What a reservoir simulation model is and why it is needed

Quick refresher

A model is a simplified likeness of a reservoir, simple enough to calculate with. The question to be answered and the amount of reliable data decide how complex it is — not a wish to “take everything into account”.
The reservoir is divided into cells. Inside a cell the properties are the same everywhere, and fluids flow between neighbouring cells. Wells are described by separate sub-models.
At each time step the simulator keeps a balance for every cell. The change in the amount of fluid in the cell equals what comes in from the neighbours, minus what goes out and minus what a well takes out. Darcy's law gives the flows between cells.
The operation of the wells drives the run: rates, bottomhole pressures (the pressure at the bottom of the well) and injection. The output is production profiles (rate against time), pressures, water cut (the share of water in the produced liquid), the producing gas–oil ratio (gas produced per unit of oil) and maps of saturation through time.
Material balance and simulation do not compete. We do a material balance almost always, and we build a model when differences across the area and through the thickness matter.
A match with the production history means the model reproduces the pressures and rates that were recorded. This raises confidence but does not prove that the forecast is right: different models can describe the past equally well.

Key relationships

\dfrac{m_i^{n+1} - m_i^{n}}{\Delta t} = \sum_j F_{ji} - W_i

balance in a cell over one step: accumulation (the change in the amount of fluid in the cell) = flows from the neighbours − withdrawal by a well

q_{ij} = \dfrac{T_{ij}}{\mu}\,(p_j - p_i)

flow between neighbouring cells (single-phase, Darcy's law)

p^{n+1} = p^{n} - \dfrac{\Delta t}{\tau}\,(p^{n} - p_{wf})

explicit time step for the “tank” model; stable (errors do not grow) for $\Delta t < 2\tau$

Common pitfalls

Assuming that the more cells there are, the more accurate the forecast. The data limit the accuracy, not the number of cells.
Taking the properties of a cell for the properties of a core (a cylinder of rock taken from a well). A cell holds effective values — averages over an enormous volume.
Treating a good history match as proof that the model is right.
Presenting the result of a single run as a forecast without a range of uncertainty.

After this lesson you can

explain what a reservoir model is, how it differs from the real reservoir and why we choose a model to fit a specific question
tell an analytical solution (a formula) from a numerical one (step-by-step arithmetic) and explain the role of the grid and the time step
describe in words what the simulator calculates in each cell: the mass balance (what flows in and out), the flows between cells by Darcy's law (the flow is proportional to the pressure difference), the fluid properties and how the wells work
list the input data of a model, where they come from and the typical results of a run
compare material balance (a simple balance for the whole reservoir) with reservoir simulation and choose a tool for the task
name the main limits of models and the ways to allow for the uncertainty of a forecast

The model as a simplified likeness of a reservoir

Here the word “model” does not mean a scale replica. It means a system of equations that reproduces the main features of how the reservoir behaves and is still simple enough to solve. Every model is a compromise. A model that is too simple will not answer the question. A model that is too complex needs data that do not exist, and takes time that will not pay off.

A good model is fit for purpose. To estimate the average reservoir pressure in five years' time, a single balance equation may be enough. To decide where to drill an infill well (an extra well between existing ones) and what kind it should be — vertical or horizontal — we need a detailed three-dimensional model. A practical rule: start from the decision to be made, and choose the simplest tool that is enough for that decision.

There are two ways to solve a model. An analytical solution is a formula derived by mathematics. It gives the answer at once for any moment in time, like the radial inflow formula in the lesson on permeability. A numerical solution divides time into steps and the reservoir into cells, and repeats simple arithmetic many times. Analytical solutions exist only for simple geometry and uniform properties. For a real reservoir only the numerical way is left — reservoir simulation — and the many repetitions are a job for a computer.

A single “tank” — by formula and step by step

We compare both ways on the simplest model — a “tank” (material balance): a closed volume of the reservoir with one well. Assume that the oil stays undersaturated: the pressure is above the bubble point pressure, the pressure at which the first gas bubbles come out of the oil. Assume also that the withdrawal comes only from the elastic expansion of the oil, the connate water (the water that has stayed in the pores since the reservoir formed) and the rock.

