treatments (e.g., gel treatments) that are intended to plug water strata are often placed without zone isolation. Therefore, DPR can be applied using bullhead injection due to excessive water production instead of mechanical zone isolation, and the injected fluids and chemicals penetrate into both hydrocarbon and water zones. However, the operator should be concerned about possible damage to hydrocarbon productivity.

1.2.

Required Conditions for DPR Treatment Application

As a rule of thumb, DPR gel treatment is not advised for fully drawn down wells. Exceptions are short-term water shut-off operations and long-term water shut-off operations with several layers which have no cross-flow. The wells which are not fully drawn down are good candidates for gel treatments, especially when a reservoir with multiple zones exists (Sydansk & Seright, 2007).

1.3.

Literature Review

The application of polymer gels for conformance control and DPR usage was introduced by White, et al. (1973). Liang, et al. (1992) examined how different types of gels reduce oil and water permeability in Berea Sandstone. Liang, et al. (1995) examined several explanations for why some gels reduce water permeability more than oil permeability. Later, Nilsson, et 4

al. (1998) in a mechanistic study showed experimentally that DPR effect can be interpreted by a mechanism with segregated pathways for oil and water. Zitha and Vermolen (1999) presented a model for the flow of water and oil in granular porous media treated with polymer and gels. Liang and Seright (2000) considered a potential explanation for DPR mechanisms which is based on a combined wall-effect and gel-droplet model. Al-Sharji, et al. (2001) proposed that adsorption and lubrication effect are the main reasons for DPR when polymers (without cross-linker) are used. Denys, et al. (2001) studying the behavior of cationic polyacrylamide (CPAM) concluded that it would be attractive for water shut-off applications. Botermans, et al. (2001) and Grattoni, et al. (2001) studied the testing procedure to find a model for relative permeability curves. . Kabir (2001) provided a brief review of all chemical water/gas shut-off (WGSO) options available. Kuzmichonok, et al. (2007) verified the DPR effect of polyacrylamide gels in carbonate reservoirs. Qing, et al. (2009), Saeedi, et al. (2007) and Al-Dhafeeri, et al. (2005) studied different field applications of gel treatment and the results obtained. Sengupta (2012) introduced an organic gel and claimed it is much more economical, especially in high temperature, high salinity cases. Shen, et al. (2014) modeled the effect of temperature on gel treatment and Lee (2014) used a technical model to assess and optimize the gel treatment. Lashari et al. (2014) recognized the beneficial use of colloidal dispersion polymer gel (CDPG) in the injection well., Han et al. (2014) reviewed indepth fluid diversion (IFD) technologies such as weak gels, sequential injection for in-situ gels, colloidal dispersion gels, microgels and preformed particle gels. Brattekas, et al. (2015) worked on cross-linked HPAM gels in open fractures. Pham and Hatzignatiou (2016) studied the use of sodium silicates in naturally fractured reservoirs. Vernáez et al (2016) introduced an oil-based self-degradable gel. He tested that on lab and argued that it can be used in field cases. Jia (2016) studied the gel treatment for water shut-off in an ultralow temperature reservoir; he suggested using some kind of salts in these cases. Maki et al (2017), Tournier et

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al (2017) and Zhanget al (2017) used x-ray, image processing and ultrasonic on their studies respectively which seems new and interesting.

As can be seen, there is a huge number of studies about gel treatment in the literature. They have investigated different aspects of gel treatment from basic concepts such as relative permeability curves. Botermans, et al. (2001) and Grattoni, et al. (2001) studied the modern matters such as the effect of ultrasonic on the gel treatment process. But a great vacancy (if not defficieny) of most of the previous works is their simple models. In none of the above works a three-phase, three-dimensional model has been assumed and then the effect of different parameters on gel treatment, three-phase, three-dimensional that has discussed. Most of the above models are two-phase, and a comprehensive study about the effective parameters on gel treatment and interaction between them has not been represented until now. In most above cases there is need to an expensive and time consuming simulator while the need to a fast and accurate and inexpensive model is felt. In addition rarely a work has discussed the economic overview of the process.

Here, first some candidates have been selected and using some test data and statistical analysis the most effective ones have been determined. Then, again a comprehensive discussion about the selected parameters interaction is represented and using them an accurate and fast model is created. Finally, the effect of these parameters no economic factors is discussed and using a genetic algorithm, the optimum point for maximizing the profit is represented.

