Reservoir Formation Damage Episode 3 Part 3 ppt

Output that can be requested: • Best estimates of the unknown parameters • Predicted versus measured data • Simulation of pressure; various species concentrations in theflowing fluid and

• Effluent species concentrations versus pore volume injected

9 Output that can be requested:

• Best estimates of the unknown parameters

• Predicted versus measured data

• Simulation of pressure; various species concentrations in theflowing fluid and the pore surface; porosity and permeability asfunctions of pore volume injected or time

Numerical Solution of Formation Damage Models

Depending on the level of sophistication of the considerations, theoreticalapproaches, mathematical formulations, and due applications, formationdamage models may be formed from algebraic and ordinary and partialdifferential equations, or a combination of such equations Numericalsolutions are sought under certain conditions, defined by specific appli-cations The conditions of solution can be grouped into two classes:(1) initial conditions, defining the state of the system prior to any orfurther formation damage, and (2) boundary conditions, expressing theinteractions of the system with its surrounding during formation damage.Typically, boundary conditions are required at the surfaces of the system,through which fluids enter or leave, such as the injection and productionwells or ports, or that undergo surface processes, such as exchange orreaction processes

Algebraic formation damage models are either empirical correlationsand/or obtained by analytical solution of differential equation models forcertain simplified cases Numerical solution methods for linear andnonlinear algebraic equations are well developed

Ordinary differential equation models describe processes in a singlevariable, such as either time or one space variable However, as demonstrated

in the following sections, in some special cases, special mathematicaltechniques can be used to transform multi-variable partial differentialequations into single-variable ordinary differential equations Amongstthese special techniques are the methods of combination of variables andseparation of variables, and the method of characteristics The numericalsolution methods for ordinary differential equations are well developed.Partial differential equation models contain two or more independentvariables There are many numerical methods available for solution

of partial differential equations, such as the finite difference method(Thomas, 1982), finite element method (Burnett, 1987), finite analyticmethod (Civan, 1995), and the method of weighted sums (the quadratureand cubature methods) (Civan, 1994, 1994, 1995, 1996, 1998; Malikand Civan, 1995; Escobar et al., 1997) In general, implementation of

numerical methods for solution of partial differential equations is achallenging task.

In the following sections, several representative examples are presentedfor instructional purposes They are intended to provide some insight intothe numerical solution process Interested readers can resort to manyexcellent references available in the literature for details and sophisticatedmethods For most applications, however, the information presented inthis chapter is sufficient and a good start for those interested in specializ-ing in the development of formation damage simulators

Although numerical simulators can be developed from scratch asdemonstrated by the examples given in the following sections, we cansave a lot of time and effort by taking advantage of ready-made softwaresavailable from various sources For this purpose, the spreadsheet programsare particularly convenient and popular Various softwares for solvingalgebraic, ordinary, and partial differential equations are available Com-mercially available reservoir simulators can be manipulated to simulateformation damage, such as by paraffin deposition as demonstrated by Ring

et al (1994)

Ordinary Differential Equations

In this section, several examples are given to illustrate the numericalsolution of ordinary differential equation models Specifically, the simpli-fied formation damage and filtration models, developed in previouschapters, are solved

Example 1: Wojtanowicz et al Fines Migration Model

a Derive a numerical solution for the following modified Wojtanowiczetal (1987, 1988) fines migration model

A simultaneous solution of Eqs 16-2 and 5 as a function of time, subject

to the initial conditions given by Eq 16-4, can be readily obtained using

an appropriate method, such as by the Runge-Kutta-Fehlberg four (five)method available in many ordinary differential equation solving software(IMSL, 1987, for example) Then, the porosity variation is calculated by

Eq 16-3 A typical numerical solution is presented in Figure 16-1

0 0.25 0.5 0.75

Time, min.

Figure 16-1 Particle concentration and porosity vs time.

Example 2: Ceriiansky and Siroky Fines Migration Model

The numerical solution is carried out for T^ = 0 Here, the numericalsolution approach presented by Cernansky and Siroky (1985) is described.Define the dimensionless time and distance, respectively, by:

/(a, e) = kpLa (ty0 - e) - k'eLeuyi/K

The characteristics are given by:

11 are solved by means of the fourth-order Runge-Kutta method, subject

to the conditions given by Eqs 16-15 and 16 along the characteristicrepresented by Eq 16-14

Figure 16-2 shows the dimensionless effluent particles concentration

as a function of the cumulative volume injected per unit area Figures16-3 and 16-4 show typical suspended particle concentration and theparticles retained in porous media as a function of distance along theporous media at different times

Example 3: Civan's Incompressive Cake Filtration

without Fines Invasion Model

The equations of Civan's (1998, 1999) incompressive cake filtrationmodel are given in Chapter 12 As described in Chapter 12, the ordinarydifferential equations of this model have been solved by the Runge-Kutta-Fehlberg four (five) numerical scheme (Fehlberg, 1969), subject to theinitial condition given by Eq 12-14

