Nigerian Journal of Technology (NIJOTECH) Vol 36, No 1, January 2017, pp 461 468 Copyright Faculty of Engineering, University of Nigeria, Nsukka, Print ISSN: 0331-8443, Electronic ISSN: 2467-8821 wwwnijotechcom http://dxdoiorg/104314/njtv36i120 ANALYSIS OF PRESSURE VARIATION OF FLUI IN BOUNE CIRCULAR RESERVOIRS UNER THE CONSTANT PRESSURE OUTER BOUNARY CONITION I Erhunmwun 1,* and J A Akpobi 2 1, 2 EPARTMENT OF PROUCTION ENGINEERING, UNIVERSITY OF BENIN, BENIN CITY, EO STATE NIGERIA E-mail addresses: 1 irediaerhunmwun@unibenedu, 2 johnakpobi@unibenedu ABSTRACT In this work, we have investigated the well pressure distribution in a bounded circular reservoir under the condition of constant pressure outer boundaries The diffusivity equation was used in the analysis The finite element technique, using Lagrange quadratic shape elements was employed to carry out the analysis over the cross-section of the reservoir which involves discretizing the domain into finite element, analysing these finite element, assembling the results from the analysis of the analysed finite element, imposing the boundary conditions and finally, getting the results that represent the entire domain The results obtained where shown in a log log plot (dimensionless pressure against dimensionless time) for dimensionless radii of 1 to 1,000,000 in log cycles It was shown that the relationship between dimensionless pressure and dimensionless time was linear whose slope was zero The result obtained at the wellbore was compared with the results obtained by Van Everdigen and Hurst It was shown that there was a strong positive correlation between the results The result obtained from the analysis also shows the pressure variation outside wellbore of the same reservoir It is important to note that solutions from existing literature only state the pressure at the wellbore at a particular time but this work predicts the pressure variation in the entire reservoir from the wellbore to the external boundary at the same time Keywords: Reservoir, Constant Terminal Rate, imensionless Variables, iffusivity Equation, Wellbore And Weak Formulation Nomenclature English Letters B Formation volume factor, RB/STB c Compressibility, psia -1 h Thickness, ft K Stiffness matrix M Mass matrix n Number of elements P Pressure, psi P imensionless pressure P imensionless pressure rate P i Initial reservoir pressure, psi Q Terminal flow rate q Volumetric flow rate, STB/ r Radius, ft r imensionless radius r External radius, ft e r imensionless externalradius e r w Wellbore radius, ft s Time step, hr t Time, hr t imensionless time w Weight function For all Greek letters t Time increment, hr Family of approximation Porosity, fraction k Permeability, md Viscosity, cp Pi Interpolation function 1 INTROUCTION There are several methods of evaluating the reservoir parameters [1] It was shown that solutions to differential equations describing flow in petroleum reservoir for given initial and boundary conditions can be expressed compactly using dimensionless variables and parameters Several of these solutions are important in reservoir engineering applications [2 7] Transient pressure response for a well producing from a finite reservoir of circular, square, and rectangular drainage shapes has been studied by [2, 8 14] among others Everdingen And Hurst [2] presented the solution to the diffusivity equation in eq (8) in the form of infinite series of exponential terms and Bessel functions The authors evaluated this series for several values of r e over a wide range of values for t Chatas [15] and Well * Corresponding author, tel: +234 807 072 8898
