Grids in reservoir simulation models

Quick refresher

The engineer chooses the grid, not the reservoir: we choose the dimension, geometry and cell size for the question the model has to answer.
Everything inside a cell is the same. Anything smaller than a cell — a front, a cone, a thin layer — the model sees only on average.
Numerical dispersion: coarse cells smear the front and cause an early breakthrough. For a tracer (a marker substance carried by the flow) or a miscible agent (a fluid that mixes with the oil), the smearing grows as the square root of the cell size. To halve it, we have to make the cells along the flow direction four times smaller.
The orientation effect: with the usual five-point scheme (each cell is linked to its four side neighbours), the result depends on whether the flow runs along the grid lines or diagonally. The effect is strongest at an unfavourable mobility ratio (the displacing fluid moves much more easily than the oil), for example in gas injection.
We need small cells where pressure and saturation change quickly — near wells and at fronts. There we refine the grid locally.
Convergence: we refine the grid until the answer stops changing noticeably. We compare development options on a grid where the numerical error is smaller than the difference between them.

Key relationships

r_i = r_w\,\alpha^{i},\quad \alpha = (r_e/r_w)^{1/N}

a logarithmically spaced radial grid of $N$ rings

\sigma_x \approx \sqrt{\Delta x\, x_f}

scale of the smearing of a tracer front with upstream weighting (the flow takes its values from the cell it comes from)

R_\infty \approx R_{2N} + \dfrac{R_{2N} - R_N}{2^{p} - 1}

extrapolation of the result to an infinitely fine grid ( $p$ is the order of convergence)

Common pitfalls

Comparing development options on a grid where the numerical error is comparable with the difference between them.
Adjusting the permeability to match an early water breakthrough in the model without checking whether the coarse cells have created it.
Studying water and gas cones on a full-field grid with cells hundreds of metres across.
Changing the size of neighbouring cells abruptly and distorting their shape strongly.

After this lesson you can

explain what spatial and temporal discretisation (dividing space and time into pieces) are, and what the properties of a cell mean
choose the dimension and type of grid — one-dimensional, cross-sectional, areal, radial or three-dimensional — for a specific task
describe Cartesian, radial, corner-point and unstructured grids, local refinement and connections across faults
build a logarithmically spaced radial grid near a well
explain numerical dispersion and the grid orientation effect, estimate the smearing of a front and name ways of reducing these errors
test whether the result converges as the grid is refined (stops changing), and estimate its error by extrapolating to an infinitely fine grid

The grid: the reservoir divided into cells

We cannot solve the flow equations exactly for a real reservoir. So we divide the reservoir into a finite number of cells and the time into steps (What a reservoir simulation model is and why it is needed). Dividing space is called spatial discretisation, and dividing time temporal discretisation. In each cell the simulator stores one pressure and one set of saturations. For each pair of neighbouring cells it calculates the flow by Darcy's law across their common face (Interblock flow and transmissibility).

The simplest is the Cartesian grid of rectangular cells (boxes) with dimensions $\Delta x$ , $\Delta y$ , $\Delta z$ . The dimensions need not be equal: along each direction the spacing may vary, and the thicknesses of the cell layers usually follow the geological layers. The number of cells is the product of the numbers along the three directions, $N_x N_y N_z$ . It grows quickly as the grid is refined: if we halve all three dimensions, there are 8 times as many cells.

We give each cell properties: the depth of its centre, the porosity and the net-to-gross ratio (the share of the cell that is reservoir rock), the permeability in three directions, and a rock type with its relative permeability and capillary pressure curves. These are effective values for a volume of tens or hundreds of thousands of cubic metres of rock. We average porosity by volume, and permeability according to the direction of flow: along the layers the average is closer to the arithmetic mean, and across them closer to the harmonic mean (Rock permeability). Transferring properties from a detailed geological model to the simulation grid is called upscaling (Permeability upscaling for single-phase flow).

Dimension: 1D, 2D or 3D

The first decision is how many dimensions we need to answer the question. A full-scale three-dimensional model is not always the best choice. To study the mechanism of a process, a fast model of lower dimension, on which we can make hundreds of runs, is often more useful.

