Analytical Solution of the Diffusivity Equation

FAQReferencesSummaryInfo Learning Objectives Introduction Analytical Solution Linear Systems Radial Systems Program Exercise Resources Home HOME Radial SystemLinear System Introduction Programming Exercise Resources Analytical Solution

FAQReferencesSummaryInfo Learning Objectives Introduction Analytical Solution Linear Systems Radial Systems Program Exercise Resources Learning Objectives Learning objectives in this module: 1.Develop problem solution skills using computers and numerical methods 2.Review flow equations and methods for analytical solution the equations 3.Develop programming skills using FORTRAN No new FORTRAN elements are introduced in this module, you should, from what you have learnt earlier, be able to solve this problem without any problems

FAQReferencesSummaryInfo Learning Objectives Introduction Analytical Solution Linear Systems Radial Systems Program Exercise Resources Introduction In analysis of fluid flow in petroleum reservoirs, we need partial differential equations that describe the fluids flowing and the reservoir they are flowing in. Then we need to be able to solve the equations for the conditions of flow that we are interested in. Derivation of the equations normally involves the following elements:  Continuity equations  Darcy’s equations  PVT relationships for the fluids  Compressibility of reservoir rock Examples of such equations are the simplest forms of the diffusivity equations for linear and radial flow

FAQReferencesSummaryInfo Learning Objectives Introduction Analytical Solution Linear Systems Radial Systems Program Exercise Resources Introduction Below, the geometries of the two simple reservoir systems and the corresponding partial differential equations are shown: r Linear flow Radial flow

FAQReferencesSummaryInfo Learning Objectives Introduction Analytical Solution Linear Systems Radial Systems Program Exercise Resources Analytical Solution In order to solve the partial differential equations shown earlier, we need to have initial conditions, i.e. initial pressure distribution in the system, and boundary conditions, i.e. rates or pressures at for instance left and right sides of the systems. We will examine two of the most common sets of conditions and analytical solutions for these Linear System Radial System

FAQReferencesSummaryInfo Learning Objectives Introduction Analytical Solution Linear Systems Radial Systems Program Exercise Resources Linear System For the linear system, we have a horizontal porous rod, where fluid is being injected into the left face at a flow rate Q. The injected fluid will be transported through the rod and eventually be produced out of the right face of the rod. The one-phase partial differential equation (PDE) for this system, in it’s simplest form, is called the linear diffusivity equation. It is valid for one-dimensional flow of a liquid in a horizontal system, where it is assumed that porosity (  ), viscosity (  ), permeability (k ) and compressibility (c ) all are constants. Q in x=0 x=L Q out PLPL PRPR

FAQReferencesSummaryInfo Learning Objectives Introduction Analytical Solution Linear Systems Radial Systems Program Exercise Resources Linear System The linear diffusivity equation may be written as: (1) If the initial pressure of the rod is P R, and we assume constant pressures at the end faces, P L and P R for left and right faces, respectively, we have the following analytical solution: (2) Continue

FAQReferencesSummaryInfo Learning Objectives Introduction Analytical Solution Linear Systems Radial Systems Program Exercise Resources which is the expression for a straight line Linear System The pressure solution is dependent on position, x, as well as time, t. As time increases, the exponential term becomes smaller, and eventually the solution reduces to the steady-state form: Click to see what the equation reduces to as time increases ? (2) (3)

FAQReferencesSummaryInfo Learning Objectives Introduction Analytical Solution Linear Systems Radial Systems Program Exercise Resources Linear System The corresponding steady state differential equation is obtained by setting the right hand side of Eq. (1) equal to zero: Graphically, the solution may be presented as : P x Left side pressure Initial and right side pressure Steady state solution Transient solution As can be observed from the figure, the pressure will increase in all parts of the system for some period of time (transient solution), and eventually approach the final distribution (steady state), described by a straight line between the two end pressures (4)

FAQReferencesSummaryInfo Learning Objectives Introduction Analytical Solution Linear Systems Radial Systems Program Exercise Resources Radial System For the radial system below (one-dimensional cylindrical coordinates), we have a horizontal porous disk, where fluid is being injected at the outer boundary and produced at the center. The one- phase one-dimensional (radial) flow equation (PDE) in this coordinate system becomes: For an infinite reservoir at an initial pressure P i and with P(r  ∞)=P i and well rate q from a well in the center (at r=r w ) the analytical solution is: where is the exponential integral (6) (5) r rwrw Continue