ANALYSIS OF PRESSURE VARIATION OF FLUI IN BOUNE CIRCULAR RESERVOIRS UNER THE I Erhunmwun & J A Akpobi [16] conveniently tabulated these solutions for the following two cases: Infinite-acting reservoir and Finiteradial reservoir Mishra and Ramey [17] presented a build-up derivative type curve for a well with storage and skin, and producing from the centre of a closed, circular reservoir Their type-curve applies for large producing times The work by [18] presents drawdown and build-up pressure derivative type-curves for a well producing at a constant rate from the centre of a finite, circular reservoir The outer boundary may be closed, or at a constant pressure The differences between the responses for a well in a closed, circular reservoir (fully developed field), and a well in a circular reservoir with a constant-pressure outer boundary (active edge water drive system, or developed five-spot fluid-injection pattern) were discussed esign relations were developed to estimate the time period which corresponds to infinite-acting radial flow, or to a semi-log straight line on a pressure vs logarithm of time graph Producing time effects on buildup responses were studied using the slope of a dimensionless build-up graph proposed in [19] In all the literatures reviewed so far, the researchers focused on predicting the wellbore pressure [2, 15, 16], etc Sometimes, it is important to know the pressure history outside the reservoir is scarce in the literature This study therefore seeks to look at the reservoir pressure both within and outside the wellbore of a bounded circular reservoir 2 THEORY The law of conservation of mass, arcy s law and the equation of state has been combined to obtain the following partial differential equation: with the assumptions that compressibility, c is small and independent of pressure, P; permeability, k, is constant and isotropic; viscosity,, is independent of pressure; porosity,, is constant; and that certain terms in the basic differential equation (involving pressure gradients squared) are negligible This equation is called the diffusivity equation and the term is the inverse of the diffusivity constant, In this work, the diffusivity equation was analysed for bounded circular reservoirs, the case in which the well is assumed to be located in the centre of a cylindrical reservoir under the condition of constant external boundary 3 GOVERNING EQUATION Initial and boundary conditions: i at t ii ( ) for iii ( ) The above equations incorporate physical parameters such as permeability, and it would be futile to solve this problem for a particular combination of values for these parameters imensionless variables are designed to eliminate the physical parameters that affect quantitatively, but not qualitatively, the reservoir response The above equations are in arcy units, and the dimensionless terms will render the system of units employed irrelevant For this line source model, 3 dimensionless variables are required In US Oilfield units, distance, time and pressure are replaced as follows: imensionless time: imensionless distance: imensionless pressure: By defining dimensionless variables this way, it is possible to create an analytical model of the well and reservoir, or theoretical type-curve, that provides a global description of the pressure response that is independent of the flow rate or actual values of the well and reservoir parameters Eq1 can be transformed by substituting the following dimensionless variables in Eqs 5-7 into eq 1 and this becomes: and the initial and boundary conditions becomes: 1 imensionless initial condition (uniform pressure in the reservoir):, 2 imensionless inner boundary condition (constant rate at the wellbore): 3 imensionless Outer Boundary Conditions: Constant pressure outer boundary, Eq 8 can also be written in a condensed form as:, ( ) Nigerian Journal of Technology Vol 36, No 1 January 2017 462