Depletion of a closed “tank” at constant bottomhole pressure

c_t V_p\,\frac{\dd p}{\dd t} = -J\,(p - p_{wf})\quad\Rightarrow\quad p(t) = p_{wf} + (p_i - p_{wf})\,\e^{-t/\tau},\qquad \tau = \frac{c_t V_p}{J}

where:

$c_t$ total compressibility of oil, connate water and rock together (how much their volume changes when the pressure changes), MPa⁻¹
$V_p$ pore volume of the “tank”, m³
$p$ pressure in the “tank”, the same throughout the volume, MPa
$p_i$ initial reservoir pressure, MPa
$p_{wf}$ bottomhole flowing pressure of the well (the pressure at the bottom of the flowing well), constant, MPa
$J$ productivity index of the well (the rate per unit of drawdown), in reservoir volumes of liquid, m³/(d·MPa)
$t$ time since the start of withdrawal, d
$\tau$ characteristic depletion time (the time scale of the pressure decline), d

Units: metric oilfield units: MPa, m³, d (any consistent system will do)

Holds when

The pressure is the same everywhere in the volume: a pressure change caused by the well spreads instantly (this is what makes it a “tank”).
The flow is single-phase (one fluid only): the pressure is above the bubble point pressure and no gas comes out of the oil.
$c_t$ and $J$ are constant, and there is no inflow from beyond the boundary.
The rate at reservoir conditions (volumes measured at the pressure and temperature of the reservoir) is proportional to the drawdown $p - p_{wf}$ , the pressure difference between the tank and the well.

The left side is the loss of the elastic storage of fluid (the fluid released as the oil, water and rock expand). The right side is the rate of the well. The pressure approaches the bottomhole pressure along an exponential curve — fast at first, then more slowly. Over a time $\tau$ the drawdown falls by a factor of $\e \approx 2.7$ .

The numerical way: replace the derivative (the rate of change of pressure) by the pressure change over a step $\Delta t$ . Then find the pressure at the next step from the pressure at the current one.

Explicit time step for the “tank”

p^{n+1} = p^{n} - \frac{\Delta t}{\tau}\,(p^{n} - p_{wf})

where:

$p^{n}$ pressure at step $n$ (known), MPa
$p^{n+1}$ pressure at the next step (to be found), MPa
$\Delta t$ time step, d
$\tau,\ p_{wf}$ as in the exact solution, d, MPa

Units: any consistent units ( $\Delta t$ and $\tau$ in the same units of time)

Holds when

The derivative is replaced by a difference. The rate over the whole step is taken from the pressure at the start of the step (an explicit scheme).
The formula becomes the exact solution only in the limit $\Delta t \to 0$ (a step that shrinks to nothing).

This is a computing procedure that we repeat step after step: from the current pressure find the rate, subtract what has been withdrawn, move on to the next step. The simulator does the same thing — only for millions of cells and several phases (oil, gas, water) at once.

CalculatedThe pressure in the “tank” of the example below: the exact solution and the explicit scheme with steps of 50 and 150 d (the points are the calculated steps).

Worked exampleExact and numerical solutions for the “tank”

Given (assumed data)

Initial pressure $p_i$: 25 MPa
Bottomhole pressure $p_{wf}$: 12 MPa (above the bubble point pressure)
Pore volume $V_p$: $2.0\cdot10^{6}$ m³
Total compressibility $c_t$: $2.0\cdot10^{-3}$ MPa⁻¹
Productivity index $J$: 20 m³/(d·MPa)

Solution

Characteristic time: $\tau = c_t V_p/J = 2.0\cdot10^{-3}\cdot2.0\cdot10^{6}/20 = 200$ d; the initial rate is $J\,(p_i - p_{wf}) = 260$ m³/d at reservoir conditions.
Exact pressure after 200 d: $p = 12 + 13\,\e^{-200/200} = 16.78$ MPa.
Explicit scheme with a 50 d step: each step multiplies the drawdown by $1 - 50/200 = 0.75$ . After 4 steps $p = 12 + 13\cdot0.75^{4} = 16.11$ MPa, which is 0.67 MPa below the exact value.
With a step of 10 d: $p = 12 + 13\cdot0.95^{20} = 16.66$ MPa. The error is 0.12 MPa, 5.5 times smaller.
Withdrawn over 200 d: $c_t V_p\,(p_i - p) = 32{,}870$ m³ of reservoir oil, which is 1.6% of the pore volume.

Answer: After 200 d the pressure is 16.78 MPa; the explicit scheme gives 16.11 MPa with a 50 d step and 16.66 MPa with a 10 d step.

What it means. The numerical solution gets closer to the exact one as the step gets smaller, and the error falls roughly in proportion to the step. A real reservoir has no exact solution. So in simulators we choose the step and the grid so that the result hardly changes when we make them finer still. The second conclusion is a physical one: the elastic storage of undersaturated oil is small. Without pressure maintenance (injecting water or gas to keep the pressure up), such a reservoir recovers only a few per cent of its oil. The data in this example are assumed for illustration.

From the “tank” to a detailed grid: which tool to choose

Forecasting tools form a series, from the least to the most detailed. Each next one answers a wider range of questions, but needs more data and time.

