INTRODUCTION TO RESERVOIR SIMULATION - Leoben

1 PHDGP rofessor HeinemannsDoktorandengruppeVerein zur F rderungvon wissenschaftlichen Arbeitenin Reservoircharakterisierung und -simulationTEXTBOOK SERIESVOLUME 4 INTRODUCTION TO RESERVOIR SIMULATIONbyZolt n E. HEINEMANNP rofessor for RESERVOIR EngineeringLeoben, October 2005actualized byDr. Georg MittermeirTehran, February 2013 Roseggerstr. 11a | 8700 Leoben | Austria | Phone: +43 (0)3842/4331611 | E-Mail: AT78 1200 0100 0531 8976 | BIC: BKAUATWWFor kind Attention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s Textbooks available at.]

2 Flow in Porous of the RESERVOIR Flow to RESERVOIR Fractured RESERVOIR EngineeringPHDG Textbooks in preparation, intended to be issued during 2015 and Gridding in RESERVOIR RESERVOIR Fluid CharacterisationSupplementary scripts used at the Montanuniversit t up to the retirement of ProfessorZolt n E. Heinemann in July Recovery No part of this publication may be reproduced in any applicable as teaching material at universities or any other kind of courses without prior, writtenpermission of the PHDG association. Students of the following universities can ask for free copies forpersonal use: Sharif University of Technology, Tehran University, Iran University of Science andTechnology, Shiraz University, University of Miskolc, Montanuniversit t Leoben . iTable of Contents1 INTRODUCTION .. What does SIMULATION mean .. What does RESERVOIR SIMULATION mean .. About the Contents.

3 52 Basic Concept of a RESERVOIR Simulator (IMPES Models) .. Derivation of a Two-Dimensional Two-Phase Black Oil Model .. The Gas Equation .. Cartesian Coordinate System .. Three-Dimensional Three-Phase IMPES-Equation .. Formulation of a Fully Implicit Black Oil Model .. Balance Equations .. Generation and Linearization of the Equations .. Rate .. Term .. Variable Bubble Point .. Solution Methods .. 393 2 -Dimensional Grid Models .. Cartesian Coordinate System .. Local Grid Refinement .. 2 -Dimensional Full-Scale Grid Construction .. Aquifer Grid .. Area Grid .. Vertical Extension .. Cylindrical Coordinate System .. Construction .. Curvilinear Grid .. Curvilinear Grid .. Tube Grid .. Point Geometry .. 684 Initialization of a Grid Initial Pressure and Saturation Distribution in a RESERVOIR .. Properties at Initial State.

4 The most simple Variation of Bubble Point Pressure with Depth .. Variation of Salinity with Variation of Temperature with Depth .. Variation of Oil Density with Depth .. of Hydrocarbon Reservoirs .. and Gas-Oil Contacts .. Vertical Pressure Distribution .. Vertical Saturation Distribution .. Assigning Pressure and Saturation Values to the Blocks .. Based Initialization .. Equilibrium Initialization .. Condition .. Flow .. Practical Remarks .. 815 Wells in RESERVOIR SIMULATION .. The Well Models .. The Peaceman Well Model .. Peaceman Results for Different Well Geometries .. Well Model for Horizontal Wells .. 876 of FiguresFigure : Use mathematical models in analytical and SIMULATION 2 Figure : The nature of numerical 3 Figure : Workflow for building a SIMULATION model .. 4 Figure :Two dimensional block model of a RESERVOIR .

5 8 Figure : Block I is divided by a side through the block center .. 9 Figure : The neighboring blocks I and J .. 10 Figure : Block with its neighbors in a two dimensional system .. 11 Figure : Demonstration of the instability in explicit methods .. 13 Figure : Block model with closed and constant pressure boundaries .. 16 Figure : Equation system corresponding to Figure .. 16 Figure : Cartesian coordinate system ..20 Figure : Construction of a block centered Cartesian grid .. 21 Figure : Construction of a point distributed Cartesian grid .. 22 Figure : Three-dimensional Cartesian grid .. 23 Figure : Steam tube approach in cross section .. 23 Figure : Three dimensional Cartesian 24 Figure : Dip correction of the 24 Figure : Cartesian coordinate system ..43 Figure : Construction of a block centered Cartesian grid .. 44 Figure : Construction of a point distributed Cartesian grid.