ANALYSIS OF PRESSURE VARIATION OF FLUI IN BOUNE CIRCULAR RESERVOIRS UNER THE I Erhunmwun & J A Akpobi 4 FINITE ELEMENT FORMULATION 41 Weak Formulation In the development of the weak form, we assumed a quadratic element mesh and placed it over the domain and apply the following steps: From eq 12, we have:, and Substituting eq 21 into eq 20, we have: ( ) Multiply eq 13 by the weight w function and integrate the final equation over the domain Eq 14 becomes, [ ( )] [ ( )] Integrating eq 15 with respect to z, then, over the limits, we have: [ ( )] Eq 16 can be exploded into: ( ) Factor out where In matrix form we can represent the semi-discrete finite element model as thus, { } { } { } Where Integrating eq 17 by part, we have: [ ] Grouping eq 18 into linear and bilinear components, we have: [ ] Where 5 INTERPOLATION FUNCTION The weak form in eq 20 requires that the approximation chosen for P should be at least quadratic in r so that there are no terms in eq 20 that are identically zero Since the primary variable is simply the function itself, the Lagrange family of interpolation functions is admissible We proposed that P is the approximation over a typical finite element domain by the expression: Using Quadratic Lagrange Interpolation functions for a quadratic element: ( ) The coefficient matrix can be easily derived by substituting the Lagrange interpolation functions into eq 25 respectively The matrices are shown below: [ ] [ ] Also, the mass matrices can be easily derived by substituting the Lagrange interpolation functions into eq 26 respectively The matrices are shown below: [ ] [ ] Using four quadratic elements, In this analysis, we have withheld the computational details of the shape assembly of the finite element Nigerian Journal of Technology Vol 36, No 1 January 2017 463
ANALYSIS OF PRESSURE VARIATION OF FLUI IN BOUNE CIRCULAR RESERVOIRS UNER THE I Erhunmwun & J A Akpobi analysis (FEA) used However, the authors would be glad to interact with researchers who may want to refer to the computational mathematics involved 6 TIME APPROXIMATION For a given time step s, eq 24will be written as { } { } { } For the next time step s+1, eq 24 becomes { } { } { } Multiply eq33 by 1 and eq 34 by, then we add the two resulting equations, [ ] [ { } { } ] [ ] [ { } { } ] { } { } The family of interpolation for time consideration is given as: { } { } { } { } Substitute eq36 into eq35 and using the Crank- Nicholson Scheme where, [[ ] [ ]] { } [[ ] [ ]] { } [{ } { } ] From the initial condition given in eq 9 for a constant terminal rate case, it implies that when s 0, all dimensionless pressure in the reservoir will be zero Also, the flow rate was constant at the wellbore all through operation This means that [{ } { } ]Hence, eq 37 becomes: [[ ] [ ]] { } Where [[ ] [ ]{ } { }] { } [[ ] [ ]] [[ ] [ ]] { } { } 7 RESULTS AN ISCUSSION The steady state condition applies, after the transient period, to a well-draining a cell which has a completely open outer boundary It is assumed that, for a constant rate of production, fluid withdrawal from the well will be exactly balanced by fluid entry across the outer boundary and therefore, onstant at i e,, and t and This condition is appropriate when pressure is being maintained in the reservoir due to either natural water influx or artificially by the injection of some displacing fluid The semi-steady state flow equations are frequently applied when the rate, and consequently the position of the no-flow boundary surrounding a well, is slowly varying functions of time If the production rate of an individual well is changed, for instance, due to closure for repair or increasing the rate to obtain a more even fluid withdrawal pattern throughout the reservoir, there will be a brief period when transient flow conditions prevail followed by stabilized flow for the new distribution of individual well rates