2. An Overview of Experimental Designs In an experiment, we deliberately change one or more process variables (or factors) in order to observe the effect of the changes on one or more response variables. An Experimental 6

Design is the laying out of a detailed experimental plan in advance of doing the experiment to maximiz the data extraction from minimum experiments (NIST/SEMATECH, 2013). Common types of experimental designs are Screening Designs.

2.1.

Screening Designs

In screening designs, the primary purpose of the experiment is to select or screen out the few important main effects from the many less important ones. There are mainly two designs used for screening analysis: two-level fractional factorial designs and Plackett-Burman Designs.

2.1.1. Two-level Fractional Factorial Designs These types of designs are used when the number of factors are less than 5 and consist of 2k-p runs, in which k is the number of parameters that are processed and p represents the size of the fraction of the full factorial used. These values determine the resolution of designs (Ou & Cai, 2013).

2.1.2. Plackett-Burman Designs (PB)

Plackett-Burman designs are used for screening experiments because, in a PB design, main effects are, in general, heavily confounded with two-factor interactions. These designs have run numbers that are a multiple of 4. However, PB designs are very useful for economically detecting large main effects, assuming all interactions are negligible when compared with the few important main effects (Quinlan & Lin, 2015).

2.1.3. Box-Behnken Design (BB)

This design is an independent two level design. The parameters are inserted in middle points of the borders. There are three levels for each border. Box-Behnken needs slightly more runs 7

than two level factorial designs, but it makes a more accurate and more pervasive response surface (Al-Dousari & Garrouch, 2013; Rai, et al., 2013). A three parameter Box-Behnken Design is shown in Figure 3.

2.1.4. Central Composite Design (CCD)

The Box-Wilson central cube composite design consists of a fractional or full factorial design and the central points as well as some star points. If assuming the distance between center of design to factorial points be equal to ±1, then the distance between center of design to star points is equal to ±α. α can be calculated using below equation:

  [2k ]

1

4

(1)

k is the number of data points. There are different kinds of Central Composite Design such as Inscribed Central Composite (CCI) design which will be used in this paper later (Panjalizadeh, et al., 2014). This design is a contracted one and is shown in Figure 4.

3. Model Description

Static properties: The simulations were conducted on a ¼ five spot carbonate reservoir with two productive layers and considering that the permeability of the bottom layer is twice the permeability of the top layer. Also, there is a narrow layer between these two layers which has a low horizontal permeability and is impermeable in vertical direction to represent nocross-flow situation, and thus is a suitable candidate for gel injection. For simplicity, it is assumed that the water source is a water injection well and not an aquifer, because adding an aquifer to model makes it very complex. Figure 5 shows a 3D schematic of the reservoir.

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The simulations were conducted on a Cartesian grid system with 8000 grid blocks, 20 grids in the X, Y and Z directions, respectively. Each grid-block has the dimension of 10 ft. There are two wells in two corners of the pattern, one producer at (1,1,1) and one injector at (20,20,1). The conditions used for the simulations are summarized in Table 1. Figure 6 shows the oil and water relative permeability curves. The oil and water curves in this figure intersect in a point with Sw < 50%. Thus reservoir is oil wet.

Recurrent injection/production (well schedule): Simulation starts with production of 200 bbl/day oil from the producer and injection of 200 bbl/day water from the injector simultaneously. Since surface production facilities are capable of handling a limited amount of produced water, production from well is ceased when water cut reaches a specific value (here 60%). When excessive water production problem emerges, the operator should decide what option to take in order to prolong the production period and postpone the water breakthrough to a later time. Based on the application conditions mentioned before, DPR polymer gel treatment is a desirable operation for combating high water cut.

Steps required for gel treatment are:

1) Pre-flush: To form a stable gel, there is a need to have a complete reaction between HPAM polymer and cross-linker in the near wellbore pores. However, the existence of impurities in the reservoir water (Cl-, Ca2+, etc.) disturbs this reaction and causes undesirable results (Cr3+ reacts with Cl-). Also, cation exchange capacity of clays causes Cr3+ to be exchanged with Na+ in an exchange reaction. Thus, a pre-flush with fresh water is required before gel placement (Pham & Hatzignatiou, 2016). Although the fresh water may contain trace amounts of chloride and calcium itself.

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2) Treatment fluid injection: The main fluid consists of polymer (HPAM), cross-linker (Cr3+), chloride (Cl-) and Hydrogen ion (H+) which are dissolved in water and injected in the porous medium. 3) Shut-in period: The shut-in time allowed after injection, before the well is put back on production, is critical to the success of a gel job. If the gel does not reach most of its strength, its efficacy in plugging the high-permeability layer will suffer.

