Reservoir Formation Damage Episode 3 Part 4 pps

Model Assisted Analysis and Interpretation of Laboratory Field Tests 557dependent on various factors, including 1 the adequacy of the model,2 the accuracy of the input data, and 3 the ac

Model Assisted Analysis and Interpretation of Laboratory Field Tests 557

dependent on various factors, including (1) the adequacy of the model,(2) the accuracy of the input data, and (3) the accuracy of the solutiontechnique Various sources of uncertainties affect the reliability of thepredictions of models, as described in Figure 17-1 by Bu and Damsleth(1996) Experimental measurements taken under controlled test conditions

to determine the input-output (or cause-and-affect or the parity relationship)response of systems (such as core plugs undergoing a flow test) alsoinvolve uncertainties

In general, solutions of models, called model predictions, and theresponse of the test systems under prescribed conditions can be repre-sented numerically or analytically by functional relationships, mathe-matically expressed as:

f- f ( r r r "1 (17-6)

INPUT DATA

Uncertainty interval Model parameters Equations/Model

Figure 17-1 Sources of errors and uncertainty associated with mathematical

modeling (after Bu and Damsleth, ©1996 SPE; reprinted by permission ofthe Society of Petroleum Engineers)

in which / is a system response and x l ,x 2 ,x 3 , denote the various

input variables and parameters Uncertainties involved in actual lations (predictions) or measurements (experimental testing) lead to esti-mated or approximate results, the accuracy of which depend on the errorsinvolved Therefore, the actual values are the sum of the estimates and

calcu-the errors Thus, if /, x {, x2, , x^ indicate the estimated values of the function and its variables, and Af,Ax l,Ax2, ,Axn represent the errors

or uncertainties associated with these quantities, the following equations,expressing the actual quantities as a sum of the estimated values and theerrors associated with them, can be written:

= x

n ^L L\J^,f\

/ = / ± A /

(17-7)(17-8)

(17-9)(17-10)The estimation of the propagation and impact of errors is usually based

on a Taylor series expansion (Chapra and Canale, 1998):

Then, applying Eqs 17-7-10 into Eq 17-12, the error or the uncertainty

in the function value can be estimated by (Chapra and Canale, 1998):

(17-13)1=1

Model Assisted Analysis and Interpretation of Laboratory Field Tests 559

Ax,1/2

Thus, substituting Jc = 0.5 and A* = 0.1 into Eqs 17-20 and 21 yields/ = 0.6 and A/ = 0.1 Therefore, the calculated value is expressed accord-ing to Eq 17-10 as:

7 1*1 \y

A similar result is obtained for / = /(jc, y) = xy.

It can be shown for / = /(jc, y) = x + y that

Model Assisted Analysis and Interpretation of Laboratory Field Tests 561Applying Eq 17-14 for Eq 17-23 results in

A similar result for / = xy is obtained.

Bu and Damsleth (1996) consider Darcy's law as an example

Thus, they express relative error in the calculated K value as a function

of the measurements involving errors as (apply Eq 17-14):

AK

K

~ \ 2 ~ \ 2 ~ \ 2 1/2

(17-33)

Sensitivity Analysis—Stability and Conditionality

Sensitivity analysis is an important tool for systematic evaluation ofmathematical models (Lehr et al., 1994) Sensitivity analysis can be usedfor various purposes, including model validation, evaluating modelbehavior, estimating model uncertainties, decision making using uncertainmodels, and determining potential areas of research (Lehr et al., 1994).Sensitivity analysis provides information about the effect of the errorsand/or variations in the variables and/or parameters and models on thepredicted behavior Sensitivity of a model to changes in its input datadetermines the condition of the model (Chapra and Canale, 1998).The sensitivity of a system's outcome or response to changes in avariable is defined by the partial derivative (Lehr et al., 1994):

(17-34)

Relative sensitivity (Lehr et al., 1994) or the condition number (Chapraand Canale, 1998) is defined as the ratio of the relative change or error

in the function to the relative change or error in the variable or parametervalue Thus, for a single parameter function, the relative sensitivity can

be expressed by means of Eq 17-12 as (Lehr et al., 1994; Chapra andCanale, 1998):

A

Thus, the condition number or relative sensitivity can be used as a criteria

to evaluate the effect of an uncertainty in the x variable on the condition

of a system as (Chapra and Canale, 1998):

< 1, effect in the function is attenuated

= 1, effect in the function is same as the variation in the variable

> 1, effect in the function is amplified

(17-36)

Given the differential equations of a model, the sensitivity equationscan be formulated for determining the sensitivity trajectory The followingexample by Lehr et al (1994) illustrates the process

