Comparative Predictive Behaviour Of Two Numerical Techniques To . - IJSR
International Journal of Science and Research (IJSR)ISSN (Online): 2319-7064Impact Factor (2012): 3.358Comparative Predictive Behaviour of TwoNumerical Techniques to Simulator Design of aReservoir with Surfactant Mixture in Enhanced OilRecovery ProcessKamilu Folorunsho Oyedeko1, Alfred AkpovetaSusu21Department of Chemical & Polymer Engineering, Lagos State University, Epe, Lagos, Nigeria2Department of Chemical Engineering, University of Lagos, Lagos, NigeriaAbstract: We consider here the application of orthogonal collocation and finite difference approximation to simulator design for areservoir with surfactant mixture in enhanced oil recovery process to the solution of the applicable equations for the multidimensional,multicomponent and multiphase system. In this work, we report on the effect of significant reservoir parameters and the amount ornature of surfactant mixture on reservoir simulator design. Some of the novel aspects of this study stem from the actual formulation ofthe development of the simulator, in particular, the choice of dependent variables, and the treatment of boundary conditions. Numericalresults obtained using orthogonal collocation and finite difference computations are used to control oscillatory overshoot. In bothorthogonal collocation and finite difference method, general multi-dimensional schemes were applied in the flow simulations. Matlabcomputer programs were used for the numerical solution of the model equations. The results of the orthogonal collocation solution werecompared with those of finite difference solutions. The results indicate that the concentration profiles of surfactants for orthogonalcollocation showed more features than the predictions of the finite difference, offering more opportunities for further understanding ofthe physical nature of this complex problem. Also, comparison of the orthogonal collocation solution with computations based on finitedifference method offers possible explanation for the observed differences especially between the methods and the two reservoirs. Wefound that the effect of surfactant in enhanced oil recovery process in surfactant flooding is in fact the dominant factor in reservoirsimulator design.Keywords: Reservoir Simulator Design; Multidimensional, Multicomponent and Multiphase Systems; Surfactant Mixture; OrthogonalCollocation Technique; Finite Difference Approximation1. IntroductionThe development of a simulator of a reservoir in a surfactantassisted water flood required the understanding of the porousformation of complex reservoir and multiphase andmulticomponent flow taking place in the reservoir. Theunderstanding of the multiphase, multicomponent flowtaking place in any displacement process is essential forsuccessful design of simulator in a reservoir. The worldenergy demand continues to increase significantly and crudeoil still remains the major source.It is very important to at least, maintain or indeed, increasethe current production levels of crude oil. These objectivescan be accomplished by further investing in exploration andproduction of new fields or optimizing production fromexisting fields. Bringing new fields online is very expensive,while recovery from existing fields by conventional methods(i.e. primary and secondary recovery) will not fully providethe necessary relief for global oil demand.On an average, only about a third of the original oil in placecan be recovered by primary and secondary recoveryprocesses. The rest of the oil is trapped in reservoir poresdue to surface and interfacial forces. This trapped oil can berecovered