Hubbert Logistic Curve

The logistic curve, and its derivative the hubbert’s curve, has been widely used to model population growth. And it has been applied to model oil production by M. King Hubbert. The model comes from the following differential equation:
dQ/dt=kQ(1-Q/URR)

where Q(t) is a function of time (measured in years) and it is defined as the cumulative production of a region until the end of year t. The parameter URR is the “Ultimately Recoverable Resources” or the maximum cumulative production that can be reached. K is the Malthusian parameter or the maximum cumulative production growth.

ยป Source: GraphOilogy

The value dQ/dt for a specific year’s, can be approximated by
(Q(s)-Q(s-1))/(s-(s-1))= Q(s)-Q(s-1)

which is the production of year’s, so let us define P(t)=Q(t)-Q(t-1). Let us assume that we have a region where the oil production follows strictly the logistic model, and that we have the information of Q(t) for all years. If we had P(t) for all years then we could obtain Q(t), since Q(t)=P(t)+P(t-1)+P(t-2)+…. What happens if we place a point at position (Q(t),P(t)) in the plane for all years t? Well, since we have seen that P(t) is approximately KQ(t)(1-Q(t)/URR), then the points would follow very closely the parabola:

KQ(1-Q/URR) = KQ - (K/URR)*Q^2.

This parabola passes through the origin (0,0) and through (URR,0). Now we are going to experiment with 50 different regions of the world. For each one we are going todo two plots. In the first one we will place all the data points (Q(t),P(t)) until year 2005 (both measured in Giga-barrels) and then find the parabola that passes through the origin that better approximates the data points (by the least squares fitting method). The intersection of this parabola with the x-axis will give us the estimated URR. In the second plot we want to show how this estimated URR has changed through time. For this plot, we define the function URR(t) as the estimated URR by the prior method if we had used the data points up until year t, and discarding later years. In the second plot we place points at (Q(t),URR(t)). Clearly Q(t)Q(t),URR(t)) lies above this line, then according to the logistic model t is before the peak year (as calculated at year t), i.e. Q(t). If (Q(t),URR(t)) lies below the dashed line then we are after peak year.

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