Thus, this solution of the diffusivity equation models radial flow of slightly compressible liquid in a homogeneous reservoir of uniform thickness; reservoir at uniform pressure before production; unchanging pressure at the outer boundary; and production at constant rate from a single well (centred in the reservoir) with wellbore radius The results obtained from this analysis were shown in the form of graphs of dimensionless pressure against dimensionless time This was shown in Fig 1 Fig 1 is a log log plot of dimensionless pressure against dimensionless time The graph shows for different dimensionless radii ranging between 1 and 1000000 in log cycles It was seen from the graph that the dimensionless pressure history of the reservoir was not captured at the initial stage between the dimensionless time of zero and the respective dimensionless times in Fig 1 This was due to the fact that, within these regions, the reservoir was at the infinite acting state After these infinite acting period, it was observed that the dimensionless pressure increases and later becomes uniform became the withdrawn fluid has been completely replaced This condition remains throughout the entire life of the reservoir presumably To test for the degree of accuracy of the results, a percentage error computation was done between the FEM and the results published by [2] Table 1 shows the percentage error between the FEM solutions and the Van Everdigen and Hurst solutions to show the level of discrepancies between the two results It was shown that there was a strong correlation between the two results Nigerian Journal of Technology Vol 36, No 1 January 2017 464
Log imensionless Pressure ANALYSIS OF PRESSURE VARIATION OF FLUI IN BOUNE CIRCULAR RESERVOIRS UNER THE I Erhunmwun & J A Akpobi 10 1 000001 00001 0001 001 01 1 10 100 1000 10000 100000 1000000 01 001 0001 00001 000001 0000001 00000001 1E-08 1E-09 1E-10 Log imensionless Time re=1 re=10 re=20 re=80 re=100 re=200 re=500 re=1000 re=10000 re=100000 re=1000000 Fig 1: Log-Log plot of P against t for various r in the infinite acting regime Table 1: Percentage error of FEM and Van Everdigen 15, n 4 2, n 4 2 5, n 4 3, n 4 3 5, n 4 4, n 4 6, n 4 t % error t % error t % error t % error t % error t % error t 005 04348 02 00000 03 00000 05 01621 05 04839 1 00000 4 02353 0055 04167 022 00000 035 00000 055 01563 06 04511 12 01167 45 01515 006 00000 024 00000 04 00000 06 00000 07 04255 14 01105 5 01470 007 00000 026 00000 045 00000 07 00000 08 02699 16 00000 55 01431 008 00000 028 00000 05 00000 08 00000 09 03876 18 01014 6 01397 009 10274 03 00000 055 00000 09 00000 1 02488 2 00000 65 01368 01 00000 035 00000 06 00000 1 00000 12 02331 22 00951 7 01342 012 00000 04 00000 07 00000 12 00000 14 01106 24 00000 75 01319 014 00000 045 01748 08 01374 14 00000 16 01058 26 05425 8 01300 016 00000 05 00000 09 01325 16 01079 18 01019 28 00000 85 01281 018 00000 055 00000 1 00000 18 00000 2 00988 3 00000 9 01266 02 00000 06 00000 12 01227 2 00000 22 01921 34 00000 10 01238 022 00000 065 00000 14 00000 22 00000 24 00939 38 00000 12 01200 024 00000 07 00000 16 00000 24 00000 26 00920 45 00000 14 01174 026 00000 075 00000 18 00000 26 00000 28 00904 5 00000 16 00578 028 00000 08 00000 2 00000 28 00000 3 00000 55 00000 18 01144 03 00000 085 00000 22 00000 3 00000 35 00864 6 00755 20 01135 035 00000 09 00000 24 00000 35 00000 4 00845 7 00742 22 01129 04 00000 095 00000 26 00000 4 00000 5 00000 8 00000 24 01125 045 00000 1 00000 28 00000 45 00000 6 00000 9 00000 26 01123 05 00000 12 00000 3 00000 5 00000 7 00000 10 00000 28 00561 % error Nigerian Journal of Technology Vol 36, No 1 January 2017 465