by reducing the capillary forces that prevent oilfrom flowing within the pores of reservoir rock and into thewell bores.Due to high oil prices and declining production inmany regions around the globe, the application of advancePaper ID: 020141243technologies called "Enhanced Oil Recovery"(EOR) hasbecome very attractive for exploration and production of thetrapped oil.This technology requires the injection of a fluidor fluids or materials into a reservoir to supplement thenatural energy present in a reservoir, where the injectedfluids interact with the reservoir rock /oil /brine system tocreate favourable conditions for maximum oil recovery.Surfactants are injected to decrease the interfacial tensionbetween oil and water in order to mobilize the oil trappedafter secondary recovery by water flooding.In a surfactant flood, a multi-component multiphase systemis involved. The theory of multi- component, multiphaseflow has been presented by several authors[1].The surfactantflooding is a form of chemical flooding and is representedby a system of nonlinear partial differential equations: thecontinuity equation for the transport of the components andDarcy’s equation for the phase flow. The system ofequations is completed by the equations representingphysical properties of the fluids and the rock. From aphysico-chemical point of view, there are three components- water, petroleum and chemical. They are in fact, pseudocomponents, since each one consists of several purecomponents. Petroleum is a complex mixture of manyhydrocarbons. Water is actually brine, and containsdissolved salts. Finally, the chemical contains different kindsof surfactants. These components are distributed betweentwo phases –the oleic phase and the aqueous phase. Thechemical has an amphiphilic character. It makes the oleicVolume 3 Issue 11, November 2014www.ijsr.netLicensed Under Creative Commons Attribution CC BY861
International Journal of Science and Research (IJSR)ISSN (Online): 2319-7064Impact Factor (2012): 3.358phase at least partially miscible with water or the aqueousphase, partially miscible with petroleum.Interfacial tension depends on the surfactant partitionbetween the two phases. Residual phase saturation decreaseas interfacial tension decreases. Relative permeabilityparameters depend on residual phase saturations. In addition,phase viscosities are functions of the volume fraction of thecomponents in each fluid phase. Therefore, the success orfailure of surfactant flooding processes depends on phasebehaviour. Phase behaviour influences all other physicalproperties, and each of them, in turn influences oil recovery.The two different mathematical techniques are to be utilizedin identifying a particular type of physical behaviour andthus enabling the understanding of the propagationphenomena. More so, the techniques will in particular beutilized to predict what happens in EOR process and showhow the complexity of the problem can be reduced. Systemsof coupled, first-order, nonlinear hyperbolic partialdifferential equations (p.d.e.s) govern the transient evolutionof a chemical flooding process for enhanced recovery. Themethod of characteristics (MOC) provides a way in whichsuch systems of hyperbolic p.d.e.s can be solved byconverting them to an equivalent system of ordinarydifferential equations. In some cases, the characteristicsolution has been used to track the flood-front in twodimensional reservoir problems [2]. Besides, anotherapproach combines the characteristic method with a finiteelement approach [3]. The MOC and an adjustable numberof moving particles to track three-dimensional solute frontshas been used in