Forecasting tools and what they can do

Tool: Material balance (“tank”)
What it answers: how the average reservoir pressure depends on the withdrawals; the initial oil in place and the water influx (water flowing in from the aquifer, the water-filled rock around the reservoir), found from the pressure and production history; the drive mechanism (what pushes the oil out)
What it cannot do: differences in pressure and saturation across the area and through the thickness; the position of the water front; choosing well locations
When to use it: almost always — as a first and independent analysis; most reliable in well-connected reservoirs

Tool: Analytical solutions (inflow to a well, displacement in a uniform layer)
What it answers: quick estimates of rates, of the time when water breaks through to the wells, and of displacement efficiency (the share of oil that the injected fluid pushes out)
What it cannot do: complex geometry, uneven (heterogeneous) properties, the way many wells affect each other (interference)
When to use it: screening of options (a quick first sorting), checking a numerical model

Tool: Sector, cross-section or single-well model (a piece of the field, a vertical slice, or one well)
What it answers: the mechanism of a process: the effect of layering and thin interbeds (thin layers of other rock), water and gas coning (water or gas pulled towards the well in a cone shape), the choice of well design
What it cannot do: the behaviour of the field as a whole
When to use it: studying the mechanism; preparing simplified relationships for a full-field model

Tool: Full-field three-dimensional model
What it answers: production by well and for the field under different development schemes; maps of pressure and remaining oil
What it cannot do: be more accurate than the initial data
When to use it: a field development plan, drilling new wells, waterflooding (injecting water to push the oil to the producers), gas injection, enhanced oil recovery methods (special methods that raise the recovery)

Tool	What it answers	What it cannot do	When to use it
Material balance (“tank”)	how the average reservoir pressure depends on the withdrawals; the initial oil in place and the water influx (water flowing in from the aquifer, the water-filled rock around the reservoir), found from the pressure and production history; the drive mechanism (what pushes the oil out)	differences in pressure and saturation across the area and through the thickness; the position of the water front; choosing well locations	almost always — as a first and independent analysis; most reliable in well-connected reservoirs
Analytical solutions (inflow to a well, displacement in a uniform layer)	quick estimates of rates, of the time when water breaks through to the wells, and of displacement efficiency (the share of oil that the injected fluid pushes out)	complex geometry, uneven (heterogeneous) properties, the way many wells affect each other (interference)	screening of options (a quick first sorting), checking a numerical model
Sector, cross-section or single-well model (a piece of the field, a vertical slice, or one well)	the mechanism of a process: the effect of layering and thin interbeds (thin layers of other rock), water and gas coning (water or gas pulled towards the well in a cone shape), the choice of well design	the behaviour of the field as a whole	studying the mechanism; preparing simplified relationships for a full-field model
Full-field three-dimensional model	production by well and for the field under different development schemes; maps of pressure and remaining oil	be more accurate than the initial data	a field development plan, drilling new wells, waterflooding (injecting water to push the oil to the producers), gas injection, enhanced oil recovery methods (special methods that raise the recovery)

The level of detail of a model must match both the question and the amount of reliable data. A detailed model built on almost no data only gives an appearance of accuracy.

Material balance and a numerical model complement each other. A balance does not need the geometry of the reservoir. The history of pressure and production and the fluid properties are enough. So it gives an independent estimate of the oil in place and of the water influx — exactly what the model then uses as input. Unlike a balance, a model forecasts where and when things will happen. We do a material balance before simulation and alongside it, because it is cheap and does not depend on the geological interpretation.

From the geological model to the simulation grid

A real reservoir is uneven (heterogeneous) at all scales. Permeability (how easily fluid flows through the rock) can change by orders of magnitude, many times over, within tens of centimetres. Sand bodies give way to shales (clay-rich rock), and faults (breaks where the rock has slipped) cut the layers. Geologists describe all this with a geological (static) model — a three-dimensional grid of millions of small cells. Each cell has its own lithology (rock type), porosity (the share of pore space in the rock), permeability and saturation (Reservoir architecture and performance, Spatial correlation and the variogram).

For a flow calculation such a grid is usually too large. We must run many options, and each run should take hours, not weeks. So we build a coarser simulation grid (Grids in reservoir simulation models) and transfer the properties onto it. This is upscaling (Permeability upscaling for single-phase flow). A cell of the simulation grid holds tens of thousands of cubic metres of rock. Its porosity and permeability are not measured values but effective values, which must reproduce the flow through that whole volume.

