Patent Application: US-201113229319-A

Abstract:
a method is provided for simulating a physical process such as fluid flow in porous media by performing a fine - grid calculation of the process in a medium and re - using the fine grid solution in subsequent coarse - grid calculations . for fluid flow in subsurface formations , the method may be applied to optimize upscaled calculation grids formed from geologic models . the method decreases the cost of optimizing a grid to simulate a physical process that is mathematically described by the diffusion equation .

Description:
the invention will be described in connection with its preferred embodiments . however , to the extent that the following detailed description is specific to a particular embodiment or a particular use of the invention , this is intended to be illustrative only , and is not to be construed as limiting the scope of the invention . on the contrary , it is intended to cover all alternatives , modifications and equivalents that may be included within the spirit and scope of the invention , as defined by the appended claims . the invention disclosed herein is a new scaleup process to allow the reuse of fine - scale ( global ) solutions for multiple coarse - scale grids . referring to fig1 , the process includes the following steps : step 1 . calculate a set of fine - scale solutions to the appropriate equations describing the physics of a diffusive process in a region of interest . linear pressure boundary conditions may be used for generating the flow solutions . other boundary conditions can be used , as discussed by wu et al . ( 2002 ). step 2 . construct a coarse grid suited to each region of interest . step 3 . from this coarse grid , form scaleup volumes to calculate particular physical properties of interest on the coarse grid . step 4 . construct a mapping between the fine - scale model and the scaleup volumes . step 5 . for each scaleup volume , retrieve the fine - scale solutions for each fine - scale cell that is associated with the scaleup volume through the mapping . step 6 . using these fine - scale solutions , calculate the upscaled property for each scaleup volume . step 7 . repeat steps 2 to 6 for a new coarse grid . to perform a scaleup on a new coarse grid , the fine scale solution is not recalculated . rather , it is retrieved from a storage device , such as a computer memory or disc . one difference between the approach disclosed herein and previous approaches is the ability to reuse the fine - scale ( global ) solutions on different coarse grids . previous approaches calculate a fine - scale solution for each new coarse grid . in one exemplary embodiment of the present invention , the upscaling of permeability in a model of darcy flow through a porous media is described . it should be understood that though the method is applied to permeability and fluid flow , it is applicable to other physical processes described by the diffusion equation , which is : where a , { right arrow over ( b )}, c , and d are known functions of space and time . the physical meaning of the coefficients depend on the context in which the equation is used . for darcy flows , a is related to rock compressibility and porosity and d is the permeability tensor . the variable u in eq . 1 is the unknown to be solved from the equation ; it corresponds to pressure , saturation , or concentration in porous media flows . fig2 demonstrates step 1 — the calculation of the fine - scale solutions for a model of fluid flow through porous media in three directions . fine grid or geo - cellular model 20 includes rock properties of porosity and permeability for each cell . results of flow calculations for three directions are illustrated at 21 , 22 and 23 . for the three dimensional (“ 3 - d ”) model of darcy flow , three solutions are necessary to calculate the upscaled permeability . these flow solutions can be calculated through any method desired , although often numerical methods such as finite difference or finite element are used , both of which are well known in the art of reservoir simulation . in this case , the region of interest is the entire model , but it may be desirable to divide the model into several regions to make the computations feasible . the solutions for the parts of the volume are combined through the use of appropriate boundary conditions to form a global solution covering the entire volume , i . e . the model or region of interest . the fine - scale solutions are then stored for later use . the method disclosed herein will work in its most efficient mode if the fine grid is structured and orthogonal . structured grids allow simpler and more efficient ways for manipulating the information compared to unstructured grids . in particular , the mapping algorithm used in step 4 can be simplified and made more efficient . if the fine grid is both structured and orthogonal , one can take advantage of simpler and more efficient methods for obtaining the solution to the physical problem . the importance of these considerations rapidly increases with the number of cells ( i . e . the resolution ) of the fine grid , especially in 3 - d applications . as stated previously , if the size of fine grid representation of the physical process is so big that it becomes impractical to compute the solution on the entire model , then the model may be split into several regions and solutions may be obtained on each region separately . preferably , regions overlap and the size of the regions will be chosen much bigger than the size of a coarse grid cell . such choice will help reduce the effect of the boundary conditions on the local ( regional ) solutions and also will enable the regions to encompass features of larger scale ( wu et al . 