groundwater systems; adjusting the numberof particles serves to maintain an accurate material balanceand save computational time [4]. This front-trackingapproach has been used in the present work to trace themovement of coherent waves, of both the diffuse and shockvariety.At the simple level, the results of simulation using the twotechniques are analogous to the Buckley-Leverett theory forwater flooding, the latter being evident in the case ofpolymer flooding [5], Also for dilute surfactant flooding[6],For carbonated water flooding, [7] and For miscible [8] andimmiscible surfactant flooding[9]. For isothermal,multiphase, multicomponent fluid Flow in permeable media[10].While Case studies for the feasibility of sweepimprovement in surfactant-assisted water flooding.[11]High oil prices and declining production in many regionsaround the globe make enhanced oil recovery (EOR)increasingly attractive. As evident in the work for a newclass of viscoelastic surfactants for EOR[12], Formicrobially enhanced oil recovery at simulated reservoirconditions by use of engineered bacteria[13], for cooptimization of enhanced oil recovery and carbonsequestration[14],while for development of improvedsurfactants and EOR methods for small operators[15] andmany others.The present work describes the design of a simulator for anEnhanced Oil Recovery process using surfactant assistedwater flooding by applying two different mathematicalmethods, orthogonal collocation and finite differencePaper ID: 020141243method, to solve the basic model transport equations. Theapproach is multidimensional and involves at least threeindependent variables for mapping the composition routes ofthe system components.2. MethodologyThis work considered the solution of a multidimensional,multicomponent and multiphase flow problem associatedwith enhanced oil recovery process in petroleumengineering. The process of interest involves the injection ofsurfactant of different concentrations and pore volume todisplace oil from the reservoir.The methodology used here is illustrated by the stepsutilized in executing the solution using the developedmathematical models describing the physics of reservoirdepletion and fluid flow in which one of the main aims is thedetermination of the areal distribution of fluids in theflooded reservoir. The system is for two or three dimensions,two fluid phases (aqueous, oleic) and one adsorbent phase,four components (oil, water, surfactants 1 and 2).The reservoir may be divided into discrete grid blocks whichmay each be characterized by having different reservoirproperties. The flow of fluids from a block is governed bythe principle of mass conservation coupled with Darcy’slaw. The following are taken into consideration in themodeling effort:(i) The simultaneous flow of oil, gas, and water in threedimensions(ii) The effects of natural water influx, fluidcompressibility, mass transfer between gas and liquidphases and(iii) The variation of such parameters as porosity andpermeability, as functions of pressure.The model is developed from the basic law of conservationof mass with assumptions[16] .The developed partial differential equation is converted toordinary differential equation using finite difference andorthogonal collocation methods.The finite difference method is a technique that convertspartial differential equations into a system of linearequations. There are essentially three finite differencetechniques. The explicit, finite difference method convertsthe partial differential equations into an algebraic equationwhich can be solved by stepping forward (forwarddifference), backward (backward difference) or centrally(central difference).The orthogonal collocation method converts partialdifferential equations into a system of ordinary differentialequations using the Lagrangian polynomial method. This setof ordinary differential equations generated is then solvedwith appropriate numerical technique such as the RungeKutta.The rock and fluid properties such as density, porosity,viscosity, oil and water etc, and other parameters are listedVolume 3 Issue 11, November 2014www.ijsr.netLicensed Under Creative Commons Attribution CC BY862
