Reservoir Formation Damage Episode 3 Part 4 pps

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 solution technique. Various sources of uncertainties affect the reliability of the predictions 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) also involve uncertainties. In general, solutions of models, called model predictions, and the response 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) / — /I A, , Xj, , JC n I \i I W 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 of the Society of Petroleum Engineers). 558 Reservoir Formation Damage in which / is a system response and x l ,x 2 ,x 3 , denote the various input variables and parameters. Uncertainties involved in actual calcu- lations (predictions) or measurements (experimental testing) lead to esti- mated or approximate results, the accuracy of which depend on the errors involved. Therefore, the actual values are the sum of the estimates and the errors. Thus, if /, x { , x 2 , , x^ indicate the estimated values of the function and its variables, and Af,Ax l ,Ax 2 , ,Ax n represent the errors or uncertainties associated with these quantities, the following equations, expressing the actual quantities as a sum of the estimated values and the errors 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): df(x } ,x 2 , ,x n ) dJt, + higher order terms (17-11) Neglecting the higher order terms for relatively small errors, Eq. 17-11 can be written in a compact form as: / = 7 + £(-*,-*,)|^ (17-12) /=! 3x i 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 or by a norm as (Reilly, 1992) ~ \2 1/2 (17-14) The uncertainty associated with summation and/or subtraction of numbers, defined by Eqs. 17-7 through 9, is the square root of the squares of the uncertainties in these numbers (Reilly, 1992). Thus, if = ±x l ±x 2 ± ±x n then 1/2 (17-15) (17-16) The relative uncertainty in a multiplication or division of numbers is the square root of the sum of the squares of the relative uncertainties in these numbers (Reilly, 1992). Thus, if x r x x (17-17) then Ax, Ax, Ax, 1/2 (17-18) and = f±Af = f 1± Ax. 1 = 1 V •*! 1/2' (17-19) For example, given a function like (17-20) the error in the function value as a result of using an erroneous measured value of * = 0.5±0.1 can be estimated by applying Eq. 17-13 as: A/ = Ax -4xe- (17-21) 560 Reservoir Formation Damage 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: / = 0.6 ±0.1 As another example, consider f = f(x,y) = x/y Thus (17-22) (17-23) -jc/y 2 (17-24) Eq. 17-13 can be applied as: f^ (17-25) 5y Thus, Eqs. 17-23 through 25 lead to the following relative error expression: (17-26) A/ = A* | Ay 7 1*1 \y A similar result is obtained for / = /(jc, y) = xy. It can be shown for / = /(jc, y) = x + y that A/ _ A x + Ay 7 ~ *I+|51 For / = /(jc) = x", we can derive A/ A£ f x For / =/(jc) = Injc, we obtain A7 AJc 7 (17-27) (17-28) (17-29) Model Assisted Analysis and Interpretation of Laboratory Field Tests 561 Applying Eq. 17-14 for Eq. 17-23 results in 7 ~\2 AJc | | Ay x 1/2 (17-30) A similar result for / = xy is obtained. Bu and Damsleth (1996) consider Darcy's law as an example A Ap Let b = q/Ap (17-31) (17-32) 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 of mathematical models (Lehr et al., 1994). Sensitivity analysis can be used for various purposes, including model validation, evaluating model behavior, estimating model uncertainties, decision making using uncertain models, and determining potential areas of research (Lehr et al., 1994). Sensitivity analysis provides information about the effect of the errors and/or variations in the variables and/or parameters and models on the predicted behavior. Sensitivity of a model to changes in its input data determines the condition of the model (Chapra and Canale, 1998). The sensitivity of a system's outcome or response to changes in a variable is defined by the partial derivative (Lehr et al., 1994): (17-34) 562 Reservoir Formation Damage Relative sensitivity (Lehr et al., 1994) or the condition number (Chapra and 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 parameter value. Thus, for a single parameter function, the relative sensitivity can be expressed by means of Eq. 17-12 as (Lehr et al., 1994; Chapra and Canale, 1998): A - L\ „ — (17-35) 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 equations can be formulated for determining the sensitivity trajectory. The following example by Lehr et al. (1994) illustrates the process. Consider a mathematical model given by an ordinary differential equation as: df(x,t) ~ (17-37) A differentiation of the g(f,x,t) function with respect to the variable (or parameters) x leads to: dx dx dx (17-38) Substituting Eqs. 17-34 and 37 into Eq. 17-38 and rearranging yield the following sensitivity trajectory equation: 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 to determine 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 =d 2 T B /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 17-2 clearly indicates that there is a strong correlation between the sensitivity with 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 the evaporation curve. Sensitivity Tg/Tb (273) o.o •0.2- -o.e-i •o.s- -1.0 300 350 400 450 500 Parameter, Tb 550 600 Figure 17-2. Dependence of the sensitivity with respect to T G on the initial bubble point T B (modified after Lehr, W., Calhoun, D., Jones, R., Lewandowski, A., and Overstreet, R., "Model Sensitivity 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). 