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Dr. K. H. Coats

 

 

Finite Difference vs. Finite Element Models

Also see:

 

From: Reservoir Simulation General Discussion discussion <SIM@communities.spe.org>

Discussion: Reservoir Simulation General Discussion

Author: Mr. Brian K Coats (Mr. Brian K Coats)

Date: 2006-06-11

Subject: Re: Future Directions in Reservoir Simulation

While I realize we are talking about future directions, any statements or conclusions that current technology is inaccurate or insufficient or that some newer technology is superior should be accompanied by (the simplest possible) demonstrable and reproducible evidence, or reference to such.

It is through such evidence that we are able to improve our empirical or partly empirical macroscopic (block-average) representations of pore-scale structure, phenomena, and properties.  These models and properties include permeability, relative permeability, capillary pressure, pvt, compaction/fracturing/geomechanics, and generally represent things that we are not able to accurately characterize or model or define distributions of on the microscopic scale, at which the true physics apply, but are able to sample, observe, and tune our models to on a macroscopic scale.  When these block-average representations are found to be inadequate, we respond by defining multiple systems, such as in dual porosity models for naturally fractured reservoirs.

For any given case, any given simulator may or may not have functional dependencies or options/couplings available for sufficient representation of block-average porosities and permeabilities.  I have no reason to believe that finite element models offer any advantages in simulating geomechanical effects.

With regard to Mr. McCracken’s statement about finite difference models not describing radial wellbore flow properly in rectilinear coordinate systems, I constructed the following test:

The second SPE comparative solution project gives a single well radial coning problem in a 10 x 15 r-z grid.  Outer radius of the reservoir is 2050 ft.  3 cartesian grid systems were created with constant areal grid spacing, computed to give the same total reservoir volume as the radial case.  Cartesian grids of 11x11x15, 51x 51x15, and 101x101x15 were used, the areally square blocks having sides of lengths 330.32, 71.25, and 35.98 ft, respectively.  The well was placed at the center of each of the Cartesian grids.  Well indices are computed internally using the Peaceman formula.  All 4 runs in Sensor give virtually the same results in terms of production up to 50 days of simulation.  The original well constraints were retained, and change from an initial oil rate of 1000 stb/d to 100 stb/d at 10 days. Comparison of bottomhole pressures shows that the Cartesian cases give 1 to 2% error in bhp at 10 days relative to the radial case, but bhp is otherwise essentially the same in all runs out to 50 days.  The 51x51 and 101x101 cases give virtually identical results out to 900 days (end of run), with water and gas cumulatives about 10% higher than in the radial case.  This difference is attributed mainly to the different representation of the outer boundary, at which a pressure response is evident.  The 11x11 case exhibits some numerical dispersion reflected in cumulatives at 900 days, with water and gas about 10% lower than in the finer Cartesian cases.  Use of the 9 point differencing option did reduce this difference somewhat, but had no effect on results up to 50 days (The x-y vs. radial nature of the flow has a more pronounced effect on the 11x11 case).  The radial (spe2.dat) and 11x11 Cartesian cases are reproducible using the downloadable Sensor version from our Coats Engineering website.

Perhaps Mr. McCracken would like to construct a finite element model for the rectangular reservoir boundary case, and demonstrate the problem with representation of well flow, say compared to our 51x51x15 case.  I would be glad to provide any output files, or even short term license files, for use in making comparisons.  As our cases are taking initial 10 day steps, perhaps it would be desirable to take smaller steps to more accurately capture the transient, and then examine the effect on production of any differences in the transient.  Our models show no such effect or differences.

Also of interest would be FDM vs FEM efficiency.  In Sensor, these 4 cases take 0.06, 1.39, 91.3, and 554 cpu seconds (in the order described), on a 2.4 Ghz Windows desktop pc, using the implicit formulation.  For the Cartesian cases, a 1/8 element of symmetry could have been used to obtain the same results with significantly reduced run times.  If it is desired to eliminate linear solver times from those timings (since it is claimed that improvements in linear solver technology will make FEM more viable), linear solver time in the radial case is too small to be measured accurately, but in the 3 Cartesian cases is 0.9, 80.5, and 497.4 cpu seconds.

