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An explicitly coupled two-dimensional (2D) multiphysics finite element method (FEM) framework comprised of thermal, phase field, mechanical and electromagnetic (TPME) equations was developed to simulate the conversion of solid kerogen in oil shale to liquid oil through in-situ pyrolysis by radio frequency heating. Radio frequency heating as a method of <i> in-situ </i> pyrolysis represents a tenable enhanced oil recovery method, whereby an applied electrical potential difference across a target oil shale formation is converted to thermal energy, heating the oil shale and causing it to liquify to become liquid oil. A number of <i> in-situ </i> pyrolysis methods are reviewed but the focus of this work is on the verification of the TPME numerical framework to model radio frequency heating as a potential dielectric heating process for enhanced oil recovery. Very few studies exist which describe production from oil shale; furthermore, there are none that specifically address the verification of numerical models describing radio frequency heating. As a result, the Method of Manufactured Solutions (MMS) was used as an analytical verification method of the developed numerical code. Results show that the multiphysics finite element framework was adequately modeled enabling the simulation of kerogen conversion to oil as a part of the analysis of a TPME numerical model.

The term, “oil shale” itself is not geologically defined by a specific chemical formula but in general refers to fine-grained sedimentary rocks that yield shale oil upon pyrolysis or retort [

Several technologies have been investigated in order to evaluate the production potential of oil shale. Given oil price uncertainty and increased U.S. desire for energy independence in the last couple of decades, the need for more reliable yet economic oil production techniques has increased. Challenges associated with production from oil shale include water use, net energy usage, carbon dioxide emissions, and commercial scalability [

The goal of this study is to demonstrate the verification of an explicitly coupled 2D TPME code that was developed using a general-purpose finite element framework, leveraging the TalyFEM libraries, in order to analyze kerogen conversion to liquid oil. Upon successfully verifying the code it is intended that numerical modeling studies be undertaken to address parametric uncertainty analysis, subsurface formation description, solid-liquid conversion rates and mechanical formation response due to kerogen to oil conversion using the multiphysics TPME framework. While dimensionality and parametric description of TPME quantities are fundamental to the understanding of in-situ pyrolysis for a target formation, this work is focused on the mathematical verification of the underlying coupled equations. An explicit coupling scheme, analogous to the description provided by Dean et al. [

Much of the work that has been conducted in developing in-situ pyrolysis has been proprietary thus details describing field trials have not been readily accessible in the public domain. As a result, there has been no attempt to verify the derived results with those obtained during field trials. Instead verification of the coupled governing equations describing TPME processes specific to radio frequency heating are undertaken by way of the Method of Manufactured Solutions (MMS). The executed study is primarily mathematical and only substantively related to the actual physical parametric description oil shale in-situ pyrolysis by radio frequency heating. Be that as it may, the main contribution of this work is the analytical verification of the developed multiphysics finite element method simulator, so that modeling the in-situ pyrolysis of oil shale by radio frequency heating can be achieved. The ability to model in-situ pyrolysis using radio frequency heating by a TPME approach is anticipated to enable environmental, economic, and technical analysis of production from oil shale formations; in this way, the associated advantages and disadvantages can be more accurately assessed.

Disparate methods have been undertaken by oil and gas operators to evaluate the technical feasibility of commercial oil shale production. The more prominent processes that have been considered for commercial scale in-situ pyrolysis have included: Shell In-situ Conversion Process (ICP) [

The ExxonMobil Electrofrac™ process, as outlined in Hoda et al. [

The Shell ICP process involves heating the oil shale formation with resistive heating elements surrounded by a hexagonal pattern of heating wells with the converted oil and gas produced through a production well in the center of the ring [

The Chevron CRUSH process involves using pressurized and heated carbon dioxide to raise the temperature of the oil shale; however, it depends on large quantities of water and leads to negative impact on the environment [

