In geology, numerical modeling is a widely applied technique to tackle complex geological problems by computational simulation of geological scenarios.
Numerical modeling uses mathematical models to describe the physical conditions of geological scenarios using numbers and equations. Nevertheless, some of their equations are difficult to solve directly, such as partial differential equations. With numerical models, geologists can use methods, such as finite difference methods, to approximate the solutions of these equations. Numerical experiments can then be performed in these models, yielding the results that can be interpreted in the context of geological process. Both qualitative and quantitative understanding of a variety of geological processes can be developed via these experiments.
Numerical modelling has been used to assist in the study of rock mechanics, thermal history of rocks, movements of tectonic plates and the Earth's mantle. Flow of fluids is simulated using numerical methods, and this shows how groundwater moves, or how motions of the molten outer core yields the geomagnetic field.
History Prior to the development of numerical modeling, analog modeling, which simulates nature with reduced scales in mass, length, and time, was one of the major ways to tackle geological problems, for instance, to model the formation of thrust belts. Simple analytic or semi-analytic mathematical models were also used to deal with relatively simple geological problems quantitatively.
In the late 1960s to 1970s, following the development of finite-element methods in solving continuum mechanics problems for civil engineering, numerical methods were adapted for modeling complex geological phenomena, for example, folding and mantle convection. With advances in computer technology, the accuracy of numerical models has been improved. Numerical modeling has become an important tool for tackling geological problems, especially for the parts of the Earth that are difficult to observe directly, such as the mantle and core. Yet analog modeling is still useful in modeling geological scenarios that are difficult to capture in numerical models, and the combination of analog and numerical modeling can be useful to improve understanding of the Earth's processes.
Components
A general numerical model study usually consists of the following components: Mathematical model is a simplified description of the geological problem, such as equations and boundary conditions. These governing equations of the model are often partial differential equations that are difficult to solve directly since it involves the derivative of the function, for example, the wave equation. Discretization methods and numerical methods convert those governing equations in the mathematical models to discrete equations. These discrete equations can approximate the solution of the governing equations. Common methods include the finite element, finite difference, or finite volume method that subdivide the object of interest into smaller pieces (element) by mesh. These discrete equations can then be solved in each element numerically. The discrete element method uses another approach, this method reassembling the object of interest from numerous tiny particles. Simple governing equations are then applied to the interactions between particles. Algorithms are computer programs that compute the solution using the idea of the above numerical methods. Interpretations are made from the solutions given by the numerical models.
Properties A good numerical model usually has some of the following properties: Consistent: Numerical models often divide the object into smaller elements. If the model is consistent, the result of the numerical model is nearly the same as what the mathematical model predicts when the element size is nearly zero. In other words, the error between the discrete equations used in the numerical model and the governing equations in the mathematical model tends to zero when the space of the mesh (size of element) becomes close to zero. Stable: In a stable numerical model, the error during the computation of the numerical methods does not amplify. The error of an unstable model will stack up quickly and lead to an incorrect result. A stable and consistent numerical model has the same output as the exact solution in the mathematical model when the spacing of the mesh (size of element) is extremely small. Converging: The output of the numerical model is closer to the actual solution of the governing equations in the mathematical models when the spacing of mesh (size of element) reduces, which is usually checked by carrying out numerical experiments. Conserved: The physical quantities in the models, such as mass and momentum, are conserved. Since the equations in the mathematical models are usually derived from various conservation laws, the model result should not violate these premises. Bounded: The solution given by the numerical model has reasonable physical bounds with respect to the mathematical models, for instance mass and volume should be positive. Accurate: The solution given by the numerical models is close to the real solution predicted by the mathematical model.
Computation
The following are some key aspects of ideas in developing numerical models in geology. First, the way to describe the object and motion should be decided (kinematic description). Then, governing equations that describe the geological problems are written, for example, the heat equations describe the flow of heat in a system. Since some of these equations cannot be solved directly, numerical methods are used to approximate the solution of the governing equations.