Brian Coats

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From: Reservoir Simulation General Discussion discussion <SIM@communities.spe.org>

Discussion: Reservoir Simulation General Discussion

Author: Mr. Brian K Coats (Mr. Brian K Coats)

Date: 2006-11-16 17:25:45

Subject: Part 1: Re: Effect of cell dimensions on Kv/Kh ratio

Kes, your last posting is indeed controversial!

Firstly, there is no link between the Effect of Cell Dimensions on Kv/Kh Ratio issue to the issue of FD vs FEM modeling. The observation that vertical perms often need reduction below measured values to match behavior was I believe correctly explained as a sampling issue by Mr. Hicks in his postings last August 24. It is in no way related to the numerical model formulation as has been suspected and implied. The opposite problem can also occur - you might capture a portion of a discontinuous shale layer in your core that yields a lab measurement of zero Kv, when in fact, on the gridblock scale, there is vertical communication.

But since the issue has been raised... There are many reasons that the industry has not substantially moved in the direction of unstructured FEM modeling, but I would argue that they do not include, as in Kes's loaded statement, tradition or inertia. The main reason is that FD models are the still the simplest and most efficient models that are sufficient for full field simulation.

Major problems with unstructured grids include upscaling (alignment of coarse and fine scale grid boundaries), conformance to boundaries (small partial cells along boundaries), very complex control volume shapes in 3D space-filling grids, and substantially increased coefficient generation and possibly linear solver cpu times (as opposed to structured grids with same number of cells). Put more simply, gravity has affected nature to provide reservoirs with features and properties that are most closely and easily represented by rectilinear shapes (blocks) and smooth but possibly discontinuous surfaces (faults and layer boundaries).

Mr. McCracken's comment about minimization of grid orientation effects applies only to high mobility ratio displacements such as in steam flooding, and even then, nine-point difference approximations generally reduce those effects in FD models to acceptable levels. And I disagree with his comment that FEM models flow in the true flow direction, unless he means with a dynamic grid.

The full tensor permeability issue is simply not a significant problem with conventional methods. We can construct our grids along the bedding planes to approximate as best we can the k-orthogonal condition for a diagonal permeability tensor. We can measure or infer permeability along and perpendicular to the bedding plane (have you ever seen measurements of the off diagonal perm tensor components?). We can stairstep any faults or fluid contacts with sufficient grid resolution such that the stairstepping has no adverse effect on results. In an unstructured grid, control volume interfaces that are not perpendicular to one of the 3 principal axes of any permeability tensor create the REQUIREMENT for tensorial treatment of flow at those interfaces. Conventional gridding techniques and FD models at least have the ability to approximate and assume k-orthogonality.

 <continued in Part 2>

From: Reservoir Simulation General Discussion discussion <SIM@communities.spe.org>

Discussion: Reservoir Simulation General Discussion

Author: Mr. Brian K Coats (Mr. Brian K Coats)

Date: 2006-11-16 17:27:43

Subject: Part 2: Re: Effect of cell dimensions on Kv/Kh ratio

And perhaps the biggest justifications here are that 1) we are going to history match away any differences that might be caused by the differences in flow representation, and 2) there is so much uncertainty in the local and block-scale property distributions (as indicated by the Kv/Kh ratio discussions - perhaps there's a link after all!) that anything more detailed than the conventional gridding and FD flow treatment is unnecessary.

There have been several publications we are aware of demonstrating the advantages of FEM over FD. In all such cases using reproducible examples, we have found that our FD model was capable of producing results equal to or better than the FEM results presented. In one such recent paper, an FEM model was shown to give significantly less numerical dispersion for a 1D miscible, zero cap pressure Buckley Leverett displacement. But the authors either were unaware of or ignored the fact that the FD model exactly reproduces the analytical (shock front) solution if it is run at the impes stable step limit! Any desired level of increasing numerical dispersion (and increased cpu time) can be introduced into the FD model results by running at increasingly smaller timesteps below the stable value.

We would be glad to compare our FD model with FEM model results for those or other cases that can be equivalently and easily represented. I suggest that FEM model results and performance might be presented for some of the SPE comparative solution problems, spe1, spe9, and spe10 (black oil) and/or spe3 and spe5 (compositional) (our results are available on our website). The fact that no FEM model results have ever been submitted in these projects or presented subsequent to them is a clue to the results that we believe these comparisons would show. Or we might start with some very simple single phase tests for unidirectional (x-direction) flow in a 2d areal cross section of a rectangular reservoir, with constant pressure boundary conditions on either end, and diagonal permeability tensor. Our model can get the correct q/dp ratio using any value of nx>2, and the results are completely independent of choice of Ky. It's not clear to me that this would be the case using an unstructured grid using, for example, hexagonal prisms. The problem could then be made progressively more complex to demonstrate diagonal flow issues, etc.