The coupled TPME computational framework developed for this investigation reflects the Raytheon Radio frequency with Critical Fluids Technology process described by Allix et al. [

numerical description without consideration of grade or constituent mineral components. Here the electromagnetic energy from the electrode is converted to thermal energy so that it heats the subsurface oil shale, enabling in-situ pyrolysis of kerogen to take place; then critical CO_{2} is injected to displace the oil to the production well. The CO_{2} is separated and recycled for further usage as a part of the process. While the injection of CO_{2} is not considered in this investigation, the focus of this work is instead the numerical modeling of radio frequency dielectric heating in the subsurface given specific mathematical considerations. Of particular interest in the use of the multiphysics framework is the conversion of solid kerogen to liquid oil for production. The advantage of the radio frequency heating considered in the Raytheon process is that oil and gas can be produced in months compared to other methods which have a longer conversion and production times, order of years. One of the main disadvantages of radio frequency heating that has been identified is the reduction in efficiency that occurs as half of the generated electromagnetic energy is lost to the formation surrounding the installed electrodes in the conversion process [

Radio frequency heating represents a dielectric heating process where electromagnetic energy is converted to thermal energy in the oil shale. Molecules in the kerogen of the shale formation experience a change in electrical polarity as they are introduced to an alternating electromagnetic field from electrodes that are placed in the subsurface rock; this leads to increased thermal energy. Once conversion temperatures are reached, the solid kerogen in the oil shale is converted to liquid oil. As a result, kerogen conversion can be characterized as a moving interface, or phase field, defined by the conversion of a solid to a liquid. As the solid kerogen is converted to liquid, the converted zone is anticipated to become mechanically unstable as it may no longer support the stress of the overburden rock. Given this outcome, the TPME was developed to specifically model thermal, phase field, mechanical, and electromagnetic processes that constitute the radio frequency heating process.

A complex electrostatic potential solution is obtained by solving a quasi-static Maxwell equation for “lossy” dielectric material [

− ∇ ⋅ [ ( σ + j 2 π f ε 0 ε ′ ) ∇ V ] = 0 (1)

where ε ″ is the loss factor of oil shale (imaginary part) describing the material’s ability to convert the electromagnetic field energy to heat, ε ′ is the relative dielectric constant of oil shale (real part) describing the lossless energy interaction of the material, j is equal to (−1)^{0.5} and V is the electric potential difference. The power conversion term P is expressed as:

P = 2 π f ε 0 ε ″ | E | 2 = σ | − ∇ V | 2 (2)

which describes the conversion of electromagnetic to thermal energy.

The power conversion term P is used to couple the quasi-static Maxwell equation to the enthalpy and Allen-Cahn phase field equations in order to compute the temperature field. The radio frequency power conversion term causes the temperature of the kerogen to increase and eventually leads to kerogen upgrading to oil. The physics of the actual TPME numerical framework is defined such that interpolation of all intrinsic rock type properties -i.e. thermal conductivity, occurs as a function of the Allen-Cahn phase field and temperature solutions. This method is typically applied in capturing parametric transitions across an interface within phase field models.

The characterization of the fluid conversion interfaces during the modeled pyrolysis of kerogen to shale oil is performed using an Allen-Cahn phase field description. In mathematical physics, the Allen-Cahn equation represents a reaction-diffusion equation which can describe physical processes like liquefaction, such as what takes place as a result of dielectric heating of oil shale. The results illustrate a moving interface (spatio-temporal) expressed through a highly localized area of the domain. The Allen-Cahn phase field equation of Dyja et al. [_{m}) of the reference material. The Allen-Cahn equation is described as:

∂ ϕ ∂ t − D C n 2 ∇ 2 ϕ + 2 D A ϕ ( 1 − 3 ϕ + 2 ϕ 2 ) − D k − P K m = 0 (3)

By convention, the phase field term, ϕ , and D, a scalar term, are dimensionless. As for the remaining terms; C_{n} is a diffusion coefficient, A and k are respectively frequency parameters for the solid-liquid interface, with A being a scaling factor and k being a bulk adjusting term.

The enthalpy equation described in Belhamadia et al. [

The modified enthalpy equation for this multiphysics model was required to include the power conversion term, describing electromagnetic to thermal energy conversion, and the solid-liquid phase transition during pyrolysis. Belhamadia et al. [

α ( ϕ ) ∂ T ∂ t + ρ l L s ∂ ϕ ∂ t − κ ( ϕ ) ∇ 2 T = P (4)

Here α ( ϕ ) is the volumetric heat capacity defined by density times the specific heat capacity as a function of the phase variable ϕ . It should be noted that phase related physical properties across the phase transition interface are interpolated between the respective pure solid and liquid phases. As for the remaining terms, the density of the liquid phase is ρ l , the latent heat of fusion is given by L_{s} and the thermal conductivity, as a function of the phase variable, is expressed as κ ( ϕ ) .