International Journal of Science and Research (IJSR)ISSN (Online): 2319-7064Impact Factor (2012): 3.358in Tables 1, 2, 3 and 4. Table 1 is the reservoircharacteristics from previous work [16]. Table 2 is thereservoir characteristics used for the simulation work [17].Parameter values used in Trogus adsorption model and forverification runs are shown in Table 3[17], while Table 4contains additional reservoir parameters presented for thework [16].In considering the more general form of the multiphase,multicomponent problem, the explicit Runge-Kutta methodis chosen for the solution of the problem. The motivation forthis explicit method is its simplicity and computationalefficiency with regard to the reduction of truncation errorsmore effectively than other methods. The MATLABcomputer program was used to obtain the solutions.φ SwThe termFor most of the simulated cases in the work, the reservoirconsisted of a rectangular composite of horizontal oilbearing strata, sandwiched above and below by twoimpervious rocks [16]. Oil is produced from the reservoir bymeans of water injection at one end and a production well atthe other. Data for the hypothetical reservoir simulated aregiven in Table 1and the model developed [16] is Ci , w Ci , w C i , w Ci ρ (1 φ ) φ vx f w φvy fw ri ( i 1, 2 ) t t x yri represents the rate of loss of surfactant due toprecipitation: for a one-to-one reaction stoichiometry,r1 r2 . Since reaction occurs instantaneously at a sharpinterface, this term may be ignored away from the singularregion of the interface.2.5 Adsorption ModelIt is possible to approximate the adsorption isotherm of apure surfactant on a mineral oxide by use of a simple model.At low concentration the adsorption obeys Henry’s law,while above the critical micelle concentration (CMC), thetotal adsorption remains constant. The Trogus adsorptionswThe model encompasses two fluid phases (aqueous andoleic), one adsorbent phase (rock), and four components (oil,water, surfactants 1 and 2). The oil is displaced by waterflooding. In-situ interaction of surfactant slugs may occur,with consequent phase separation and local permeabilityreduction. The model accommodates two (or three) physicaldimensions and an arbitrary, nonisotropic description ofabsolute permeability variation and porosity.(1)model [18], [19] is used in this work. The followingassumptions are made:3.Application of Finite Difference toSolution of Model EquationsFirst-order, finite-difference expressions for the spatialderivatives were substituted into the hyperbolicchromatographic transport equations (Eq. 1), yielding 2 x mcoupled ordinary differential equations which may then beintegrated simultaneously (also known as the ‘numericalmethod of lines’). Ci ,w 2 Ci ,w C (τ , ε h ) Ci , w (τ , ε h 1 ) mij f w (τ , ε h ) i ,w0 τ τεj 1 where i 1,2 and h 1,2,. .m.Eqn.2 is the finite-difference form of Eqn.1written for onespatial dimension ε , where mij are the adsorptioncoefficients, τ is dimensionless time (injected volume/ porevolume), and ε is dimensionless distance (pore volumestravelled). In two dimensions, the finite-difference terms aremultiplied by