Kinematic descriptions
In numerical models and mathematical models, there are two different approaches to describe the motion of matter: Eulerian and Lagrangian. In geology, both approaches are commonly used to model fluid flow like mantle convection, where an Eulerian grid is used for computation and Lagrangian markers are used to visualize the motion. Recently, there have been models that try to describe different parts using different approaches to combine the advantages of these two approaches. This combined approach is called the arbitrary Lagrangian-Eulerian approach.
Eulerian The Eulerian approach considers the changes of the physical quantities, such as mass and velocity, of a fixed location with time. It is similar to looking at how river water flows past a bridge. Mathematically, the physical quantities can be expressed as a function of location and time. This approach is useful for fluid and homogeneous (uniform) materials that have no natural boundary.
Lagrangian
The Lagrangian approach, on the other hand, considers the change of physical quantities, such as the volume, of fixed elements of matter over time. It is similar to looking at a certain collection of water molecules as they flow downstream in a river. Using the Lagrangian approach, it is easier to follow solid objects which have natural boundary to separate them from the surrounding.
Governing equations
Following are some basic equations that are commonly used to describe physical phenomena, for example, how the matter in a geologic system moves or flows and how heat energy is distributed in a system. These equations are usually the core of the mathematical model.
Continuity equation
The continuity equation is a mathematical version of stating that the geologic object or medium is continuous, which means no empty space can be found in the object. This equation is commonly used in numerical modeling in geology.
One example is the continuity equation of mass of fluid. Based on the law of conservation of mass, for a fluid with density at position in a fixed volume of fluid, the rate of change of mass is equal to the outward fluid flow across the boundary :
where is the volume element and is the velocity at .
In Lagrangian form:
In Eulerian form:
This equation is useful when the model involves continuous fluid flow, like the mantle is over geological time scales.
Momentum equation The momentum equation describes how matter moves in response to force applied. It is an expression of Newton's second law of motion.
Consider a fixed volume of matter. By the law of conservation of momentum, the rate of change of volume is equal to: external force applied on the element plus normal stress and shear stress applied on the surface bounding the element minus the momentum moving out of the element on that surface
where is the volume element, is the velocity.
After simplifications and integrations, for any volume , the Eulerian form of this equation is:
Heat equation The heat equations describe how heat energy flows in a system.
From the law of conservation of energy, the rate of change of energy of a fixed volume of mass is equal to: work done at the boundary plus work done by external force in the volume minus heat conduction across boundary minus heat convection across boundary plus heat produced internally Mathematically:
where is the volume element, is the velocity, is the temperature, is the conduction coefficient and is the rate of heat production.
Numerical methods
Numerical methods are techniques to approximate the governing equations in the mathematical models.
Common numerical methods include finite element method, spectral method, finite difference method, and finite volume method. These methods are used to approximate the solution of governing differential equations in the mathematical model by dissecting the domain into meshes or grids and applying simpler equations to individual elements or nodes in the mesh.
The discrete element method uses another approach. The object is considered an assemblage of small particles.
Finite element method The finite element method subdivides the object (or domain) into smaller, non-overlapping elements (or subdomains) and these elements are connected at the nodes. The solution for the partial differential equations are then approximated by simpler element equations, usually polynomials. Then these element equations are combined into equations for the entire object, i.e. the contribution of each element is summed up to model the response of the whole object. This method is commonly used to solve mechanical problems. The following are the general steps of using the finite element method: Select the element type and subdivide the object. Common element types include triangular, quadrilateral, tetrahedral, etc. Different types of elements should be chosen for different problems. Decide the function of displacement. The function of displacement governs how the elements move. Linear, quadratic, or cubic polynomial functions are commonly used. Decide the displacement-strain relation. The displacement of the element changes or deforms the element's shape in what is technically called strain. This relation calculates how much strain the element experienced due to the displacement. Decide the strain-stress relation. The deformation of the element induces stress to the element, which is the force applied to the element. This relation calculates the amount of stress experienced by the element due to the strain. One of the examples of this relation is Hooke's law. Derive equations of stiffness and stiffness matrix for elements. The stress also causes the element to deform; the stiffness (the rigidity) of the elements indicates how much it will deform in response to the stress. The stiffness of the elements in different directions is represented in matrix form for simpler operation during calculation. Combine the element equations into global equations. The contributions of every element are summed up to a set of equations that describe the whole system.