It is anticipated that with suitably weak formation properties the upgraded zone becomes unable to support the stress of the overburden once the in-place solid kerogen upgrades to liquid oil. In this structurally unstable situation, as the liquid can no longer support the overburden stress, compaction of the target formation ensues. The complexity of such poro-elastic and plastic deformation exhibited by oil shale as a result of in-situ pyrolysis has been extensively discussed [

3 ( 1 − υ ) ( 1 + υ ) ∇ 2 σ m + ∇ ⋅ F − 2 ( 1 − 2 υ ) 1 + υ ( 3 β K m ∇ 2 T ) = 0 (5)

Expressed within this equation is the Poisson’s Ratio υ , a force density term F which is taken to be the force density due to the overburden and the linear thermal expansion coefficient is β . Within this equation the Poisson’s Ratio, linear thermal expansion coefficient and bulk modulus are functions of the phase variable.

The description of the explicitly coupled TPME formulation is shown in

Few studies have been published about radio frequency heating as an enhanced

oil recovery method or about numerical modeling of the same. Furthermore, results of field trials, recoverable volumes of hydrocarbon and economic viability of in-situ recovery methods remain closely guarded. The intellectual property that has been developed in conjunction with these methods are overwhelmingly reserved to maintain strategic advantage by the respective corporate entities which developed or acquired the specific upgrading technology. This lack of published production and numerical modeling results was the main motivation for at least verifying the mathematical and algorithmic implementation of the multiphysics framework.

The employed technique for code verification in this study was the MMS. While this technique was developed some time in the 1980’s or 1990’s there is no exact attribution as many have claimed to use it without specifically referencing it. Be that as it may, a description of the technique may be reviewed in Salari and Knupp [

According to Salari and Knupp [

1) The manufactured solutions should be comprised of smooth analytical functions so that solutions may be easily computed. These would include trigonometric or polynomial functions. This guideline is designed to ensure that theoretical order-of-accuracy is attainable.

2) The solution should sample every term in the governing equation being evaluated.

3) An appropriate number of non-trivial derivatives for the solution should exist.

4) The solution derivatives should be constrained by a small constant and not contain singularities.

5) Successful execution of the code should not be impeded by the imposed solution.

6) The solution domain should be defined in a connected subset of the modeled space.

7) The differential operators in the partial differential equations should be formed in a manner that is logical.

In each case the MMS was used to verify the coupled equations that were incorporated into the multiphysics TPME solution. The verification was performed on the coupled code so that coupling terms, and non-solution variables were modified to be constants in this way the primary variables are independent of implemented coupling terms. In each case the MMS was applied to a 2D rectilinear FEM mesh with dimensions [0:1] × [0:1] and 20 elements in the X-direction and Y-direction, respectively. Essential boundary conditions were enforced on each of the boundaries in the quasi-static Maxwell and mechanical equilibrium equations. Conversely, in the Allen-Cahn and enthalpy equations, essential boundary conditions were applied to the left and right boundaries while natural boundary conditions were applied on the top and bottom boundaries. Appendix A contains the list of the parameters used in the underlying code verification performed by MMS. It is important to note that the listed parameters have no physical significance with respect to oil shale in-situ pyrolysis but are considered mathematically relevant to the verification of the code using MMS.

Given the aforementioned guidelines a solution to V for the quasi-static Maxwell equation in Equation (1) was set to the following analytical spatial function for m ( x , y ) :

m ( x , y ) = sin π x sin π y (6)

An illustration of Equation (6) is shown in

Γ ( x , y ) = [ σ + j 2 π f ε 0 ε ′ ] ( 2 π 2 sin π x sin π y ) (7)

Equation (1) is modified to include Equation (7) in its right-hand side. It is then solved for V ( x , y ) using the scalable linear equation Krylov solver (KSP) in the PETSc framework [

solution to m ( x , y ) shown in ^{2} coefficient equal to unity; this suggests that the solutions at each node are consistent with the analytical solution. This is further evidenced by the slope and y-intercept of the linear trend line equation that has values of 0.9938 and 4e−07, respectively. As a corollary, the anticipated slope of an exact match in solution is unity and the y-intercept is zero. The identical solution is obtained for the imaginary part of the electric potential difference in

near the center of the domain.

