History matching

With the model built and initialized, Ethan runs it through the pool's history: fourteen years of primary production and eighteen months of pilot injection, with each producer controlled by its measured liquid rate. The first run is disappointing. The simulated pool pressure falls too fast, the first pilot producer sees water five months late and two of the other producers water out in the model before they have in reality. 'Good,' says Lena. 'Now we know what the model has to learn. History matching is not about making the curves line up. It is about finding out which parts of our description of the reservoir are wrong, and fixing those.'

What is matched

A history match compares the model's predictions with field observations over the period for which both exist. The observations come in a hierarchy. At the top are pool-wide quantities: average pressure from shut-in surveys and the total production of oil, water and gas. Below them are well-level data: each well's water cut, gas-oil ratio and flowing or static pressure. Below those are local measurements that test the vertical description: injection and production profiles from logs, pressure tests and repeat formation tests in new wells that show pressure by layer, tracer arrival times and, in some fields, saturation logs or time-lapse seismic that show where water has gone.

Matching generally proceeds from the top down. The pool pressure is matched first, by adjusting quantities that control the overall energy of the system: pore volume, aquifer size and strength, compressibility and the connection to neighbouring areas. Once the energy balance is right, the fluid movement is matched: breakthrough times, water cut development and gas-oil ratio, by adjusting the properties that control where fluids go, such as layer permeabilities, vertical permeability, fault transmissibilities and relative permeability. Matching the details before the totals usually wastes effort, because changing the pressure match later changes everything else.

The objective function

To judge a match objectively, the misfit between simulated and observed values is summarized in an objective function. The most common is a weighted sum of squared differences:

Here m is the vector of uncertain model parameters, d are the data, σ_i is the measurement uncertainty of each observation and w_i is a weight that expresses how much the engineer trusts or cares about each data type. Dividing by the measurement variance has an important consequence: the match is good when misfits are about the size of the measurement errors, not when they are zero. A model that matches noisy data perfectly has probably been tuned to the noise and will forecast badly.

The weights are a matter of judgement. Pressure surveys from permanent gauges are precise; allocated well rates in a pool with shared test separators are not. Water cuts from monthly tests may scatter by several percent. Lena assigns weights that reflect both the data quality and the purpose of the model: the waterflood forecast depends most on breakthrough and water cut development, so those get a high weight once the pressure is matched.

A pressure match

The first step of Ethan's match is the pool pressure. The code below reduces it to its simplest form. The pool is a tank with an analytical Schilthuis aquifer, whose influx is proportional to the pressure drop from the initial pressure. Seven pressure surveys with gauge and averaging errors are available. A golden-section search finds the aquifer constant that minimizes the misfit. The code then repeats the search for several values of the pore volume, to see whether a different combination of oil in place and aquifer strength fits just as well.

# History matching a pool tank model: tune the aquifer constant to fit measured pressures
import random
SI = True   
N, Boi, ce, pi = 5.11e6, 1.294, 1.4e-5, 15300.0     # m3, rm3/m3, 1/kPa (effective, includes rock and water), kPa
q_oil, Bo = 60.0, 1.296                             # m3/d oil, rm3/m3
def run(kaq, months=60, pv_mult=1.0):
    p = pi; out = []
    for m in range(months):
        for _ in range(30):                          # daily steps
            influx = kaq * (pi - p)                  # Schilthuis steady-state aquifer, rm3/d
            p -= (q_oil * Bo - influx) / (N * pv_mult * Boi * ce)
        out.append(p)
    return out
random.seed(3)
truth = run(0.020)
obs = {m: truth[m] + random.gauss(0, 40) for m in (5, 11, 17, 23, 35, 47, 59)}   # gauge surveys
def misfit(kaq, pv_mult=1.0):
    sim = run(kaq, pv_mult=pv_mult)
    return sum((sim[m] - v) ** 2 for m, v in obs.items()) / len(obs)
a, b = 0.0, 0.1; g = 0.6180339887                    # golden-section search on the aquifer constant
for _ in range(40):
    c, d = b - g * (b - a), a + g * (b - a)
    if misfit(c) < misfit(d): b = d
    else: a = c
k_best = (a + b) / 2
f, u = (1, "kPa") if SI else (0.145038, "psi")
ka, au = (1, "rm3/d per kPa") if SI else (6.28981 / 0.145038, "rb/d per psi")
print(f"best aquifer constant {k_best * ka:.4f} {au}; RMS misfit {misfit(k_best) ** 0.5 * f:.0f} {u}")
print(f"final pressure: simulated {run(k_best)[-1] * f:.0f}, last survey {obs[59] * f:.0f}")
# non-uniqueness: a bigger pool with a weaker aquifer fits almost as well
for pvm in (0.8, 0.9, 1.0, 1.1, 1.25):
    best = min((misfit(kk / 2000, pvm), kk / 2000) for kk in range(0, 201))
    print(f"pore volume x{pvm:4.2f}: best aquifer {best[1] * ka:.4f}, RMS misfit {best[0] ** 0.5 * f:4.0f} {u}")