Grid dimension and typical tasks

Grid: One-dimensional horizontal
What it is suited for: displacement in a uniform reservoir, checking numerical settings against an analytical solution (Buckley–Leverett theory and the Welge method)
What it does not account for: heterogeneity through the section and over the area, gravity

Grid: One-dimensional vertical (column)
What it is suited for: gravity-stable displacement of oil by gas from above, gravity drainage (oil draining down under its own weight)
What it does not account for: the areal picture

Grid: Two-dimensional cross-section (x–z)
What it is suited for: the effect of layering and of the $k_v/k_h$ ratio (vertical to horizontal permeability) on vertical sweep (how much of the thickness the displacing fluid reaches), breakthrough through high-permeability layers, pseudo-functions (adjusted curves) for a coarse model
What it does not account for: areal heterogeneity and the well pattern

Grid: Two-dimensional areal (x–y)
What it is suited for: areal sweep (the share of the area that the displacing fluid reaches) in a waterflood or gas injection, well pattern and infill drilling
What it does not account for: layering and gravity, if the reservoir is represented by a single layer of cells

Grid: Two-dimensional radial (r–z)
What it is suited for: inflow to a single vertical well, water and gas cones
What it does not account for: other wells and areal heterogeneity: axial symmetry is assumed (the reservoir looks the same in every direction around the well)

Grid: Three-dimensional
What it is suited for: full-scale models of fields and sectors, forecasts of production by well
What it does not account for: it accounts for everything listed, but is the most expensive

Grid	What it is suited for	What it does not account for
One-dimensional horizontal	displacement in a uniform reservoir, checking numerical settings against an analytical solution (Buckley–Leverett theory and the Welge method)	heterogeneity through the section and over the area, gravity
One-dimensional vertical (column)	gravity-stable displacement of oil by gas from above, gravity drainage (oil draining down under its own weight)	the areal picture
Two-dimensional cross-section (x–z)	the effect of layering and of the $k_v/k_h$ ratio (vertical to horizontal permeability) on vertical sweep (how much of the thickness the displacing fluid reaches), breakthrough through high-permeability layers, pseudo-functions (adjusted curves) for a coarse model	areal heterogeneity and the well pattern
Two-dimensional areal (x–y)	areal sweep (the share of the area that the displacing fluid reaches) in a waterflood or gas injection, well pattern and infill drilling	layering and gravity, if the reservoir is represented by a single layer of cells
Two-dimensional radial (r–z)	inflow to a single vertical well, water and gas cones	other wells and areal heterogeneity: axial symmetry is assumed (the reservoir looks the same in every direction around the well)
Three-dimensional	full-scale models of fields and sectors, forecasts of production by well	it accounts for everything listed, but is the most expensive

A cross-section model is a “slice” between an injection well and a production well. We often carry the results of such detailed calculations over to a coarse three-dimensional model through pseudo relative permeabilities (Two-phase upscaling: pseudo-functions).

Cell geometry: Cartesian, radial, corner-point, unstructured

SchematicTypes of grids: (a) Cartesian with spacing decreasing towards the centre; (b) radial r–z with rings widening away from the well, and a bottom-water cone; (c) corner-point geometry: the cells follow the dip of the layers, and across a fault with a throw (one side has slipped down) layer 2 of the left block borders layer 1 of the right one — this is a non-neighbour connection (NNC); (d) local refinement — a nested fine grid around the well. The sketches are not to scale.

The Cartesian grid is simple to build and to compute with, and for a long time we built full-scale models mainly on it. Its weakness is the geometry of real reservoirs: it shows dipping and curved layers as a “staircase”, and faults that are not parallel to the grid lines as steps.

The radial grid (r–z) describes inflow to a single vertical well. The cells are rings around the well axis, divided vertically into layers. Near the well the pressure changes in proportion to the logarithm of the distance (Steady-state radial flow), so the sharpest changes are in the first metres. We make the rings thin near the well and wide far from it, so that the ratio of the radii of neighbouring boundaries stays constant.

Logarithmically spaced radial grid

r_i = r_w\,\alpha^{i}, \qquad \alpha = \left(\frac{r_e}{r_w}\right)^{1/N}, \qquad i = 0, 1, \dots, N

where:

$r_i$ radius of the $i$ -th boundary between rings; $r_0 = r_w$ , $r_N = r_e$ , m
$r_w$ wellbore radius, m
$r_e$ outer radius of the model, m
$N$ number of rings, —
$\alpha$ ratio of the radii of neighbouring boundaries, —

Units: any identical units of length

Holds when

A vertical well at the centre of the model, with axial symmetry.
The calculation point of a ring is often placed at its “logarithmic midpoint”, at the radius $\sqrt{r_{i-1} r_i}$ . The rules differ slightly between simulators.

