Applied Reservoir Simulation Course Workshop Problems Reservoir Simulation Application Training Course and (Eclipse) Workshop GeoQuest Training and Development, And NExT Denver and Houston March 06 Applied Reseervoir Simulation Day 1 1
Outline of Workshop Problems Problem 1: A. IMPES and Implicit Comparison B. Time Truncation Tests C. Numerical Dispersion March 06 Applied Reseervoir Simulation Day 1 2
Outline of Workshop Problems 2. Problem 2: A. Water Coning Critical Coning Rate B. Water Influx History Matching K h C. Water Coning I. History Matching K v II. Creation of Pseudo K rw in Coarse Grid to Match Coning D. Vertical Equilibrium Comparison March 06 Applied Reseervoir Simulation Day 1 3
Outline of Workshop Problems 3. Problem 3: History Matching Primary Production A. Sensitivity Simulations B. History Match C. Predictions March 06 Applied Reseervoir Simulation Day 1 4
Problem 1 March 06 Applied Reseervoir Simulation Day 1 5
Problem 1 Part A: IMPES Fully Implicit Comparison Coarse Grid IMPES Sub-directory March 06 Applied Reseervoir Simulation Day 1 6
Reservoir Details We have a linear reservoir 5000 x 100 x 30 feet. Grid is 50 x 1 x 1 Water injector at one end and producer at other end Two phase (oil-water) system Oil viscosity = 2 cp and water viscosity = 0.5 cp Simulate 2000 days. March 06 Applied Reseervoir Simulation Day 1 7
FloViz View March 06 Applied Reseervoir Simulation Day 1 8
Choice in Solution of the Equations IMPES Implicit Pressure Explicit Saturation Linear problem smaller Through put limitations: 5 10% PV Stability problems Timestep size - small March 06 Applied Reseervoir Simulation Day 1 9
Choice in Solution of the Equations Fully Implicit Unconditionally Stable large timesteps Timesteps controlled by time truncation error Few long timesteps March 06 Applied Reseervoir Simulation Day 1 10
Comparison IMPES and Fully Implicit Solution Amount of Numerical Dispersion (Error) in the results Work to obtain solution Number of linear iterations Number of non-linear (Newton) iterations CPU time per time step Total CPU time March 06 Applied Reseervoir Simulation Day 1 11
Iteration Process in Reservoir Simulation Example of linear and non-linear iteration process: 4 non-linear iterations Usually a non-linear iteration requires 10 to 30 linears to converge pressure and saturations March 06 Applied Reseervoir Simulation Day 1 12
Data Set Names IMPES.DATA uses IMPES solution FULLIMP.DATA uses fully implicit solution Location sub-directory: Problem 1/IMPES March 06 Applied Reseervoir Simulation Day 1 13
View Results: Comparison Oil Production Rate Production Water Cut Map of Water Saturations At 500 Days At 1000 Days Number of Linear Iterations in Each Timestep Sum of Linear Iterations March 06 Applied Reseervoir Simulation Day 1 14
View Results: Comparison Newton (non-linear) Iterations in each timestep Sum of Newton Iteration Timestep Length Note: Fully Implicit takes 31 timesteps IMPES takes 207 timesteps CPU Time Per Timestep Total CPU Time March 06 Applied Reseervoir Simulation Day 1 15
Your Task View the.data sets Difference IMPES (in Schedule Section) Fully Implicit (E100 Default) Run the data sets with ECLIPSE 100 Plot the results Study the results and answer the following questions March 06 Applied Reseervoir Simulation Day 1 16
Questions Which technique takes more linear iterations per timestep? Which technique takes more Newton iterations per timestep? Which technique requires more work to solve the simulation? Why? We will discuss the results in class. March 06 Applied Reseervoir Simulation Day 1 17
Problem 1 Part A: IMPES Fully Implicit Comparison Fine Grid IMPES 2 Sub-directory March 06 Applied Reseervoir Simulation Day 1 18
Reservoir Details We have the same reservoir, fluids, etc. except that now we will subdivide the grid into 2000 blocks in the x-direction instead of the previous 50. The length of the grid blocks are now 2.5 feet instead of the previous length of 100 feet. March 06 Applied Reseervoir Simulation Day 1 19
Run and Compare Results Run the IMPES.DATA and FULLIMP.DATA in the IMPES2 subdirectory and run Graf in the IMPES2 sub-directory. With these smaller grid blocks the problem is numerically more difficult, but the numerical dispersion is less. March 06 Applied Reseervoir Simulation Day 1 20
Questions Answer the same questions from the previous part. Note the differences in the IMPES and Fully Implicit runs. Magnitude of the numerical dispersion in the water cut, etc. Instabilities in the IMPES run. Number of linear and non-linear iterations Sum of the linear and non-linear Total CPU time for these cases. March 06 Applied Reseervoir Simulation Day 1 21
