MOL & BoundaryConditions

[B-LIE]

** Does MOL support BCs including Algebraic Equations❓**

I'm trying to solve a "realistic"/"engineering" type PDE system with MOL. Most (every?) example I have seen of the use of MOL just contains PDEs for equations, and simple boundaries for BCs. In my case, I consider a two phase system of liquid and gas; a gas-lift system as used in oil production. I have 3 PDEs + numerous AEs (algebraic equations) within the system volume.

Then I have boundary conditions, typically represented by pressures outside of the pipe and flow of mass through valves governed by pressure differences.

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The documentation is not very clear to me whether MOL allows for a mixture of PDEs and AEs, and BCs with AEs.

Here is a sketch of the system I consider with boundaries:

• Liquid (oil+water) enters at x=0 from an oil reservoir at pressure p_f.
• Gas enters at x=0 from an annulus pipe surrounding the tubing pipe that I consider; annulus pressure is p_a.
• The mixture (two phase) leaves the pipe at x=L towards a production manifold pressure p_p.

The gas and liquid flows as two separate phases, and may take up varying cross sectional areas given by variable alpha:

When I run my code, I get an error message when I try to discretize the model into DAEs: I'm told there is an illegal boundary condition.

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To make my problem clearer, here are the PDEs valid within x=(0,L):

In addition, I have the following AEs which are also valid within x=(0,L):

In my model, in general, m is mass, fracture m is momentum, so two mass balances (liquid, gas) and a total momentum balance. The check decoration ("v") indicates "per volume". A dot on top (".") indicates flow rate. p is pressure, etc. For initial simplification, I have assumed that liquid density is constant (in reality, slightly compressible), and I have used a simplified gas EquationOfState which is almost an ideal gas but with compressibility factor z_g set to constant; in a more realistic model, z_g is the real root of a cubic polynomial. That will come in later.

My BCs are a mixture of traditional BCs and some AEs:

The last of the BCs I've listed (pressure gradient) is probably incorrect... because of gravity and friction, I don't think the gradient should be zero. Also, I already have 3 BCs related to the three external pressures p_f, p_a, p_p, and I have 3 PDEs with first order derivatives -- not sure if the pressure gradient should be counted in.

When I try to create the problem, I get an error message:

> prob = discretize(gas_lift,discr_scheme);
AssertionError: Boundary condition ṁ_a2t(t) ~ Y_a2t(t)*sqrt((max(Δp_a2t(t), 0)*ρ_a(t)) / (p_at2__ς*ρ_a2t__ς))*f_v(u_a2t(t))*ṁ_a2t__c is not on a boundary of the domain, or is not a valid boundary condition

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NOTE: If someone knowledgeable would like to test what is wrong with my code, let me know.

[cmhyett]

This is not an appropriate place to post confidential code. Anyone can view.

[ruidanielgm]

Hello. Were you able to solve the problem? I'm having the same issue with another problem.

[B-LIE]

No. I found a mistake in one of my boundary conditions, and started to check how I could compute steady state of a simpler problem using a boundary value solver. Then I got caught up in other things, and have put it at rest for the time being... will return to it RSN... ["Real Soon Now", an expression used by writer Jerry Pournelle in Byte Magazine back in the days...].

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Note: my problem with formulating the model using MethodOfLines was not due to the mistaken boundary condition, but is a structural thing with restriction on the form of boundary conditions. Reason for sidestepping into the steady state study was in a hope that the system is simpler to solve if I start close to steady state.

I haven't checked lately whether the possibilities with boundary values have been improved.

Anyways, I have some ideas of how to get around the problem, but it will be extremely wasteful wrt. unknowns.