Both wells will be opened in the post-treatment period with rates similar to pre-treatment period. Again the water cut starts to rise. At last, the curve of water cut versus time will match that of pre-treatment after the effect of gel disappeared.

4. Effects of Gel Treatment Results of using a DPR gel for a high water cut well are illustrated in Figure 7, Figure 8, Figure 9 and Figure 10. Figure 7 shows the water cut after the gel treatment. As this figure shows, when water cut increased to more than a specific value (here 60%) the operation started and the production well was shut down, converted to an injection well. Subsequently, gel was injected and given some time to stabilize. This period is shown with zero water cut (the line water cut without treatment shows the water cut changes of the well when no gel injection is implemented). After the operation, again the well was converted to a production well. In the very immediate step, a peak in water cut is seen which is mainly because of the remained water existence after gel reaction in the gel-affected area after full placement of the gel in the reservoir ends. Finally the true effect of gel treatment is shown by a considerable reduction in water cut. Afterward, as time passes the gel effect deteriorates and the water production increases. This continues and the gel of the layer with higher permeability is weakened more. Just after its gel is completely destroyed, water trapped behind that will be released and causes a high water production. This process is repeated for the layer with lower 10

permeability but this time, it has lower water production and also lower trapped water which is obvious in the figure.

Figure 8 is similar to Figure 7 but it shows the oil cut. Again after the treatment and converting the production well to an injection well the oil cut becomes zero, and in the beginning times of the production its value is lower than the state in which no treatment has occurred (because of the production of the remaining water after gel placement and reaction). Nevertheless, the oil cut is much higher than the state of no treatment for a certain period. The reason of the oscillation of the oil cut at the time of about 400 days is the change of the water cut which explained earlier. Figure 9 shows the cumulative water production. This figure shows that despite the oscillation in water cut in various times, the outcome of the gel treatment is a huge reduction in water production. Similar to Figure 9, Figure 10 confirms the considerable effect of gel treatment in increasing the cumulative oil production. The simulation values for polymer concentration (Cp), cross-linker concentration (Cc), chloride concentration (CCl), treatment fluid injection rate (Qp), treatment fluid injection time (tinj), shut-in time (tshut), permeability multiplier of the layers (multp) and temperature (T) are all set to the mid-point of the specified range of parameter.

These figures clearly show water production reduction by DPR gels. Not only does water cut decrease, but also oil cut increases with respect to the case which represents the normal oil recovery and no treatment. Also, cumulative water production falls below the curve of cumulative water production without treatment and has a reduction of 2607 barrels. Cumulative oil production is higher after gel treatment during which the well produces 1829 barrels of more oil. Hence, conformance control will be achieved if gel treatment is applied in a suitable candidate reservoir properly. 11

As far as DRP is concerned, different mechanisms have been proposed by different researchers. Researchers have not selected a certain mechanism for DPR action of gels in reducing the relative permeability of water more than oil. The discussion on finding the mechanism for DPR is not the subject of this paper. However, the following mechanisms are suggested to be the main ones (Liang, et al., 1995; Liang & Seright, 2000): 1) Gravity 2) Lubrication effects 3) Gel shrinking and swelling 4) Segregated flow pathways 5) Competition between capillary forces and gel elasticity 6) Wall effect 7) Gel droplet model In this study UTCHEM was used for simulation. Here, the basic mechanisms proposed for this treatment are still vague. At the same time that a mechanism is valid for one case, it may not verify another one. However, wall effect model (Zaitoun, et al., 1998) and gel droplet model (Nilsson, et al., 1998) together can probably be the governing rules of DPR. These models are selected in this study, because one is used for water wet and another for oil wet. Thus selecting both of them in a study can provide generality for that (Ealesa, et al., 2015).

5. Sensitivity (Screening) Analysis Finding the most effective parameters on gel treatment is of great importance. It is just by knowing them that a full control over the gel treatment is possible. In this part using statistical methods a comprehensive study over them will be conducted and the most effective parameters will be determined. Afterward, these parameters will be discussed and the

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interaction between them and their effect on the gel treatment operation will be discussed. Based on vast investigations about DPR gel treatment, 8 parameters were detected to be analyzed. These parameters are referred as the effective parameters in similar cases (Hajilarya & Vafaie Seftia, 2015; Baia, et al., 2015; Ye, et al., 2010). These parameters are: A: Cp (polymer concentration in the treatment fluid, wt%), B: Cc (cross-linker concentration in the treatment fluid, ppm), C: Qp (treatment fluid injection rate, ft3/day), D: ti (treatment fluid injection time (duration), day), E: ts (shut-in time (duration) after treatment, day), F: permeability multiplier (fraction), G: T (reservoir temperature, °F), H: CCl (chloride concentration in the treatment fluid, meq/ml).

