History matching · MunaiHub

Quick refresher

History matching is an inverse problem: the response of the reservoir (pressures, production, water cut, producing gas–oil ratio) is known, and the task is to find a description of the reservoir that explains it.
In a history run, we give the wells the actual withdrawals and injection, and the model calculates the rest. A pressure mismatch points first of all to errors in the volumes and energy of the reservoir. A mismatch in water cut and producing gas–oil ratio points to errors in the paths along which the fluids move.
We adjust the parameters that are poorly known and strongly affect the calculation: the aquifer (the water-bearing rock around the reservoir), the pore volume, the permeability of layers, the transmissibility of faults and interbeds, the relative permeabilities, and well productivity. Every change must have a geological explanation.
We judge the quality of a match from plots and from the misfit function. This is the mean square of the discrepancies, each divided by its measurement error. The discrepancies should be comparable with the error and show no systematic trend.
History matching is non-unique (it has no single answer): different combinations of parameters reproduce the history equally well but give different forecasts. Additional kinds of data and geological constraints reduce this non-uniqueness, and we show what remains as a forecast range.
The history tests the model only for the processes and conditions that have already occurred. A forecast of a new process, such as gas injection or drilling in an undrilled area, remains an extrapolation (a step beyond the data).

Key relationships

M = \dfrac{1}{N}\sum_{k=1}^{N}\left(\dfrac{d_k^{\text{meas}} - d_k^{\text{calc}}}{\sigma_k}\right)^{2}

the misfit function; $M \approx 1$ means discrepancies at the level of the measurement error

p(t) = p_i - \dfrac{q\,t}{C}, \qquad C = c_t V_p

a closed reservoir with undersaturated oil (a “tank”) under constant withdrawal

p(t) = p_i - \dfrac{q}{J_a}\left(1 - \e^{-J_a t/C}\right)

the same “tank” with an aquifer. For small $t$ it looks the same as the closed one

Common pitfalls

Taking a match with the history as proof that the forecast is right.
Seeking a match for every well with local multipliers that have no geological explanation.
Fitting the model to erroneous production records without checking them.
Correcting a mismatch in average pressure with relative permeabilities. The volumes and energy of the reservoir decide the pressure first of all, so we tune these first and the paths of fluid movement afterwards.
Showing a forecast from a single model when several models describe the history equally well.

After this lesson you can

explain the purpose of history matching and its place in work with a reservoir simulation model
list the data that the model is compared with, and explain how we set the operation of the wells when we calculate the history
choose the parameters to adjust from the nature of the mismatch and justify the changes geologically
calculate the misfit function and use it to judge the quality of a match
explain the difference between manual and assisted history matching, and the reasons why the answer is not unique
assess what a history match does and does not tell us about the forecast

Why match the model to the history

We build a reservoir simulation model from static data: seismic data, core, well logs and fluid samples (What a reservoir simulation model is and why it is needed). At the wells we know the properties of the reservoir fairly well. But the wells reach only a negligible fraction of its volume, and between them the properties are interpolated (estimated from the nearby wells) or modelled geostatistically (with random fields that follow the statistics of the data). Static data often cannot tell how the layers are actually connected, whether the faults let fluid through (conduct), or how large the aquifer is. And these are the things that govern the movement of water and gas.

When a field is developed, a new kind of evidence appears: the history. It shows how the pressure fell, when and in which wells water appeared, and how the gas–oil ratio changed. These data reflect the behaviour of the whole reservoir at once. History matching uses them to test and refine the model. We run the model with the actual operating regime of the wells, compare the calculation with the measurements, and change the uncertain parameters until the discrepancy is acceptable. In mathematical terms this is an inverse problem: we recover the properties of a system from its response.

The first model almost always disagrees with the facts in some respect. The pressure in one of the blocks falls differently from the measurements, or gas reaches the wells at the wrong time. This is not a verdict on simulation but information. Suppose that, to get a match, we have to accept that the fault between two blocks hardly lets fluid through. Then the blocks are being developed independently, and each needs its own pressure maintenance system (injection to hold the pressure up). History matching teaches us to understand the reservoir. It does not merely fit curves.

