Modeling Dependencies Between Geologic Risks in Multiple Targets
- Pierre Delfiner (TotalFinaElf)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- February 2003
- Document Type
- Journal Paper
- 57 - 64
- 2003. Society of Petroleum Engineers
- 5.7 Reserves Evaluation, 4.1.2 Separation and Treating, 5.7.2 Recovery Factors, 1.10.1 Drill string components and drilling tools (tubulars, jars, subs, stabilisers, reamers, etc), 5.6.3 Deterministic Methods, 5.1.5 Geologic Modeling, 5.1.1 Exploration, Development, Structural Geology, 5.2.1 Phase Behavior and PVT Measurements, 4.1.5 Processing Equipment, 1.6 Drilling Operations
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A single-target prospect can be characterized probabilistically by two elements: the risk or probability of success and the distribution of reserves given success. But there is no obvious way of combining the results of multiple targets into a single representative assessment.
This paper presents a simple approach for aggregating multiple targets for the purpose of prospect ranking. Reserves given success are assumed independent, and the novelty lies in the modeling of dependencies between geologic risk factors. Although interdependence can be represented by conditional probabilities, the selection of these values is a problem for most explorationists. In this paper, a more practical method is proposed. A matrix is completed for multiple targets, specifying which geologic factors are common and which are independent. An outcome tree with dependent targets is created, and the probability distribution of aggregate reserves is determined either analytically or by Monte Carlo simulation.
A case study with six targets illustrates the method. Dependencies reduce the probability of at least one success from 92% to 65% and the number of possible outcomes from 64 to 25. This allows a useful grouping by development cases. The method can be used for stacked reservoirs or portfolio analysis.
In deepwater exploration, a costly wildcat may require multiple potential targets to be justified. The probabilistic assessment of each individual target is standard work. However, to evaluate the prospect as a whole and compare it with others, some form of aggregation is required.
Far from being straightforward, aggregation is perhaps the most controversial task in probabilistic reserves assessment. When there is uncertainty about the reserves A and B, their sum A+B depends on how A and B are related. For example, if A and B are fully dependent, the proved aggregate reserves (P90) are the sum of the proved reserves; the usual math 2+2=4 applies. Now, if A and B are independent, it is unlikely that they both are at their low values so that the P90 of the sum is greater than the sum of the P90s; the math becomes something like 2+2=4.3, but it is not possible to identify where the additional .3 comes from. In multitarget exploration, this difficulty is aggravated by geologic risk because we are not even sure that reserves A and B will be there to be added. Risk may be regarded as an on/off switch: success or failure. If the switch is on, the reserves take on a value drawn from the distribution specified for the target. If the switch is off, the reserves of that target are zero. In this paper, only geologic success is considered, as opposed to commercial or economic success (e.g., Rose1). When we speak of reserves, we make the simplifying assumption that recovery factors are included in the uncertainty analysis just like other parameters, rather than create a more complex logic to link recovery-factor ranges to development plans that depend on the size of the accumulation. Readers uncomfortable with this may replace the word "reserves" throughout the paper with "volume in place."
A method is proposed for combining risks and reserves into a single description incorporating geologic dependencies between targets in a practical yet mathematically sound manner. If all targets are equivalent from a development standpoint, reserves can be pooled into a single distribution of aggregate reserves. Conversely, if there are significant development constraints, as would be the case with oils of different densities and viscosities at different depths, outcomes can be regrouped by development plan. The two approaches are demonstrated by a deep offshore case study with stacked reservoirs, and the results are summarized by way of graphs and tables.
The assessment of single-target prospects is a well-established process documented in the reference paper by Otis and Schneidermann.2 While there are variations from company to company, the basic ingredients stay the same: an evaluation of reserves and an evaluation of geologic risk.
Hydrocarbon reserves are calculated as the product of individual volume parameters such as gross pay volume, net/gross ratio, porosity, hydrocarbon (HC) saturation, formation volume factor, and recovery factor. If we were sure of the value of each parameter, we could compute a single reserves figure (deterministic case). Because of uncertainty, individual parameters are better represented by a range of values, or a probability distribution, which then leads to a probability distribution for reserves.
Monte Carlo simulation can be used for propagating the uncertainty of individual parameters to reserves. Parameter values are drawn randomly from their respective distributions and multiplied to produce a histogram of reserves. This approach has maximum flexibility, and in particular can handle correlation among inputs, but it requires the specification of a distribution model for each individual parameter.
An alternative to Monte Carlo simulation is an analytic approach. For example, if all inputs are log-normal and independent, reserves also follow a log-normal distribution with parameters that can be determined analytically (Bourdaire et al.3). In this model, the mode of the product is equal to the product of the modes, which provides us with a simple "base case." However, it forces all parameters to be log-normal even though net/gross, porosity, and HC saturation tend not to be.
Ref. 2 proposes a less constraining analytic approach, known as "the three-point method," that only assumes reserves to be log-normally distributed. This model finds a theoretical justification in the central-limit theorem for the product of independent random variables and is confirmed by observations from basins around the world. Uncertainty on individual parameters is captured by specifying a low (P95), a medium (P50), and a high value (P5). A formula from Pearson-Tukey4 allows the estimation of the mean and mean square of the input-parameter distribution by a weighted average of 95%, 50%, and 5% quantiles with respective weights (0.185, 0.63, 0.185). The beauty of this formula is that it requires no assumption on the underlying distribution and works remarkably well for all cases of practical interest. Assuming statistical independence of the inputs, it then suffices to multiply all means and all mean squares to obtain the first two moments of the lognormal distribution of reserves, which is thus completely determined.
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