Solution to a radial diffusion model with polynomialdecaying flux at the boundary Yarith del Angel, Mayra N´ un˜ez-L´ opez and Jorge X. Velasco-Hern´andez Instituto Mexicano del Petr´ oleo Programa de Matem´ aticas Aplicadas y Computaci´ on March 13, 2012 Yarith del Angel, Mayra N´ un˜ ez-L´ opez and Jorge X. Velasco-Hern´ andez (IMP) March 13, 2012 1 / 16 2012 2 / 16 . Mayra N´ un˜ ez-L´ opez and Jorge X. Velasco-Hern´ andez (IMP) (1) (2) March 13.Introduction In order to plan a successful secondary recovery project sufficient reliable information concerning the nature of the reservoir-fluid system must be available. Heterogeneous medium: ∂ 2 p2D ∂tD ∂p1D (1 − ω) ∂tD ω = ∂ 2 p2D 1 ∂p2 ∂p1D + − (1 − ω) 2 rD ∂rD ∂tD ∂rD = λ(p2D − p1D ) Figure: Heterogeneous porous mo Yarith del Angel. Homogeneous medium: φµct ∂p ∂ 2 p 1 ∂p + = . Velasco-Hern´ andez (IMP) (4) March 13. 2012 3 / 16 . 2 ∂r r ∂r k ∂t Yarith del Angel.Introduction Diffusion in fractal medium: ∂p k 1 ∂ c = ∂t µ r Df −1 ∂r r Df −1−θ ∂p (3) ∂r where k = k0 r −θ . Mayra N´ un˜ ez-L´ opez and Jorge X. it therefore depletes soon due to the relative limited volume that the sandstone body can hold. Yarith del Angel. each cluster is embedded in a particular body of sandstone which in itself is surrounded by very low permeability structures. Mayra N´ un˜ ez-L´ opez and Jorge X.Chicontepec Chicontepec shows low production rates Chicontepec is characterized not so much by a fractal network that extends along particular formations. In Chicontepec. but rather by an extreme compartmentalization of clusters of the network. 2012 4 / 16 . Whenever oil is extracted from a sandstone. Velasco-Hern´ andez (IMP) March 13. rw kt = . Mayra N´ un˜ ez-L´ opez and Jorge X. stratification. 2012 5 / 16 .The basic model The observed behavior in the flow-rate production or in the pressure can be attributed to many different factors other than fractures such as faults. φµct rw2 2πhk . + ∂r 2 r ∂r k ∂t rD tD pD (5) r . and heterogeneities. = (pi − p) qBµ = Yarith del Angel. Velasco-Hern´ andez (IMP) March 13. Fluid flow: ∂ 2 p 1 ∂p φµct ∂p = . The basic model With this variables. tD ) = 1 . 2 rD ∂rD ∂tD ∂rD (6) We consider either of the following boundary conditions in the well radius pD (1. we have rD = 1 and our main equation becomes ∂ 2 pD 1 ∂pD ∂pD + = . ∂pD . . = −1 ∂rD . production at constant pressure (7) production at constant rate (8) rD =1 Usually this condition is expressed as . ∂pD . . ∂rD . = 0. rD =RD Yarith del Angel. 2012 6 / 16 . Velasco-Hern´ andez (IMP) March 13. Mayra N´ un˜ ez-L´ opez and Jorge X. Boundary approximates the waiting times of a sub-diffusive flow outside the sandstone: .The basic model Chicontepec: The oil depletes before the fluxes equilibrate albeit with the flux at the boundary being very small but non-zero. ∂pD . . = −(1 − tD−α ). (9) ∂rD . Mayra N´ un˜ ez-L´ opez and Jorge X. Velasco-Hern´ andez (IMP) (10) March 13.rD =RD Finally. Yarith del Angel. 2012 7 / 16 . let pD (rD . 0) = 0. pD (1. 2012 8 / 16 .0 Dimensionless Time 0 5 10 15 20 25 30 Figure: pD vs tD for a closed boundary reservoir located at RD = 10.5 1.Dimensionless Pressure Closed reservoir 2. Velasco-Hern´ andez (IMP) (11) March 13. J12 (RD ui ) − J12 (ui ) Yarith del Angel. Mayra N´ un˜ ez-L´ opez and Jorge X. tD ) = +2 RD2 2 1 t + ln (RD ) − D R2 − 1 2 RD2 − 1 2 ∞ X e −ui tD i=1 ui2 J12 (RD ui ) .0 1. tD > 0 . Mayra N´ un˜ ez-L´ opez and Jorge X.A model with variable flux at the boundary Interior boundary condition ∂pD = −1 . s 2 I1 (RD s)K1 ( s) − I1 ( s)K1 (RD s) Yarith del Angel. s) = 3 √ √ √ √ + f (s). tD > 1 (12) √ √ √ √ K1 (RD s)I0 (rD s) + I1 (RD s)K0 (rD s) ¯ p¯D (rD . tD ≤ 1 ∂pD = −α ∂rD rD =RD (1 − tD ). 