dimensionless velocities. The distortion of thesolution in the τ direction may be neglected by using a 4thorder Runge-Kutta method and a sufficiently small timestep.mi, j C i' C 'j , w(5)Again, recall that differentiation of a function of anotherfunction (chain rule) is of the form y y u x u x(6)The above equation is now transformed to the original formof Eqns. 1 using the following defined variables:C i' , w φ C i , w(3)C i' ρ (1 φ ) C i'Paper ID: 020141243(2)(4)Volume 3 Issue 11, November 2014www.ijsr.netLicensed Under Creative Commons Attribution CC BY863
International Journal of Science and Research (IJSR)ISSN (Online): 2319-7064Impact Factor (2012): 3.358Applying the chain rule above, Eqn.2 becomes ' Ci', w (τ , ε h ) Ci', w (τ , ε h 1 ) Ci C1', w Ci' C 2' , w ' . 0 ' . f w (τ , ε h ) τ ε C1, w τ C 2, w τ Ci', wSwEliminating the primes (') and bars (-) and introducing m i , jterms yield(S w m11 ) C1, w(S w m22 ) C 2, w τ τ m12 m21 C 2, w τ C1, w τ fw fw C1, w ε C 2, w ε 0(8)(S w m11 )(S w m22 ) m12 τ C2,w τ m21 C 2, w τ C1, w τ fw fwC1, w(τ ,ε h ) C1, w(τ ,ε h 1) εC2, w(τ ,ε h ) C2,w(τ ,ε h 1) ε(9)(S w m22 ) C1, w (τ ,εh) τ C2,w(τ ,ε h ) τ m12 m21 C2, w(τ ,ε h ) τ C1, w(τ ,ε h ) τ 0 0(10)(11)(S w m11 )(S w m22 )h) τ C2,w(τ ,ε h ) τ m12 m21 C2, w(τ ,ε h ) τ C1, w(τ ,ε h )where τ[] C 2, w(τ ,ε h ) C 2 C1, w(τ ,ε h )f w C 2 , w(τ ,ε ) C 2 ,w(τ ,ε 1) 0 hh C ττε1,w Sw C1,w (τ ,ε τ C1,w (τ ,ε τh) f C1 C1, w (τ ,ε h ) C1 C 2 ,w(τ ,ε h ). w C1, w(τ ,ε ) C1, w(τ ,ε 1) 0hh C1, w τ C 2, w τ εh) C1 C1 f wC1,w(τ ,ε ) C1, w(τ ,ε 1) 0 hh τ τ ε[[(17) C1,w (τ ,ε]][(18)] Cfh) 2 1 w C1,w(τ ,ε h ) C1, w(τ ,ε h 1) 0 τ τ εsimilarly C 2, w (τ ,ε ) CfhSw 2 2 w C 2, w(τ ,ε h ) C 2 ,w(τ ,ε h 1) 0 τ τ εSw[][]fwC1, w(τ ,ε h ) C1, w(τ ,ε h 1) 0 ε(12)[]fwC2, w(τ ,ε h ) C2, w(τ ,ε h 1) 0 ε(13)[]f w C1, w(τ ,ε h ) C1, w(τ ,ε h 1) 0 ε(14) ](19)From the Trogus model,Since we have a set of simultaneous ODE’s, we will attemptto solve the equations C1, w (τ ,ε[These on simplification yieldSwThis can also be written as follows(S w m11 ) C1,w (τ ,ε h ) C 1 C 2 , w( τ , ε h )f w C1, w(τ ,ε ) C1, w(τ ,ε 1 ) 0 hh τ ετC 2 ,w (16)and C 2 Sw C 2,w Applying the method of lines, a partial transformation to adifference equation, to the equations above yield: C1, wSubstitution of these terms in Eqs.14 and 15 yield C1 Sw C1, w 0(7)[]fwC2, w(τ ,ε h ) C2, w(τ ,ε h 1) 0 ε(15)C1 k1C1, wC 2 k 2 C 2,wA final substitution results in the equation below:SwSw C1, w (τ ,ε τ C1, w (τ ,ε τ( S w 2 k1 ) (k1C1, w )h) 2h) 2 k1 C1, w τ τ C1, w τ [ []fwC1, w(τ ,ε h ) C1, w(τ ,ε h 1) 0 ε[]fwC1, w(τ ,ε h ) C1, w(τ ,ε h 1) 0 ε]fwC1, w(τ ,ε h ) C1, w(τ ,ε h 1) 0 εand C 2, w (τ ,ε ) ( k 2 C 2,w ) f whSw 2 C 2, w(τ ,ε ) C 2, w(τ ,ε 1) 0hh τ τ ε C 2, w f w( S w 2k 2 ) C 2, w(τ ,ε h ) C 2, w(τ ,ε h 1) 0 τ ε(21)[[]]3.2 Application of Orthogonal Collocation to Solution ofModel EquationsEquation7can be written as:SwPaper ID: 020141243 C i', w (τ , ε h ) C i', w (τ , ε h 1 ) C i'(22) 2 f w (τ , ε h ) 0 τ τ ε C i', wVolume 3 Issue 11, November 2014www.ijsr.netLicensed Under Creative Commons Attribution CC BY864
International Journal of Science and Research (IJSR)ISSN (Online): 2319-7064Impact Factor (2012): 3.358SwφS w C i ,w τ [φC i , w ] τ [φC i , w ](τ , ε h ) [φC i , w ](τ , ε h 1 ) [ ρ (1 φ ) C i ] 2 f w (τ , ε h ) 0 τ ε 2 ρ (1 φ ) C i , w (τ , ε h ) C i , w (τ , ε h 1 ) Ci φf w (τ , ε h ) 0 τ ε Now, from the Trogus model,φS w C i , wφS wφS w C i , w τ C i , w τ 2κ i ρ (1 φ ) τ[φS w 2κ i ρ (1 φ )] 2 ρ (1 φ ) τ τ τ 2κ i ρ (1 φ ) C i , w C i , w (κ i C i , w ) φf w (τ , ε h ) φf w (τ , ε h )(24)Ci κ i Ci , w (25) C i , w (τ , ε h ) C i , w (τ , ε h 1 ) φf w (τ , ε h ) 0 