What the model is compared with

Observed data for history matching

Data: Average reservoir pressure (from pressure build-up curves, which show the pressure rise after a well is closed, and from measurements in shut-in wells)
What they show: the total volume and energy of the reservoir: pore volume, compressibility, aquifer, the balance of withdrawals and injection
Features: measurements are referred to the datum depth (one chosen reference depth); we compare them with the pressure in the cells, not with the bottomhole pressure of a producing well

Data: Rates and cumulative production (the total produced to date) by well and for the whole reservoir
What they show: they set the controls of the history run
Features: well rates are often shared out from one common measurement; errors in this sharing pass into the match

Data: Water cut
What they show: when and where water arrived: flow paths, the aquifer, relative permeabilities
Features: water breakthrough time is one of the most informative features and one of the hardest to reproduce

Data: Producing gas–oil ratio
What they show: gas coming out of the oil, gas breakthroughs and coning (gas drawn towards the well in a cone shape), the behaviour of the gas cap (free gas on top of the oil)
Features: a rise above the initial solution gas–oil ratio means that free gas is reaching the well

Data: Pressure profiles through the section (wireline formation tester, a tool on a cable that measures pressure at points in the formation)
What they show: isolation of the layers, the behaviour of interbeds and faults
Features: a pressure difference between layers is a strong argument for or against impermeable interbeds

Data: Production logging (measuring the flow along the well), tracers (marker chemicals added to the injected water), repeat seismic surveys
What they show: which intervals are producing, where the water came from, where the front is moving
Features: expensive data, but they greatly narrow the number of possible models

Data	What they show	Features
Average reservoir pressure (from pressure build-up curves, which show the pressure rise after a well is closed, and from measurements in shut-in wells)	the total volume and energy of the reservoir: pore volume, compressibility, aquifer, the balance of withdrawals and injection	measurements are referred to the datum depth (one chosen reference depth); we compare them with the pressure in the cells, not with the bottomhole pressure of a producing well
Rates and cumulative production (the total produced to date) by well and for the whole reservoir	they set the controls of the history run	well rates are often shared out from one common measurement; errors in this sharing pass into the match
Water cut	when and where water arrived: flow paths, the aquifer, relative permeabilities	water breakthrough time is one of the most informative features and one of the hardest to reproduce
Producing gas–oil ratio	gas coming out of the oil, gas breakthroughs and coning (gas drawn towards the well in a cone shape), the behaviour of the gas cap (free gas on top of the oil)	a rise above the initial solution gas–oil ratio means that free gas is reaching the well
Pressure profiles through the section (wireline formation tester, a tool on a cable that measures pressure at points in the formation)	isolation of the layers, the behaviour of interbeds and faults	a pressure difference between layers is a strong argument for or against impermeable interbeds
Production logging (measuring the flow along the well), tracers (marker chemicals added to the injected water), repeat seismic surveys	which intervals are producing, where the water came from, where the front is moving	expensive data, but they greatly narrow the number of possible models

We check the data before matching. Otherwise we carefully fit the model to an error. Typical problems are these. The rates of individual wells are shared out in proportion from one common measurement. There are periods without a reliable measurement of gas or water. Part of the injection goes into other layers through casing leaks or flow behind the casing. Pressure gauges fail. Sometimes history matching itself reveals such errors: to reproduce the pressure decline, the model has to “withdraw” more than the records show, or “inject” less. The material balance (General material balance equation for oil reservoirs) gives an independent check of the volumes.

How we set the well regime. In a history run, we give the model the actual withdrawals and injection, and it calculates everything else itself. We usually give producers the actual oil or liquid rate, and injectors the actual injection rate. Suppose we give an oil rate, and the model calculates the water cut with an error. Then the model withdraws the wrong volume of liquid from the reservoir, and the error passes into the pressure. That is why, when we tune the pressure, we often give the liquid rate or the withdrawal at reservoir conditions instead. Sometimes the model cannot deliver the rate we set, because the pressure in the well hits the limit we set. That, too, is a signal: the productivity of the well, or the pressure in the reservoir, is too low in the model.

