Reservoir simulation
A reservoir simulation model forecasts production under different development options. Its value depends on understanding the equations, the simplifications and the quality of the input data.
The section runs from the purpose of models and how they are set up to grids, well models, equations and numerical schemes, and then to upscaling and petrophysics.
Purpose, types and construction of models
- 01What a reservoir simulation model is and why it is neededThe model as a forecasting tool: from the geological model to the simulation grid, what the simulator solves in each cell, input data and results, the stages of the work, limitations and uncertainty. Basic
- 02Types of reservoir simulation modelHow the models differ: black oil, compositional, thermal and chemical; dual porosity for naturally fractured reservoirs; full-field, sector and single-well models. What each cell keeps track of, when each type is needed and what the calculation costs. Intermediate
- 03Building a simulation model: input data and resultsWhat the model needs and in what form — cell arrays, tables, regions, schedule events; the sections of the input file and the keyword format; initialisation by equilibration and the free water level; wells, controls and report steps; output of results; checks on a new model — pore volume, volumes in place, equilibrium, units, productivity — on a 10 × 8 × 3 sector model. Intermediate
- 04History matchingWhat data the model is compared with, what is adjusted and in what order, how to measure the quality of the match, how manual history matching differs from assisted history matching, and why a good match does not guarantee a good forecast. Intermediate
- 05Reservoir simulation in practice: case studiesHow a model supports decisions at different stages of the life of a field: a set of geological realisations ranked by production rather than by volumes in place; designs for sensitivity runs; infill drilling — incremental recovery versus acceleration; gas injection and limits on gas-handling capacity; a mature field — remaining oil and isolation of a swept layer; checking a forecast against what actually happened. The examples are synthetic. Intermediate
Grids and well models
- 06Grids in reservoir simulation modelsCartesian, radial, corner-point and unstructured grids, local grid refinement, choosing the dimensionality and the cell size; numerical dispersion, the grid orientation effect and convergence testing. Intermediate
- 07Interblock flow and transmissibilityHow the simulator calculates flow between neighbouring cells: transmissibility as two half-cells in series and the pressure at the face, geometric factors of Cartesian, radial and corner-point cells, vertical communication of thin layers, faults with throw and non-neighbour connections, upstream weighting of mobility for each phase and what arithmetic and harmonic averages lead to; the limits of the two-point formula and the grid orientation effect. Advanced
- 08Well model in a grid blockWhy the well-block pressure equals neither the bottomhole pressure nor the average pressure in the cell; the Peaceman equivalent radius for a square, rectangular and anisotropic cell, the well index with skin and its check against the Dupuit equation, the distribution of rate between layers and the well controls in the model. Advanced
Equations and numerical methods
- 09Pressure equation for single-phase flowThe pressure equation from the mass balance of a cell, Darcy's law and compressibility: the nonlinear and linear forms, initial and boundary conditions, the cell equation with transmissibilities, explicit and implicit schemes and the limiting time step, a calculation on a four-cell grid. Advanced
- 10Two-phase flow equationsMass balance of oil and water and Darcy's law for each phase, the closure relations, the pressure equation with total mobility and the saturation equation in terms of total velocity, the role of relative permeability and capillary pressure, the finite-difference cell equations and a calculation of waterflooding a strip of four cells. Advanced
- 11Finite differences: from equations to computationFinite-difference formulas from the Taylor series, truncation error and order of accuracy; consistency, stability and convergence; von Neumann stability analysis; the weighted scheme and the Crank–Nicolson scheme; convergence studies in grid size and in time step. Advanced
- 12IMPES and fully implicit schemesHow the simulator takes a time step for the multiphase equations: the IMPES scheme and its step limit from the Courant–Friedrichs–Lewy (CFL) condition, the fully implicit scheme — residuals, the Jacobian, Newton iterations and convergence criteria, time step selection, the sequential implicit and adaptive implicit methods; calculations on a strip of four cells. Advanced
- 13Solving systems of linear equations in simulatorsHow the matrix of a reservoir model is structured and why its conditioning worsens as the time step increases; Gaussian elimination, LU decomposition, band fill-in and the role of ordering; the Jacobi, Gauss–Seidel and successive over-relaxation methods with a calculation on a four-cell section; Krylov methods and preconditioning (ILU, CPR, multigrid); stopping criteria, the link between residual and material balance and why the solver determines the run time. Advanced
- 14Numerical dispersionWhy the upstream scheme smears the Buckley–Leverett front: the modified equation and the numerical diffusion coefficient, a self-sharpening front a couple of cells wide, the effects of cell size, time step and the choice of explicit or implicit scheme; early breakthrough and underestimated oil production compared with the exact solution, the grid orientation effect, refinement, TVD schemes with limiters and extrapolation over grids. Advanced
Upscaling and petrophysics
- 15Permeability upscaling for single-phase flowGoing from a detailed geological model to a coarse grid: exact averaging of porosity and saturation, average permeabilities and their bounds (Wiener, Cardwell–Parsons), flow calculation in a block with no-flow lateral boundaries, periodic and linear boundary conditions with an example on a 4 × 4 grid, upscaling of transmissibility, cells containing wells and checking the coarse model by running it on fine and coarse grids. Advanced
- 16Two-phase upscaling: pseudo-functionsWhy equivalent permeability and core curves are not enough for a coarse cell; pseudo relative permeability and pseudo capillary pressure, steady-state methods (capillary equilibrium, viscous limit) and dynamic ones (Kyte–Berry, pore-volume-weighted averaging); a layered section on a fine grid and on a coarse one — with core curves, pseudo-functions and a non-uniform grid; where pseudo-functions stop working. Advanced
- 17Special topics in upscalingWhen upscaling fails: cells as large as the correlation length and scale separation, discontinuous shale lenses and vertical permeability on fine and coarse grids, stepwise upscaling of laminae (geopseudo), the full tensor and the two-point scheme, the well index from a local calculation, the balance of forces along the path from the well and the choice of pseudo-functions, checking against the flow capacity–storage capacity (F–Φ) curve. Advanced
- 18Pore-scale flow modellingPores, throats and the coordination number; the capillary bundle and its limits; permeability of a pore network from balance equations; drainage as invasion percolation and the flow threshold; piston-like filling and snap-off during imbibition, residual oil and contact angle; what networks contribute to relative permeability; digital rock — resolution versus volume. Advanced
- 19Petrophysical input for simulation: relative permeability and capillary pressureA special core analysis programme for the model, the capillary end effect and processing of an unsteady-state experiment by the JBN method, normalisation and averaging of relative permeability curves, rock types and the Leverett J-function, end-point scaling and Land trapping, Stone's three-phase models, assembling and checking the tables for the simulator — with a calculation from laboratory data to the table. Advanced
This section is growing: new topics are published after a technical review.