Regards,

Brian Coats

Coats Engineering

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From: Reservoir Simulation General Discussion discussion <SIM@communities.spe.org>

Discussion: Reservoir Simulation General Discussion

Author: Mr. Brian K Coats (Mr. Brian K Coats)

Date: 2006-11-20 09:48:43

Subject: Re: FEM vs FD

Deepankar,

I did not dismiss FEM, but it's interesting that you think so. I'm sure it can be valuable for detailed near-well modeling, and it definitely has many valuable applications in other areas, although I think the problems I noted for reservoir simulation still apply. My opinions are not based on conjecture. I developed a 2.5D unstructured simulator in the mid 90's that used a full perm tensor. Sait Kocberber did the CVFE and Pebi gridding. He was often able to construct k-orthogonal grids, but not always, in which case he computed the 'transmissibility matrix' for each connection representing our form of the multipoint flux approximation. However, your implementation could be better than ours was.

The main thrust of my post was that FD models are the 'simplest and most efficient models that are sufficient for FULL FIELD simulation'. I'll extend that a little and say they are also the most suitable for constructing simple prototype models as engineers often do to answer basic questions. You specify dx,dy,dz,depth,perms directly to the simulator and run. There's nothing simpler or faster in gridding and modeling these simple approximate systems.

Perhaps you, or others, could point us to or provide us with the simplest possible reproducible examples, and FEM results and performance, using diagonal perm tensors wrt Cartesian coordinates, demonstrating any or all of the following:

1. That hybrid gridding is necessary in full field modeling.

2. Your model's ability, using hexagonal prisms, to give the correct answer to the unidirectional flow problem I described, independent of Ky. I've described a test problem and given our data and output files on the web page http://www.coatsengineering.com/unidirectional.htm . The correct steady state rate is 474.7 stb/d. Sensor can obtain this same result using as little as 2 total grid cells (without demonstrating independence of results on Ky, that requires 4 cells). An additional test - in your model, does a tracer injected in the middle of the high pressure boundary disperse in the y direction?

3. The need or benefit of using dynamic grids. We have not been able to obtain such, despite requests.

4. Your solutions to any of the SPE comparative solution problems.

We could endlessly trade claims and arguments, but no progress or conclusions can be made without simple reproducible examples and results.

Thanks,

Brian@CoatsEngineering.com

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From: Reservoir Simulation General Discussion discussion <SIM@communities.spe.org>

Discussion: Reservoir Simulation General Discussion

Author: Mr. Brian K Coats (Mr. Brian K Coats)

Date: 2006-12-02 20:41:58

Subject: Unstructured Grid Orientation Effects, part 1

Unstructured grid orientation effects

We've all heard many times about the problems with Cartesian grid orientation effects that unstructured gridding can eliminate. But somehow, it does not seem to be recognized that some commonly used types of unstructured grids introduce orientation effects that can be as bad or worse than the ones that they eliminate for near-well flow in Cartesian grids, and that make them inappropriate for use in full field modeling. This is one of the things that the unidirectional flow problem I posed is intended to illustrate.

I wonder if anyone has run that case using a hexagonal grid? It is the simplest possible test that the model either uses a k-orthogonal grid or implements a robust multipoint flux approximation, or both. Without going into detail, the need to conform to boundaries and permeability discontinuities adds great complexity to the already complex requirements. And 3D is very much more difficult than 2.5D. But I won't at all be surprised if the correct flow rate can be obtained. Random Ky would be the best test, followed by added internal conformation to a zero-throw 100% conductive angled fault. Solutions using more advanced unstructured grids would also be very interesting.