In steady-state radial inflow (flow towards the well that no longer changes with time) in a uniform reservoir, the pressure drop across a ring is proportional to $\ln(r_i/r_{i-1}) = \ln\alpha$ and is therefore the same for all rings. A logarithmic grid divides the drawdown equally among the rings and describes the pressure equally well near the well and far from it.

Worked exampleA radial grid for a water cone model

Given (assumed data)

Wellbore radius $r_w$: 0.108 m (215.9 mm bit)
Outer radius of the model $r_e$: 500 m
Number of rings $N$: 12

Solution

Ratio of the radii: $\alpha = (500/0.108)^{1/12} = 2.021$ .
Ring boundaries: $r_1 = 0.108\cdot 2.021 = 0.218$ m, $r_2 = 0.441$ m, $r_3 = 0.891$ m … $r_{11} = 247$ m, $r_{12} = 500$ m. The first ring is 0.110 m wide, the last 253 m; 6 of the 12 rings lie within 10 m of the well.
In steady-state inflow each ring carries $1/12$ of the drawdown, that is, 8.3%.
For comparison — 12 rings of equal width, $(500 - 0.108)/12 = 41.7$ m: the first would carry $\ln(41.8/0.108)/\ln(500/0.108) = 0.71$ of the whole drawdown.

Answer: The ratio of neighbouring radii is 2.021; the rings widen from 0.110 m at the well to 253 m at the edge, and each carries 8.3% of the drawdown.

What it means. In a uniform grid, 71% of the pressure drop would fall on a single cell. The model could show neither the shape of the drawdown funnel nor the rise of the water cone towards the well. A logarithmic grid spends cells where the changes take place. For a cone model, we add thin layers at the oil–water contact and at the perforated interval. The data in this example are assumed for illustration.

Corner-point geometry keeps the logical structure of the Cartesian grid: every cell has indices $(i, j, k)$ and six faces. But the coordinates of its eight corners give the position of a cell. The corners lie on “pillars” (lines that can be inclined), so the cells follow the top and base of the layers, the bends of the structure and the planes of faults. Across a fault with a throw, the cells of one fault block border cells of the other block that are not neighbours by index at all: layer 2 may touch layer 1 or 3. We specify such connections as non-neighbour connections (NNC). Fluid crosses the fault from one layer to another, if the fault is conductive.

Corner-point grids have become the standard where we build the simulation grid from a geological model of complex structure. The flexibility has a price. One part of the price is the labour of building: we cannot build such a grid without specialised software. The other part is accuracy. On strongly skewed, non-orthogonal cells (cells whose faces do not meet at right angles), the simple formula for the flow between two neighbouring cells becomes inaccurate. So we avoid strong distortions of cell shape.

Unstructured grids do without the indices $(i, j, k)$ . The best-known example is Voronoi grids, or PEBI grids (perpendicular bisector). Around chosen calculation points we build polygons whose faces are the perpendicular bisectors of the segments between neighbouring points. Such grids are easily refined towards wells, follow faults and reservoir boundaries, and need fewer cells. But they are harder to build, and calculation on them needs more complex algorithms.

Numerical dispersion: why the front is smeared

Imagine a one-dimensional model: we inject water or gas into the first cell and take production from the last. The agent (the injected fluid) that enters a cell at once “dissolves” throughout its volume, because the model does not know where inside the cell the front is. Already at the next step, part of the agent passes into the second cell, then into the third. In reality, the front would by this time have passed only a small fraction of the first cell. But in the model the agent is already in several cells. A sharp front is smeared, and the agent arrives at the producing well earlier than in reality. This purely numerical error is called numerical dispersion.

It is easy to estimate in the simplest case: the transport of a tracer, which moves with the flow and does not change it. Suppose the time step is very small, and the flow between cells takes the concentration of the cell the flow comes from (upstream weighting). Then a model of $N$ cells turns into a chain of well-mixed “tanks” one after another, and we can write the outlet concentration exactly.

