What is the difference between numerical solution and approximate solution. Also when we can say that the method is numerical or approximate method?
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All numerical solutions are approximate.
All approximate solutions are numerical.
When trying to make a model of the actual physical reservoir, all models are approximations, and all models have to be solved numerically.
A numerical solution is when you take the physical equations and solve them using numerical methods or algorithms, i.e use approximation for differential operators assuming small increments. dx/dt = (x1-x2)/dt
An approximate solution is when you solve the physical equations exactly but use approximate values to get the final answer i.e solve the differential equation and then use approximate values. i.e solve dx/dt=2 therefore x=2t +C and use approximate values for x and/or t.
numerical solution are usually applied to highly non-linear equation, e.g complex flow partial differential equation. It might involve the application of finite difference approximation using Taylors series or other discretization algorithm.
Approximate solution applies when you try to obtain a direct solution to a complex polynomial or equation which may involve unequal fractional powers. I always use approximate solution anytime I am dealing with Juhasz or Waxman-Smith water saturation equations.
There can be numerical and analytical solutions. Analitical solutions can be both precise and approximate. Numerical solutions are always approximate.
Problem of analitical solutions is that they can only be applied to very simple problems (i.e. simple geometry of the well and reservoir). So even if the analitical solution is precise the answer you get at the end of the day is approximate because in most cases before using analytical solytion you had to simplify the problem.
When it comes to the numerical solutions in flow simulation there are two subtypes of the models:
“Full physics” models – they pretend to be as precise as it can be. I.e. in flow simulators we try to account for all physical effects we know how to account for. But even if this case it is hard to say that any particular model does “everything possible”. For example you can always refine the grid i a hope to improve solution.As well Full Physics models are known for long simulation times – that may be a problem for uncertainty modeling and assisted history matching in particular.
On the other end there are reduced physics models when we neglect some effects (i.e. gravity, heterogeniety PVT etc.). One simple way of getting “reduced physics model” is coarsening “reduced physics model”. Those models can be very fast but if chosen incorrectly may introduce unrealistic results (especially when soing uncertainty quantification and assisted history matching).
Numerical and approximate solutions are based on the same physics of developing the mathematical equations needed to solve a problem. A coarse model of a reservoir can be considered as an approximate solution (approximate analysis) where a fine gridded model for the same reservoir is considered as a numerical solution (numerical analysis). However, mathematical equations used in both cases are still approximate as they are developed based on assumptions that not necessarily representing the real case e.g. modelling of homogeneous and heterogeneous reservoirs are based on the same mathematical models and amount and quality of structure and fluid data i.e. both are approximate. Reservoirs are systems that are mathematically mimicked based on the available data that by all means will always result in approximate solutions regardless of the amount and accuracy of data used to model those reservoirs.
I think we have may be in danger of making it too complicated.
We have a physical reservoir. We have a mathematical model of the reservoir.
The mathematical model, normally encoded in software, is an approximation to the physical reservoir.
We use the mathematical model to perform studies, and hope the results of these studies are close enough to the physical model to make sound decisions.
We can solve the mathematical model in many ways. If it is a very simple model, we can solve it analytically. In most cases, we have to solve it numerically.
By analytically, I mean you can write out all the equations and the solution can be expressed as a mathematical equation which can (in principle) be solved exactly. Plug the numbers into your equation and you have the solution.
But even here, you may have an analytical solution, but it may be expressed with complex functions, or even simple functions like sin, cos, exp, Bessel etc. and these are still solved using approximate iterative methods, so it is still numerical even if it appears to be analytical.
The analytical solution is an exact solution to the model, but the model is an approximation to the physical reservoir.
The numerical solution is an approximate solution to the model, and the model is also an approximation to the physical reservoir.
In most cases the important difference is not between analytical or numerical solutions (provided care is taken in the numerical solution, which it is in all commercial reservoir simulations), but between the mathematical model and the physical reservoir.
As an example, for a given SPE test problem, different commercial simulators (should) give very similar results, which shows that the numerical solution methods have high accuracy.
But nobody would say those SPE test problems bear much relationship to any known physical reservoir.
As an aside, I don’t know whether they do or not, but it would be comforting if there were models for which the analytical solution was known, and the commercial simulators modelled it numerically to reproduce the analytical solutions. The fact that they give similar results to the same model is not sufficient to demonstrate they give the correct results.
Uncertainty quantification is quite important to verify the numerical solutions by testing against known analytical problems (e.g. variance of simple correlated Gaussians in high dimensions), but I am not aware of any software which does this, hence the big differences we see in uncertainty quantification between commercial vendors.
If your numerical solution cannot replicate the analytical solution on simple problems, or you have not attempted to replicate, go away and come back when it does.
Is the question about the proxy models derived by exact numerical models. ?
If so the the answer could be different.
In order to make it simply, all models of any mathematical kind or physical models are not exacts to a reservoir as it is well known, but numerical or analytical equations of a mathematical model, if they are simple can give exact solutions to the models but not to the reservoirs(even if with feeding by stochastic reservoir data).
Using analytical, numerical or finite difference methods of solutions of complex system or equations are subject to the type of that system and equations(complexity of fluid and flow) of our mathematical model towards exact or approximate solution