6 45 Figure : 3D Cartesian grid in cross section .. 46 Figure : 2 -dimensional Cartesian 46 Figure : Dip correction of the transmissibilities (must be corrected, D should point to the middle of the block)47 Figure : Cross section of a Cartesian layered 48 Figure : Determination of the sub-coordinates .. 49 Figure : Not recommended Cartesian grid refinements.. 50 Figure : Definition of the subdivided area. a) not recommended, b) recommended .. 51 Figure : Incorrect and correct block interfaces for refined grid .. 52 Figure : No orthogonal refinement possible by stretched 53 Figure : Global mesh with two PA s and different spacing in the 54 Figure : Block model constructed from the global mesh .. 54 Figure : Productive Area 55 Figure : Possibilities offered for the highest permeability directions .. 56 Figure : Definition of the areal : Cartesian grid with partially diagonal main permeability direction and increasing anisot-ropy ratio (from left to right)57 Figure : Productive area with channels.

7 57 Figure : PA with two refined zones ..58 Figure : Vertical column of blocks if (a) anisotropy is uniform and (c) changes over the layers .59 Figure : Block-centered cylindrical block 61 Figure : Point-distributed cylindrical block model .. 62 Figure : Three-dimensional radial coordinate 65ivFigure : Streamlines and orthogonal streamtube grid after Mlacnik et al.. 66 Figure : Orthogonal and non-orthogonal stream tube grid for five-spot pattern .. 67 Figure : Use of non-rectangular grid to approximate stream tube grid (after Wadsley[136]) .. 68 Figure : Grid construction with corner point 68 Figure : Formation of two phase 72 Figure : Formation of oil 72 Figure : Primary drainage capillary pressure curves for oil/water and 73 Figure : Different interpretations of the phase contact .. 74 Figure : Initial pressure distribution in the RESERVOIR .. 75 Figure : Initial saturation distribution in the 76 Figure : Initial saturation of the blocks using INITM 77 Figure : Initial saturation of the blocks using INITD 78 Figure : Segregated flow initialization (INITSF option).

8 81 Figure : Numerical representation of drainage and imbibition type capillary pressure curves. 821-11 IntroductionThis volume is the fourth within the series of the RESERVOIR engineering textbooks provided by the Association of Professor Heinemann Doctorate accompanying lecture presumes that the reader possesses profound knowledge of RESERVOIR engineering . A moderate level in applied mathematics and computer application will also be expected. However, this text book is written for petroleum engineers and not for scientists. Wherever if it is possible, application of higher mathematics will be avoided. On the other hand no relation will be made to any given commercial simulator and no special methods will be discussed. The task is to achieve profound understanding rather than to write a manual or a guide line for the SIMULATION textbook is only one of the tools to teach the RESERVOIR SIMULATION techniques at the university and in post graduate courses efficiently.

9 It is important to learn by doing. The readers have to work with a sophisticated RESERVOIR simulator to deepen their theoretical knowledge too. A collection of exercises with growing complexity was worked out and will be used. The exercises are based on the simulator PRS, but they can easily be adapted to any other similar What does SIMULATION meanFor developing an engine, a prototype of it will be built after some preliminary computation and examinations on different modules. The prototype will be tested and improved step by step. Knowledge about the already used product can be gathered and serves as basis for further improvements. This approach is not applicable to every real system, due to one of the following circumstances: the system is unique, it is inaccessible, its dimensions are too large or too small, its life cycle is hydrocarbon reservoirs all four limitations are valid.

10 For such cases three principle possibilities for modeling are given: physical, analogous and numerical. The criteria for all of them is to be able to formulate every or a least the most important physical and chemical processes in a mathematical model. This makes it possible to deduce the similarity conditions for a physical model or to replace the real processes with analogous, but easy realizable 1-2 Introductionprocesses. When dealing with a hydrocarbon RESERVOIR the possibilities for modeling would be a three dimensional sand pack laboratory model or an electrical model consisting of a network of resistivity and electrical capacities. Both were tried but without or with very limited success. The only remaining possibility is numerical mathematical model can be complete. The mathematical formulae are more or less approximations of the physical phenomena Furthermore, to be able to calculate with the formulae they have to be simplified.