Worked exampleScales: core, cell, field

Given (assumed data)

Size of the reservoir: 8 × 4 km, total thickness 60 m
Geological model cell: 50 × 50 × 0.5 m
Simulation grid cell: 100 × 100 × 2 m
Core sample: diameter 3.8 cm, length 5 cm

Solution

Simulation grid: $80\times40\times30 = 96{,}000$ cells.
Geological model: $160\times80\times120 = 1{,}536{,}000$ cells; each simulation cell combines 16 geological ones.
Volume of a simulation cell: $100\cdot100\cdot2 = 20{,}000$ m³; volume of a core sample: $\pi\cdot0.019^{2}\cdot0.05 = 5.671\cdot 10^{-5}$ m³.
One simulation cell holds about $3.5\cdot10^{8}$ such samples.

Answer: About 96,000 simulation cells; one cell is equal in volume to about

3.5\cdot10^{8}

core samples.

What it means. We carry properties measured on core samples over to volumes hundreds of millions of times larger. So the main task in building a model is not to carry over every detail. It is to keep the heterogeneities that control the flow: continuous shale layers, high-permeability streaks (thin layers where the rock lets fluid through very easily) and faults. The data in this example are assumed for illustration.

For each cell we give the model the porosity and the net-to-gross ratio (the share of the cell that is reservoir rock), the permeability in three directions $k_x$ , $k_y$ , $k_z$ and the rock compressibility. Through the rock type we also set the curves of relative permeability (how well each fluid flows when several fluids share the pores) and capillary pressure (the pressure difference across the boundary between two fluids in small pores). We calculate the initial pressures and saturations from the positions of the fluid contacts (the surfaces between oil and water, or between gas and oil) and from capillary equilibrium. This is the initialisation of the model. The sum of the oil over the cells gives the initial oil in place of the model, and we always check it against the volumetric calculation.

What the simulator solves: a balance in every cell

A simulator rests on two laws applied to each cell, plus tables of properties. The first is conservation of mass: over one time step the amount of each fluid in a cell changes by exactly what came in from the neighbours, minus what went out to them and what a well withdrew. The second is Darcy's law: the flow between neighbouring cells is proportional to the pressure difference and to the transmissibility (how easily fluid passes) of the rock between their centres.

SchematicWhat the simulator calculates for each cell: the flows through its faces, the withdrawal by a well and the change in the amount of fluid over a time step.

Fluid balance in a cell over a time step

\frac{m_i^{n+1} - m_i^{n}}{\Delta t} = \sum_{j} F_{ji} - W_i

where:

$m_i^{n}$ amount of fluid (a phase or a component) in cell $i$ at step $n$ ; for oil in a black oil model it is convenient to express it in standard cubic metres (the volume the oil takes at surface conditions): $V_{p,i}\,S_{o,i}/B_{o,i}$ , that is, pore volume × oil saturation ÷ the formation volume factor, which turns a reservoir volume into a surface volume, kg or standard m³
$F_{ji}$ flow from the neighbouring cell $j$ into cell $i$ (negative if the fluid leaves), kg/d or standard m³/d
$W_i$ withdrawal by a well from cell $i$ (negative for injection), the same
$\Delta t$ time step, d

Units: any consistent units

Holds when

Within a cell the pressure, saturations and properties are uniform.
We write the balance separately for each component: oil, gas (free gas and gas dissolved in the oil) and water.
We calculate the flows and withdrawals from the pressures and saturations of the new step (an implicit scheme) or of the start of the step (an explicit one).

This is the same material balance as for the “tank”, but written for every cell and with the flows between cells added. A model with a single cell is the “tank”. The expression $V_{p}\,S_{o}/B_{o}$ is the same formula as in the volumetric calculation of oil in place, applied to one cell.

Flow between neighbouring cells (single-phase, by Darcy's law)

q_{ij} = \frac{T_{ij}}{\mu}\,(p_j - p_i),\qquad T_{ij} = \frac{A}{\dfrac{\Delta x_i}{2k_i} + \dfrac{\Delta x_j}{2k_j}}

where:

$q_{ij}$ volumetric rate from cell $j$ into cell $i$ at reservoir conditions, m³/s
$T_{ij}$ transmissibility of the connection between the cells (how easily fluid passes between them; this is its geometric part, without viscosity), m³
$\mu$ fluid viscosity (the resistance of the fluid to flow), Pa·s
$p_i,\ p_j$ pressures at the cell centres, Pa
$A$ area of the common face of the cells, m²
$\Delta x_i,\ \Delta x_j$ sizes of the cells in the direction of flow, m
$k_i,\ k_j$ permeability of the cells in that direction, m²

Units: SI

Holds when

The flow is single-phase and horizontal. In an inclined reservoir we add the weight of the fluid column to the pressure difference.
The cells are rectangular and their centres lie on a line perpendicular to the common face.
With several phases, we multiply the rate of each phase by its relative permeability, usually taken in the cell the flow comes from.