564 Reservoir Formation Damage Sensitivity Tg/Tg (273) o.o- •0.2- •0.6 - •0.8 - -1.0 100 200 300 400 500 600 700 800 900 1000 Parameter, Tg Figure 17-3. Dependence of the sensitivity with respect to T G on T G (after Lehr, W., Calhoun, D., Jones, R., Lewandowski, A., and Overstreet, R., "Model Sensitivity 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 estab- lished without validating it by experimental data. However, the micro- scopic phenomena is too complex to study each detail individually. Thus, a practical method is to test the system for various conditions to generate its input-output response data. Then, determine the model parameters such that model predictions match the actual measurements within an acceptable tolerance. However, some parameters may be directly measurable. A general block diagram for parameter identification and model development and verification is given in Figure 17-4 (Civan, 1994). Experimental System The experimental system is a reservoir core sample subjected to fluid flow. The input variables are injection flow rate or pressure differential and its particles concentration, temperature, pressure, pH, etc. The output ' Measured Input Variables Porous media properties 2. Influent conditions 1. Assumptions based on the observational evidence and common sense. 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 >-*> r 8- o CD 566 Reservoir Formation Damage variables are the measured pressure differential, pH, and species con- centration of the effluent. Frequently, data filtering and smoothing are required 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 at the pore throat may cause temporary permeability improvements, which are real and not just a noise. Baghdikian et al. (1989) reported that accumulation and flushing of particle floccules can cause an oscillatory behavior during permeability damage. Parity Equations An integration of the model equations over the length of core yields the equations of a macroscopic model called the "parity equations." However, for a complicated model of rock-fluid-particle interactions in geological porous formations it is impractical to carry out such an integration analytically. Hence, an appropriate numerical method, such as described 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., the conditions of the influent, confining stress, temperature, pressure, pH, etc.). Parameter Estimation with Linearized Models Luckert (1994) points out that estimating parameters using linearized model equations obtained by transformation is subject to uncertainties and errors because of the errors introduced by numerical transformation of the experimental data. Especially, numerical differentiation is prone to larger 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 = \ogK + qlog — (17-41) [...]... 0 .41 0 .36 0. 24 0.25 0.06 0.10 . 10.00 SO 4 270 130 0 2.5 30 4 690 1200 990 9.5 930 43 0 1.8 720 600 31 30 510 660 36 18 49 0 NO 3 0.62 0.08 0.10 0.60 <0. 04 <0.01 0.27 0 .41 0 .36 0. 24 0.25 0.06 0.10 <0.02 0.29 0.90 0 .38 0.51 0.10 0.29 CO 3 0. 04 0.10 0.86 0.05 1.00 0. 13 0 . 34 0.00 0.05 0 .39 0.01 0.11 0.26 1 .30 0 .45 0.21 0. 03 0 .36 0.07 <1 0. 03 HCO 3 140 110 46 0 60 170 180 230 64 90 210 60 120 140 170 120 110 79 30 0 180 156 120 Na 52100 46 400 18160 43 8 30 35 560 42 100 45 940 44 850 50700 36 200 44 210 48 540 41 050 2 045 0 42 990 37 810 537 80 1 745 0 30 910 30 220 45 950 Ca 40 10 5170 892 5 34 0 37 70 37 50 34 90 5820 40 50 4 030 537 0 47 10 44 90 1580 43 4 0 5200 5120 892 245 0 2058 6020 Mg 137 0 131 0 601 1210 1 040 1180 1 140 1950 1270 1960 2510 1720 133 0 5 64 131 0 1 34 0 136 0 46 8 1 43 0 11 64 1900 After . no. 01120 0 031 2 2 34 65 2 34 91 0 039 2 2 34 56 237 44 237 50 237 53 237 68 237 69 01070 015 43 2 34 99 00078 01862 01972 01950 01 935 0 049 6 0 048 8 0 049 0 00090 0 045 9 0 130 1 01 131 00795 0 131 9 01169 00291 0 133 0 00788 0 032 5 02187 00551 01266 01812 01165 Well. Cyp. AuxV. Wallersbrg Cyp. Cyp. AuxV. Average depth (ft.) 2825 33 60 967 32 36 32 34 31 02 2 940 31 72 2959 28 64 236 4 2 744 1820 1808 2611 28 04 2909 1159 NA 2629 Resis. (ohm-m) 0.0601 0.0 639 0. 138 2 0.0676 0.0782 0.0692 0.0670 0.0615 0.0621 0.0 746 0.0 632 0.0621 0.0692 0.1252 0.0707 0.07 23 0.06 13 0. 140 6 0.0877 0.0877 0.0610 Eh (mV) -186 - 2 34 -31 8 -150 -33 5 -287 -285 - 1 34 -270 -2 13 -102 -287 -276 -31 4 -156 -2 84 - 144 -59 -178 - 73 -111 pH 6 .44 6.89 7.01 5.92 7. 24 6.56 7. 24 5 . 34 6. 24 6.56 6. 63 6.81 6.85 7 .36 5.92 6. 74 6. 13 7.22 6 .47 6.68 6. 24 TDS (mg/1) 14 533 3 132 780 49 667 12 546 0 100768 116926 1 242 54 137 44 4 140 537 108598 132 040 141 529 117069 56100 1166 03 11 642 9 146 456 4 432 0 87529 85 236 137 329 Cl 87000 78000 29000 740 00 59000 68000 72000 840 00 830 00 65000 79000 85000 69000 33 000 67000 71000 85000 25000 52000 51276+ 82000 Br