Consider a mathematical model given by an ordinary differentialequation as:

Model Assisted Analysis and Interpretation of Laboratory Field Tests 563

(17-39)

dt\dx dt a/

One of the practical applications of the sensitivity analysis is todetermine the critical parameters, which strongly effect the predictions

of models (Lehr et al., 1994) Lehr et al (1994) studied the sensitivity

of an oil spill evaporation model Figures 17-2 and 17-3 by Lehr et al.(1994) depict the sensitivity of the fractional oil evaporation,/, from an

oil spill with respect to the initial bubble point, T B, and the rate of bubble

point variation by the fraction of oil evaporated (the slope of the

evapor-ation curve), T G =d2TB/dtdf, respectively Examination of Figures 17-2 and 17-3 reveals that the initial bubble point T B is the critical parameter,influencing the sensitivity of the oil spill evaporation model Figure 17-2clearly indicates that there is a strong correlation between the sensitivitywith respect to the slope of the evaporation curve and the bubble point.Whereas, Figure 17-3 shows that the sensitivity with respect to the slope

of the evaporation curve cannot be correlated with the slope of theevaporation curve

Lehr, W., Calhoun, D., Jones, R., Lewandowski, A., and Overstreet, R., "ModelSensitivity Analysis in Environmental Emergency Management: A Case Study

in Oil Spill Modeling," Proceedings of the 1994 Winter Simulation Conference,

J D Tew, S Manivannan, D A Sadowski, and A F Seila (eds.), pp

1198-1205, ©1994 IEEE; reprinted by permission)

Model Validation, Refinement, and Parameter Estimation

As stated by Civan (1994), confidence in the model cannot be lished without validating it by experimental data However, the micro-scopic phenomena is too complex to study each detail individually Thus,

estab-a prestab-acticestab-al method is to test the system for vestab-arious conditions to generestab-ateits input-output response data Then, determine the model parameters suchthat model predictions match the actual measurements within an acceptabletolerance However, some parameters may be directly measurable Ageneral block diagram for parameter identification and model developmentand verification is given in Figure 17-4 (Civan, 1994)

Experimental System

The experimental system is a reservoir core sample subjected to fluidflow The input variables are injection flow rate or pressure differentialand its particles concentration, temperature, pressure, pH, etc The output

2 Averaged differential balance equations.

3 Averaged rate equations.

4 Boundary/Initial Conditions.

5 Numerical solution.

Understanding of Mechanism ofFormation Damage Processes

1 Evaluation

2 Sensitivity Analysis with respect to various assumptions and parameters.

o

8-I

Figure 17-4 Steps for formation damage process identification and model development (after Civan, 1994; reprinted by

permission of the U.S Department of Energy)

3 O

>-*>

r8-

o

CD

variables are the measured pressure differential, pH, and species centration of the effluent Frequently, data filtering and smoothing arerequired to remove noise from the data as indicated in Figure 17-4.However, some important information may be lost in the process Millan-Arcia and Civan (1990) reported that frequent breakage of particle bridges

con-at the pore throcon-at may cause temporary permeability improvements, whichare real and not just a noise Baghdikian et al (1989) reported thataccumulation and flushing of particle floccules can cause an oscillatorybehavior during permeability damage

Parity Equations

An integration of the model equations over the length of core yieldsthe equations of a macroscopic model called the "parity equations."However, for a complicated model of rock-fluid-particle interactions ingeological porous formations it is impractical to carry out such anintegration analytically Hence, an appropriate numerical method, such asdescribed in Chapter 16, is facilitated to generate the model response(pressure differential across the core or sectional pressure differentials,and the effluent conditions) for a range of input conditions (i.e., theconditions of the influent, confining stress, temperature, pressure, pH, etc.)

Parameter Estimation with Linearized Models

Luckert (1994) points out that estimating parameters using linearizedmodel equations obtained by transformation is subject to uncertainties anderrors because of the errors introduced by numerical transformation ofthe experimental data Especially, numerical differentiation is prone tolarger errors than numerical integration Luckert (1994) explains this

problem on the determination of the parameters K and q of the following

filtration model:

dV 2

(17-40)

where t and V denote the filtration time and the filtrate volume,

respec-tively This equation can be linearized by taking a logarithm as

log d 2 t

dV 2

Model Assisted Analysis and Interpretation of Laboratory Field Tests 567

Thus, a straightline plot of Eq 17-41 using a least-squares fit provides

the values of log K and q as the intercept and slope of this line,

respec-tively Luckert (1994) points out, however, this approach leads to highlyuncertain results because numerical differentiation of the experimentaldata involves some errors, second differentiation involves more errors thanthe first derivative, and numerical calculation of logarithms of the secondnumerical derivatives introduce further errors Therefore, Luckert (1994)recommends linearization only for preliminary parameter estimation, whenthe linearization requires numerical processing of experimental data fordifferentiation Luckert (1994) states that "From a statistical point of view,experimental values should not be transformed in order that the errordistribution remains unchanged."

