Schrodinger’s Equation in the Oil and Gas Industry:
Sean Huber

Abstract:
Quantum mechanics is a fundamental theory in physics that describes the behavior of matter and energy on a microscopic scale. The theory is essential in various fields, including chemistry, electronics, and materials science, and it has applications in the oil and gas industry. In this paper, I will discuss Schrödinger’s equation, one of the most crucial equations in quantum mechanics, and its relevance to the oil and gas industry.

Schrödinger’s equation:
Schrödinger’s equation is a mathematical equation that describes the behavior of quantum systems. It was first introduced by Austrian physicist Erwin Schrödinger in 1925 and is central to the development of quantum mechanics. The equation is written as:
ĤΨ = EΨ
where Ĥ is the Hamiltonian operator, Ψ is the wave function of the quantum system, and E is the energy of the system. The equation is a differential equation that describes the evolution of the wave function with time. The wave function of a quantum system contains all the information about the system’s properties, including its energy and momentum. The square of the wave function, |Ψ|², gives the probability density of finding the system in a particular state. Schrödinger’s equation can be used to calculate the wave function of a quantum system for a given set of initial conditions.
Relevance in the oil and gas industry:
The oil and gas industries are complex and challenging industries that requires the use of advanced technologies to extract hydrocarbons from the ground. Quantum mechanics has several applications in the industry, and Schrödinger’s equation is particularly relevant in the following areas:
1. Reservoir modeling:
The behavior of oil and gas reservoirs is governed by the laws of physics, and quantum mechanics provides a fundamental understanding of these laws. Schrödinger’s equation can be used to model the behavior of fluids in the reservoir and predict their flow rates and pressures. The equation is used to describe the behavior of the electrons in the fluid, which is essential in understanding the fluid’s properties. Schrödinger’s equation can be used to describe the behavior of fluids in the reservoir through the wave function of the system. The wave function contains all the information about the system, including its energy, momentum, and position. The square of the wave function, |Ψ|², gives the probability density of finding the system in a particular state. In the case of fluid flow in a reservoir, the wave function represents the probability density of finding the fluid in a particular location at a particular time. The behavior of the fluid can be described by the time-dependent Schrödinger equation:
iħ∂Ψ/∂t = ĤΨ
where ħ is the reduced Planck constant, Ĥ is the Hamiltonian operator, and Ψ is the wave function of the fluid. The Hamiltonian operator for a fluid in a reservoir considers the fluid’s potential energy, kinetic energy, and interactions with other particles. The operator can be written as:
Ĥ = -ħ²/2m∇² + V(r,t)
where m is the mass of the fluid particle, ∇² is the Laplacian operator, and V(r,t) is the potential energy of the fluid particle. By solving Schrödinger’s equation for the fluid, the wave function can be obtained, which describes the fluid’s probability density in the reservoir at any given time. This probability density can then be used to calculate the flow rate and pressure of the fluid in the reservoir. For example, consider a simple case of fluid flow in a one-dimensional reservoir. The wave function for the fluid can be written as:
Ψ(x,t) = Aexp[i(kx – ωt)]
where A is a normalization constant, k is the wave vector, and ω is the angular frequency. The probability density of finding the fluid in a particular location x at time t is given by |Ψ(x,t)|² = |A|².
The flow rate of the fluid can be calculated using the continuity equation:
∂ρ/∂t + ∇·(ρv) = 0
where ρ is the fluid density and v is the fluid velocity. By integrating this equation over the reservoir and using the wave function to describe the fluid density, the flow rate of the fluid can be obtained. Similarly, the pressure of the fluid in the reservoir can be calculated by solving the fluid dynamics equations using the wave function as the initial condition. In summary, Schrödinger’s equation can be used to model the behavior of fluids in a reservoir and predict their flow rates and pressures by solving for the wave function of the system. This approach provides a quantum mechanical description of fluid flow in the reservoir and can be used to optimize oil and gas extraction processes.

2. Enhanced oil recovery:
Enhanced oil recovery (EOR) techniques are used to extract more oil from reservoirs than can be recovered using traditional methods. EOR techniques involve injecting fluids into the reservoir to increase the fluid’s mobility and reduce its viscosity. Quantum mechanics can be used to understand the behavior of the injected fluids in the reservoir and optimize the EOR process. Schrödinger’s equation is used to calculate the wave function of the injected fluids and predict their behavior in the reservoir. Enhanced oil recovery (EOR) techniques are designed to increase the amount of oil that can be extracted from a reservoir beyond the primary and secondary recovery stages. One such technique is the use of surfactants to reduce the interfacial tension between the oil and water phases in the reservoir. This reduces the capillary forces that hold the oil in place and allows it to be more easily extracted. Schrödinger’s equation can be used to model the behavior of surfactant molecules in the reservoir and predict their ability to reduce the interfacial tension. The wave function of the surfactant molecules contains information about their energy, momentum, and position, and can be used to calculate their interactions with the oil and water phases in the reservoir. The Hamiltonian operator for a surfactant molecule in a reservoir considers its potential energy and interactions with the oil and water phases. The operator can be written as:

