Simulate the Reservoir: Flow, History Matching, Forecasting
The static model is a photograph; the simulator makes it move. Darcy to the pressure equation to a running flood, then the two jobs every simulation is hired for: matching the history you have and forecasting the field you do not. It ends at the frontier, where ensembles and proxies take over the forecasting.
You can walk from Darcy's law to the discretized pressure equation, initialize a model in capillary equilibrium, predict a waterflood with Buckley-Leverett before running it, put a well in a grid block with the Peaceman model, choose IMPES or fully implicit and hold the timestep, read a convergence report for lies, treat a history match as the non-unique inversion it is, and forecast with scenarios, ensembles, and a proxy you know when not to trust.
Flow physics and the initial state
Darcy's law is the only physics the simulator ever solves; relative permeability and mobility decide which phase gets to use it, and every rate on every report descends from here.
Mass conservation plus Darcy gives the pressure equation, and pressure is a diffusion: it moves first, everything else in the reservoir follows it.
Buckley-Leverett is the one displacement you can solve by hand; it is the analytic answer every waterflood run gets checked against, and the mobility ratio is its one dial.
The PDE becomes neighbor-to-neighbor arithmetic between grid blocks; transmissibility is the doorway geology walks through to enter the flow equations.
Day zero must be quiet: a model out of capillary equilibrium starts flowing before the first well opens, and no history match can be trusted on top of that.
Wells and displacement
The drive mechanism sets what the reservoir gives up for free; sweep efficiency decides how much of the rest your wells ever touch. Recovery factor is the product of those two honesties.
Wells are how the model meets the field: the productivity index turns pressure into a rate, and the Peaceman model fits a real wellbore into a grid block a thousand times its size.
When the waterflood plateaus, the question becomes what else the rock and fluids will respond to; EOR screening is reservoir engineering as matchmaking, and it is done in the simulator.
Run, match, forecast
IMPES is cheap and jumpy, fully implicit is stable and expensive, and the CFL limit is the border between them; the timestep report tells you which side of it your run lives on.
The simulator confesses in its convergence report and its material-balance error; reading the output is how you catch a bad run before it grows up to become a forecast.
The match is an inverse problem with many right-looking answers; non-uniqueness is not a flaw to hide, it is the true width of your forecast's error bar.
A forecast is a decision wearing a production profile; running the scenarios side by side is how the model earns its seat at the development meeting.
Frontiers
When phase behavior or temperature drives recovery, black-oil bookkeeping runs out: flash calculations and thermal balances take over, and the same compositional machinery carries CO2 storage, which has a path of its own.
One matched model is an anecdote; an ensemble updated by the data is a forecast with an honest spread, and watching that spread is a skill in itself.
A proxy trades physics for speed so uncertainty and optimization loops can afford thousands of evaluations; the craft is knowing exactly where it stops being trustworthy.