Decline curve analysis is the process of fitting curves to production rate data from oil and gas wells. These best fit decline curves characterize the natural fall-off of production over time as oil and gas in the reservoir is depleted. They can be projected beyond the available data, extrapolating production rates and volumes out until the well is no longer viable. It is relatively simple to sum the production rates over time and get an estimate of the ultimate recovered volume of oil and natural gas that a well will yield (EUR: Estimated Ultimate Recovery). These estimates feed public companies’ filings of reserves with the markets, and directly affect the valuation / stock prices of the companies.
Estimates that are found to be too high have resulted in huge drops in corporate stock prices, and the firing of leadership all the way to the top of the organization, as happened at Shell Oil in 2004.
Estimates that are too low leave the company under-valued and often in a poor position to compete for necessary capital, and therefore prone to takeover or dissolution.
14 monthly production rates (y-axis) vs. elapsed days from first production. Deterministic best-fit decline curve is shown in red. Note that the curve is a compromise fit with data points above and below it
Decline curve analysis is difficult for the same reasons that affect any other sorts of regression or curve fitting processes: the data on which these analyses depend, is imperfect and noisy. Along with mechanical failures or geologic processes which curtail production rates, there are frequently interventions like workovers and recompletions that tend to enhance production rates, sometimes for a relatively short period of time, and other times for more extended time horizons.
Markov Chain Monte Carlo Estimate – A Walk on the Beach
What is Markov Chain Monte Carlo (MCMC) estimation? To describe how it works, let’s use an example. One of the parameters of importance in decline curve analysis, is the decline rate – it is a number that describes the percent fall off in production over a year. Let’s say that it is our job to estimate the EUR of a well that has been producing oil for the past 24 months. To do so, we need an estimate of the decline rate for this well. We go and talk to Loretta, the Senior Engineer for this well, who tells us that based on her decades of experience in the area, that the decline rate for this well is approximately 65%, give or take 5%.
Bayesian statistics is a wonderfully intuitive approach to problems like this. It starts with some prior belief (e.g. Our expert Loretta says the decline rate is around 65% more or less), and then uses actual data to update this prior estimate in a way which is consistent with the data. The resulting estimate is a range of likely values for the decline rate called a ‘posterior’ distribution. It is the product of the prior belief, and decline rates that are estimated using only the data. Note that it is a distribution instead of a single estimate. We can draw boundaries around this distribution and attach confidence values around the likelihood that the decline rate is greater than x, or less than y. These boundaries in the decline rate can be extrapolated to boundaries in EUR, providing best-case and worst-case estimates that are consistent with the data, and also informed by Loretta’s expertise and experience in the area!
But how do we go about measuring the extent to which a particular decline rate explains the actual production data? We need these measurements to update our prior beliefs!
Imagine that we have a decline curve savant who randomly chooses possible decline curve values from a hat. Once he draws a value, he can look at the data, and tell us the likelihood that the decline rate he just drew randomly, explains the data! For simplicity, let’s say that he reports this likelihood with 3 possibilities:
- Not Likely – not likely that this decline rate explains the data
- Possible – this decline rate is a moderately-good explanation of the data
- Very Likely – this decline rate is a very compelling explanation of the data
With such a genius available we could do the following. Imagine we place our genius on a beach where decline rates are inscribed in the sand, getting smaller to the west, and larger to the east. We place our savant at some arbitrary starting point on the beach, let’s say at the location corresponding to Loretta’s prior estimate of 65%. We then ask our savant to draw a new decline rate from his hat that must be nearby, say within a few steps east or west. He draws 63.5% and reports on how well it explains the data – let’s say it is a 3 (explains the data very well). Since the draw is a good explanation of the data, our genius takes a few steps west to that location. He draws again, this time 59.3%, and reports that is a 1 – very poor explanation of data. Being it is a poor move, our savant does not step over to that location further west on the beach but stays put at 63.5%. Now imagine that we repeat this process several thousand times. . .