Outlet concentration of a chain of N well-mixed cells

c_N(W_{iD}) = 1 - \e^{-N W_{iD}} \sum_{k=0}^{N-1} \frac{(N W_{iD})^k}{k!}

where:

$c_N$ fraction of the injected agent in the flow at the outlet of the last cell, —
$W_{iD}$ injected volume as a fraction of the pore volume of the model, —
$N$ number of cells between the injection and production wells, —

Units: dimensionless

Holds when

One-dimensional flow, cells of equal size, constant porosity and Darcy velocity.
A tracer or a miscible agent of the same viscosity is transported, so the flow does not depend on the concentration.
Upstream weighting, with a time step much smaller than the time to cross a cell. An implicit scheme with a large step smears the front even more.
There is no physical dispersion: the exact solution is a step at $W_{iD} = 1$ .

The arrival of the agent is spread out in time: the spread (standard deviation) is about $1/\sqrt{N}$ of a pore volume. In space this corresponds to a smearing scale $\sigma_x \approx \sqrt{\Delta x\, x_f}$ , where $x_f$ is the distance travelled by the front. Over a length of $2\sigma_x$ around the true position of the front, the concentration changes from about 16 to 84%. The smearing falls only as the square root of the cell size: to halve it, we need four times as many cells along the flow direction.

CalculatedFraction of the agent in the production against the injected volume for a chain of 5, 20 and 100 cells (calculated from the mixing-cells formula) and the exact solution — a step at one injected pore volume. The fewer the cells, the earlier and more spread out the breakthrough.

Worked exampleHow much the grid smears the front between wells

Given (assumed data)

Distance between the injection and production wells: 600 m
Cell size: 50 m (coarse grid) and 10 m (fine grid)
Process: tracer injection; the exact breakthrough comes after 1 pore volume is injected

Solution

Number of cells between the wells: $600/50 = 12$ and $600/10 = 60$ .
Scale of the front smearing at the production well: $\sqrt{50\cdot 600} = 173$ m (29% of the distance) and $\sqrt{10\cdot 600} = 77$ m (13%).
Fraction of tracer in the production after 0.7 pore volumes injected (by the chain-of-cells formula): 14% on the coarse grid and 0.5% on the fine one; in reality, zero.
The tracer fraction reaches 5% on the coarse grid after 0.58 pore volumes injected, on the fine one after 0.80.

Answer: On the coarse grid the front-smearing scale is 173 m, and a noticeable breakthrough (5%) comes at 0.58 pore volumes instead of 1; on the fine grid it is 77 m and 0.80 pore volumes.

What it means. A fivefold refinement reduced the smearing by a factor of only $\sqrt{5} \approx 2.2$ . In waterflooding the picture is milder: the front of oil displacement by water “sharpens” itself, so the smearing does not grow without limit. But the early breakthrough on a coarse grid remains. So if a model shows breakthrough earlier than the facts, check whether the coarse cells created it before you change the permeability. The data in this example are assumed for illustration.

How numerical dispersion is reduced:

refining the grid along the flow direction — reliable, but expensive
using schemes of higher order of accuracy (more exact formulas), which smear the front less
calculating the transport along streamlines (see the last section)
in coarse models, choosing pseudo relative permeabilities that compensate for the smearing (Two-phase upscaling: pseudo-functions)

The grid orientation effect

Take an injection well and two production wells at the same distance from it: one along a row of cells, the other along the diagonal. The reservoir is uniform, so physically the two wells are equivalent. The model, however, often gives them different breakthrough times and different production. This is the grid orientation effect.

The cause lies in how the flows are calculated. In the usual five-point scheme of a two-dimensional model, a cell exchanges fluid only with its four neighbours across the faces. Flow along a row runs along a straight chain of cells. A diagonal flow has to go in a “staircase”, and the numerical dispersion is different for these two directions. At a favourable mobility ratio, when the displacing agent is no more mobile than the oil, the difference is small. At an unfavourable one the front is unstable and forms tongues, and the scheme pushes them to grow along the grid lines. An unfavourable ratio means gas, whose viscosity is tens of times lower than that of oil, or water displacing viscous oil. Then a well on a grid line gets breakthrough noticeably earlier, and a diagonal one later. Neither result is the correct one.

What helps:

nine-point schemes, which also take the diagonal neighbours into account — they reduce the effect considerably
refining the grid
orienting the grid along the main direction of flow, for example along the lines between injection and production wells
unstructured grids and calculation along streamlines
a simple check: repeat the calculation on a grid rotated by 45° and compare the results

How many cells are needed: convergence testing

The cell size is the easiest thing to get wrong: a coarse grid gives itself away only by a distorted result. We set the initial grid from experience — from the dimensions of the reservoir, the number of wells, the thickness of the layers and the available computing resources. Then we check the choice by calculation. We solve the same problem on a refined grid and compare the quantities we need: production, breakthrough time, pressures. When the next refinement changes them by less than the problem allows, we call the result grid-converged: the numerical error is no longer the main one among the uncertainties of the model.