The code for the Allen-Cahn equation, from Equation (3), was verified by assuming the phase variable ϕ as equal to the following spatio-temporal analytical function for g ( x , t ) :

g ( x , t ) = sin π x sin π t (8)

A contour plot of Equation (8) is shown in

χ ( x , t ) = π sin π x cos π t + D C n 2 π 2 sin π x sin π t + 2 D A sin π x sin π t [ 1 − 3 sin π x sin π t + 2 ( sin π x sin π t ) 2 ] − D κ − 1 × 10 − 1 (9)

Equation (3) was then revised so that Equation (9) became the right-hand side of the equation and then solved for the phase field variable ϕ ( x , t ) . The solution for the phase field variable was then obtained in the numerical model by using the Newton-Raphson method of the SNES solver in the PETSc framework [

term of P K m in the coupling between the quasi-static Maxwell equation and the

Allen-Cahn equation was fixed to a constant value of 1 × 10^{−1} in order to independently verify the implementation of the Allen-Cahn equation in the coupled

framework in the absence of complimentary coupling terms. The cross-plot of node values for ϕ ( x , t ) -as the variable “u” in the x-axis, and the function g ( x , t ) -as the variable “u_a”, is shown in Figures 9(a)-(d) for select times. An alternating positive-negative response is observed in these results and is attributed to the time-dependent sinusoidal analytical function g ( x , t ) used in the MMS. In addition to ^{2} in both cases equals to 0.9997. This is compared to t = 63 where R^{2} equals 0.9983 andt = 75 where R^{2} equals 0.9661. Even though the t = 75 solution has a trend line slope that is closer to unity it is important to note that R^{2} value is more indicative of the relationship between ϕ ( x , t ) and g ( x , t ) than is the slope of the trend line. Similar results are illustrated in the absolute error maps of | ϕ ( x , t ) − g ( x , t ) | in Figures 10(a)-(d) where changes in the solution at the node are observed over time but appear to

be consistent att = 50 and t = 68. Solutions with the greatest accuracy occur near the left and right-side boundaries where essential boundary conditions were defined in contrast to the top and bottom side boundaries of the domain where natural boundary conditions were defined.

The code for the enthalpy equation, Equation (4), remained coupled, by definition, to Equation (3) through the phase field variable. As a result, this code was not verified in the absence of or the explicit fixing of a constant phase variable value using MMS. The assumption was made that the temperature variable T was set to Equation (8) to provide a solution to the enthalpy equation.

Substitution was performed for Equation (8) into Equation (4) and upon rearranging the following result was derived:

γ ( x , t ) = α ( ϕ ) π sin π x cos π t + ρ l L s ∂ ϕ ∂ t + κ π 2 sin π x sin π t − 1.0 (10)

The updated Equation (4) was then modified so that P was subtracted from each side of the equation, then Equation (10) was set to be the right-hand side of the equation then solved for temperature. Here the power conversion term, used to couple the quasi-static Maxwell equation to the enthalpy equation, was set to a constant value of unity solely for the purpose of code verification. This minimized the impact of the solution to the electromagnetic equation by isolating the temperature term. The phase field term was maintained as it was a component of the enthalpy equation that was used as a starting point the numerical model. The temperature solution, obtained using the MMS, is illustrated in ^{2} coefficient of unity. While the y-intercept in each plot of

increase in absolute node error with time fromt = 50 until t = 68 but then the absolute node error decreases oncet = 75. While additional times were not analyzed it is anticipated that due to the sinusoidal behavior of g ( x , t ) the response of the absolute node error would continue to alternate with time.