Ĥ = -ħ²/2m∇² + V(r,t)

where m is the mass of the surfactant molecule, ∇² is the Laplacian operator, and V(r,t) is the potential energy of the surfactant molecule. The wave function of the surfactant molecule can be obtained by solving Schrödinger’s equation for the Hamiltonian operator. This wave function can then be used to calculate the probability of the surfactant molecule being in a particular location in the reservoir. The reduction in interfacial tension can be calculated using the Gibbs adsorption equation:

Δγ = γ_oil-water – γ_oil-surfactant – γ_water-surfactant

where γ_oil-water is the interfacial tension between the oil and water phases, γ_oil-surfactant is the interfacial tension between the oil and surfactant phases, and γ_water-surfactant is the interfacial tension between the water and surfactant phases. By using the wave function of the surfactant molecule to calculate the Gibbs adsorption equation, the reduction in interfacial tension can be predicted. This reduction in interfacial tension increases the mobility of the oil in the reservoir, allowing it to be more easily extracted. Schrödinger’s equation can be used to model the behavior of surfactant molecules in a reservoir and predict their ability to reduce the interfacial tension between the oil and water phases. This quantum mechanical approach can be used to optimize EOR techniques and increase the amount of oil that can be extracted from a reservoir.

3. Carbon capture and storage:
Carbon capture and storage (CCS) is a process that involves capturing carbon dioxide (CO2) emissions from industrial processes and storing them underground. Quantum mechanics can be used to understand the behavior of CO2 in the reservoir and predict its long-term storage potential. Schrödinger’s equation is used to calculate the wave function of the CO2 molecules and predict their behavior in the reservoir. The wave function of a CO2 molecule contains information about its position, energy, and momentum, and can be used to calculate its interactions with the rock and fluid phases in the reservoir. The Hamiltonian operator for a CO2 molecule in a reservoir considers its potential energy and interactions with the other molecules in the reservoir. The operator can be written as:

Ĥ = -ħ²/2m∇² + V(r,t)

where m is the mass of the CO2 molecule, ∇² is the Laplacian operator, and V(r,t) is the potential energy of the molecule. The wave function of the CO2 molecule can be obtained by solving Schrödinger’s equation for this Hamiltonian operator. This wave function can then be used to calculate the probability of the CO2 molecule being in a particular location in each reservoir. The interaction of the CO2 molecule with its surroundings can be modeled using molecular dynamics simulations. These simulations use the wave function of the CO2 molecule to calculate its interactions with other molecules in the reservoir. The capture of CO2 from power plants and other industrial processes is typically done using solvents, such as amines, that react with the CO2 to form a solution. The CO2 is then separated from the solvent and compressed for transportation to the storage site. Schrödinger’s equation can also be used to model the behavior of the solvent molecules and predict their interactions with the CO2 molecules. The reduction in interfacial tension between the CO2 and the solvent can be calculated using the Gibbs adsorption equation:

Δγ = γ_CO2-solvent – γ_CO2 – γ_solvent

where γ_CO2-solvent is the interfacial tension between the CO2 and solvent phases, γ_CO2 is the interfacial tension between the CO2 and the surrounding rock and fluid phases, and γ_solvent is the interfacial tension between the solvent and the surrounding rock and fluid phases. By using the wave function of the solvent molecule to compute the Gibbs adsorption equation, the decrease in interfacial tension can be predicted. This reduction in interfacial tension improves the efficiency of the CO2 capture process. Schrödinger’s equation can be used to model the behavior of CO2 and solvent molecules in a reservoir and predict their interactions with the surrounding rock and fluid phases. This quantum mechanical approach can be used to optimize the Carbon Capture and Storage process and increase the amount of CO2 that can be captured and stored in geologic formations.

Conclusion:
In conclusion, Schrödinger’s equation is a fundamental equation in quantum mechanics that has many applications in the oil and gas industry. The equation is used to model the behavior of fluids in reservoirs, optimize EOR techniques, and predict the behavior of CO2 in CCS processes. Quantum mechanics is an essential tool in the oil and gas industry, and its applications will continue to grow in the future.

Sources:
1. Atkins, P. W., & De Paula, J. (2006). Physical chemistry. Oxford: Oxford University Press.
2. Stone, M. (2012). Quantum mechanics for scientists and engineers. Mineola, N.Y.: Dover Publications.
3. Tomaselli, Nick “Oil and Gas” Interview by Sean Huber. 08 Mar. 2023
4. Broughton, Matt “Oil and Gas” Interview by Sean Huber. 09 Mar. 2023