What is adjusted, and why

We do not adjust everything. We adjust only the parameters that are poorly known and also have a noticeable effect on the calculation. We leave alone what has been measured reliably, unless we have strong reasons: the bubble point pressure (the pressure at which gas starts to come out of the oil) from good-quality samples, or the porosity from core and well logs at the wells. Every change must have a physical or geological explanation. Reducing the vertical transmissibility is justified if the core shows tight interbeds. A permeability multiplier around a single well, with no such explanation, is a fudge factor (an arbitrary correction) that will spoil the forecast.

Mismatches and the parameters checked first

Mismatch: The average pressure falls faster (slower) than the actual
Probable causes: the pore volume or the aquifer is too small (too large); errors in the records of withdrawals and injection
What is adjusted: size and activity of the aquifer, pore volume, rock compressibility; the records are checked first

Mismatch: Pressure differences between the blocks of the reservoir do not match the measurements
Probable causes: faults conduct differently from what is assumed in the model
What is adjusted: fault transmissibility multipliers, block boundaries

Mismatch: Water arrives earlier (later) than the actual
Probable causes: high-permeability layers, fractures and conducting faults are overestimated (missing); inaccurate relative permeabilities
What is adjusted: permeability of layers, fault transmissibility, $k_v/k_h$ (the ratio of vertical to horizontal permeability), end points of relative permeabilities

Mismatch: The producing gas–oil ratio does not rise as it does in the wells
Probable causes: gas coning, continuity of the interbeds between the gas cap and the perforated interval, critical gas saturation (the gas saturation at which gas starts to flow)
What is adjusted: vertical transmissibility, relative permeability to gas, flow behind the casing in individual wells

Mismatch: The rate or bottomhole pressure of an individual well does not match although the pressure around it is right
Probable causes: skin factor (a number that shows how much the zone around the well helps or hinders flow), hydraulic fracture, an inaccurate well model
What is adjusted: the productivity index of the well in the model (Well model in a grid block)

Mismatch	Probable causes	What is adjusted
The average pressure falls faster (slower) than the actual	the pore volume or the aquifer is too small (too large); errors in the records of withdrawals and injection	size and activity of the aquifer, pore volume, rock compressibility; the records are checked first
Pressure differences between the blocks of the reservoir do not match the measurements	faults conduct differently from what is assumed in the model	fault transmissibility multipliers, block boundaries
Water arrives earlier (later) than the actual	high-permeability layers, fractures and conducting faults are overestimated (missing); inaccurate relative permeabilities	permeability of layers, fault transmissibility, $k_v/k_h$ (the ratio of vertical to horizontal permeability), end points of relative permeabilities
The producing gas–oil ratio does not rise as it does in the wells	gas coning, continuity of the interbeds between the gas cap and the perforated interval, critical gas saturation (the gas saturation at which gas starts to flow)	vertical transmissibility, relative permeability to gas, flow behind the casing in individual wells
The rate or bottomhole pressure of an individual well does not match although the pressure around it is right	skin factor (a number that shows how much the zone around the well helps or hinders flow), hydraulic fracture, an inaccurate well model	the productivity index of the well in the model (Well model in a grid block)

Relative permeability (how well each fluid flows when several fluids share the pores) is covered in detail in the lesson Relative permeability.

A special place among the parameters belongs to transmissibility multipliers. These are coefficients that multiply the calculated transmissibility between cells (how easily fluid passes from one cell to the next). A multiplier of 0 makes a fault or interbed impermeable, 0.01 makes it weakly conducting, and 1 leaves the connection as it is. This is how we test hypotheses about parts of the reservoir being isolated, without rebuilding the grid (Grids in reservoir simulation models). Good practice is to set up several options (“the fault conducts”, “conducts partly”, “is impermeable”) and choose the one that agrees with all kinds of data, not just one.