No results are needed to predict what the answer is to the additional question I asked regarding dispersion of an injected tracer, which would give an indication of the modeled fluid flow path in the regular or irregular hexagonal (or any shape other than rectangular) grid. The unstructured solution will exhibit massive numerical dispersion in the y direction, since flow will necessarily occur from the point of injection on the left towards the right across all interfaces not parallel to the x direction, i.e. flow will occur across diagonal hexagonal interfaces (those not oriented in the x or y directions), obviously not in the true flow direction, even if the grid is k-orthogonal! This is purely a gridding effect, caused by having staggered blocks in the direction of flow.

<continued in part 2>

A pair of pictures here would beat a thousand words. A unique tracer injected in each cell on the left boundary would show that the FD fluid flow paths (streamlines) consist of a set of straight horizontal lines from left to right, one starting at each injection cell. The unstructured flow paths would make a very messy picture. If you took a map of the grid and started from each cell on the left and scribbled up and down randomly and diagonally towards the right, 500 times or so for each cell (assuming 10 hexagons in the x direction), and added a single straight horizontal line from each injection cell on the left to the right, you would be pretty close! For near-well radial flow, use of hexagonal grids will similarly result in severe theta direction dispersion. I believe that this near-well unstructured dispersion is even greater than would be observed with 5-point Cartesian radial flow, because in the unstructured grid those staggered blocks are directly connected (even though the unstructured grid tracks fronts better, particularly for high mobility ratio displacements). Knowing this behavior, it would be easy to construct any number of other cases that clearly demonstrate that the incorrect answer is obtained when using unstructured grids, where FD models can easily obtain the correct solution through appropriate selection of the grid. For anisotropic near-well flow, structured radial grids can of course also be shown to have problems (since the principal axes of the permeability tensor are most certainly not generally in the r and theta directions!), unless perhaps a radial multipoint flux approximation is implemented. This might make a very good near-well model, but it seems to me that all this indicates that either corner-point or unstructured (rectilinear!) gridding based on approximate streamlines (near the wells, the these might look similar to a radial grid and would be k-orthogonal, and these can also be run in FD models) would theoretically be, for deterministic systems, the best generalized single-grid full field gridding solutions, and would also make very good detailed near-well models. It also suggests that hybrid or dynamic grids might be called for. But the real questions that we need to answer are (1) what level of complexity in gridding is required, considering the uncertainty in our descriptions, and (2) does detailed modeling of near-well effects have significant effect on full field recovery. These questions might be addressed with respect to current capabilities through the other examples I suggested regarding the need for refined, hybrid, or dynamic grids in full field simulation. If anyone has a simple case demonstrating any of these needs, please send it to me.

Occam and I disagree with the suggestion that complicated situations should be considered before capabilities and needs are demonstrated through simple tests.

Regards,
Brian
@CoatsEngineering.com

 

From: Reservoir Simulation General Discussion discussion <SIM@communities.spe.org>

Discussion: Reservoir Simulation General Discussion

Author: Mr. Brian K Coats (Mr. Brian K Coats)

Date: 2006-12-06 13:56:46

Subject: Re: Unstructured Grid Orientation Effects, part 2

Can anyone demonstrate a more advanced mesh-based FE solution to the unidirectional flow problem? Does avoidance of transverse dispersion require that the nodes be aligned in the direction of flow and/or transverse to it? The pictures of 3D triangulated FE meshes that I've seen don't appear to do that.

 

From: Reservoir Simulation General Discussion discussion <SIM@communities.spe.org>

Discussion: Reservoir Simulation General Discussion

Author: Mr. Brian K Coats (Mr. Brian K Coats)

Date: 2006-12-13 12:17:01

Subject: Re: Unstructured Grid Orientation Effects, part 2

Since no one has responded, I'll again answer my own question with the generalized prediction that avoidance of transverse numerical dispersion in any numerical solution to the unidirectional flow problem, using a block-centered or a point-centered grid, requires a 5-point Cartesian differencing scheme or its equivalent point-centered RECTANGULAR node and node connection pattern, since, even in the isotropic case, modeled flow will occur through any existing diagonal connections due to non-zero flow potential. If this is correct, global use of either unstructured block-centered grids (of similar block shapes, for example hexagons) or triangulated FE meshes is inappropriate for full field modeling.

