The routine and classical reservoir engineering divides the composite systems as either parallel, series, or a mix of both cases. In these systems, the overall systems permeability and porosity will be determined by writing the Darcy's law equation for each layer and equating either the flow rate (in series systems) or overall pressure drop (in parallel systems). However, this is not the case when the interface between layers is inclined or curved. An example is shown in the attached file.
I appreciate receiving any suggestion that helps calculating average porosity and permeability for this system.
Dear Reza,
It seems that it is a 1D system and in that case, the permeability and length of each plug or sector would affect the total permeability of the system. So, it could be crucial in this case to calculate the average length of each plug based on its shape and curvature.
Kind regards
In this example, we can see 2D flow,also pressure will be changes along the each face regarding to hydro static pressure, So in this case I think we can only use the reservoir simulation and based on matching or try and error we can find a equal system with an average permeability.
But for simpler cases, we can solve the problem based on node analysis and considering some assumptions.for more information please find the attached file.
Dear Mehdi
thanks a lot for your answer. The whole system may be considered steady-state. However, the end of this structure does not have a constant and sharp surface area perpendicular to flow. This way I think one needs to split the flow vector into two elements, one vertical and the other horizontal.
what's your suggestion?
That is an interesting question, Reza. I agree with Mehdi the the flow field is 2D. But, if you wish to assume a pure 1D field, in theory, the average permeability for a puzzle shape composite medium with an arbitrary combination of pieces can be calculated using the common parallel and series layers approach provided that we know the geometry and spatial location of all pieces. However, this approach becomes very cumbersome if we deal with a complex puzzle!
In such a case of a complex system, the best approach is to use numerical simulations to find kxx, kyy, and possibly other elements of permeability tensor.
you may change your continuum field into a grid block system, even though it will be an estimation but the approach is conceivable. then, you can write an analytical solution based on the number of grid blocks in x and y-axis assuming them parallel-series blocks (if you assume total m blocks in x axis and n blocks in y axis, in the first row all blocks are K1 which is in series configuration to m-1 blocks of K1 parallel to 1 block of K2, ...). you can also build the grid block system in eclipse and perform a black oil simulation and write the Darcy law for the system and calculate the overall permeability (you may also calculate off diagonal components of the permeability tensor).
Dear Sir,
please read the attached paper which may help.
Regards
Dominique
Dear Dr. Azin
Thank you for your question. This case is 2D flow system, horizontal and vertical. Please use numerical reservoir simulation to find the overall transmissibility.
Regards,
Amin
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