ε C i , w (τ , ε h ) C i , w (τ , ε h 1 ) φf w (τ , ε h ) 0 τ ε C i , w ε C i , w εa JI 0 (28)R 0 (29)R [φS w 2κ i ρ (1 φ )](30)Using the method of orthogonal collocation, let C beapproximated by the expression(31)I 1Equation 31 can now be expressed as follows:I(τ ) X J (ε I ) 0(32)N 1 C B [C I (τ ) X J (ε I )] 0 τI 1 ε(33)N 1 C B [ X J (ε I )].C I (τ ) 0 τI 1 ε(34)RRI 1Paper ID: 020141243N 1 C J B a JI C I 0 τI 1(36) C JB N 1 a JI C I τR I 1(38)Therefore, CJ B [aJ1C1 aJ2C2 aJ3C3 aJ4C4 . aJN 1CN 1] (39)R τAgain J 1, 2, 3, 4 N 1Therefore the following system of ODE’s can be generatedN 1 C(35)For I 1, 2, 3, 4 N 1where C is a function of both ԑ (dimensionless distance) andτ (dimensionless time).C (τ , ε ) C I (τ ) X J (ε I ) X J (ε I ) ε(37)The above equations now become: C C B 0R ε τ(27) C J B N 1 a JI C I 0 τR I 1B φf wN 1(26) C i , wLet C BR τ ε(23) C1B [a11C1 a12 C 2 a13C3 a14 C 4 . a1N 1C N 1 ]R τ C 2B [a 21C1 a 22 C 2 a 23C3 a 24 C 4 . a 2 N 1C N 1 ]R τ C3B [a31C1 a32C 2 a33C3 a34C 4 . a3 N 1C N 1 ] τR C 4B [a 41C1 a 42 C 2 a 43C3 a 44 C 4 . a 4 N 1C N 1 ]R τ::::Volume 3 Issue 11, November 2014www.ijsr.netLicensed Under Creative Commons Attribution CC BY865
International Journal of Science and Research (IJSR)ISSN (Online): 2319-7064Impact Factor (2012): 3.358 C N 1B [a N 11C1 a N 12 C 2 a N 13C3 a N 14 C 4 . a N 1N 1C N 1 ] τRIn matrix form, we have the following expression:Also, the following recurrence relations are defined below. C1 τ C2 a11 a12 a13 a14 . . . . . a1N 1 C1(τ ) τ a a2N 1 21 a22 a23 a24 C2(τ ) C 3 a31a3N 1 C3(τ ) (41): τ a4N 1 C4(τ ) : C a41 4 B : : τ : : R : : :: : : : : :: : aN 11 aN 12 . . . . . . . aN 1N 1 CN 1(τ ) CN 1 τ Similarly, the following expression defines aJI[20], [21]a JI 1 PN( 2 )1 (ε I )For J I (1)2εP() N 1I PN(1 )1 (ε I ) 1For I J ε ε P (1) (ε )JN 1J I(42)PJ (ε ) (ε ε J ) PJ 1 (ε ); J 1,2,3,., N 1PJ(1) (ε ) (ε ε J ) PJ(1 )1 (ε ) PJ 1 (ε )PJ( 2) (ε ) (ε ε J ) PJ( 21) (ε ) 2 PJ(1 )1 (ε )P0(1) (ε ) P0( 2 ) (ε ) 0(43)Recall that the elements of the matrix can be generated fromthe following Lagrange polynomial dl j ( x i ) a ij dx x i1 PN( 2 )1 ( x i )2 PN(1 )1 ( x i ) j i1PN(1 )1 ( x i ) x j PN(1 )1 ( x j ) i j(44)For i j, the elements here refer to the leading diagonal ofthe matrix to be generatedFor i j, the elements here refer to all other elements of thematrixPaper ID: 020141243po ( x) 1P j ( x ) ( x x j ) P j 1 ( x )P j( 1 ) ( x ) ( x x j ) P j( 1 1) ( x ) P j 1 ( x )P j( 2 ) ( x ) ( x x j ) P j( 21) ( x ) 2 P j( 1 1) ( x )(45)For j 2, 3, 4, ., N 1The following substitutions and manipulations will now bemade to redefine Eqn.44. Substituting the recurrencerelations into Eqn.44 yields: 1 (xi xj )Pj( 21) (xi ) 2Pj( 1)1(xi ) (1) 2 (xi xj )Pj 1(xi ) Pj 1(xi ) j iaij (1) 1 (xi xj )Pj 1(xi ) Pj 1(xi ) xi xj (xj xj )Pj( 1)1(xj ) Pj 1(xj ) i j (46)Now, some terms will be cancelled out.Since j i,(xi – xj) 0and(xj – xj) 0whereP0 (ε ) 1(40)(1) 1 2Pj 1(xi ) 2 Pj 1(xi ) j i aij (1) 1 (xi xj )Pj 1(xi ) Pj 1(xi ) xi xj Pj 1(xj ) i j(47)The above becomes: Pj( 1)1(xi ) Pj 1(xi ) j iaij (1) (xi x j )Pj 1(xi ) 1 Pj 1(xi ) (xi x j )Pj 1(x j ) xi xj Pj 1(x j ) i j(48)This becomes: a ij (1) P j 1 ( x i ) P (x ) j 1 jVolume 3 Issue 11, November 2014www.ijsr.net P j( 11) ( x i ) j i P j 1 ( x i ) P j 1 ( x i ) 1 x i x j P j 1 ( x j ) Licensed Under Creative Commons Attribution CC BYi j866(49)