The usual order of manual history matching

1
Check the data and the model
Check the production and injection records. Refer the pressures to the datum depth. Compare the initial volumes in place of the model with the volumetric calculation and the material balance (Volumetric estimation of hydrocarbons in place).
2
Match the pressure
Match the average pressure for the whole reservoir first, then the pressure in each block. Adjust what sets the volumes and energy of the reservoir: the aquifer, the pore volume and the communication between blocks.
3
Match the movement of fluids
Match the arrival time of water and gas and their share in the production. Do this for the whole reservoir, then for groups of wells, then for individual wells. Adjust the heterogeneity, interbeds, faults and relative permeabilities.
4
Refine the wells
Match the productivity and the bottomhole and wellhead pressures.
5
Check the result
Check that all the changes are physically plausible. If possible, test the forecast against data that were not used in the matching.

Pressure is usually the easiest to reproduce: the balance of volumes mainly decides it, and in a well-connected reservoir it evens out quickly. The hardest to reproduce is the arrival time of water and gas at individual wells. It depends on a few conducting layers and on details of heterogeneity that the model may not see, and on how much the grid smears the displacement front (the sharp boundary between the injected fluid and the oil).

How to measure the quality of a match

The first tool is plots of the calculated and measured values for the reservoir, blocks and wells. We accept a match if the discrepancy does not exceed the measurement error and the tolerances set in advance. We set the tolerances from the accuracy of the data and the purpose of the model: for example, a few tenths of a megapascal for pressure, and a few percentage points for water cut. A model used to choose the location of a new well needs a stricter match at the neighbouring wells than a model that estimates total production.

The character of the discrepancies matters too. A random scatter of points around the calculated curve is normal. A systematic discrepancy that grows with time means that the model describes something essential wrongly: the volumes, the energy of the reservoir or the displacement mechanism. Adjusting secondary parameters cannot correct it.

To compare options, and to hand the search over to a computer, we reduce the discrepancy to a single number — the misfit function (objective function):

Misfit function of history matching

M = \frac{1}{N}\sum_{k=1}^{N} w_k\left(\frac{d_k^{\text{meas}} - d_k^{\text{calc}}}{\sigma_k}\right)^{2}

where:

$M$ misfit function — the mean square of the normalised discrepancy, —
$d_k^{\text{meas}}$ measured (observed) value of the $k$ -th measurement: pressure, rate, water cut, gas–oil ratio, units of the measurement
$d_k^{\text{calc}}$ calculated value of the same quantity at the same moment, units of the measurement
$\sigma_k$ measurement error (standard deviation, the typical size of the random error), including the inaccuracy of tying the measurement to the model, units of the measurement
$w_k$ weight of the measurement; 1 by default, changed to balance kinds of data with different numbers of points, —
$N$ number of measurements, —

Units: dimensionless: each discrepancy is divided by the error in the same units

Holds when

The measurement errors are independent and random.
We must estimate the errors $\sigma_k$ honestly, because what counts as a good match depends on them.
The engineer chooses the weights, and the result depends on them. The misfit function puts judgement into a formula, but it does not replace judgement.

Dividing by the error puts different quantities on a common scale — the number of errors. At $M \approx 1$ , the model departs from the data no more than the accuracy of the data allows. $M \gg 1$ is a systematic mismatch. $M \ll 1$ with a large number of measurements means that the errors are overestimated, or that the model has been fitted to random noise. With few measurements, $M$ fluctuates noticeably by chance, so these guides are rough.