Conclusion For a blackoil simulation with ECLIPSE we will prefer to use the default solution Fully Implicit since the results are much more stable and the CPU time is less. March 06 Applied Reseervoir Simulation Day 1 22
Problem 1 Part B: Time Truncation Test: Quarter 5-Spot Water Flood March 06 Applied Reseervoir Simulation Day 1 23
March 06 Applied Reseervoir Simulation Day 1 24 We have the accumulation term in our equations, for example When it is discretized we get terms as follows Time Truncation Error o o B S t φ ( ) t P - P - t P - P d ) B (1/ d ) S (1 - + t P - P C B ) S (1 - ) ( - t S - S B ) ( cow n cow n+1 n o n+1 o i p i w oi i o n i o n+1 i f wi oi i z y x n o n+1 o i w i i z y x φ φ φ 1
Time Truncation Error This discretization process yields and error that is O( t) in the solution of the equations. This error is called Time Truncation Error March 06 Applied Reseervoir Simulation Day 1 25
Simulation Situation 11x11x1 quarter 5-spot involving water injection Diagonal Grid 1452 x 1452 x 50 meter flow field K = 830 md, ϕ = 0.17 Water Injector on water rate control Producer on liquid rate control Simulate 3660 days March 06 Applied Reseervoir Simulation Day 1 26
Grid March 06 Applied Reseervoir Simulation Day 1 27
Time stepping Three data sets are provided with different time stepping regimes maxts.data initial timestep = 10 days, grows to max timestep = 183 days, 53 total timesteps smallts.data Initial timestep = 1 day, max timestep = 10 days, 389 total timesteps mints.data Initial timestep = 1 day, maximum timestep = 1 day, 3666 total timesteps March 06 Applied Reseervoir Simulation Day 1 28
Questions to Consider While Viewing the Results What is the effect of solving the flow field with a few large timesteps? What is the most accurate solution? Which solution requires the most CPU time? Given a balance between CPU time and errors in the solution which time stepping system would you prefer? March 06 Applied Reseervoir Simulation Day 1 29
Results to be Viewed with Graf Timestep length Water cut effect of time truncation error Oil production rate effect of time truncation error Water saturation maps CPU time per timestep Total CPU time March 06 Applied Reseervoir Simulation Day 1 30
Your Task View the.data sets Difference in the TUNING Keyword timestep control Run the data sets with ECLIPSE 100 Plot the results Study the results and answer the following questions We will discuss the results in class March 06 Applied Reseervoir Simulation Day 1 31
My New Computer and ECLIPSE 2004a runs so fast has trouble storing timestep lengths Following are results from old Pentium III computer March 06 Applied Reseervoir Simulation Day 1 32
CPU Time per Timestep in Seconds March 06 Applied Reseervoir Simulation Day 1 33
Total CPU Time March 06 Applied Reseervoir Simulation Day 1 34
Problem 1 Part C: Numerical Dispersion: Radial Model with Gas Coning March 06 Applied Reseervoir Simulation Day 1 35
Numerical Dispersion Definition: The error that occurs as we discritize the partial differential equations on a grid to create the finite difference equations. March 06 Applied Reseervoir Simulation Day 1 36
Finite Difference Approximations to First Derivatives forward difference f = x Error term: forward error f x + x f x ( ) ( ) x x 2! 2 x f 2 x 3! 2 3 x f 3... March 06 Applied Reseervoir Simulation Day 1 37
Objective To see the effect of number of grid block on a simulation model, a 2-D radial (r, z) model was set up The reservoir has constant properties; see data set RAD4.DATA (rad4.data) March 06 Applied Reseervoir Simulation Day 1 38
Description Metric Units Oil, Water, Gas, Dissolved Gas The reservoir is 200 meters thick with the top 50 meters in the gas cap Flat Top at 2950 meters The width (radius) of the model is 500 meters March 06 Applied Reseervoir Simulation Day 1 39
Description The well is completed in the bottom 50 meters of the model The oil production rate is set at 6000 Sm 3 / day March 06 Applied Reseervoir Simulation Day 1 40
Description The flow field was sub-divided into grids that were 4x1x4, 8x1x8, 16x1x16, 32x1x32 and 64x1x64 blocks The well completions are located in the proper layers so that the simulation problems are identical except for the nr and nz (number of grid blocks in the r and z directions) values March 06 Applied Reseervoir Simulation Day 1 41
Data Sets Provided Five data sets are provided for you: RAD4.DATA 4x1x4 grid RAD8.DATA 8x1x8 grid RAD16.DATA 16x1x16 grid RAD32.DATA 32x1x32 grid RAD64.DATA 64x1x64 grid 128 x 1 x 128 grid takes half a day to run so not provided March 06 Applied Reseervoir Simulation Day 1 42