Permeability multiplier is defined to show the heterogeneity of the reservoir and the permeability contrast between the two layers. Its range is between 0.5 to 1.5.

In addition, the simulator shows the dependency of the operation to these parameters. The most effective parameters are then extracted by performing screening analysis. The whole parameters (factors), their ranges (minimum and maximum) and their average values are given in Table 2. This range is selected in a way that it covers the applicable range of the parameters in the operation. In fact, beyond this range rarely is selected in real cases. Also, similar works confirm this (Hajilarya & Vafaie Seftia, 2015; Baia, et al., 2015; Ye, et al., 2010).

Two-level full factorial design for 8 parameters requires 256 runs. This number of runs selected after examining some different designs. A 2-level fractional factorial design of resolution V with 64 simulation runs (running using UTCHEM) is considered to find the influential uncertain parameters rather than performing a 2-level full factorial design which is time-consuming and is not cost-effective. The effect of each parameter is investigated on DCOP and DCWP at the end of a 600-day period. DCOP is the magnitude of difference 13

between the curves of cumulative oil production with and without treatment on the vertical axes. Similarly, DCWP is the magnitude of difference between the curves of cumulative water production with and without treatment on the vertical axes.

5.1.

Screening by Pareto Chart

The statistical task of doing screening analysis to find the most significant parameters is done by use of Pareto chart; Pareto chart, is a type of chart that represents the desired data with both bars and line schematics; in fact, the values are represented by bars in descending order, while the line is a representation of the cumulative total. The purpose of the Pareto chart is to represent the most effective factors among different factors. It represents the absolute value and a reference line. An effective value is one that extends beyond the reference line (Wilkinson, 2006). Here, t-values are used for drawing the reference line.

An enhanced tool which takes a minimum t-value to put aside the effective factors; in statistic, t-value is a parameter to show the departure of an estimated value. T-values are calculated using t-tests and it is just a ratio; a ratio of signal to noise. For a series of data it divides the difference between sample mean and null hypothesis over standard error. A tvalue of zero shows that the sample results is exactly equal to null hypothesis (null hypothesis refers to a situation that there is no relation between two measured parameters. (Martsynyuk, 2013).

Pareto chart is prepared for both DCOP and DCWP in Figure 11 and Figure 12, respectively. As can be seen, the first five factors which highly influence the result of a DPR gel treatment are Qp (treatment fluid injection rate), ti (injection time), Cc (cross-linker concentration in treatment fluid), T (reservoir temperature) and Cp (polymer concentration in treatment fluid). All these parameters result in a more desirable effect (higher COP and lower CWP) when

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increased. Qp ranks first in both Figure 11 and Figure 12, and an operator should include this factor as the most influential in a treatment design. The positions of the bars related to Cc and ti are changed in the two figures, which means that a given change in Cc has a greater effect on the amount of oil production increase than on the amount of water production decrease. Similarly, a given change in ti has a greater effect on the amount of water production decrease than on the amount of oil production increase. It should be noted that ts (shut-in time) behaves as an influencing parameter in Figure 12. Nevertheless, it is not set in the most influential parameters list and would be eliminated, because it has a negligible effect on the oil production increase. In addition, interactions will be analyzed until BD bar.

6. Description and Illustration of Interactions There are some interactions between these influential parameters. Possible effective interactions again are selected based on their t-values. This shows that interactions between some parameters are important. We therefore did not use Plackett-Burman Design for this study because in this design main effects generally are heavily confounded with two-factor interactions. This may lead to an error in finding influential parameters. Whenever interaction between two parameters is investigated, the other parameters are in their average.

6.1.

Interaction between Cc and Qp

Figure 13 shows interaction between treatment fluid injection rate and cross-linker concentration.

As it can be seen in this figure, when cross-linker concentration in the injected fluid increases at low rates, the magnitude of the positive effect is not very high and it is negligible. However, at high rates the slope of rise becomes considerably high. This is because the

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amount of cross-linker available for the reaction with polymer increases greatly and a complete reaction occurs.