The check is especially important when the model compares options. A coarse grid distorts different processes differently: it may “add” oil to one option and “take” it away from another. Sometimes a new enhanced oil recovery method gives an impressive increment on a coarse grid, and the increment almost disappears when we refine the grid. We can compare options only when the numerical error is smaller than the difference between them.

The requirements for vertical detail are often unexpectedly high. Gas or water may separate from the oil under gravity and spread as a thin “tongue” along the top or base of the reservoir. A model with layers of 5–10 m will not see a tongue one metre thick. Such problems need tens and sometimes hundreds of layers. The same difficulty arises with thin high-permeability layers through which water breaks through.

Extrapolation of the result to an infinitely fine grid

R_\infty \approx R_{2N} + \frac{R_{2N} - R_N}{2^{p} - 1}, \qquad p = \log_2\frac{R_N - R_{N/2}}{R_{2N} - R_N}

where:

$R_N$ result of the calculation (for example, oil recovery) on a grid with $N$ cells in the direction being refined; $R_{N/2}$ and $R_{2N}$ are the same on a grid twice as coarse and twice as fine, %
$R_\infty$ estimate of the result on an infinitely fine grid, %
$p$ observed order of convergence: when the grid is refined twofold, the error falls by a factor of $2^{p}$ , —

Units: in the units of the result

Holds when

The grid is refined by a factor of two each time, with nothing else changed.
The result changes monotonically (in one direction only), and the error already falls as $h^{p}$ , where $h$ is the cell size. We need at least three grids to check this.
For $p = 1$ this is a linear extrapolation in $1/N$ : $R_\infty \approx 2R_{2N} - R_N$ .

The difference $R_\infty - R_N$ estimates the error of the chosen grid. We compare it with the difference between the options that the model has to show.

Worked exampleOil recovery on three grids

Given (assumed data)

Model: a two-dimensional cross-section between an injection well and a production well, waterflood
Oil recovery after 1.5 pore volumes injected: 5 layers — 37.1%; 10 layers — 40.2%; 20 layers — 41.8%

Solution

Changes on refinement: $40.2 - 37.1 = 3.1$ and $41.8 - 40.2 = 1.6$ percentage points (pp). The order of convergence $p = \log_2(3.1/1.6) = 0.95$ is close to one.
Extrapolation at $p = 1$ : $R_\infty \approx 2\cdot 41.8 - 40.2 = 43.4$ %.
Error of the grids: for the model with 10 layers, $43.4 - 40.2 = 3.2$ pp; with 20 layers, 1.6 pp.

Answer: On an infinitely fine grid the oil recovery would be about 43.4%; the model with 10 layers underestimates it by 3.2 pp, the one with 20 layers by 1.6 pp.

What it means. If the development options we compare differ by 10 pp or more, the error of the model with 10 layers (3.2 pp) is still acceptable. If they differ by 1–2 pp, even 20 layers (1.6 pp) are too few, and we need a finer grid or a correction for its error. Halving the vertical spacing doubled the number of cells. In a three-dimensional model, refining in all directions increases the number 8 times, so we usually do convergence studies on sector and cross-section models. The data in this example are assumed for illustration.

The calculation time grows faster than the number of cells: we solve more equations at each step, and small cells usually force us to reduce the time step as well. So the number of cells of a full-scale model is always a compromise between detail, the time of one run and the number of runs needed to analyse uncertainty (What a reservoir simulation model is and why it is needed).

Streamlines: transport without smearing

Another way of fighting numerical dispersion is streamline simulation. We still calculate the pressure on the grid. From the pressure field we find the velocities and build streamlines — the paths along which the fluid goes from the injection wells to the production wells. Along each streamline, the transport of saturation is a one-dimensional problem. We solve it exactly or on a very fine one-dimensional grid, so there is no smearing from coarse cells. Then we map the saturations back onto the grid and update the phase mobilities. After some time we recalculate the pressure and the streamlines.