Creation of the Radial Grid Spacing In ECLIPSE radial grid Have the option of letting simulator calculate the radial grid spacing Required input: INRAD internal (well) radius and OUTRAD outside radius and NR number of grid blocks in the radial direction ECLIPSE will then calculate the radial grid spacing with exponential growth March 06 Applied Reseervoir Simulation Day 1 43
r i The outer cell radius for cells logarithmically spaced can be calculated by: = r w Equations for Calculating Exponential Growth Grid Spacing n exp n i c r ln r e w OR r i = r w r r e w n n i c r i r w r e n c n i = outer radius of cell i = well radius = external radius = number of cells = number of cell i March 06 Applied Reseervoir Simulation Day 1 44
Creation of the Radial Grid Spacing For our case: INRAD = 0.22 meters OUTRAD = 500 meters The r results for the 5 grids follow March 06 Applied Reseervoir Simulation Day 1 45
r Values for NR = 4 Increasing Exponentially Next to the well 1.299 8.969 61.928 427.584 Outer most grid block March 06 Applied Reseervoir Simulation Day 1 46
r Values for NR = 8 Increasing Exponentially Next to the well 0.358 0.941 2.472 6.497 17.071 44.857 117.868 309.716 Outer most grid block March 06 Applied Reseervoir Simulation Day 1 47
r Values for NR = 16 Increasing Next to the well Exponentially 0.137 0.221 0.359 0.582 0.943 1.529 2.479 4.018 6.513 10.558 17.114 27.742 44.971 72.897 118.167 191.549 Outer most grid block March 06 Applied Reseervoir Simulation Day 1 48
r Values for NR = 32 Increasing Next to the well Exponentially 0.060 0.077 0.097 0.124 0.158 0.201 0.256 0.326 0.415 0.528 0.673 0.856 1.090 1.388 1.768 2.250 2.865 3.648 4.645 5.913 7.529 9.586 12.204 15.538 19.783 25.187 32.068 40.829 51.983 66.184 84.265 107.285 March 06 Applied Reseervoir Simulation Day 1 49 Outer most grid block
r Values for NR = 64 Increasing Next to the well Exponentially 0.0282 0.0319 0.0360 0.0406 0.0458 0.0516 0.0583 0.0658 0.0742 0.0837 0.0945 0.1066 0.1203 0.1357 0.1531 0.1728 0.1950 0.2200 0.2482 0.2801 0.3161 0.3566 0.4024 0.4540 0.5123 0.5781 0.6523 0.7360 0.8305 0.9371 1.0574 1.1931 1.3462 1.5190 1.7140 1.9340 2.1822 2.4623 2.7784 3.1350 3.5374 3.9914 4.5037 5.0818 5.7341 6.4701 7.3006 8.2376 9.2950 10.4880 11.8342 13.3532 15.067 17.0012 19.1834 21.6457 24.4240 27.5590 31.0963 35.0877 39.5914 44.6732 50.4073 56.8773 March 06 Applied Reseervoir Simulation Day 1 50 Outer most grid block
z Values for the 4 Grids nz = 4 z = 50 meters nz = 8 z = 25 meters nz = 16 z = 12.5 meters nz = 32 z = 6.25 meters nz = 64 z = 3.125 meters March 06 Applied Reseervoir Simulation Day 1 51
Grids for Radial Coning Problem March 06 Applied Reseervoir Simulation Day 1 52
64 x 1 x 64 Grid March 06 Applied Reseervoir Simulation Day 1 53
RAD4.DATA from FloViz March 06 Applied Reseervoir Simulation Day 1 54
RAD8.DATA from FloViz March 06 Applied Reseervoir Simulation Day 1 55
RAD16.DATA from FloViz March 06 Applied Reseervoir Simulation Day 1 56
RAD32.DATA from FloViz March 06 Applied Reseervoir Simulation Day 1 57
RAD64.DATA from FloViz March 06 Applied Reseervoir Simulation Day 1 58
Numerical Dispersion Numerical Dispersion can occur in both the r and z direction By decreasing r and z the numerical dispersion from both discretization errors will decrease March 06 Applied Reseervoir Simulation Day 1 59
Your Task View the 5 data sets with an editor Run the 5 data sets with ECLIPSE Plot the results Analyze the effect of numerical dispersion on the Gas-Oil Ratio We will discuss the results in class March 06 Applied Reseervoir Simulation Day 1 60
Results with 128x1x128 Grid March 06 Applied Reseervoir Simulation Day 1 61
Question? Which refinement is more important? Refinement in the radial direction? OR refinement in the vertical direction? For this gas coning situation? March 06 Applied Reseervoir Simulation Day 1 62
To Answer the Question We assume that the 64 x 1 x 64 numerical is close to the solution. We will run two cases: radial-fine.data refined in the radial direction only 64 x 1 x 8 grid vertical-fine.data refined in the vertical direction only 8 x 1 x 64 grid March 06 Applied Reseervoir Simulation Day 1 63
Guess Which Refinement is More Important Before you run the 2 cases try to estimate / guess which refinement case will be closer to the 64 x 1 x 64 very fine grid results. Then run the cases, plot with fine.grf Did you guess correctly? March 06 Applied Reseervoir Simulation Day 1 64
End of Workshop Problem 1 March 06 Applied Reseervoir Simulation Day 1 65