6.2.

Interaction between Qp and ti

When injection time is at its minimum, increasing the treatment fluid injection rate causes an improvement in the operation. The bbl/bbl/day gradient increases gradually after rising the duration as indicated in Figure 14.

6.3.

Interaction between T and ti

The gel reaction at low temperatures is slow. Hence, inserting a huge volume of treatment fluid will not help in having a stable gel. Yet, when there is a high temperature reservoir this reaction is fast. Therefore, if a voluminous amount of gel is transported through the reservoir the reaction is so fast that the whole polymer will be converted to gel in a short time. These findings are shown in Figure 15.

6.4.

Interaction between Qp and Cp

If the treatment fluid volume is small (economic considerations) the fluid progresses a short distance in the reservoir. The blockage of water related to this amount of progression produces a little resistance to water. Therefore, changing the polymer concentration in the injection fluid in fact would have no effect on results, as illustrated in Figure 16. Also, there is possibility of an unwanted early reaction in the well when the injection rate is low, which can decrease the efficacy of the treatment. But, in cases that injection rate is high polymer concentration represents itself as a dominant parameter. It should be noted that injection rate should not be so high that polymer solution undergoes a shear degradation process.

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6.5.

Interaction between Cc and Cp

Reactants which enter into a reaction have a coefficient that determines their molar ratio. A given amount of one reactant with a coefficient equal to 1 requires this amount from the other reactant multiplied by the coefficient of the other reactant. This rule governs the gelation reaction too. In addition, it seems that cross-linker is the limiting reactant. In other words, when a little cross-linker exists in the treatment fluid this amount leads to creation of a certain amount of gel and the excess polymer remains unreacted. Nevertheless, increasing the polymer concentration in the existence of a substantial amount of cross-linker can have a good result, since polymer can utilize the available cross-linker in an efficient manner. This is shown in Figure 17.

At low concentrations of cross-linker, increasing polymer concentration produces no effect but wasting a great deal of money. For the gelation reaction to complete, there is a need to enough amount of cross-linker to be available.

7. Response Surface Construction Response surface is considered as a substitute for the simulation tool at higher levels of reservoir study including uncertainty analysis, risk analysis and production optimization. Using simulator is a time-consuming and expensive method. Thus, there is a need to introduce a method which is fast and inexpensive (Mahdiani & Khamehchi, 2016; Khamehchi & Mahdiani, 2017). For this reason, in this part a fast and accurate proxy model is created. Here, proxy models are designed and structured for the two main objective parameters, namely DCOP and DCWP. Using proxy models is a common method for modeling different engineering parameters (Mahdiani, et al., 2015; Mahdiani & Khamehchi, 2014; Mahdiani & Kooti, 2016).

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A couple of models are built for the entire 600-day period using artificial neural network (ANN). The ANN models are constructed using two schemes: 1) combined Box-Behnken (BB) and Inscribed Central Composite (CCI) designs 2) combined two level full factorial, Box-Behnken (BB) and Inscribed Central Composite (CCI) designs. The ANN models are able to predict the values of objective parameters at the end of the period. The structure of neural network is introduced based on relative error values obtained after validation runs. It is determined by undergoing a trial and error process. Utilizing back propagation feed forward network, the response surfaces are established. Subsequent to a considerable number of trial and error steps, the resulted network possesses one output layer and two hidden layers with tansig transfer function. In addition, trainlm function has been used to train the network. The properties of the artificial neural network used in this study are shown in Table 3. The network structure is depicted in Figure 18. As shown in Figure 18, five inputs accept the values of the most influential parameters which were screened and specified before. To the most effective structure of the network (the best number of nodes in the first and second layers) a sensitivity analysis is done over the number of nodes in each layer which is shown in Figure 19. The basis is selected as 10 nodes in first layer and 10 nodes in second layer. Then, a sensitivity analysis is done over the node number of each layer. As this figure shows in first and second layers 20 and 15 number of nodes results the minimum error respectively. Decreasing the number of nodes increases the average relative error of the final model. Increasing the number of nodes results a small change in the error of the final model which is too small and therefore, it is not economic to increase the number of nodes. It should be mentioned with respect to vertical axis the amount of change in errors is small. 18