Transmissibility is Darcy's law for the two “halves” of the cells, joined one after the other (in series). Their resistances add up, so the less permeable half decides the flow. In a model we represent faults and impermeable layers by multipliers applied to $T$ (Interblock flow and transmissibility).

In other unit systems (1)

Metric oilfield units

q_{ij}\,[\text{m}^3/\text{d}] = 0.0853\,\frac{T_{ij}\,[\text{mD·m}]\;(p_j - p_i)\,[\text{MPa}]}{\mu\,[\text{mPa·s}]}

We calculate $T_{ij}$ by the same formula, with $k$ in mD, $A$ in m² and $\Delta x$ in m (this gives mD·m). Pressures are in MPa, viscosity in mPa·s.

0.0853 is the same constant as in Darcy's law in metric oilfield units. It converts mD to m², MPa to Pa, mPa·s to Pa·s and seconds to days.

Worked exampleFlow between two cells

Given (assumed data)

Cells: 100 × 100 × 4 m, adjoining along a face of 100 × 4 m
Permeability: $k_1 = 300$ mD, $k_2 = 30$ mD
Pressures at the centres: $p_1 = 20.4$ MPa, $p_2 = 20.0$ MPa
Oil viscosity: 1.5 mPa·s

Solution

Face area $A = 100\cdot4 = 400$ m²; transmissibility $T = \dfrac{400}{50/300 + 50/30} = 218$ mD·m.
Flow from cell 1 into cell 2: $q = 0.0853\cdot218\cdot0.4/1.5 = 4.96$ m³/d at reservoir conditions.
Effective permeability of the connection: $T\,L/A = 218\cdot100/400 = 55$ mD, where $L = 100$ m is the distance between the cell centres. This is much closer to the smaller of the two permeabilities.
If we mistakenly use the arithmetic mean, 165 mD: $T = 165\cdot400/100 = 660$ mD·m and $q = 15.0$ m³/d, which is 3.0 times more.

Answer: The flow is about 4.96 m³/d at reservoir conditions.

What it means. The simulator makes such a calculation for every pair of neighbouring cells, for every phase, at every iteration of every step. The way we average permeability between cells is not a technical detail: it changes the flows several times over. The data in this example are assumed for illustration.

To these two laws we add tables of fluid properties: formation volume factors (the ratio of a fluid's volume in the reservoir to its volume at the surface), the solution gas–oil ratio (the gas dissolved in the oil) and viscosities as functions of pressure (Black oil model: solution gas–oil ratio and formation volume factors). We also add the curves of relative permeability and capillary pressure, and well models. A well model relates the pressure in a cell to the bottomhole pressure and the rate (Well model in a grid block). As a result each cell has several equations with unknowns — the pressure and the saturations — and the flows couple the equations of neighbouring cells. For a model of a million cells this is a system of several million nonlinear equations (equations that cannot be solved in one direct step). At each step we solve it by iterations, that is, by repeated rounds of improving a first guess (IMPES and fully implicit schemes, Solving systems of linear equations in simulators).

What goes in and what comes out

Input data of a reservoir simulation model

Group of data: Geometry and structure
What it includes: tops and bases of layers, faults, thicknesses, the simulation grid
Where it comes from: seismic surveys, layer tops picked in wells, the geological model

Group of data: Rock properties by cell
What it includes: porosity, net-to-gross ratio, permeability $k_x$ , $k_y$ , $k_z$ , rock compressibility
Where it comes from: well logging, core, well tests; upscaling of the geological model

Group of data: Fluid properties (PVT: pressure, volume, temperature)
What it includes: formation volume factors, solution gas–oil ratio, viscosities, densities as functions of pressure
Where it comes from: laboratory analysis of samples (Laboratory PVT tests), correlations

Group of data: Flow of several phases together
What it includes: relative permeabilities, capillary pressure
Where it comes from: special core analysis (laboratory tests on cores)

Group of data: Initial state
What it includes: oil–water and gas–oil contacts (OWC, GOC), pressure at the datum depth (the reference depth), temperature
Where it comes from: well logs, formation tester (a tool that measures the formation pressure in the well), pressure measurements

Group of data: Aquifer
What it includes: size and properties of the region beyond the oil zone
Where it comes from: geology; pressure history (material balance)

Group of data: Wells
What it includes: trajectories, perforated intervals (where holes let fluid into the well), skin factor (a number that shows how much the zone around the well helps or hinders flow), limits on rate and pressure
Where it comes from: drilling and completion data, well tests

Group of data: History and operating plan
What it includes: actual oil, gas and water rates, injection, pressure measurements; the schedule of the wells in the forecast
Where it comes from: field production records, the field development plan

