**Applications**** of Electrodynamic theory in Reservoir Engineering**

*Written by Kittipong Limchuchua*

*Edited by Sriniketh Sukumar*

Electrodynamic theory is a fundamental theory used to describe how electric and magnetic forces interact in our universe. Even though most applications in petroleum engineering rely on the principles on fluid mechanics, Electrodynamic theory contributes several interesting applications in the hydrocarbon industry. Petroleum engineers rely on electronic instrumentations to obtain data on important petrophysical parameters of the formation. Specifically, Reservoir engineers draw analogies from electromagnetism to predict and model reservoir depletion phenomena and forecast oilfield production given changing production rates, how reservoirs will be depleted and how much hydrocarbon would be produced, accounting for changes in production rates. We will conclude our findings with a qualitative interpretation of all comparisons made between the two disciplines.

**Depletion of Rectangular Reservoirs**

The first part of this article will discuss about the depletion of the rectangular reservoirs (simple box-shape) as shown in **Fig.1**. Note that we choose this type of reservoir for simplicity. We will assume a transient state (parameters change respect to time or time-dependent) solution to approximate curves of production rate and accumulated oil produced. Exponential decay production rate equations can then be derived and plotted against production time. Interestingly, this model matches in form perfectly with the classical model for charge flow in a resistor-capacitor circuit (RC circuit) , and can therefore serve as a useful analogue that helps analyzing this system.

**Mathematical Derivation of Reservoir Depletion Model **

We can begin to develop our model for reservoir depletion from the mass continuity and mass conservation equations. We get the general form of a partial differential equation for pressure in reservoir depletion. **Eq.1** is a diffusivity equation (simplified for one dimension). Its physical meaning is that mass is conserved, but the equation shows pressure as an independent variable. It shows how pressure relates to radius from wellbore and time. The other petrophysical parameters are permeability (κ), porosity (φ), compressibility (c), and viscosity (μ).

We summarize the equivalent relations between the reservoir depletion and RC circuit model as follows; See **Fig 3(a)**. and **Fig 3(b)** for electric charge curve for RC circuit and oil produced curve respectively:

Our model shows that when we arrange a direct current (DC) generator, a resistor and a capacitor in series and close the loop, the battery immediately discharges a current that decreases exponentially over time. As time passes, charges get accumulated in the capacitor until the loop is opened, at which time the capacitor has accumulated charge. At that point, the charge flow completely stops. This is analogous to our reservoir depletion model. The producing well is a driving force for liquid petroleum to move upward. The initial rate (of production) is a maximum point, which decreases exponentially as hydrocarbon in the reservoir is depleted through production. The charge in the capacitor is analogous to the oil produced during the production phase, and current is analogous to flowrate. Note that this model is intentionally simplified to show how oil depletion could be recognized as an RC charging circuit. Indeed, the system is a lot more complex to analyze in any real formation.

Fig 3 time-dependent solution for (a) electric charge in a capacitor and (b) oil produced curve

**Applications in Economic Sense**

The big question is why we need a production rate equation as a function of time. You might think why oil and gas companies are doing all this hard work. Simple answer! To make money, they must know not just where, but also when to produce. If you think about monthly or yearly cash flow, knowing the rate equation is crucial information. Why? Remember that:

**Revenue ($)=Production Rate×Price**

Production rate gives us an insight into how much oil/gas we can produce in a given time. The equation gives some clues on why a volumetric calculation and a material balance equation are tied with accounting and money. Yes! Even this multi-disciplinary concept, as well as most concepts in engineering in general is an application of the universal accounting equation, which fundamentally states that any quantity, tangible or otherwise must be conserved in any process, and that it is impossible to create something out of nothing.

Isn’t it amazing how petroleum engineers, geologists and economists work together to make oil and gas companies accomplish their goals in an extremely-competitive economic world?

**References**

Mian, M. A. (2010). *Project economics and decision analysis*. Tulsa, OK: PennWell.

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