The first and the second hidden layers have 20 and 15 neurons respectively. Furthermore, the output layer contains 20 neurons which are divided in two halves. The first 10 neurons of the output layer yield a value of DCOP at the end of the 600-day period and the remaining 10 neurons yield a value of DCWP at the end of the 600-day period. 20 validation runs with random values for parameters (in their specific ranges) were prepared to test the response surfaces with respect to the simulator. Table 4 shows values of relative errors associated with training and validation steps. Comparison between errors of two designs clearly demonstrates that first design is not as exact as the second one. However, modelling of BB+CCI+2LF (combined two level full factorial, Box-Behnken (BB) and Inscribed Central Composite (CCI) designs) includes 32 runs (39%) in excess of BB+CCI (combined Box-Behnken (BB) and Inscribed Central Composite (CCI) designs). Therefore, it seems logical to overlook the negligible difference in errors and deploy the time-efficient BB+CCI model. Also, comparison between these models and their little difference with the simulator response for validation data is illustrated in Figure 20 and Figure 21 for DCOP and DCWP separately.

8. Treatment Optimization The benefit of any operation depends on its efficiency in terms of economic issues. If a specific operation can offset its operational costs in the future, and it can be profitable as well, it will be justifiable to be performed. While an operation proves to be economically explainable, it should be implemented so that the corresponding financial gain reaches its maximum level. Hence, the concept of net present value (NPV) is employed here. One of the main applications of NPV pertains to engineering economics studies as wells as technical and

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economical evaluation of projects, and its positive value brings testimony to the costeffectiveness of an operation. Since operation beginning is the same in all cases of the process of gel treatment, outgoing cash flows and incoming cash flows are identical before reaching 60% water cut. Because simulation runs have been produced to the end of 600-day period, the value of NPV when no treatment is done must be subtracted from the value of NPV when treatment is done in order to get the value of treatment net present value, namely ΔNPV. It can be said that ΔNPV is the difference between cash inflows and cash outflows when absorbed gel still maintains its effect. For water resistance effect of the gel attenuates after a while and the production curves corresponding to ‘with treatment’ and ‘without treatment’ cases lie on each other. Thus, equation of ΔNPV is as follows: (2) The benefits include cost of the incremental oil produced and cost of the process on the reduced water. Nevertheless, because the value of DCWP has been calculated as negative in the response surfaces, the resulted benefit of production water reduction is included in the costs. (3) The costs include reduced produced water treatment expenditure, the cost of polymer and cross-linker, and also the cost of treatment fluid injection.

(

)

[ ] 20

(

)

(4)

All parameters that were screened in the sensitivity analysis process show their influence in this equation except for temperature. Temperature affects DCOP and DCWP. However, it cannot be optimized as a parameter, because the temperature is determined by the geothermal condition of the reservoir and cannot be altered. Incorporating the constructed proxy model (BB+CCI+2LF) coupled with genetic algorithm, the final equation of ΔNPV calls the response surface each time to input DCOP and DCWP values and allows the genetic algorithm to find the optimized cases which have the maximum ΔNPV. Genetic algorithm is a heuristic method for finding the global optimum. In the first step, it assumes some possible solutions and finds their fitness. Then using some operations such as crossover (combining two possible solutions) and mutation (inserting a small change in a possible solution), the population is extended. Subsequently, by evaluating the fitness of individuals the individuals with unsuitable fitness will be omitted. Again the population is increased and the steps are repeated until a solution with satisfying fitness will be found. This algorithm is widely used in engineering problems to find the optimum solutions (Mahdiani & Khamehchi, 2015; Mahdiani & Khamehchi, 2015). Table 5 shows three cases in which the specified combination of parameters results the highest amount of ΔNPV. Table 6 shows the parameters of the genetic algorithm which has been used in this paper. Following figures show behavior of ΔNPV versus the four influential parameters near these three points. As far as Cp and Qp are concerned in Figure 22 and Figure 23 respectively, ΔNPV experiences a maximum almost near the end of the specified range. It can be virtually understood that as Cp and Qp increase, ΔNPV increases too. Nevertheless, overuse of polymer may have a negative impact on injection pressure, which can add to injection costs.