Group of data: Numerical parameters
What it includes: time step, convergence tolerances of the iterations (how close the answer must be before the iterations stop)
Where it comes from: chosen by the engineer

Group of data	What it includes	Where it comes from
Geometry and structure	tops and bases of layers, faults, thicknesses, the simulation grid	seismic surveys, layer tops picked in wells, the geological model
Rock properties by cell	porosity, net-to-gross ratio, permeability $k_x$ , $k_y$ , $k_z$ , rock compressibility	well logging, core, well tests; upscaling of the geological model
Fluid properties (PVT: pressure, volume, temperature)	formation volume factors, solution gas–oil ratio, viscosities, densities as functions of pressure	laboratory analysis of samples (Laboratory PVT tests), correlations
Flow of several phases together	relative permeabilities, capillary pressure	special core analysis (laboratory tests on cores)
Initial state	oil–water and gas–oil contacts (OWC, GOC), pressure at the datum depth (the reference depth), temperature	well logs, formation tester (a tool that measures the formation pressure in the well), pressure measurements
Aquifer	size and properties of the region beyond the oil zone	geology; pressure history (material balance)
Wells	trajectories, perforated intervals (where holes let fluid into the well), skin factor (a number that shows how much the zone around the well helps or hinders flow), limits on rate and pressure	drilling and completion data, well tests
History and operating plan	actual oil, gas and water rates, injection, pressure measurements; the schedule of the wells in the forecast	field production records, the field development plan
Numerical parameters	time step, convergence tolerances of the iterations (how close the answer must be before the iterations stop)	chosen by the engineer

The lesson Building a simulation model: input data and results shows in detail how to write these data into the input file of a simulator.

The reservoir data alone do not “start” anything. The calculation runs because wells produce and inject fluids on a given schedule, with set rates or bottomhole pressures and limits on them. The results of the calculation are:

oil, gas and water production and injection for the field, for layers and for individual wells — daily and cumulative (the total to date)
average reservoir pressure and pressures in the wells: at the bottom of the well and, through lift performance tables (tables of how the fluid rises up the well), at the wellhead
water cut and producing gas–oil ratio of each well through time
the distribution of pressure and saturations over the cells at any moment — maps, cross-sections and three-dimensional images that show where the water is going and where oil remains
pressure profiles with depth in the wells, for comparison with formation tester measurements: they show whether the layers are isolated from each other by impermeable interbeds

The volume of results is enormous, so we use visualisation programs to analyse them: maps and cross-sections of saturation, plots by well, and comparisons of the options with each other and with the actual data.

How we work with a model: stages and decisions

A typical order of work

1
Pose the question
Which decision has to be made, and what accuracy is enough for it. The type of model, the size of the grid and the time needed for the work all depend on this.
2
Collect and check the data
Seismic data, well logs, core, PVT (fluid properties), well tests and the production history. It is better to find contradictions between the data before building the model.
3
Build the geological model
The structure, faults, the distribution of lithology and properties — often in several versions.
4
Build the simulation grid
Choose the cell size, transfer the properties (upscaling), define the wells.
5
Initialise the model
Calculate the initial pressures and saturations from the contacts and check the volumes in place of the model against the volumetric calculation.
6
Match the history
If the reservoir is already on production, adjust the uncertain parameters until the model reproduces production, pressures and water cut (History matching).
7
Run the forecasts
A base case (the reference option) and development options, each with a range of uncertainty.
8
Evaluate the economics and decide
Compare the options by net present value (the value of future income in today's money) and by risks (Net present value (NPV)).
9
Update the model
As wells are drilled and oil is produced, we refine the model: it lives together with the field.

Typical decisions that a model helps to make: the number, locations and types of wells (vertical, deviated, which means inclined, or horizontal); the waterflood pattern and rate; gas or water injection; infill drilling and sidetracks (new branches drilled from an existing well); converting wells to injection; capacity for liquid production and water treatment; the order in which layers are brought on stream; the choice of enhanced oil recovery methods.

The role of a model changes as the field ages. At the appraisal stage (after a discovery, while the size of the field is still being checked) there are few data and the model is crude. We use it to compare major concepts — waterflooding or gas injection, how many wells and platforms are needed — and to see which uncertainties matter most. In a mature field the model is tuned to many years of history and helps with finer decisions: where oil is left, where to drill infill wells, which wells to convert to injection.