In Chapter 12, detailed examples of constructing diagnostic charts fordetermining the parameters of the incompressive cake filtration model byCivan (1998) have been presented It has been demonstrated that themodel parameters can be determined from the slopes and intercepts ofthe straight line plots of the experimental data according to the linearizedforms of the various equations, describing the linear and radial filtrationprocesses Using the parameter values determined this way, Civan (1998)has shown that the model predictions compared well with the measuredfiltrate volumes and cake thicknesses The advantage of this type of directmethod is the uniqueness of the parameter values as described in Chap-ter 12 As described in Chapter 10, Wojtanowicz et al (1987, 1988) alsoused linearized diagnostic equations given in Table 10-1 for determiningthe parameters of their single-phase fines migration models

History Matching for Parameter Identification

The model equations contain various parameters dealing with the rateequations They are determined by a procedure similar to history matchingcommonly used in reservoir simulation In this method, an objectivefunction is defined as

(17-42)

where Y m is the measured values, Y c is the calculated values, W is the weighting matrix, W = V"1, V is the variance-covariance matrix of the measurement error, and n is the number of data points Szucs and Civan

(1996) facilitated an alternative formulation of the objective function,called the p-form, which lessens the effect of the outliers in the measureddata points

Then, a suitable method is used for minimizing the objective function

to obtain the best estimates of the unknown model parameters (Ohen andCivan, 1990, 1993)

Note that the number of measurements should be equal or greater thanthe number of unknown parameters When there are less measurementsthan the unknown parameters, additional data can be generated by inter-polation between the existing data points However, for meaningfulestimates of the model parameters the range of the data points shouldcover a sufficiently long test period to reflect the effect of the governingformation damage mechanisms

The above described method has difficulties First, it may require alot of effort to converge on the best estimates of the parameter values.Second, there is no guarantee concerning the uniqueness of the parametervalues determined with nonlinear models However, some parameters can

be eliminated for less important mechanisms for a given formation andfluid system The remaining parameters are determined by a historymatching procedure In this method, the best estimates of the unknownparameters are determined in such a way that the model predictions matchthe measurements obtained by laboratory testing of cores within a reason-able accuracy (Civan, 1994)

Frequently used optimization methods are: (1) trial-and-error [tediousand time consuming]; (2) Levenberg-Marquardt (Marquardt, 1963) method[requires derivative evaluation, computationally intensive]; (3) simu-lated annealing [algebraic and practical (Szucs and Civan, 1996; Ucan

6 by Gadiyar and Civan (1994) shows the effect of the acidizing reactionrate constants on permeability alteration The solid line represents the bestfit of the experimental data using the best estimates of the model para-meters obtained by history matching When the parameter values were

Model Assisted Analysis and Interpretation of Laboratory Field Tests 569

Figure 17-5 Sensitivity analysis for the tertiary reaction at different

tempera-tures: (a) 22°C and (b) 66°C (after Ziauddin et al., ©1999; reprinted bypermission of the Society of Petroleum Engineers)

Cumulative Volume (cm ) u>00

Figure 17-6 Dependence of the sensitivity with respect to the model

param-eters (after Gadiyar and Civan, ©1994 SPE; reprinted by permission of theSociety of Petroleum Engineers)

perturbed in a random manner, a significantly different trend, represented

by the dashed line, was obtained

Determination of the Formation Damage

Potential by Simulation

Reservoir exploitation processes frequently cause pressure, temperatureand concentration changes, and rock-fluid and fluid-fluid interactions,which often adversely effect the performance of these processes Prior

to any reservoir exploitation applications, extensive laboratory, field andsimulation studies should be conducted for assessment of the formationbrine and mineral chemistry and the formation damage potential of thereservoir Consequently, optimal strategies can be designed to effectivelymitigate the adverse effects and improve the oil and gas recovery.Typical examples of such detailed studies have been presented in aseries of reports by Demir (1995), Haggerty and Seyler (1997), and Seyler(1998) for characterization of the brine and mineral compositions and theinvestigation of the formation damage potential in the Mississippian AuxVases and Cypress formations in the Illinois Basin Demir (1995) usedthe chemical data on the formation brines and minerals to interpret the