21

With regard to Cc and tinj, Figure 24 and Figure 25 both have a maximum value of ΔNPV in the specified range and this point is far from the minimum and maximum caps for the two parameters. These figures show that ΔNPV vs. Cc or tinj reaches its global maximum at a point in the parameter limit range, provided all other parameters are in their optimized values. 9. Conclusions 1. Using a 2-level fractional factorial design, the most crucial parameters which have the highest effect on the gel treatment process have been indicated in a carbonate reservoir. These are ranked based on their importance as: 1. Qp (injection rate) 2. ti (injection time) 3. Cc (cross-linker concentration) 4. T (reservoir temperature) 5. Cp (polymer concentration). 2. Interactions between parameters have been recognized in the screening results (Pareto chart). Hence, the influence of one parameter should be investigated in relation to other parameters (and not individually) in gel treatment design. 3. The two constructed response surfaces (BB+CCI+2LF and BB+CCI) may not be the proxy models with the least error. However, they proved to work efficiently in terms of the corresponding relative error. 4. Use of BB+CCI+2LF model although proved to have less error, more time consumption on its construction should be considered too. 5. Analyzing the best three cases in optimization shows important issues. First, the more we increase Cp and Qp the more we get a value for ΔNPV. This demonstrates the role of polymer in strengthening the gel structure and prolonging its working life in reservoir. Second, Cc and tinj exhibit a maximum in their range. It can be inferred that overuse of cross-linker agent decrease the profitability, because the agent costs a great deal.

Nomenclature 22

Cc: cross-linker concentration in the treatment fluid, ppm CCl: chloride concentration in the treatment fluid, meq/ml COP: cumulative oil production, bbl Cp: polymer concentration in the treatment fluid, wt% Cr: reservoir rock compressibility, psia-1 CWP: cumulative water production, bbl DCOP: difference between cumulative oil production with and without treatment after 600 days, bbl DCWP: difference between cumulative water production with and without treatment after 600 days, bbl Di: the depth at which initial reservoir pressure is reported, ft Dtop: depth of the top layer of the reservoir, ft k: the number of data points kr: relative permeability, fraction Multp: permeability multiplier, fraction Φ: porosity, fraction Pi: initial reservoir pressure at a specified depth, psia Pref: reference pressure for rock compressibility, psia Qp: treatment fluid injection rate, ft3/day Sw: water saturation, fraction Swi: initial water saturation, fraction T: reservoir temperature, °F ti: treatment fluid injection time (duration), day ts: shut-in time (duration) after treatment, day α: distance between center of design to star points

References 23

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Figure 1. A cross-linked polymer which forms polymeric gel to control water production in reservoir.

27

1

0.8

0.6 kro

kr

kro* 0.4

krw krw*

0.2

0 0

0.2

0.4

0.6

0.8

1

Sw Figure 2. A typical relative permeability curve, showing water and oil relative permeability before (k rw and kro) and after (krw* and kro*) DPR gel treatment.

Figure 3. Three parameters Box-Behnken Design (Jun, et al., 2003)

28

Figure 4. Inscribed Central Composite design (Prasad, et al., 2012)

Figure 5. 3D view of the reservoir.

29

1 0.9 0.8 0.7 0.6 kr 0.5

kro

0.4

krw

0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

Sw

Figure 6. The oil and water relative permeability curves of the reservoir of this study

1.0 0.9 0.8

Water Cut

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0

100

200

300

400

500

Time (days) Water Cut With Treatment

Water Cut Without Treatment

Figure 7. Water cut with and without gel treatment.

30

600

1 Oil Cut With Treatment

0.9

Oil Cut Without Treatment

0.8

Oil Cut

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

100

200

300

400

500

600

Time (days) Figure 8. Oil cut with and without gel treatment.

70,000 60,000

CWP (bbls)

50,000 40,000 30,000 20,000

Cumulative Water Production With Treatment

10,000 0 0

100

200

300

Cumulative Water Production Without Treatment 400 500 600

Time (days) Figure 9. Cumulative water production with and without gel treatment.

31

70,000 60,000

COP (bbls)

50,000 40,000 30,000 20,000 Cumulative Oil Production With Treatment

10,000 0 0

100

200

300

Cumulative Oil Production Without Treatment 400 500 600

Time (days) Figure 10. Cumulative oil production with and without gel treatment.

Figure 11. Pareto chart for DCOP. The influence priority of each parameter on DCOP can be determined.

32

Figure 12. Pareto chart for DCWP. The influence priority of each parameter on DCWP can be determined.

Figure 13. Interaction between cross-linker concentration and injection rate.

33

Figure 14. Interaction between injection rate and injection time.

Figure 15. Interaction between reservoir temperature and injection time.

Figure 16. Interaction between injection rate and polymer concentration.

34

Figure 17. Interaction between cross-linker concentration and polymer concentration.