Worked exampleMisfit function for a simple history match

Given (assumed data)

Model: a “tank”: a closed reservoir with undersaturated oil (bubble point pressure below 9 MPa), the pressure uniform throughout the volume
Initial pressure $p_i$: 30.0 MPa at the datum depth
Withdrawal at reservoir conditions $q$: 1000 m³/d, constant
Average pressure measurements: after 0.5, 1.0, 1.5 and 2.0 years: 28.9, 27.6, 26.8 and 25.6 MPa
Measurement error $\sigma$: 0.3 MPa
Unknown: the storage capacity of the reservoir $C = c_t V_p$ — how many cubic metres of fluid the reservoir releases when the pressure falls by 1 MPa

Solution

Calculated pressure: $p(t) = p_i - q\,t/C$ , with the time $t$ in days (1 year = 365 d).
$C = 120$ thousand m³/MPa: the differences “measured − calculated” are 0.42, 0.64, 1.36 and 1.68 MPa, $M = 14.7$ . All the differences have the same sign and grow with time: the model loses pressure faster than the reservoir does.
$C = 180$ thousand m³/MPa: −0.09, −0.37, −0.16 and −0.34 MPa, $M = 0.80$ , which is within the error.
The best value by the least-squares method (the value that makes the sum of the squared differences smallest): $C^{*} = q\sum t_k^2 \big/ \sum t_k\,(p_i - p_k)$ $= 165.4$ thousand m³/MPa, $M = 0.14$ .
All values of $C$ from 151 to 182 thousand m³/MPa satisfy the condition $M \le 1$ .

Answer: The storage capacity of the reservoir is about 165.4 thousand m³/MPa. Two years of pressure history fix it to within about ±9%.

What it means. Even for a model with one parameter, the result of history matching is not a number but a range. The measurement error and the length of the history set its width. Also, the pressure history fixes only the product $c_t V_p$ . It could equally well be a larger pore volume with a smaller compressibility, or the reverse. Independent data help to separate them: the laboratory compressibility of the rock, and the volumetric estimate of the oil in place. A full-scale model has thousands of parameters, but the logic is the same. The data in this example are assumed for illustration.

Non-uniqueness: one history, many models

An inverse problem usually has many solutions. A model has thousands of parameters, but the history holds far fewer independent pieces of information: the pressure at a few dozen points, and the water cut and gas–oil ratio of each well. The parameters make up for one another. A too-high permeability can be offset by a weaker aquifer, wrong relative permeabilities by wrong heterogeneity, and an error in the oil in place by the size of the aquifer. Models that describe the past equally well can predict the future differently.

Worked exampleTwo models, one history

Given (assumed data)

History: the same as in the previous example: four pressure measurements over 2 years, $\sigma = 0.3$ MPa
Model A: a closed reservoir, $C_{\text{A}} = 165.4$ thousand m³/MPa — the best one from the previous example
Model B: a smaller reservoir, $C_{\text{B}} = 150$ thousand m³/MPa, but with an aquifer: the water influx is proportional to the pressure drop, $J_a = 50$ m³/(d·MPa)
Forecast: the same withdrawal of 1000 m³/d for another 6 years — up to 8 years of development

Solution

Model B: $p(t) = p_i - \dfrac{q}{J_a}\left(1 - \e^{-J_a t/C_{\text{B}}}\right)$ . The differences from the measurements are 0.08, −0.11, 0.14 and −0.08 MPa, $M_{\text{B}} = 0.12$ . This is no worse than model A ( $M_{\text{A}} = 0.14$ ).
After 8 years model A gives $p = 12.3$ MPa and model B 17.6 MPa: a difference of 5.2 MPa.
The difference between the models exceeds twice the measurement error, $2\sigma = 0.6$ MPa, only 3.2 years after the start of development — more than a year after the end of the history.
By year 8, the aquifer of model B replaces 36% of the volume withdrawn: 1.05 million m³ of water enters the reservoir. Model A has no such water.

Answer: Both models reproduce the history within the error, but after 8 years their pressures differ by 5.2 MPa.

What it means. Two years of pressure history cannot tell a large closed reservoir from a smaller reservoir with an aquifer: the difference shows up later. Yet the main decision depends on the choice between them — whether we need water injection, and how much. What helps is more observation, signs of water in wells outside the oil contour, geological data on the aquifer, and the material balance. Until we have them, it is more honest to show the forecast for both models. The data in this example are assumed for illustration.

