Figures 4(a) and 4(b) illustrate the stock-flow and causal diagrams for the second step in the recreation of Naill’s model of U.S. natural gas discovery and production. The model is merely an extension of the model presented in Figure 1.

A comparison of Figure 4(a) to Figure 1 reveals that the right-hand side of the model (ProvenReserves, UsageRate, UsageCoef) has not changed. The left-hand side of the model, on the other hand, has had a significant piece of system dynamics structure added to it. More precisely, the simple discovery rate equation has been replaced by a balancing feedback loop (negative feedback) that works its way through the fraction of unproven reserves remaining and the cost of exploration.

Refering to Figure 4(b), the logic of the balancing loop is as follows: a fall in unproven reserves (UnprovenReserves), ceteris paribus, causes a fall (i.e., a move in the Same direction) in the fraction of unproven reserves remaining (FracRemaining). A fall in the FracRemaining, ceteris paribus, causes a rise (i.e., a move in the Opposite direction) in the effect of the fraction remaining on the cost of exploration (EffFracRemainCost). A rise in the EffFracRemainCost, ceteris paribus, causes a rise in the cost of exploration (CostOfExploration). A rise in the CostOfExploration, ceteris paribus, causes a fall in the indicated discovery rate (IndicDiscovRate) which, in turn, ceteris paribus, causes a fall in the discovery rate (DiscoveryRate). A fall in the DiscoveryRate closes the loop and ensures, ceteris paribus, that unproven reserves will be higher than they otherwise would have been.

Three other features of the model presented in Figure 4a are important to note. The first is that the balancing loop preserves first order control in the model. That is, although the feedback from the stock of unproven reserves to its outflow (DiscoveryRate) is not direct as it was in the first cut of the model, the equations in the loop still causes the computer to shut-off the discovery rate when the stock of unproven reserves is zero.

The second feature of the model presented in Figure 4a that is important to note is the delay symbol that appears in the discovery rate icon. The purpose of embedding a delay in the discovery rate is to account for the lag that exists between the time a decision is made to invest in gas exploration and the time actual gas discoveries occur. This delay could also have been modeled explicitly with a simple stock and flow stucture.

The last feature of the model that is important to note is its table functions. Represented by the zig-zag-like symbols, table functions allow the modeler to externally impose relationships(linear and nonlinear) between system variables.

Figure 5 lists the system dynamics equations that directly correspond to the icons shown in Figure 4. Three characteristics of these equations are particularly noteworthy. The first is that, in addition to the stocks(rectangles), flows (pipe and faucet assemblies) and constants (diamonds) presented in the first cut of the model, there are now auxiliary equations (circles) in the model.

The second noteworthy characteristic is that the model is again dimensionally correct. In particular, the measurement units of each variable in the balancing loop correspond to the arithmetic embodied in each equation in the loop. To provide just one example, the indicated discovery rate (Cubic Feet/Year) is equal to investment in exploration (Dollars/Year) divided by the cost of exploration (Dollars/Cubic Foot).

The third noteworthy characteristic is that two of the auxiliaries contain table functions (EffFracRemainCost and InvestInExplor) in nongraphical form. The graphical versions of these functions are shown in Figure 6.

An examination of Figure 6 reveals that the table functions both represent nonlinear relationships in the model and help to define the system’s limits. The effect of the fraction of gas remaining on the exploration cost function (EffFracRemainCost) indicates that as gas depletion occurs, the cost of gas exploration increases monotonically. The investment in exploration function (InvestInExplor) indicates that, over time, the number of dollars per year allocated to gas exploration increases and then saturates. In future cuts of the model, this relationship will be made endogenous.

Figure 6: Table Functions in the Second Cut of Naill’s Model of U.S. Natural Gas Discovery and Production

Figure 7 shows time series plots from a simulation of the model presented in Figures 4 and 5. As with the previous version of the model, this cut produces dynamic behavior consistent with the real system. That is, the UsageRate again rises from 1900 to the mid-1970s, peaks, and then declines. UnprovenReserves again fall monotonically due to depletion and ProvenReserves again rise during the early part of the century, peak during the mid-1970s, and then decline.

The only behavior in this version of the model that is new involves the DiscoveryRate. A comparison of Figures 3 and 7 reveals that the DiscoveryRate now rises and falls whereas, in the previous version of the model, it simply fell monotonically. The reason for this difference is that, in this cut of the model, the InvestInExplor table function exogenously increases investment in exploration over time. Early in the system’s evolution, the increasing amount of investment dollars causes a rise in natural gas discovery. Later on, however, the effects of depletion overwhelm the investment dollars and the discovery rate falls. An interesting simulation experiment that could be done is to systematically alter the InvestInExplor function (its shape, its saturation level, etc.) and resimulate the model to ascertain the results of the changes.

Figure 7: Time Series Plots from a Simulation of the Second Cut of Naill’s Model of U.S. Natural Gas Discovery and Production