Example 4: Civan's Compressive Cake Filtration

Including Fines Invasion Model

The equations of Civan's (1998, 1999) compressive cake filtrationincluding fines invasion model are given in Chapter 12 As described in

Figure 16-2 Experimental and simulated dimensionless concentrations vs.

filtrate volume for the POP 1 material, usmg c 0 = 0.1 kg/m^ H = 5, 10, 15,

20, 50, and 100 mm, and u = 0.5 cm/s (Cerfiansky, A., & Siroky, R., 1985;

reprinted by permissioji of the AlChE, ©1985 AlChE, all rights reserved; andafter Cernansky and Siroky, 1982, reprinted by permission)

Figure 16-3 Simulated dimensionless concentration vs dimensionless

distance at different times for the FINET-PES 1 material, using c 0 = 0.1 kg/

m3, H = 100 mm, and u = 0.5 cm/s (Cerhansky, A., & Siroky, R., 1985;

reprinted by permissioji of the AlChE, ©1985 AlChE, all rights reserved; andafter Cernansky and Siroky, 1982, reprinted by permission)

120

80

0,8 1.0

Figure 16-4 Simulated mass of particles retained per unit volume of porous

material vs dimensionless distance at different times for the FINET-PES 1

material, using c 0 = 0.1 kg/m3, H = 100 mm, and u = 0.5 cm/s (Cernansk^,

A., & Siroky, R., 1985; reprintedwby permission of the AlChE, ©1985 AlChE,all rights reserved; and after Cerhansky and Siroky, 1982, reprinted bypermission)

Chapter 12, the ordinary differential equations of this model have beensolved by the Runge-Kutta-Fehlberg four (five) numerical scheme (Fehlberg,1969), subject to the initial condition given by Eq 12-14

Partial Differential Equations

In this section, the application of the finite difference method for solution

of partial differential type models is illustrated by several examples

The Method of Finite Differences

The method of finite differences is one of many methods availablefor numerical solution of partial differential equations Because of itssimplicity and convenience, the method of finite differences is the mostfrequently used numerical method for solution of differential equations.This method provides algebraic approximations to derivatives so thatdifferential equations can be transformed into a set of algebraic equations,which can be solved by appropriate numerical procedures Although thefinite difference approximations can be derived by various methods, a

simple method based on the power series approach is presented here toavoid complicated mathematical derivation Interested readers may resort

to many excellent textbooks and literature available on the finite ference method The information provided in this chapter is sufficient formany applications and for the purpose of this book Most transportphenomenological models involve first and second order derivatives.Therefore, the following derivation is limited to the development of thefirst and second order derivative formulae However, the higher orderderivative formulae can be readily derived by the same approach pre-sented in this chapter

dif-First Order Derivatives

In general, a function can be approximated by a power series as:

in which aQ ,a l ,a 2 , are some fitting coefficients To determine the

fitting coefficients, consider any set of three discrete function values

yj_j, ff, and fi+l located at the sample points *,-_,, x f , and x i+l , respectively,

as shown in Figure 16-5

Figure 16-5 Sample points considered for the finite difference method.

More points could be considered for better accuracy Higher order finitedifference formulae can be derived easily using the quadrature method

as described by Civan (1994) With three points, we can write thefollowing three quadratic approximations at / -1, /, / +1:

fi=aQ+ a^ + a2xf

+ ax 2+l

(16-20)(16-21)(16-22)

If the middle point is considered as a reference point, then the locations

of the three points are given by:

Thus, substituting Eq 16-23 into Eqs 16-20 through 22, and then solvingthe resultant three algebraic equations simultaneously yields the followingexpressions for the fitting coefficients of the quadratic expression:fln=/, (16-24)

Thus, the following forward difference formula is obtained by substituting

Eqs 16-25 and 26 into Eq 16-27 for a},a2 at x = xi_l =-A*:

The central difference formula is obtained as, by substituting Eqs

16-25 and 26 for a } ,a 2 into Eq 16-27 at x = x l; =0:

The backward difference formula is obtained as, by substituting Eqs

16-25 and 26 for a\,a2 into Eq 16-27 at x = xi+l = AJC:

f" = bl+2b2x (16-35)

Thus, the forward difference formula is obtained as, by substituting

Eqs 16-33 and 34 for b{,b2 at x = xi_l =-A;c into Eq 16-35:

2Ax (16-36)

The central difference formula is obtained as, by substituting Eqs

16-33 and 34 for b{,b2 at x = xf=0 into Eq 16-35:

The backward difference formula is obtained as, by substituting Eqs

16-33 and 34 for b^b2 into Eq 16-35 for x = xi+l =Ax:

However, only the central second order derivative formula is used in ourmodels Thus, substituting the first order forward and backward differenceformulae given by Eqs 16-28 and 30 into Eq 16-37, the central secondorder difference formula is obtained as:

and the inlet and outlet boundary conditions

(16-42)

- = <U = U > 0 (HM3)

For numerical solution purposes, the time-space solution domain isdiscretized as shown in Figure 16-6, by separating the time and space

into a number of equally spaced discrete points Ar and At denote the

grid point spacing and time increment, respectively Accurate solution with

Fictitious

Block

Fictitious Block

a uniform grid requires sufficiently small Ar and A?, and therefore ahigh level of computational effort This approach is selected here forsimplicity Computationally efficient schemes can be developed by vary-ing the grid size in space and time This is beyond the scope of this book.The grid system depicted in Figure 16-6 is implemented here by centralfinite difference formulae, derived in the previous section The spatial grid

points are denoted by the subscript indices i = 0,1, 2, , N, N +1 Because

the spatial grid points were placed in the center of the grid blocks, the

points identified by i = 0 and i = N +1 are outside the inlet and outlet

boundaries and, therefore, called fictitious points The points designated

by i = l,2, ,N are the real points, called interior points The crete times are denoted by the superscript indices n = 0,1,2, , °o n = 0

dis-denotes the initial condition, at which time the concentrations at variousdiscrete spatial points are prescribed by Eq 16-41

The central difference formulae necessary to develop a numericalsolution are given as following:

The concentration values at the inlet and outlet boundaries are estimated

by the following arithmetic averages:

2

-; + C;

(16-49)

(16-50)

Applying the Crank-Nicolson formulation (see Thomas, 1982), the centraldifference discretization of Eq 16-40 in time and space yields:

p? (Ar) - ex'n Ci+\ Ci-\ r" — r"

i I = 1 / 2 1 = 1/2

The outlet boundary condition given by Eq 16-43 is discretized as:

CN+l CN _ Q

from which the fictitious point value is obtained as:

For convenience in numerical solution, first we rearrange Eq 16-51 as:

n+icn+i + #«+ic«+i + Cf = Df , i = 1, 2, , N and

Substituting Eqs 16-55 and 59 into Eqs 16-65 and 66, respectively, yields:

and solved by Thomas algorithm (see Thomas, 1982)

Figure 16-7 shows the typical concentration profiles calculated

by Civan and Engler (1994) at different times using the parametervalues a = 0.08m3/X fe = 1.67xlO-5/i-1, / = 51.7, g = 1.25, fc = 0.5/n,rw

= 0.05m, and re=\0m.

Note that in this presentation, the dimensionless quantities were defined

as following:

3.05 4.05 5.05 6.05 7.05 Radial Distance From Wellbore, r, meters Figure 16-7 Mud filtrate concentration vs radial distance from wellbore at

different times (reprinted from Journal of Petroleum Science and Engineering,

Vol 11, Civan, F., & Engler, T., "Drilling Mud Filtrate Invasion—ImprovedModel and Solution," pp 183-193, ©1994; reprinted with permission fromElsevier Science)

Amaefule, J O., Kersey, D G., Norman, D L., & Shannon, P M.,

"Advances in Formation Damage Assessment and Control Strategies,"CIM Paper No 88-39-65, Proceedings of the 39th Annual TechnicalMeeting of Petroleum Society of CIM and Canadian Gas ProcessorsAssociation, June 12-16, 1988, Calgary, Alberta, 16 p

Baghdikian, S Y., Sharma, M M., & Handy, L L., Flow of Clay

Sus-pensions Through Porous Media, SPE Reservoir Engineering, Vol 4.,

No 2 , May 1989, pp 213-220

Burnett, D S., Finite Element Analysis, Addison-Wesley Publishing

Company, Massachusetts, 1987, 844 p

Cernansky, A., & Siroky, R., "Hlbkova Filtracia Polydisperznych Castic

z Kvapalin na Vrstvach z Vlakien," Chemicky Prumysl, Vol 32 (57),

No 8, 1982, pp 397-405

Cernansky, A., & Siroky, R., "Deep-bed Filtration on Filament Layers on

Particle Polydispersed in Liquids," Int Chem Eng., Vol 25, No 2,

Civan, F.,"Solving Multivariable Mathematical Models by the Quadrature

and Cubature Methods," Journal of Numerical Methods for Partial Differential Equations, Vol 10, 1994, pp 545-567.

Civan, F /'Rapid and Accurate Solution of Reactor Models by the

Quadrature Method," Computers & Chemical Engineering, Vol 18 No.

10, 1994, pp 1005-1009

Civan, F., Predictability of Formation Damage: An Assessment Study andGeneralized Models, Final Report, U.S DOE Contract No DE-AC22-90-BC14658, April 1994