ANALYSIS OF PRESSURE VARIATION OF FLUI IN BOUNE CIRCULAR RESERVOIRS UNER THE I Erhunmwun & J A Akpobi Table 2: imensionless Pressure istribution for r 1 5, n 4 and t 0 005 r t 1 10625 11250 11875 12500 13125 13750 14375 15000 0050 0229 0174 0127 0091 0063 0041 0025 0011 0000 0055 0241 0183 0137 0099 0070 0046 0028 0013 0000 0060 0249 0192 0145 0107 0076 0051 0031 0015 0000 0070 0266 0209 0160 0121 0087 0060 0037 0018 0000 0080 0282 0224 0174 0133 0098 0068 0043 0020 0000 0090 0295 0237 0187 0144 0107 0075 0048 0023 0000 0100 0307 0249 0198 0154 0115 0082 0052 0025 0000 0120 0328 0269 0216 0170 0129 0092 0059 0029 0000 0140 0344 0284 0231 0183 0140 0101 0065 0032 0000 0160 0356 0297 0243 0194 0149 0108 0069 0034 0000 0180 0367 0307 0252 0202 0156 0113 0073 0036 0000 0200 0375 0315 0260 0209 0161 0117 0076 0037 0000 0220 0381 0321 0265 0214 0166 0121 0078 0038 0000 0240 0386 0326 0270 0218 0169 0123 0080 0039 0000 0260 0390 0330 0274 0221 0172 0125 0082 0040 0000 0280 0393 0333 0277 0224 0174 0127 0083 0040 0000 0300 0396 0335 0279 0226 0176 0128 0084 0041 0000 0350 0400 0340 0283 0229 0179 0131 0085 0042 0000 0400 0402 0342 0285 0231 0180 0132 0086 0042 0000 0450 0404 0343 0286 0232 0181 0133 0086 0042 0000 0500 0405 0344 0287 0233 0182 0133 0087 0042 0000 0600 0405 0345 0287 0233 0182 0133 0087 0043 0000 0800 0405 0345 0288 0234 0182 0134 0087 0043 0000 Table 3: imensionless Pressure istribution for r 10, n 4 and t 0 05 r t 10000 21250 32500 43750 55000 66250 77500 88750 100000 10 1639 0923 0547 0327 0191 0107 0055 0023 0000 12 1719 0999 0615 0384 0235 0139 0076 0032 0000 14 1719 0999 0615 0384 0235 0139 0076 0032 0000 16 1845 1120 0725 0479 0311 0195 0112 0050 0000 18 1896 1170 0770 0518 0343 0219 0128 0058 0000 20 1941 1213 0811 0553 0372 0241 0142 0065 0000 25 2032 1302 0892 0624 0430 0285 0171 0079 0000 30 2099 1367 0952 0676 0474 0318 0193 0089 0000 35 2149 1415 0996 0715 0506 0342 0209 0097 0000 40 2186 1451 1029 0744 0530 0360 0221 0103 0000 45 2213 1477 1054 0766 0547 0374 0230 0107 0000 50 2234 1497 1072 0782 0560 0383 0236 0110 0000 55 2249 1511 1085 0793 0570 0391 0241 0113 0000 60 2260 1522 1095 0802 0577 0396 0245 0114 0000 65 2268 1530 1103 0809 0583 0400 0247 0116 0000 70 2274 1536 1108 0813 0587 0403 0249 0117 0000 Nigerian Journal of Technology Vol 36, No 1 January 2017 466 e e
ANALYSIS OF PRESSURE VARIATION OF FLUI IN BOUNE CIRCULAR RESERVOIRS UNER THE I Erhunmwun & J A Akpobi r t 10000 21250 32500 43750 55000 66250 77500 88750 100000 80 2282 1544 1115 0819 0592 0407 0252 0118 0000 90 2286 1548 1119 0823 0594 0409 0253 0119 0000 100 2289 1550 1121 0825 0596 0410 0254 0119 0000 110 2290 1552 1122 0826 0597 0411 0254 0119 0000 120 2291 1552 1123 0826 0597 0411 0255 0119 0000 130 2291 1553 1123 0827 0598 0412 0255 0119 0000 140 2291 1553 1123 0827 0598 0412 0255 0119 0000 160 2292 1553 1124 0827 0598 0412 0255 0119 0000 The results presented in Fig 1 are the dimensionless pressure at the wellbore at different dimensionless time for the case of constant pressure outer boundary condition When a reservoir is opened for production, a pressure disturbance is created in the reservoir from the wellbore This disturbance is not only felt at the wellbore but it travels through the entire reservoir formation to the external boundary Therefore, Tables 2 and 3 shows the dimensionless pressure at different points within and outside the wellbore of the reservoir against their corresponding dimensionless time 8 CONCLUSION This paper has been able to present the pressure distribution across a bounded circular reservoir assumed to have constant terminal rate at the wellbore The diffusivity equation was used to analyse the pressure in the system It was shown from Figs 1 that the dimensionless pressure increases drastically immediately this flow regime is attained But as time increases, the dimensionless pressure variation flattens out asymptotically The results obtained from this analysis showed that there