FAQReferencesSummaryInfo Learning Objectives Introduction Analytical Solution Linear Systems Radial Systems Program Exercise Resources Radial System A steady state solution does not exist for an infinite system, since the pressure will continue to decrease as long as we produce from the center. However, if we use a different set of boundary conditions, so that: we can solve the steady state form of the equation: By integrating twice, the steady state solution becomes: (7) (9) (8) Continue

FAQReferencesSummaryInfo Learning Objectives Introduction Analytical Solution Linear Systems Radial Systems Program Exercise Resources Program Exercise This programming exercise involves the construction of a reservoir simulation program, although in a very simple form. The following steps should be carried out: 1.Make a FORTRAN program that computes the analytical solutions of Eqs. (2) and (6). When the program is started, it should ask on the screen which geometry should be used, LIN or RAD, and the name of the input data file (where all parameters are to be read from) 2.Read from the screen which values of x (or r) and t the solution should be computed for. 3.The results should be written to the screen as well as to an output file Data set for linear system Data set for radial system Here

FAQReferencesSummaryInfo Learning Objectives Introduction Analytical Solution Linear Systems Radial Systems Program Exercise Resources Data Set for Linear System P L =2 atm  = 0.25 k=1.0 DarcyP 0 =P R =1 atm L=100 cmA=10 cm 3  =1.0 cpc= atm -1 t-intervals:t=10 -3, 10 -2, s x-intervals:x=5, 50 cm k = permeability [Darcy] L = length (of rod) [cm]  =viscosity [cp] N= porosity c = compressibility [atm -1 ]

FAQReferencesSummaryInfo Learning Objectives Introduction Analytical Solution Linear Systems Radial Systems Program Exercise Resources Data Set for Radial System h=1000 cm  = 0.25 k=1.0 Darcyc= atm -1 r w =25 cmq=10 4 cm 3 /s  =1.0 cp t-intervals:t= 1E06, 5E06, 10E06 s r-intervals:r=100, 1000, 5000 cm k = permeability [Darcy] r w =wellbore radius [cm]  =viscosity [cp] N= porosity c = compressibility [atm -1 ] q=flowrate [cm 3 /s]

FAQReferencesSummaryInfo Learning Objectives Introduction Analytical Solution Linear Systems Radial Systems Program Exercise Resources Introduction to Fortran Fortran Template here The whole exercise in a printable format here Web sites  Numerical Recipes In Fortran Numerical Recipes In Fortran  Fortran Tutorial Fortran Tutorial  Professional Programmer's Guide to Fortran77 Professional Programmer's Guide to Fortran77  Programming in Fortran77 Programming in Fortran77

FAQReferencesSummaryInfo Learning Objectives Introduction Analytical Solution Linear Systems Radial Systems Program Exercise Resources General information Title:Analytical Solution of the Diffusivity Equation Teacher(s):Professor Jon Kleppe Assistant(s):Per Jørgen Dahl Svendsen Abstract:Provide a good background for solving problems within petroleum related topics using numerical methods 4 keywords:Diffusivity Equation, Linear Flow, Radial Flow, Fortran Topic discipline: Level:2 Prerequisites:None Learning goals:Develop problem solution skills using computers and numerical methods Size in megabytes:0.7 MB Software requirements:MS Power Point 2002 or later, Flash Player 6.0 Estimated time to complete: Copyright information:The author has copyright to the module and use of the content must be in agreement with the responsible author or in agreement with About the author

FAQReferencesSummaryInfo Learning Objectives Introduction Analytical Solution Linear Systems Radial Systems Program Exercise Resources FAQ No questions have been posted yet. However, when questions are asked they will be posted here. Remember, if something is unclear to you, it is a good chance that there are more people that have the same question For more general questions and definitions try these Dataleksikon Webopedia Schlumberger Oilfield Glossary

FAQReferencesSummaryInfo Learning Objectives Introduction Analytical Solution Linear Systems Radial Systems Program Exercise Resources References See for instance: H. S. Carslaw and J. C. Jaeger: Conduction of Heat in Solids, 2nd ed., Oxford, 1985 Numerical Recipes in Fortran in pdf format online: Numerical Recipes in Fortran

FAQReferencesSummaryInfo Learning Objectives Introduction Analytical Solution Linear Systems Radial Systems Program Exercise Resources Summary Subsequent to this module you should...  be able to keep track of loops and conditional statements  have no problems handling output and input data  have obtained a better understanding on solving problems in Fortran