International Journal of Science and Research (IJSR)ISSN (Online): 2319-7064Impact Factor (2012): 3.3584. ResultsRewriting the above in terms of epsilon, (ε): (1 ) P j 1 ( ε i ) P (ε ) j 1 ja ij P j( 11) ( ε i ) j i P j 1 ( ε i ) P j 1 ( ε i ) 1 ε i ε j P j 1 ( ε j ) i j(50)The matrix now looks like this:a11 a12 a13 a21 a22 a23 a31 a32 a32 P0(1) (ε 1 )P0 (ε 1 )P1(1) (ε 1 )P1 (ε 2 )P2(1) (ε 1 )P2 (ε 3 )P (ε 2 )(1)0P0 (ε 1 ) 1 P1 (ε 1 )ε 1 ε 2 P1 (ε 2 ) 1 P2 (ε 1 )ε 1 ε 2 P2 (ε 3 ) 1 P0 (ε 2 )ε 2 ε 1 P0 (ε 1 ) 1 P2 (ε 2 )ε 2 ε 3 P2 (ε 3 ) 1 P0 (ε 3 )ε 3 ε 1 P0 (ε 1 ) 1 P1 (ε 3 )ε 3 ε 2 P1 (ε 2 )The two fluid phases consisted of a water phase and an ssible. The density of oil, the viscosity of oil, thesalinity of water, and the formation volume factor of oil andwater are listed in Table 2. All cases mentioned above wererun by using anionic sodium dodecyl sulfate (SDS) andcationic dodecyl pyridinium chloride (DPC) as surfactants.P1(1) (ε 2 )P1 (ε 1 )P2(1) (ε 2 )P2 (ε 3 )P0(1) (ε 3 )P0 (ε 1 )P1(1) (ε 3 )P1 (ε 2 )The system of equations is complete with the equationsrepresenting physical properties of the fluids and the rock.Physical properties described here are: (i) phase behaviour(ii) interfacial tension between fluid phases, (iii) residualphase saturations, (iv) relative permeabilities, (v) rockwettabiliy, (vi) phase viscosities, (vii) capillary pressure,(viii) adsorption and (ix) dispersion. From a physicochemical point of view, there are three components: water,petroleum and chemical. As stated earlier on, these are allpseudo-components, since each one consists of several purecomponents. Petroleum is a complex mixture of manyhydrocarbons. Water is actually brine, and containsdissolved salts. Finally, the chemical contains different kindsof surfactants.P2(1) (ε 3 )P2 (ε 3 ) (51)The recurrence relations below will again be used toevaluate the terms of the matrix.po (ε ) 1Pj (ε ) (ε ε j ) Pj 1 (ε )(ε ε j ) Pj(1)Pj(1) (ε ) 1 (ε ) Pj 1 (ε )P0(1) (ε ) 0Let ԑ assume the range:ԑ [0:0.01:0.09]whereԑ1 0 (53)ԑ2 0.01 (54)ԑ3 0.02 (55)Paper ID: 020141243The reservoir response, as predicted by the simulation on thebasis of orthogonal collocation is compared with thenumerical predictions obtained using traditional finitedifference method. The case studies are chosen to be bothhypothetical and using of existing Nigerian well data withsimple representative of the important elements of thesimulator. The main objective of these case studies has beento demonstrate that the mathematical techniques oforthogonal collocation and finite difference in the context ofapplication of the simulator can be used to obtain wavebehaviour in a reservoir. A gradually increasing level ofcomplexity is introduced, representing a range of systemsfrom aqueous phase flow, to surfactant chromatography intwo phase flow, to surfactant chromatography in twodimensional porous medium. The initial and injectedsurfactant compositions corresponding to cases 1,2 and3 areshown in Table 5. The rock and fluid properties are listed inTable 1, 2, 3 and4. These were taken as uniform forconvenience.(52)These three pseudo-components are distributed between twophases –the oleic phase and the aqueous phase. Thechemical has an amphiphilic character. It makes the oleicphase at least partially miscible with water or the aqueousphase at partially miscible with petroleum.Interfacial tension depends on the surfactant partitionbetween the two phases. Residual phase saturation decreasesas interfacial tension decreases. Relative permeabilityparameters depend on residual phase saturations. Phaseviscosities are functions of the volume fraction of thecomponents in each fluid phase. Therefore, the success orfailure of surfactant flooding processes depends on phasebehaviour. Phase behaviour influences all other physicalproperties, and each of them, in turn influences oil recovery.Volume 3 Issue 11, November 2014www.ijsr.netLicensed Under Creative Commons Attribution CC BY867