Current FE/unstructured methods may seem more valuable to the user because of their apparent ability to more quickly and easily represent complex geometry, but they are less valuable if the grid is inappropriate for flow modeling. I believe that for the general anisotropic case, an optimal FE mesh would be essentially identical to an optimal corner-point grid with 5-point connectivity and arbitrarily missing cells (for which at least some FD models pay no penalty), except for the point-centered/block-centered difference, in regions where the flow is locally unidirectional. Other optimally gridded regions would have unstructured cell/mesh shapes, such as the use of a 5-sided (or 3+n sided) block or its point-centered equivalent to properly represent divergence or convergence of a pair of (or n) streamlines, or the use of a many-sided cell for well blocks (looking similar to a radial grid cell with many theta-increments). FE methods do seem to offer flexibility in their ability to use different types of elements for features like fractures and conductive faults, but these can be represented I think much more easily and efficiently, and for all practical purposes just as accurately, in grids or meshes using multiple porosity/permeability systems.

I should note that for single-well radial anisotropic flow, the global and more uniform numerical dispersion in FE and unstructured (and 9-point FD) methods over FD 5-point Cartesian and corner-point methods can be and has been shown to be beneficial in tracking diagonal movement of fronts (but at the expense of creating dispersion for flow in x and y directions), and has or at least can have little adverse effect on results for Darcy flow when there is no composition variation in injected or in-situ fluids. The latter is best understood by considering the homogeneous isotropic incompressible radial case, in which the exact radius to an ideal piston displacement front is a function only of the volume of fluid injected, regardless of the flow paths.

If I were promoting superior technology, I would certainly be offering a large number of cases and results that demonstrate it. The fact that no one has referenced or provided any demonstrable evidence of FE superiority regarding the issues raised, presented results for any standard benchmarks (which obviously would exist for any model), or has offered a solution to or answered any questions related to one of the simplest possible flow problems that should not take more than a few minutes to set up and run, is I think a good indication that the superiority does not currently exist. Demonstrable evidence of superiority is usually a prerequisite for significant industry adoption of any new (or 40 year old) technology.

Regards,
Brian
@CoatsEngineering.com

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From: Reservoir Simulation TIG [specommunities@spe.org]

Title: Re: Simulation model w/non-orthogonal cells due to low angle fault

Created at 10/7/2007 6:38 PM  by Mr Brian K Coats 

JP,

 Mr. Balnaves gave excellent advice in warning to be careful.

 Errors will depend on case, method, and possibly model (vendor) implementation choices.  If possible you should evaluate sensitivity of results to your gridding (and any numerical method) options.  If that requires too much time, or if determining which solution is most accurate is a problem, then construct a prototype that exhibits the same gridding problems, preferably one to which the correct solution is known or can be easily determined.  Almost a year ago I submitted to this forum (the original post date is now (incorrectly) listed as 12/2/06, "Unstructured Grid Orientation Effects, Part 2") a very simple problem that tests the ability of the gridding/numerical method to give the correct flow rate for unidirectional (x-direction) flow in a 2D areal grid with constant pressure boundary conditions, independent of Ky.  The correct rate will be obtained if the model properly applies a method resulting in k-orthogonality.  Consider conformation to a no-throw sloping 100% conductive fault, rather than your thrown fault.  Let a vertical no-throw fault stairstepped in the x-y plane in the 2D areal (uni.dat) Cartesian grid represent your sloping fault.  If we use any arbitrarily stairstepped representation of the fault, with no changes to Kx and Ky across the fault caused by the presence of the fault, (there are no data changes required to the FD Cartesian model and) obviously this FD model will give the (same 474.7 stb/d) correct result.  The difference between this solution and those using each of your options including corner point or unstructured grids, or finite element meshes, that exactly conform to the slope of the fault, gives each of their errors due to non-orthogonality.  Sensor data and output files uni.dat, uni.out are attached and are at http://www.coatsengineering.com/unidirectional.htm (the simple case description in English is here, use it so you don’t have to interpret the Sensor data file to construct an Eclipse deck).

Nobody responded to my year-ago request for demonstration of an unstructured grid or finite element mesh solution to this problem, even though it shouldn't take more than a few minutes to set up and run.  So I can only assume (and repeat my 10-month old conclusion) that these accurate and efficient solutions, using anything but a grid or mesh forming regular rectangular prisms, don’t exist.  Actually, it was back then that I first made the suggestion that we might look specifically at the issues involved here by adding internal conformance to a no-throw angled fault.  I also suggested that we look at the flow path of a tracer injected at a point on the left boundary.