best aquifer constant 0.0238 rm3/d per kPa; RMS misfit 35 kPa
final pressure: simulated 14090, last survey 14123
pore volume x0.80: best aquifer 0.0440, RMS misfit   56 kPa
pore volume x0.90: best aquifer 0.0340, RMS misfit   39 kPa
pore volume x1.00: best aquifer 0.0240, RMS misfit   35 kPa
pore volume x1.10: best aquifer 0.0130, RMS misfit   42 kPa
pore volume x1.25: best aquifer 0.0000, RMS misfit   64 kPa

The best match has a root-mean-square misfit of about 35 kPa (5 psi), close to the scatter of the surveys themselves, and the simulated final pressure is close to the last survey. That would be a satisfactory match by most standards. The second part of the output is more important. A pore volume ten percent larger with a weaker aquifer, or ten percent smaller with a stronger one, fits nearly as well. The pressure history alone cannot separate oil in place from aquifer support, a classic example of non-uniqueness. Only more data, such as the volumetric estimate, the material balance plot and the aquifer's response to the waterflood, can narrow the range.

Matching fluid movement

With pressure matched, Ethan turns to the water. The early breakthrough at the first pilot producer and the injection profile log both point to the thin high-permeability layer. Increasing that layer's permeability between the injector and the first producer brings breakthrough forward, but it also brings water to the second producer too early. The geologist suggests a channel orientation that would connect the injector to the first producer more strongly than to the others. Ethan represents it with a permeability multiplier in a band of blocks along the channel, consistent with the facies map, and the match at all four producers improves together.

This is the essence of good history matching: changes should be geologically plausible, should be applied over regions that make physical sense and should improve the match in several places at once. A local permeability multiplier around one well that fixes that well and nothing else is a warning sign. So is a parameter pushed to an extreme value, such as a vertical-to-horizontal permeability ratio of 10⁻⁵ in rock where core shows 0.1. Lena keeps a log of every change, its justification and its effect on the objective function, so that anyone reviewing the model later can see what was done and why.

Assisted history matching

Manual matching relies on the engineer's understanding and works well for a few parameters. For many parameters, or to explore non-uniqueness systematically, assisted history matching methods use the computer to search the parameter space. Experimental design and proxy models run a planned set of cases, fit a response surface to the objective function and use it to find good regions cheaply. Evolutionary and other global optimization methods search directly with many runs.

Ensemble methods have become widely used. The ensemble Kalman filter and its smoother variants, such as ensemble smoother with multiple data assimilation (ES-MDA), start from an ensemble of perhaps 100 models drawn from the prior uncertainty in the geology and properties. They run all members through the history and then update every member's parameters together, using the correlations between parameters and simulated data across the ensemble. The result is not one matched model but an ensemble of matched models that represents the remaining uncertainty, ready for the probabilistic forecasts of Lesson 11. These methods can adjust thousands of parameters, including permeability in every block, but they need care: small ensembles can collapse to too little spread, and updates that ignore the geological model can produce unrealistic property fields.

Whatever the method, the engineer must decide when the match is good enough. A common criterion is that misfits are comparable to measurement errors and show no systematic trend, such as a simulated pressure that is too high early and too low late. Another is that further improvement would require changes that the geology does not support. The match is a means to an end: a model that forecasts the decisions it was built for with an honest range of uncertainty.

Testing the match with a blind period

One of the best tests of a history match is to hold back part of the data. Ethan matches the model to the history up to twelve months after the start of injection and then runs it forward for the remaining six months without further changes, comparing the prediction with what actually happened. The matched model predicts the water cut at the first producer within a few percent and the onset of water at the second producer within about a month. The third producer, which the model predicts will see water in month seventeen, has not yet watered out at month eighteen. That small miss suggests the channel connection to that well is slightly too strong in the model, and it is one of the parameters that the ensemble carries with a wide range.

A blind test is not proof that the model will forecast well over twenty years, but it is far better evidence than a match alone. It also guards against a common temptation: tuning the model to every new data point as it arrives. If a model needs adjustment every month to stay matched, it is describing the data rather than the reservoir. Lena tells Ethan to judge a model by how little it has to change when new data arrive, not by how closely it fits the old data.