CalculatedThe two models of the example: a closed reservoir (A) and a smaller reservoir with an aquifer (B). Both pass through the measurements of the first two years within the error of 0.3 MPa, but by year 8 they diverge by about 5 MPa.

How non-uniqueness is reduced:

More kinds of data. Pressure, water cut, gas–oil ratio, pressure profiles through the section, production logging, tracers and repeat seismic surveys react to the parameters in different ways. A model that agrees with all of them is more tightly constrained.
Geological constraints. We change the parameters only within plausible limits, and consistently: by facies (rock types formed in the same setting), layers and blocks, not cell by cell.
Several models. Instead of one “best” model, we keep several models that agree with the history, and we calculate the forecast for each of them (Sensitivity analysis and scenarios).
Hindcast on withheld data. We tune the model on the first years of the history and keep the last years for testing. If the model predicts them without further adjustment, confidence in the forecast is higher. It is equally useful to compare an old forecast with what actually happened a few years later.

Manual and assisted history matching

Manual history matching is a series of runs. In each run the engineer changes one or two parameters, relying on an understanding of the physics and geology. It is slow and subjective, but every change is thought through, and while working the engineer learns a great deal about the reservoir. A useful technique is sensitivity calculations: we see how the misfit changes as we vary each uncertain parameter within its range. They show which parameters influence the data at all, and so can be refined from them, and which ones the history does not “see”.

Assisted history matching hands the search over to the computer. The engineer specifies the uncertain parameters with their ranges, and the misfit function. An algorithm then looks for combinations with a small misfit. Several families of methods are used. One is series of runs planned by experimental design (a planned set of test runs), with fast approximating (proxy) models on which the minimum is sought. Another is gradient methods, in which the sensitivity of the misfit to the parameters is calculated together with the model. Others are evolutionary algorithms (which copy natural selection) and ensemble methods, which adjust tens or hundreds of model variants at once. Streamline models run quickly and directly give the sensitivity of the water arrival time to permeability. With them, we can first match the water breakthrough time at the wells and then the whole production performance.

The result of assisted history matching is usually not one model but a set of models with an acceptable misfit. We build a forecast range from them. But the algorithm finds only what the formulation allows. If the geological concept is wrong — say, the model lacks a fault that exists in the reservoir — no tuning of parameters will correct it. Choosing the parameters, ranges and weights remains an engineering task.

What a history match says about the forecast

A good match is necessary for a good forecast, but it is not enough. It shows that the model agrees with the processes and parts of the reservoir that have already shown themselves in the history. A forecast is most reliable where it continues these processes: the same wells, the same displacement mechanism, similar withdrawals. A forecast relies on extrapolation if it includes:

a new mechanism — gas injection, enhanced oil recovery methods, a pressure decline below the bubble point pressure
undrilled areas and new wells, especially horizontal ones, about which the history of neighbouring wells says little
regimes that have not occurred: substantially greater withdrawals, conversion of wells to injection
late stages that depend on properties which hardly affected the history, for example the relative permeability to oil at low oil saturation, which controls the “tail” of production (the slow decline at the end)

That is why we give a forecast from a history-matched model as a range, from several models or sets of parameters, and we update the model as data arrive. In a mature field, history matching is not a one-off job but part of regular reservoir management. We usually run forecasts as a continuation from the last step of the history run, rather than start anew.

Terms and symbols

History matching: Adjusting the uncertain parameters of a model until a calculation with the actual operating regime of the wells reproduces the observed history.
Inverse problem: Finding the properties of a system from its observed response. As a rule, such a problem has many solutions.
Misfit function $M$: A measure of the discrepancy between the calculation and the measurements, divided by the error. History matching makes it as small as possible.
Transmissibility multiplier: A coefficient that multiplies the transmissibility between cells. A value of 0 gives an impermeable boundary, and 1 changes nothing.
Assisted history matching: A search over the parameters, within given ranges, by an algorithm that makes the misfit function as small as possible.
Storage capacity (capacitance) of the reservoir $C = c_t V_p$: The volume of fluid that the reservoir releases when the pressure falls by one unit, owing to the expansion of the fluids and the contraction of the pores.
Hindcast: A comparison of the model's forecast with the part of the history that we did not use in the matching.