Limitations and uncertainty

The forecast of a model is no more accurate than what has gone into it. The main sources of error are:

Description of the reservoir. Between wells we interpolate the properties (we estimate them from the nearest wells). The continuity of shale layers, the transmissibility of faults and the size of the aquifer are often unknown — and they control how water and gas move.
Upscaling. The effective properties of coarse cells reproduce the flow in uneven (heterogeneous) rock only approximately, especially when one phase displaces another, for example water pushing oil (Two-phase upscaling: pseudo-functions).
Numerical effects. Coarse cells and large steps “smear” the displacement front (the sharp boundary between the injected fluid and the oil) (Numerical dispersion), and the result may depend on the orientation of the grid.
Rock and fluid properties. We measure relative permeabilities and capillary pressure on a few samples and transfer them to the whole reservoir.
Non-uniqueness of the history match. Different combinations of parameters describe the history equally well but give different forecasts.

So a single run is not yet a forecast. We allow for uncertainty in three ways. One is several versions of the geological model and parameters (pessimistic, base and optimistic). Another is sensitivity analysis, which shows which parameters affect the result most. The third is probabilistic methods, when we make tens or hundreds of runs (Sensitivity analysis and scenarios). Where possible, we choose robust decisions: ones that stay reasonable across the whole range of possible cases.

Why, then, do people use simulation everywhere, in spite of all these limits? Because no other tool takes into account, all at once, the reservoir geometry, the uneven properties, the flow of several phases and the operation of many wells. A model is not an oracle. It is a way of thinking through the consequences of decisions systematically, under clear assumptions that can be checked.

Terms and symbols

Reservoir simulation model, simulator: A numerical model of how fluids flow in a reservoir divided into cells, and the program that does the calculation.
Cell (grid block): The smallest unit of volume in the simulation grid. Inside it, the properties, pressure and saturations are the same everywhere.
Geological (static) model: A three-dimensional description of the structure and properties of a reservoir, with no calculation of flow. It is the basis of the reservoir simulation model.
Upscaling: Transferring properties from a fine grid to a coarse one in a way that keeps the flow behaviour.
Transmissibility: The coefficient that links the flow between neighbouring cells to the pressure difference. It depends on the sizes of the cells and their permeability.
Initialisation: The calculation of the initial pressures and saturations from the contacts and capillary equilibrium. We check it against the volumetric calculation of oil in place.
History matching: Adjusting the uncertain parameters of a model until it reproduces the actual development history of the field.
Time step: The interval of time by which the simulator moves from one calculated state to the next.

Common mistakes and misconceptions

Wrong

“The more cells, the more accurate the forecast.”

In fact

A fine grid reduces numerical errors, but not the uncertainty in the description of the reservoir. A model with more detail than the data only gives an appearance of accuracy.

Wrong

“The model reproduces the history well, so the forecast is right.”

In fact

History matching has no single answer (it is non-unique): different combinations of parameters give the same history and different forecasts. We raise confidence by testing the model on data that were not used in the matching, and by running several cases.

Wrong

“Now that we have a model, we do not need a material balance.”

In fact

A material balance gives an estimate of the oil in place and of the water influx that does not depend on the geological maps. We also use it to check the model itself.

Wrong

“We will take the permeability of a cell equal to the average over the core.”

In fact

A cell is hundreds of millions of times larger than a core sample. Its properties are effective ones. We average horizontal and vertical permeability in different ways, and continuous layers can change them by orders of magnitude.

Wrong

“The time step is a technical setting and does not affect the result.”

In fact

With too large a step, an explicit scheme gives errors and can even become unstable. We check how much the result changes with the step and with the cell size.

In depth: advanced detail

Stability of the time step: explicit and implicit schemes

In the “tank” example each explicit step multiplies the drawdown $p - p_{wf}$ by $1 - \Delta t/\tau$ . As long as the step is smaller than $\tau$ , the multiplier is positive and the solution is monotonic (it moves in one direction only). For $\tau < \Delta t < 2\tau$ the multiplier is negative: the calculated pressure “jumps over” the bottomhole pressure and swings above and below it. For $\Delta t > 2\tau$ the swings grow — the scheme is unstable.

Implicit time step for the “tank”

p^{n+1} = p^{n} - \frac{\Delta t}{\tau}\,(p^{n+1} - p_{wf})\quad\Rightarrow\quad p^{n+1} = \frac{p^{n} + (\Delta t/\tau)\,p_{wf}}{1 + \Delta t/\tau}

where:

$p^{n},\ p^{n+1}$ pressure at the start and at the end of the step, MPa
$\Delta t,\ \tau$ time step and characteristic depletion time, d
$p_{wf}$ bottomhole flowing pressure, MPa

Units: any consistent units

Holds when

We calculate the rate over the step from the pressure at the end of the step (an implicit scheme).
The model is linear. In a real simulator the equations are nonlinear, and at each step we solve them by iterations.

The drawdown multiplier $1/(1 + \Delta t/\tau)$ always lies between 0 and 1, so the solution does not swing, whatever the step. With a large step the accuracy is still poor, but the calculation does not “fall apart”.