Some have criticized and may criticize this example because it can be represented and solved exactly by FD Cartesian methods.  It is useful because it is simple, it is representative of a typical field-scale flow situation, and it can be solved exactly by at least one method, so that we can easily measure error resulting from use of other methods.  Maybe someone can present a simple and significant example where instead, the unstructured or FE method yields the exact representation and solution.

A side note to Andras Gilicz:

Since you are recommending both of these types of models (FEM to David Ogbe for hydraulic fracture modeling and unstructured to JP here), I assume that you have used them.  Have you performed this simple unidirectional flow test?  If so, what were the results, and if not, why not, i.e. what indications do you have that your recommendations provide the best or even good solutions to the problems described?

Regards,

Brian Coats

Coats Engineering

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From: Reservoir Simulation TIG [specommunities@spe.org]

Title: Re: Finite element modelling - a contra-challenge

Created at 7/22/2011 7:38 AM  by Mr Brian K Coats

Kes,

You’ve hit on one of the main reasons for our disagreements on the value of FD vs FE in reservoir simulation – we seem to disagree on what reality is, let alone how to simulate it!  I’m referring to your statements “...been able to reproduce … the generic stress-related directionalities observed (mainly through tracers) in the majority of real-field flooding schemes”, and “…a contra-challenge that requires the model to face reality”.

Following is a quote of Dr. Keith H. Coats from his July 5, 2005 post “Effect of pressure on fractured reservoir effective permeability” (you also discussed your opinions counter to these in that discussion):

Dr. Flavio da Silva (with Petrofina circa 1990) had over 30 years of field operational and engineering experience with many if not most of the world's naturally fractured reservoirs. A quote of his remarks appears in my paper "Engineering and Simulation", Fourth International Forum on Reservoir Simulation, Salzburg, Austria, 1992. Following is an abbreviation of his remarks: "All fractured reservoirs I have seen ... have exhibited no permeability loss due to pressure decline...(application of isochronal tests) for North Sea fractured reservoir wells indicated no effect of pressure on fracture permeability... Virtually all fractured reservoir wells over large ranges of pressure decline exhibit normal or conventional rate-decline behavior which occurs only when effective permeability is constant. ... the 10-30 year large pressure depletion .. behavior of at least eight North ! Se! a fractured reservoirs cannot be explained if one assumes any significant effect of lower pressure causing lower fracture permeability."

I’m no geologist or geophysicist, and maybe some will give their opinions, but mine is that if the effects you speak of don’t exist near wells in naturally fractured reservoirs, it’s not possible that they exist away from the wells where temperature and stress changes are much smaller (in fractured or unfractured reservoirs).  I don’t see any solid evidence that supports your self-described conjecture regarding permeability changes, and I see much against it.  One of the most common mistakes in simulation is to assume that movement of results in the direction of observations due to the addition of some dependency or option  necessarily indicates that the dependency exists or that the option is valid (another example: Eclipse TZONE).

But, just for the sake of argument, and even though I don’t believe it, let’s assume that permeability stress and temperature dependence is needed to reproduce observed flow.  The next question is “how much complexity is needed to model it?”  The simplest solution in a finite difference model (if coupling to a geomechanical model isn’t necessary) would be to use transmissibility modifiers (separate x,y,z direction values) that are any (possibly hysteretic) functions of stress and temperature we might envision or desire, similar to what is in some models.  I’ll bet that is much simpler than characterizing whatever empirical relationships you’re using in your finite element model, and it would probably work just as well if not better.  You should try it before assuming the need for additional complexity.

We also disagree in that I don’t think that any progress can be made in resolving disagreements without model comparisons (when I said this on 6/4/2007 in “Discussion of SPE papers”, you agreed with me!).  We need the simplest possible cases that are designed specifically to demonstrate the claimed advantages, either by demonstrating an improvement over an existing solution, or by giving a solution to a new completely defined problem.  These cases are most easily created by adding whatever complexity is necessary to known cases.  All papers claiming improvements should include such evidence, and willingness to do so should be a condition for acceptance by all publishers.  We are having these discussions now only because of these deficiencies.  As I’ve said many times now, the fact that those claiming improvements and! t! he need for additional complexities are not willing to present proper evidence continues to indicate that those improvements and needs don’t exist.

Regards,

Brian

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