The governing mechanical equation, Equation (5), was verified while maintaining the coupling with the temperature solution obtained by solving the enthalpy equation, Equation (4). The assumption was made that the mean normal stress σ m was equal to Equation (6) which is shown in

ζ ( x , y ) = − 6 ( 1 − υ ) π 2 sin π x sin π y 1 + υ − 2 ( 1 − 2 υ ) 1 + υ ( 3 β K m ∇ 2 T ) ( 1 )

In the next step, Equation (5) was revised so that Equation (11) was set as the right-hand side of Equation (5) then solved for the mean normal stress. In this case, the enthalpy equation was coupled to the mechanical equilibrium equation through an enthalpy-stress coupling parameter that was set to a value of 1.6e1. The result of the executed mechanical simulation code by MMS led to a solution for the mean normal stress which is shown in ^{2} coefficient as well as a trend line that is described by a slope of unity and a y-intercept in the trend line equation that is approximately zero. The cross-plot highlights the accurate match of the MMS computed solution to that of the initial analytical function. The spatial description of the deviation between the

MMS solution and the analytical solution at each node, | σ m ( x , y ) − m ( x , y ) | is shown in

A multiphysics finite element computational framework was developed which explicitly coupled thermal, phase field, mechanical and electromagnetic equations for the purpose of numerically simulating the in-situ pyrolysis of oil shale by radio frequency heating. Specific processes for in-situ pyrolysis were discussed but the multiphysics solution proposed in this work was specifically designed to address radio frequency heating as an enhanced oil recovery method. There have been limited publications that have addressed either the development of multiphysics numerical models for simulating the in-situ pyrolysis process or production from oil shale using radio frequency heating. Furthermore, very few publications have demonstrated the development or use of multiphysics finite element solutions to model in-situ radio frequency heating. As a result, verification of the developed multiphysics finite element model necessitated the use of MMS. The results from the use of MMS in this work show that the electromagnetic, phase field, enthalpy and mechanical equilibrium solutions are respectively and collectively implemented in an accurate manner in the computational TPME framework. The observed matches between the computed and analytical solutions of the coupled equations using MMS demonstrated combinations of negligible or numerically insignificant differences between the numerical solutions. As a corollary, the TPME numerical framework has been verified for use in numerically modeling oil shale in-situ pyrolysis by radio frequency heating.

The author wishes to thank Dr. Baskar Ganapathysubramanian for fruitful discussions and supervision in the development of the multiphysics TPME finite element framework.

The author declares no conflicts of interest regarding the publication of this paper.

Ramsay, T.S. (2021) Verification of an Explicitly Coupled Thermal-Phase Field-Mechanical Electromagnetic (TPME) Framework by the Method of Manufactured Solutions. Open Journal of Modelling and Simulation, 9, 1-25. https://doi.org/10.4236/ojmsi.2020.91001

A = Allen-Cahn frequency parameter, t^{−}^{1}, s^{−}^{1}

C n = Allen-Cahn diffusion coefficient, L^{2}/t, m^{2}/s

D = Allen-Cahn scaling factor, dimensionless

E = quasistatic electric field, V/L, V/m

f = frequency, t^{−}^{1}, s^{−}^{1}

F = force density, F/L^{3}, N/m^{3}

g = general spatio-temporal function

j = unit imaginary number, − 1

k = Allen-Cahn frequency parameter, t^{−}^{1}, s^{−}^{1}

K m = bulk modulus, m/Lt^{2}, Pa

L s = latent heat of fusion (solid), E/m, J/kg

m = general spatial function

P = power conversion term, t^{3}I^{2}V^{2}/L^{3}m^{3}, s^{3}∙A^{2}∙V^{2}/m^{3}∙kg^{3}

t = time, t, s

T = temperature, T, K

V = voltage, V, V

x = distance, L, m

y = distance, L, m

α = volumetric heat capacity, E/L^{3}T, J/m^{3}∙K

β = linear thermal expansion coefficient, T^{−}^{1}, K^{−}^{1}

γ = general spatio-temporal function

Γ = general spatial function

ϵ = relatively small numerical value

ε 0 = electromagnetic permittivity of free space, t^{4}I^{2}/L^{3}m, F/m

ε ′ = electromagnetic relative dielectric constant, dimensionless

ε ″ = electromagnetic loss factor, dimensionless

ζ = general spatial function

κ = thermal conductivity, P/LT, W/m∙K

ρ = density, m/L^{3}, kg/m^{3}

ρ l = density (liquid), m/L^{3}, kg/m^{3}

σ = electrical conductivity, t^{3}I^{2}/L^{3}m, s^{3}∙A^{2}/m^{3}∙kg

σ m = mean normal stress, m/Lt^{2}, kg/m∙s^{2}

υ = Poisson’s ratio, dimensionless

ϕ = Allen-Cahn phase field, dimensionless

χ = general spatio-temporal function