CalculatedA 300 d step with

\tau = 200

d: the explicit scheme swings around the bottomhole pressure, and the implicit one keeps falling steadily (the “tank” data of the example).

Worked exampleOne large step: explicit and implicit schemes

Given (assumed data)

“Tank” data: $p_i = 25$ MPa, $p_{wf} = 12$ MPa, $\tau = 200$ d
Time step: 300 d ( $\Delta t/\tau = 1.5$ )

Solution

Exact solution: $p(300) = 12 + 13\,\e^{-1.5} = 14.90$ MPa.
Explicit scheme: $p = 25 - 1.5\cdot13 = 5.5$ MPa. This is below the bottomhole pressure, so the well would seem to be injecting oil into the reservoir.
Implicit scheme: $p = (25 + 1.5\cdot12)/2.5 = 17.2$ MPa.

Answer: The explicit scheme gives 5.5 MPa, which is physically impossible. The implicit scheme gives 17.2 MPa. The exact value is 14.90 MPa.

What it means. In a real model the characteristic time is short for small cells of high permeability, above all those with wells. If we calculated the pressure explicitly, we would have to make the step tiny for the whole model. So simulators almost always calculate the pressure implicitly. They calculate the saturations explicitly, or also implicitly (IMPES and fully implicit schemes, Finite differences: from equations to computation). The data in this example are assumed for illustration.

Connections to other topics

General material balance equation for oil reservoirs — the balance equation for the “tank” is the basis of the model and an independent check of its results.
Rock permeability — Darcy's law and the averaging of the permeability of layers.
Volumetric estimation of hydrocarbons in place — we check the initial oil in place of the model against the volumetric calculation.
Types of reservoir simulation model — black oil, compositional and thermal models, streamline models.
Grids in reservoir simulation models — types of grid and the choice of cell size.
History matching — how we match a model to the development history, and why this has no single answer.
Building a simulation model: input data and results — how the input data of a simulator are organised.

Check yourself

1
Which question calls for a reservoir simulation model rather than a material balance?
1. A.What the average reservoir pressure will be for given withdrawals
2. B.What the initial oil in place is, from the pressure and production history
3. C.Whether there is water influx
4. D.Where to drill an infill well and whether it should be horizontal
Show the answer
Correct answer: D. Where to drill an infill well and whether it should be horizontal
A material balance treats a reservoir as a “tank” with a single average pressure. It answers questions about withdrawals, oil in place and water influx for the reservoir as a whole. To choose the location and type of a well, we need the distribution of pressures and saturations through the reservoir. That is a task for a reservoir simulation model.
2
In the explicit scheme for the “tank”, the time step is made 5 times smaller. What happens to the error in the calculated pressure?
1. A.It will not change
2. B.It will fall by a factor of 25
3. C.It will grow: there are more steps, and errors accumulate
4. D.It will fall by a factor of about 5
Show the answer
Correct answer: D. It will fall by a factor of about 5
The explicit scheme is first-order, which means the error is roughly proportional to the step. In the example of the lesson, going from 50 to 10 d reduced the error about 5.5 times.
3
Neighbouring cells of equal size have permeabilities of 100 and 1 mD. By what factor will the flow calculated from the arithmetic mean of the permeabilities overestimate the correct one?
Show the answer
The correct permeability of the connection is the harmonic one: $2/(1/100 + 1/1) \approx 1.98$ mD. The arithmetic one is 50.5 mD. The overestimate is about 25 times, because the less permeable cell decides the flow.
4
A model was built from geological maps and then matched to ten years of production history. Does this confirm the initial volumes in place built into the model?
Show the answer
Not necessarily. We set the volumes in place when we built the model. During the matching, other parameters could have made up for an error in them: the size of the aquifer, the permeability, the relative permeabilities. A material balance, or several versions of the model with different volumes in place, gives an independent check.
5
What mainly limits the accuracy of the forecast of a full-field model?
1. A.The uncertainty in the description of the reservoir between wells
2. B.The choice of program
3. C.The number of cells
4. D.The speed of the computer
Show the answer
Correct answer: A. The uncertainty in the description of the reservoir between wells
The equations and numerical methods are well established. The main uncertainty lies in the geology between wells (the continuity of interbeds, faults, the aquifer) and in transferring properties to coarse cells. That is why we give the forecast as a range.
6
Why check that the initial volumes in place of a reservoir simulation model agree with the volumetric calculation?
Show the answer
A difference points to an error in building the model: in the contacts, the net-to-gross ratio, the porosity, the saturations after initialisation or the formation volume factors. A forecast from a model with the wrong volumes in place is always off in the same direction, so we make this check before the history matching.

Updated: 22 September 2026