The most expensive part of the calculation is the pressure, and with streamlines we recalculate it rarely. So the method is very fast for large models with many wells, if pressure differences mainly decide the flow and the streamlines change slowly, for example in waterflooding. When gravity, capillary forces or compressibility matter, we have to account for the flows across the streamlines separately and recalculate the pressure more often, and the speed advantage shrinks. So streamlines are a tool for a certain range of tasks: fast analysis of a waterflood, ranking of geological cases, and allocation of injection among wells (Types of reservoir simulation model).

Terms and symbols

Spatial and temporal discretisation: Division of the reservoir into cells and of time into steps.
Cartesian grid: A grid of rectangular cells with dimensions $\Delta x$ , $\Delta y$ , $\Delta z$ .
Radial grid (r–z): Ring-shaped cells around a vertical well, divided into layers. We use it for inflow to a well and for cones.
Corner-point geometry: Cells defined by the coordinates of their eight corners, which lie on pillars. They follow the structure and the faults.
Non-neighbour connections (NNC): Connections between cells that are not neighbours by index, for example across a fault that shifts the layers.
PEBI grid (Voronoi grid): An unstructured grid of polygons whose faces are the perpendicular bisectors between the calculation points.
Local grid refinement (LGR): A nested fine grid inside selected coarse cells.
Numerical dispersion: Smearing of the front caused by the finite size of the cells. It gives an early breakthrough.
Grid orientation effect: Dependence of the result on the direction of flow relative to the grid lines.
Grid convergence: Further refinement of the grid does not noticeably change the result.

Common mistakes and misconceptions

Wrong

“The more cells the better: let us make the grid as detailed as possible.”

In fact

We choose the number of cells for the question and check it for convergence. Extra cells use up time that we need for running options and assessing uncertainty.

Wrong

“Water breaks through earlier in the model than in the field — so the permeability in the model is too high.”

In fact

An early breakthrough may be created by the coarse grid itself. First check how sensitive the result is to the cell size, and only then adjust the geology.

Wrong

“The model will calculate wells at the same distance from the injector identically.”

In fact

With a five-point scheme the result depends on the direction relative to the grid lines, especially in gas injection. Nine-point schemes, refinement and a check on a rotated grid help.

Wrong

“A water cone can be studied on a full-field model with 100 × 100 m cells.”

In fact

A cone develops within the first metres and tens of metres from the well. It needs a radial grid with logarithmic spacing or local refinement.

Wrong

“An oil recovery increment of 10 pp on a coarse grid is a reliable result.”

In fact

Only an increment that survives grid refinement is reliable: a coarse grid can create an effect that does not exist.

In depth: advanced detail

Where numerical diffusion comes from

The transport of a tracer at velocity $v$ is described by the equation $\partial c/\partial t + v\,\partial c/\partial x = 0$ . In the upstream scheme, we replace the derivative with respect to $x$ by a difference with the neighbouring cell upstream, $(c_i - c_{i-1})/\Delta x$ . A Taylor series expansion (a way of writing a function as a sum of ever smaller terms) shows that this difference equals $\partial c/\partial x - \tfrac{\Delta x}{2}\,\partial^2 c/\partial x^2$ plus terms of higher order. So the model actually solves an equation with an extra term. It is as if the reservoir had diffusion with a coefficient $v\Delta x/2$ , which in reality does not exist.

Numerical diffusion coefficient of the upstream scheme

D_{num} = \frac{v\,\Delta x}{2}\,(1 \pm C), \qquad C = \frac{v\,\Delta t}{\Delta x}

where:

$D_{num}$ numerical diffusion coefficient, m²/s
$v$ velocity of the front — the Darcy velocity divided by the porosity, m/s
$\Delta x,\ \Delta t$ cell size and time step, m, s
$C$ Courant number: the fraction of a cell that the front crosses in a step, —

Units: SI (any consistent system)

Holds when

Linear transport of a tracer on a uniform one-dimensional grid.
The minus sign is for an explicit scheme in time (stable only for $C \le 1$ ), and the plus sign for an implicit one (stable for any step).
In the limit of a very small step, $D_{num} = v\Delta x/2$ . The mixing-cells formula corresponds to this limit.

Numerical diffusion is proportional to the cell size. An implicit scheme with a large time step smears the front more than an explicit one. This is the price of its stability (IMPES and fully implicit schemes). The smearing over a time $t$ is $\sqrt{2D_{num}t}$ . For $C \to 0$ this gives $\sqrt{\Delta x\, x_f}$ , the estimate from the main section.