Common mistakes and misconceptions

Wrong

“The model has been history matched, so the forecast is right.”

In fact

History matching is non-unique: different models describe the history equally well and predict the future differently. Additional data, testing on withheld data and a forecast from several models raise confidence.

Wrong

“It is best to match every single well by adjusting multipliers around it.”

In fact

Local corrections without a geological explanation improve the match but worsen the forecast. We make changes consistently, by layers, facies and blocks, and only within plausible limits.

Wrong

“The pressure does not match — let us adjust the relative permeabilities.”

In fact

The volumes and energy of the reservoir decide the pressure first of all: the pore volume, the aquifer, the records of withdrawals and injection. We adjust relative permeabilities when we reproduce the water cut and the producing gas–oil ratio.

Wrong

“The production records are the truth; only the model may be adjusted.”

In fact

The data are wrong too: the sharing of rates among wells, gaps in the records, leaks of injected water into other layers. We check them before history matching, and a persistent mismatch is a reason to check the data once more.

Wrong

“Assisted history matching will find the right model by itself.”

In fact

The algorithm finds the best combinations only among the given parameters and ranges. If the geological concept is wrong, the result will be a well-fitted but wrong model.

In depth: advanced detail

Misfit function with prior knowledge

We can reduce non-uniqueness by adding to the misfit function a term that “penalises” a parameter for departing from what is known about it independently, from core, well logs and geological concepts:

Misfit function with a prior term

S(\mathbf{m}) = \sum_{k}\left(\frac{d_k^{\text{meas}} - d_k^{\text{calc}}(\mathbf{m})}{\sigma_k}\right)^{2} + \sum_{j}\left(\frac{m_j - m_j^{\text{prior}}}{s_j}\right)^{2}

where:

$\mathbf{m}$ the set of model parameters being adjusted, —
$d_k^{\text{calc}}(\mathbf{m})$ calculated value of the $k$ -th measurement at the parameters $\mathbf{m}$ , units of the measurement
$m_j^{\text{prior}}$ prior estimate of the $j$ -th parameter — before the history is taken into account, units of the parameter
$s_j$ uncertainty of the prior estimate, units of the parameter
$\sigma_k$ error of the $k$ -th measurement, units of the measurement

Units: dimensionless

Holds when

The errors of the measurements and of the prior estimates are independent and normally distributed (they follow the usual bell-shaped curve). Then the minimum of $S$ corresponds to the most probable set of parameters.
We obtained the prior estimates independently of the history. Otherwise we would count the same information twice.

The first term demands agreement with the history, the second the plausibility of the parameters. A parameter to which the history is insensitive stays close to its prior estimate instead of “drifting away” for the sake of a negligible improvement in the match.

Which pressure to compare with which

The pressure in a model cell is the average over its volume, and a cell is sometimes hundreds of metres across. The bottomhole flowing pressure of a producing well is lower than the cell pressure by the drawdown. The model calculates the drawdown through the productivity index of the well in the cell (Well model in a grid block). So we compare the measured bottomhole pressure with the calculated bottomhole pressure. We compare the average reservoir pressure found from a pressure build-up curve (Pressure build-up analysis) with the pressure in the cells of the drainage area (the area from which the well draws fluid). After a short shut-in, the pressure has not yet recovered and is lower than the reservoir pressure. If we compared it with the cell pressure, the model would seem “too optimistic” without any error in the reservoir. We refer all pressures to one datum depth.

Grid first, matching second

Coarse cells smear the displacement front and bring water breakthrough forward (Numerical dispersion). Upscaling can also smooth out high-permeability layers and anisotropy (the difference between horizontal and vertical permeability). Suppose we begin history matching on such a grid. The adjusted parameters then make up for the numerical errors, and the permeability or relative permeabilities depart from plausible values. In the forecast, for new wells for example, this compensation stops working. So we check the grid before history matching. We compare a calculation on the coarse grid with one on a fine grid for a representative part of the reservoir and, if necessary, correct the upscaling (Two-phase upscaling: pseudo-functions).