Figure 18. A schematic view of artificial neural network with an input layer, two hidden layers and an output layer.

35

0

10

20

30

Absolute Average Relative Error (%)

Absolute Average Relative Error (%)

3.2 3 2.8 2.6 2.4 2.2 2 40

No. of Nodes

3.2 3 2.8 2.6 2.4 2.2 2 0

10

20

30

40

No. of Nodes of Second Layer

(a)

(b)

Figure 19. Sensitivity of the ANN model error to the number of nodes in (a) first layer. (b) second layer

Figure 20. Comparing two models and simulator DCOP responses for the validation data.

36

Figure 21. Comparing two models and simulator DCWP responses for the validation data.

194000 0.24903, 193500 193000 0.24873, 192400 192000 191000 ΔNPV ($) 190000 189000 188000 0.24973, 187680 187000

Cp (wt%)

Figure 22. Behavior of ΔNPV vs. Cp near the best three cases.

37

7972.152332, 192716 7978.352445, 193029 ΔNPV ($)

7760.688116, 187680

Qp (ft3/day)

Figure 23. Behavior of ΔNPV vs. Qp near the best three cases.

194000 437.000, 193250 193000 192000

432.062, 192500

191000 ΔNPV ($) 190000 189000 188000 423.617, 187680 187000

Cc (ppm)

Figure 24. Behavior of ΔNPV vs. Cc near the best three cases.

38

196000 195000

2.23676, 195000

194000 193000 ΔNPV ($)

192000 191000 190000

2.22244, 191000

189000 188000

2.24545, 187680

187000

tinj (day)

Figure 25. Behavior of ΔNPV vs. tinj near the best three cases.

Table 1. Properties of the reservoir subjected to gel treatment.

Property

Value

Cr (psia-1)

4×10-6

Pref (psia)

5200

φ

0.25

Dtop (ft)

11200

Pi (psia)

5200

Di (ft)

11200

Swi

0.2

Table 2. Parameters affecting gel treatment.

Factor

Min

Max

Average

A: Cp (wt%)

0.1

0.25

0.175

B: Cc (ppm)

200

1000

600

C: Qp (ft /day)

2000

8000

5000

D: ti (day)

1

2.5

1.75

3

39

E: ts (day)

1

2.5

1.75

F: Multp

0.5

1.5

1

G: T (°F)

80

120

100

H: CCl

0.5

0.8

0.65

Table 3. The properties of used artificial neural network

Parameter Train points Test point Data division Training Performance Stop tolerance

Value 121 10 Random Levenberg-Marquart Mean Square Error 1.00E-05

Table 4. Results (corresponding errors) of training and validation steps.

Response

DCOP

Training Error Design

Validation Error

Avg. RE.%

Max RE.%

Avg. RE.%

Max RE.%

BB+CCI

0.19

0.73

2.08

6.6

BB+CCI+2LF

0.24

1.19

2.02

5.5

& DCWP

Table 5. The best 3 cases in terms of profitability for gel treatment in the reservoir.

GA+ANN Cp

Cc

Qp

tinj

(wt%)

(ppm)

(ft3/day)

(day)

1

0.24903

439.104

7978.35245

2

0.24873

432.062

3

0.24973

423.617

Case

ΔNPV

Generation

Population Size

2.23676

40

50

193029

7972.15233

2.22244

40

50

192716

7760.68812

2.24545

40

20

187680

Table 6. The parameters of the genetic algorithm

Parameter/ Property

Value/ Type

Population type Fitness Scaling

Double vector Rank

40

($)

Selection Function Elite Count Crossover Function Crossover Fraction Mutation function Mutation Probability Migration

Stochastic uniform 8 Scattered 0.8 Uniform 0.05 Forward

Highlights 1. The feasibility of DPR gel treatment in carbonate oil reservoirs is investigated by utilizing a three dimensional three phase chemical simulator. 2. Design of Experiment (DoE) is applied to extract the most influential parameters from the group of possibly effective factors and to rank them based on their magnitude of effect (In polymer injection). They are Qp (injection rate), ti (injection time), Cc (cross-linker concentration), T (reservoir temperature), Cp (polymer concentration) as the most effective ones, respectively. 3. A fast and accurate proxy model for predicting the efficiency of the gel treatment (increase in oil production and decrease in water production) introduced. 4. The created model was optimized using genetic algorithm to maximize the profit. 5. A comprehensive analysis on the effect of different gel treatment parameters on the net profit value represented.

41