2012 (13) 9 / 16 . Velasco-Hern´ andez (IMP) March 13. ∂rD rD =1 and exterior boundary condition ∂pD = H(tD − 1)(1 − tD−α ) ∂rD rD =RD ( 0. A model with variable flux at the boundary 5. Yarith del Angel.5 Ε=2.85 Dimensionless Time 2000 4000 6000 8000 10 000 Figure: The influence of on the radial model with flux at the boundary can be seen in the curve pD vs tD and external radius RD = 200.5 ---.0 4. Velasco-Hern´ andez (IMP) March 13. 2012 10 / 16 .5 Ε=0. Mayra N´ un˜ ez-L´ opez and Jorge X.Ε=0.5 Dimensionless Pressure The production rate decreases very fast and the reservoir gradually fills ----------- 4.0 Ε=0 Ε=1. tk ). 2012 Yarith del Angel.3 and 13. approximating the equation (2) using a explicit finite difference scheme we have k k pik+1 = (g − fi )pi−1 + (1 − 2g )pik + (fi + g )pi+1 ∆t 2∆r (1+i∆r ) .Numerical solution Let pik the value of the pressure drop at node (ri . Mayra N´ un˜ ez-L´ opez and Jorge X. 6 6 5 5 Dimensionless Pressure Dimensionless Pressure where fi = 4 3 2 1 4 3 2 1 0 0 0 2000 4000 6000 8000 10000 Dimensionless Time (a) 0 50 100 150 (b) Figure: Numerical solution of the radial diffusion with parametersMarch =0. g= ∆t ∆r 2 and pik ≈ p(ri . tk ). Velasco-Hern´ andez (IMP) 200 Radius 11 / 16 . 0 1 0 0 0 0 2000 4000 6000 8000 10000 -1 Dimensionless Time (a) 50 100 150 200 250 Radius (b) Figure: Numerical solution for different permeabilities for α=0. Mayra N´ un˜ ez-L´ opez and Jorge X. 2012 12 / 16 .4. Velasco-Hern´ andez (IMP) March 13.3 epsilon 0.6 6 5 5 4 epsilon 0.8 2 epsilon 1 1 Dimensionless Pressure Dimensionless Pressure Numerical solution 4 3 epsilon 0.1 epsilon 0. Yarith del Angel.3 epsilon 0.8 2 epsilon 1.1 3 epsilon 0. 2012 13 / 16 . where T is the final time of the production reservoir and the total flow in the system is Z rD =RD Z T d p(rD . tD )rD drD = (1 − RD )T − αRD T 1−α .Conservation de the diffusion equation To validate the numerical solution we calculated the analytical integral over tD and rD and verify the conservation of flux in the system Finally we integrate respect to time. Mayra N´ un˜ ez-L´ opez and Jorge X.4. 0 dt rD =1 Figure: Numerical approximation for different mesh sizes where =0. 16 14 Pressure 12 100 10 500 8 1000 2000 6 4000 5000 4 6000 2 0 0 2000 4000 andez 6000(IMP)8000 Yarith del Angel. Velasco-Hern´ 10000 March 13.15 and α=0. T = 10000.0928 Yarith del Angel.15 and α=0.067 Table: Relative errors with different mesh size with ∆t=0.60 9300. =0. Mesh size 100 500 1000 2000 4000 5000 6000 Numerical integral 37609.90 9153.Conservation of the diffusion equation We compare the numerical solution with different mesh size with the following parameters: RD = 200.0005.84 Relative error 3.18 0.09 0.35 0.00 10038. Analytical integral: 8484. 2012 14 / 16 .70 0.07 0. where ∆t =0.10 14450.27 9054.0005.4. Mayra N´ un˜ ez-L´ opez and Jorge X.80 11512.43 0. Velasco-Hern´ andez (IMP) March 13. Velasco-Hern´ andez (IMP) March 13.Concluding remarks We obtain a radial diffusion model for variable flux at the boundary to represent the behavior of Chicontepec. Future work: Obtain the analytical solution for different values of rD and compare with the numerical solution Calculate the semi-log derivative for long times to apply the model to test pressure . 2012 15 / 16 . Mayra N´ un˜ ez-L´ opez and Jorge X. Yarith del Angel. References [1] Van Everdingen. Cambridge.F. A. [5] Morton K. 320-326. (1996).S. M. Trans. 186.: Operational Methods in Applied Mathematics. [3] Bourdarot: Well Testing: Interpretation Methods. [4] Carslaw H. Vol. A.: The Application of the Laplace Transformation to Flow Problems in Reservoirs. and Cable. SPE. Yarith del Angel.: A Polynomial Approach to the van Everdingen-Hurst Dimensionless Variables for Water Encroachment.C. (1994). (1949).: Numerical solutions of Partial Differential Equations. and Mayers D.J. C. and Hurst W. 2012 16 / 16 .. (1963).A. ditions Technip.F.. Dover. [2] Klins. Bouchard. (1988). Velasco-Hern´ andez (IMP) March 13.W. 15433.L. Mayra N´ un˜ ez-L´ opez and Jorge X. and Jaeger J. AIME.