International Journal of Science and Research (IJSR)ISSN (Online): 2319-7064Impact Factor (2012): 3.358If a one-dimensional, adsorbing porous medium is initiallyequilibrated with an aqueous composition C1 0.21, C2 0.181 ( concentrations normalized as moles in solution perm3 off bed) and is then injected with a composition C1 0.17, C2 0.013 (Riemann-type problem: case 1, refer toTable 5 ), the composition upstream of this injected fluidand composition downstream of the initial or previouslyinjected fluid follows the slow “path” from the injectedcomposition to the junction with the “fast path” from thefinal composition, where it switches to this “fast” path. InFigure 1a, the profile C1 of finite difference (FD)shows asteady rise from C1 0.17 to C1 0.21 and then attainecd aconstant state. Also the profile C1 of the orthogonalcollocation (OC) increased steadily from C1 0.17 to C1 0.21 after which it started depressing from C1 0.2 atdistance 0.3 epsilon to C1 0.07 at distance 0.5 epsilonbefore rising back to attain a constant state with the finitedifference method. Similarly, the C2 of finite difference (FD)increased steadily from C2 0.017 to a constant state as forC1.The constant state is at C2 0.18. The orthogonalcollocation (OC) for C2first moves at constant state beforerising steadily to C2 0.18 and then declined from C2 0.18to a minimum of C2 0.08 before rising to a constant state.The profiles for finite difference (FD) and that of orhogonalcollocation (OC) agree except for the depressions of theorthogonal collocation profiles.Figure 1b shows the results obtained for solving Eqn.2 bythe use of orthogonal collocation (OC) and finite difference(FD) methods. The graph is for the bed composition profilefor one dimensional aqueous phase chromatography for case1 at one pore volume injected.In this case also, the adsorbingporous medium is initially equibrated with an aqueouscomposition concentrations.C1 0.21, C2 0.181(concentrations normalized as moles in solution per m3 offbed) and is then injected with a composition C1 0.17, C2 0.013 (Riemann-type problem: case 1,( refer to Table 5)).The profile C1 of finite difference (FD) indicates rise inconcentration fromC1 0.17 to 0.21 after which theconcentration maintained a constant state. The profile of C1of the orthogonal collocation (OC) also rise from C1 0.17to C1 0.21 but falls to 0.03 at distance 0.4 epsilon and thenincreased steadily to constant state as for C1 finite difference(FD). The C2 of finite difference increased steadily from C2 0.02 to attain constant state at 0.18. Also the profile of C2of the orthogonal collocation (OC) increase gradually fromC2 0.02 to C1 0.18 at distance 0.2 epsilon for shortconstant state and thendecline to C2 0.02 at distance 0.4epsilon before rising back to reach constant state with thefinite difference.The bed composition profile for one dimensional aqueousphase chromatography for case 1 at two pore volumeinjected is shown in Figure 1c. This is the result obtained forPaper ID: 0201412430.25C1-FDC2-FDC1-OCC2-OC0.2C1,C2(moles in soln/m3 bed)Figure 1a is the result obtained for solving Equation 2 usingthe numerical technique for both orthogonal collocation andfinite difference. The graph is for the bed compositi
Numerical results obtained using orthogonal collocation and finite difference computations are used to control oscillatory overshoot. In both orthogonal collocation and finite difference method, general multi-dimensional schemes were applied in the flow simulations. Matlab computer programs were used for the numerical solution of the model .
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