was a strong correlation with the results obtained from the Van Everdigen and Hurst It is important to note that the Van Everdigen and Hurst solutions only state the pressure at the wellbore at a particular time but this work predicts the pressure variation in the entire reservoir from the wellbore to the external boundary at the same time These where shown in Tables 2 and 3 and it was noticed that the pressure decreases from the wellbore to the external boundary of the reservoir Therefore the Finite element method has been used to approximate not only the values of the wellbore pressures for bounded circular reservoirs but also pressure outside the wellbore to the external reservoir boundaries 9 REFERENCES [1] Razminia K, Hashemi A, and Razminia A A Least Squares Approach to Estimating the Average Reservoir Pressure Iranian J Oil & Gas Sci Technol 2(1), 22 32 2013 [2] Van Everdingen, A F and Hurst, W The Application of the Laplace Transformation to Flow Problems in Reservoir Trans, AIME 186, 305-3241949 [3] Essa K S M, Mina A N, and Higazy M Analytical Solution of iffusion Equation in Two imensions Using Two Forms of Eddy iffusivities Rom J Phys 56, 1228 1240 2011 [4] Oane M, Medianu R, Georgescu G, Toader, and Peled A The determination of two photon thermal fields in laser-two-layer solids weak interactions using Green function method Rom Rep Phys 65, 997 1005 2013 [5] Timofte C Homogenization Results for Hyperbolic- Parabolic Equations, Rom Rep Phys 62, 229 238 2010 [6] Earlougher, RC, Jr, Ramey, HJ, Jr, Miller, FG, and Mueller, T Pressure istributions in Rectangular Reservoirs J Pet Tech, 199-208 1968 [7] Momoniat E, McIntyre R and Ravindran R Numerical inversion of a Laplace transform solution of a diffusion equation with a mixed derivative term Appl Math Comput 209(2), 222 229 2009 [8] Miller, CC, yes, AB, and Hutchinson, CA, Jr The Estimation of Permeability and Reservoir Pressure from Bottom-Hole Pressure Build-Up Characteristics Trans, AIME 189, 91-104 1950 [9] Aziz, K and Flock, L Unsteady State Gas Flow Use of rawdown ata in the Rediction of Gas Well Behaviour J Can Pet Tech, 2 (l), 9-15 1963 [10] Earlougher R C Jr Advances in Well Test Analysis, Monogragh Series, SPE, allas 1977 Nigerian Journal of Technology Vol 36, No 1 January 2017 467
ANALYSIS OF PRESSURE VARIATION OF FLUI IN BOUNE CIRCULAR RESERVOIRS UNER THE I Erhunmwun & J A Akpobi [11] Ramey, HJ, Jr and Cobb, W M A General Pressure Build-up Theory for a Well in a Closed rainage Area J Pet Tech, 1493-1505 1971 [12] Kumar, A and Ramey, H J, Jr Well-Test Analysis for a Well in a Constant-Pressure Square Soc Pet Eng J, 107-116 1974 [13] Cobb, WM and Smith, J T An Investigation of Pressure Build-up Tests in Bounded Reservoirs," J Pet Tech Vol 27, No 8, 1975, pp 991 997 [14] Chen, HK and Brigham, W E Pressure Build-up for a Well with Storage and Skin in a Closed Square J Pet Tech, 141-146 1978 [15] Chatas, A T A Practical Treatment of Non-steadystate Flow Problems in Reservoir Systems J Pet Eng, B-44 56 1953 [16] John L Well Testing, Soc Pet Eng of AIME, New York 1982 [17] Mishra, S and Ramey, H J, Jr A New erivative Type-Curve for Pressure Build-up Analysis with Boundary Effects Proc, 12th Workshop on Geothermal Reservoir Engineering at Stanford Univ, Stanford, CA, 45-47 1987 [18] Ambastha, K, Anil and Henry J Ramey, Jr Well-Test Analysis for a Well in a Finite, Circular Reservoir Proc, 13 th Workshop on Geothermal Reservoir Engineering at Stanford Univ, Stanford, Califonia, 53-57 1988 [19] Agarwal, R G A New Method to Account for Producing Time Effects when rawdown Type Curves are Used to Analyse Pressure Build-up and Other Test ata, paper SPE 9289 presented at the 55th Annual Meeting of SPE of AIME in allas, TX, 21-24 1980 Nigerian Journal of Technology Vol 36, No 1 January 2017 468