It turns out that this approach ‘learns’ where the most explanatory values of decline rate are on the beach. We could literally trace our genius’s steps in the sand (assuming low tide ;-), tally up the locations on the beach that he visited the most often, and derive a probability distribution that describes the highest likelihood values of decline rate that are consistent with the observed production data! Voila!
Line plot showing the decline-rate random walk for 2000 steps along the x-axis. The histogram shows the resulting posterior distribution of the parameter.
We have created a Markov Chain Monte Carlo estimator using the process above. The succession of steps on the beach is a Markov Chain – a random walk in decline rate space. Monte Carlo simulation drives the random walk through some specified number of draws or steps. The algorithm that prescribes whether or not our savant steps to the randomly-drawn new spot on the beach is called the Metropolis-Hastings Algorithm.
Real Life Application
In our real-life application of MCMC to decline curves, instead of deriving one parameter (decline rate), we must estimate 3 parameters simultaneously. We combine decline curves defined by the most explanatory estimates of the 3 parameters, with the prior belief of our expert Loretta, and end up with an entire family of decline curves, which determine an entire distribution of EUR’s. Most importantly, these decline curves shed light on the uncertainty around these most-important EUR forecasts, giving us boundaries that describe best case and worse case forecasts that are consistent with the data, and also informed by Loretta’s expertise.
This MCMC estimation was done using Blue River Analytics – Energy Analytics Workbench 20 (EAW_20). In this chart, we have a cross plot of EUR on the Y axis versus IP (the initial production rate of the well, at Time = 0) on the X axis. The points in the scatterplot each represent a distinct decline curve (defined by its 3 parameters). To the right, is a horizontal histogram which shows the distribution of the MCMC-estimated EUR’s for the well:
The red grid lines on both axes define the 97.5th and 2.5th percentiles of each axis, the therefore we see that the High Confidence Interval (95% of outcomes) EURs lie in the interval between 166.9 and 171.4 thousand barrels of oil. Note that these numbers can also be read from the histogram in the middle. Notice in the cross plot, that there is a slightly negative correlation between EUR’s and IP’s: as the IP’s get larger, the EUR’s are tending to get smaller. To the right of the chart, you can see histograms for 2 of the other parameters (in addition to IP) that are being estimated: Decline Rate (which we used in the example above), and a B parameter, which defines the upward curvature of the decline curve over time.
Let’s compare the deterministic best fit, and see where it fits in the range of outcomes defined by MCMC. The best fit curve has an IP = 524, and an EUR of 165.8. If we plot those parameters against the cloud of most-likely parameter values, we can see that the deterministic curve is outside of the 95% HCI for EUR, as shown here:
The MCMC average EUR across all 2000 decline curves is 169.3 thousand barrels of oil, and the average IP is much lower at 506.
This is a great example of the additional information that comes from Markov Chain Monte Carlo Estimation: without the posterior population of decline curves, a single deterministic best fit curve has no context. We don’t have any way of knowing how good it is, is it on the low side, or high side, of what is suggested by the data and our prior beliefs about the parameters.
Finally, let’s take a look at the deterministic decline curve best fit, and the average of the 2000 MCMC-derived curves. The MCMC average curve will correspond to an IP of about 506, and results in an EUR of around 169.3 thousand barrels of oil:
A question for your consideration: If you were simply manually tweaking this curve, would you be able to see that the curve fit to the right is an outlier (as defined by the EUR which is below the 95% high-confidence interval)? And conversely, could you recognize that the curve to the left represents the expected value across 2000 high-likelihood decline curves that best explain the production data and also account for our experts’ background and recommendation on the likeliest curve?
Hopefully when you are faced with estimating parameters based on noisy data, and you need to understand the variability of possible values of these parameters based on the data, that you will consider Markov Chain Monte Carlo estimation as a possible solution!