Connections to other topics

What a reservoir simulation model is and why it is needed — what a reservoir simulation model is, its input data and its limitations.
Grids in reservoir simulation models — how cell size and numerical dispersion affect breakthrough time — and history matching.
General material balance equation for oil reservoirs — an independent estimate of the oil in place and the water influx from the pressure history.
Relative permeability — relative permeabilities are one of the main parameters in matching the water cut.
Pressure build-up analysis — where the measurements of reservoir pressure and well productivity come from.
Sensitivity analysis and scenarios — how to present a forecast as a range.
Reservoir simulation in practice: case studies — how we use a history-matched model for decisions, on illustrative examples from exploration to a mature field.

Check yourself

1
The average pressure in the model falls noticeably faster than the actual one, while the water cut of the wells is reproduced well. What should we check first?
1. A.Relative permeabilities
2. B.The skin factors of the wells
3. C.The time step
4. D.The records of withdrawals and injection, the size and activity of the aquifer, the pore volume
Show the answer
Correct answer: D. The records of withdrawals and injection, the size and activity of the aquifer, the pore volume
The balance of volumes decides the average pressure: how much has been withdrawn and injected, and how much energy the reservoir has (elastic storage, the aquifer). Relative permeabilities and skin factors have almost no effect on the average pressure. The first divide the withdrawal between the phases, and the second set the drawdown at the well.
2
Which quantity is usually the easiest to reproduce in history matching?
1. A.The average reservoir pressure
2. B.All are equally hard
3. C.The producing gas–oil ratio of individual wells
4. D.The water breakthrough time at each well
Show the answer
Correct answer: A. The average reservoir pressure
The average pressure depends mainly on the volumes and energy of the reservoir and, in a well-connected reservoir, evens out. Details of heterogeneity, interbeds, relative permeabilities and the smearing of the front on the grid decide the water breakthrough time and the gas–oil ratio of individual wells. These are the hardest to reproduce.
3
After history matching, the misfit function is $M = 0.05$ with a pressure measurement error of 0.3 MPa. Is that good?
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Not necessarily. The discrepancies are much smaller than the measurement error: the root-mean-square discrepancy (the typical size of the differences) is about $0.3\sqrt{0.05} \approx 0.07$ MPa. Either the errors were estimated with a large margin, or the model has been fitted to random noise, for example by a multitude of local corrections. Such a model may forecast worse than a model with $M \approx 1$ .
4
Why can two models that describe the history equally well give different forecasts? Give an example.
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The history “tests” only those combinations of parameters that have already affected the data. For example, two years of pressure history cannot tell a large closed reservoir from a smaller reservoir with an aquifer. The difference shows up later, and after 8 years the pressures in the models differ by several megapascals. Likewise, the aquifer can make up for the permeability, and heterogeneity for the relative permeabilities.
5
A model has been matched to ten years of waterflooding. For which forecast is it the least reliable?
1. A.Switching to gas injection instead of water
2. B.Production next year with the same wells
3. C.Continuing the waterflood with a slightly higher injection rate
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Correct answer: A. Switching to gas injection instead of water
Gas injection is a new mechanism. Properties that hardly affected the history of the waterflood govern it: the relative permeability to gas, the vertical transmissibility and gravity segregation. The forecast here is an extrapolation, so we give it as a range over several variants of the model.
6
To reproduce the pressure, an engineer applied a multiplier of 5 to the permeability within a radius of 200 m around each of three wells. What is wrong?
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These are local fitting factors without a geological explanation. The match will improve, but the cause of the mismatch most likely lies elsewhere: in the volumes, the aquifer, the records of withdrawals. In a forecast, especially for new wells, such corrections act as noise. First check the data and the parameters that decide the pressure over the whole reservoir. Then make changes consistently, by layers and blocks.

Updated: 19 October 2026