Connections to other topics

What a reservoir simulation model is and why it is needed — what the simulator solves in each cell and why a model is needed.
Rock permeability — averaging the permeability of layers along and across the flow.
Interblock flow and transmissibility — how the flow between cells is calculated: transmissibility and upstream weighting.
Well model in a grid block — why the pressure of a cell with a well is not equal to the bottomhole pressure.
Permeability upscaling for single-phase flow — how we transfer properties from the geological model to the simulation grid.
Numerical dispersion — the mathematics of numerical dispersion and ways of suppressing it.
Steady-state radial flow — the logarithmic pressure profile near a well.
Types of reservoir simulation model — streamline models and other types of models.

Check yourself

1
Which grid is it more sensible to choose to study the rise of a bottom-water cone towards a single vertical well?
1. A.One-dimensional horizontal
2. B.Areal x–y with 100 × 100 m cells
3. C.Any: a cone does not depend on the grid
4. D.Radial r–z with logarithmic spacing and thin layers at the oil–water contact
Show the answer
Correct answer: D. Radial r–z with logarithmic spacing and thin layers at the oil–water contact
A cone is an axisymmetric process (the same in every direction around the well) at the well, where the pressure varies as the logarithm of the distance. A radial grid with logarithmic spacing spends cells on the first metres from the well, and thin layers at the contact let us see the rise of the water.
2
Which of the following is a manifestation of numerical dispersion?
1. A.The front in the model is smeared, and the agent reaches the production well earlier than in the exact solution
2. B.The model shows a later water breakthrough than in reality
3. C.The pressure in the well cell is not equal to the bottomhole pressure
4. D.The oil in place of the model does not agree with the volumetric calculation
Show the answer
Correct answer: A. The front in the model is smeared, and the agent reaches the production well earlier than in the exact solution
Coarse cells immediately “mix” the agent that has entered over their whole volume, so the front is smeared and the agent reaches the production well earlier. The difference between the cell pressure and the bottomhole pressure is a matter for the well model, and a discrepancy in the volumes in place is an error in building the model.
3
A radial grid from the wellbore wall ( $r_w = 0.1$ m) to 1000 m is divided into 10 rings with logarithmic spacing. What is the ratio of the radii of neighbouring boundaries, and where does the first boundary lie?
Show the answer
$\alpha = (1000/0.1)^{1/10} = 10{,}000^{0.1} \approx 2.51$ . The first boundary is $r_1 = 0.1\cdot 2.51 \approx 0.25$ m. The next boundaries are at 0.63, 1.58 and 3.98 m, and so on. In steady-state inflow each ring carries the same share of the pressure drop.
4
A tracer front has travelled 400 m on a grid with 40 m cells. What is roughly the scale of its smearing, and how small must the cells be to halve the smearing?
Show the answer
$\sigma_x \approx \sqrt{40\cdot 400} \approx 126$ m. The smearing is proportional to the square root of the cell size, so to halve it we need cells of 10 m. They are four times smaller, so there are four times as many cells along the flow direction.
5
On grids with 5, 10 and 20 layers the oil recovery increment from gas injection compared with waterflooding was 9, 5 and 3 pp. Can a decision on a gas injection project be made from these calculations?
Show the answer
The increment falls with every refinement. The changes of 4 and 2 pp indicate first-order convergence, and extrapolation gives $2\cdot 3 - 5 = 1$ pp. Most of the effect on the coarse grids is numerical. We cannot make a decision on these models. We need calculations on a finer grid or a correction for its error, and the project may turn out to be unprofitable.
6
Why is the grid orientation effect in gas injection usually stronger than in the waterflooding of light oil?
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The viscosity of gas is tens of times lower than that of oil, so the mobility ratio is unfavourable, and the front is unstable and forms tongues. In a five-point scheme it is “easier” for the tongues to grow along the grid lines, so the result depends strongly on how the grid is oriented relative to the wells. In the waterflooding of light oil the mobility ratio is close to one, the front is stable and the difference is small.

Updated: 19 October 2026

Quick refresher

After this lesson you can

The grid: the reservoir divided into cells

Dimension: 1D, 2D or 3D

Cell geometry: Cartesian, radial, corner-point, unstructured

Where small cells are needed: local refinement

Numerical dispersion: why the front is smeared

The grid orientation effect

How many cells are needed: convergence testing

Streamlines: transport without smearing

Terms and symbols

Common mistakes and misconceptions

In depth: advanced detail

Connections to other topics

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