The process of simulating petroleum reservoirs has been thoroughly described by Peaceman (24) as the:
“process of inferring the behavior of a real reservoir, the prototype system, from the performance of a model of that reservoir. The model may be physical, such as a scaled laboratory model, or mathematical.
A mathematical model of a real reservoir is a set of partial differential equations, together with an appropriate set of boundary conditions, which are believed to adequately describe the significant physical processes taking place in the real reservoir, see the below Figure.
The processes taking place in a real reservoir are basically fluid flow and mass transfer.
Up to three immiscible phases (gas, oil, and water) may flow simultaneously where gravity, capillary, and viscous forces play an important role in the flow process. Mass transfer may take place between the phases (chiefly between gas and oil phases).
The model equations must account for all forces, and should also take into account an arbitrary reservoir description with respect to heterogeneity and geometry. The equations are obtained by combining the mass conservation equation with the equation of motion (Darcy's law).
To use the mathematical model for predicting the behavior of a real reservoir, it is necessary to solve the model equations subject to the appropriate boundary conditions.
The methods of solution are basically divided into two main methods, analytical and numerical. Analytical methods are applicable only to the simplest cases involving homogeneous reservoirs and very regular boundaries. Numerical methods, on the other hand, are extremely general in their applicability and have proved to be highly successful for obtaining solutions to very complex reservoir situations.
A numerical model of a reservoir, then, is a computer program that uses numerical methods to obtain an approximate solution to the mathematical model.”
Material balance calculation is an excellent tool for estimating gas reserves. It is based on the non-ideal gas law, PV = ZnRT.
If a reservoir comprises a closed system and contains single-phase gas, the pressure in the reservoir will decline proportionately to the amount of gas produced. Unfortunately, sometimes bottom water drive in gas reservoirs contributes to the depletion mechanism, altering the performance of the non-ideal gas law in the reservoir.
Under these conditions, optimistic reserves estimates can result. The below figure is a typical material balance plot for a tank-type reservoir.
A general material balance equation that can be applied to all reservoir types was first developed by Schilthuis in 1936.
Although it is a tank model equation, it can provide great insight for the practicing reservoir engineer. It is written from start of production to any time (t) as follows:
Expansion of oil in the oil zone + Expansion of gas in the gas zone + Expansion of connate water in the oil and gas zones + Contraction of pore volume in the oil and gas zones + Water influx + Water injected + Gas injected = Oil produced + Gas produced + Water produced
Mathematically, this can be written as:
A decline curve of a well is simply a plot of the well’s production rate on the y-axis versus time on the x-axis. The plot is usually done on a semilog paper; i.e. the y-axis is logarithmic and the x-axis is linear.
When the data plots as a straight line, it is modeled with a constant percentage decline “exponential decline”. When the data plots concave upward, it is modeled with a “hyperbolic decline”. A special case of the hyperbolic decline is known as “harmonic decline”.
The most common decline curve relationship is the constant percentage decline (exponential). With more and more low productivity wells coming on stream, there is currently a swing toward decline rates proportional to production rates (hyperbolic and harmonic).
Although some wells exhibit these trends, hyperbolic or harmonic decline extrapolations should only be used for these specific cases. Over-exuberance in the use of hyperbolic or harmonic relationships can result in excessive reserves estimates. The below figure is an example of a production graph with exponential and harmonic extrapolations.
The volumetric method entails determining the physical size of the reservoir, the pore volume within the rock matrix, and the fluid content within the void space.
This provides an estimate of the hydrocarbons-in-place, from which ultimate recovery can be estimated by using an appropriate recovery factor.
Each of the factors used in the calculation have inherent uncertainties that, when combined, cause significant uncertainties in the reserves estimate. The below figure is a typical geological net pay isopach map that is often used in the volumetric method.
The analogy method is applied by comparing the following factors for the analogous and current fields or wells:
The RF of a close-to-abandonment analogous field is taken as an approximate value for another field. Similarly, the BAF, which is calculated by the following equation,
is assumed to be the same for the analogous and current field or well. Comparing EUR’s is done during the exploratory phase. It is also useful when calculating proved developed reserves.
Analogy is most useful when running the economics on a yet-to-be-drilled exploratory well. Care, however, should be taken when applying analogy technique.
For example, care should be taken to make sure that the field or well being used for analogy is indeed analogous. That said, a dolomite reservoir with volatile crude oil will never be analogous to a sandstone reservoir with black oil. Similarly, if your calculated EUR is twice as high as the EUR from the nearest 100 wells, you had better check your assumptions.
The process of estimating oil and gas reserves for a producing field continues throughout the life of the field. There is always uncertainty in making such estimates.
The level of uncertainty is affected by the following factors:
The magnitude of uncertainty, however, decreases with time until the economic limit is reached and the ultimate recovery is realized