2002 ). the coarse grid may also be an unstructured grid . fig3 a - 3b illustrate an embodiment of step 2 — construction of a coarse structured grid for the fine - scale geologic model shown in fig2 . in fig3 a , coarse grid 30 is made up of coarse grid cells 31 , which could simply be defined as the union of a specific set of structured and orthogonal fine grid cells . fig3 b shows a structured , rectangular fine grid 35 and two coarse grid cells 37 that are aligned with the fine grid 35 . for this simple case , there exists an efficient discretization , namely the two - point flux finite volume approximation . also , because of the grid structure , a solver will be more efficient . since the coarse grid cells are aligned with the fine grid , the mapping between coarse and fine grids is trivial and will not produce sampling errors . in step 3 , scaleup volumes are calculated for the coarse grid . the scale - up volumes are a particular volume of interest for the problem being solved . for reservoir simulations , these volumes are typically associated with coarse grid cells or connections . the methods of this invention work equally well for either structured or unstructured grids . fig4 a - 4c illustrate exemplary embodiments of step 3 — forming of scaleup volumes in a coarse unstructured grid . fig4 a shows traditional cell - based scaleup volume 40 and fine grid cells 41 within it . coarse volume 40 within which the upscaled property is calculated is an approximation of the coarse grid cells . for a coarse unstructured grid with unstructured or vornoi areal grid but a layered structure in the vertical dimension , if the finite difference method is used to obtain the flow solutions on the coarse grid , then the scale - up volumes 44 in fig4 b and 45 in fig4 c are preferred . however , for the finite difference method on general unstructured grids , scale - up volumes based on cells or the unions of two neighboring cells can be used . the scaleup volume allows the direct calculation of the transmissibility , a key parameter in the finite difference method . the approach disclosed in u . s . pat . no . 6 , 826 , 520 may be used to calculate transmissibility . persons skilled in the art will know other approaches . for other numerical discretization schemes , different scale - up volumes may be required . fig5 a - 5b illustrate an exemplary embodiment of step 4 — a mapping to determine which fine grid cells are associated with each scaleup volume . in fig5 a coarse scaleup volume 51 is shown superimposed on fine - scale grid 50 . in fig5 b , a preferred method is depicted for determining if fine grid cell 52 , for example , is associated with ( i . e ., will be considered to be included within ) coarse scaleup volume 51 . in this method , fine grid cell 52 is associated with scaleup volume 51 if its cell center 53 lies within the coarse scaleup volume 51 . this method or criterion for partial inclusion is discussed in durlofsky ( 2005 ) and in u . s . pat . no . 6 , 826 , 520 . other methods may be used , as is known in the art . the mapping between the fine and coarse grid can be constructed in many different ways . for example , one could use geometric algorithms that are well known in the art of computational geometry and grid generation . u . s . pat . no . 6 , 106 , 561 teaches a suitable method for creating a grid . other methods of gridding may be used , as is well known in the art . there are many references on the subject , such as the handbook of grid generation ( j . f . thompson et al ., crc press , 1999 ). as an example of step 6 of fig1 , the case of permeability of a porous medium , which is so important in the simulation of petroleum reservoirs to facilitate production of hydrocarbons from them , may be considered . in this case , both velocity and pressure gradient are components of the fine scale ( i . e ., global ) solution for darcy flow in porous media . therefore , both pressure gradient and velocity are retrieved from data storage ( step 5 ) for each of the three solutions calculated in step 1 . for the permeability property of the darcy flow equations , it has been shown by wen and gomez - hernandez (“ upscaling hydraulic conductivity in heterogeneous media ,” j . hydrology 183 , 9 - 32 ( 1996 )) that the coarse grid permeability property can be represented by : { right arrow over ( v )} =− k ∇ p , [ eq . 2 ] where { right arrow over ( v )} is the volume - weighted average of the fine - scale velocity in the scaleup volume , ∇ p is the volume weighted average of the fine - scale pressure gradient , and k * is the coarse scale permeability . these averages are calculated for each flow solution . it should be noted that the velocity and pressure gradient are vectors and the permeability is represented as a tensor . this is why three different solutions are preferred ; three solutions and three equations per solution ( one for each component of the vector ) allow the calculation of the nine components of the coarse - scale permeability tensor . there are several methods for computing the coarse - grid effective property once the fine grid solution is available . these methods are discussed by durlofsky ( 2005 ). a preferred method is to use the volume - average approach and eq . 2 . if a new coarse grid is desired to improve performance , the fine - scale solution is not re - calculated in the present inventive method . as shown in fig1 , the scaleup volumes and their mapping to the fine grid must be reconstructed , and the fine scale solution is simply re - sampled on the new scaleup volumes defined based on the new coarse grid . results of calculations with the new coarse grid can then be compared with results of calculations with the first coarse grid . results of linear or single - phase flow calculations from the different coarse grids may be compared with the global flow solutions based on a geo - cellular model to select the preferred coarse grid . this process can be repeated until the most preferred coarse grid is found . the preferred coarse grid from these comparisons may then be used in a mathematical model based on non - linear equations , as in the case of multi - phase fluid flow in porous media . a fine grid calculation was performed using a geologic model having 14 million cells , of which 580 , 000 were active cells . a global solution for velocity and pressure was obtained for single phase darcy flow within the model . using one embodiment of the present inventive method , an initial scaleup to a coarse grid required 60 minutes computing time and , by retrieving and re - using results of the fine scale solution , only 7 minutes were required to scale up to a re - gridded model . in contrast , a typical method previously used required 125 minutes to scale up both the initial model and the re - gridded model . both coarse grids had 40 , 500 active cells . a fine grid calculation was performed using a geologic model having 7 . 5 million cells , almost all of which were active . a global solution for velocity and pressure was obtained for single phase darcy flow within the model . using the present inventive method , an initial scaleup to a coarse grid required 390 minutes computing time and , by retrieving and re - using results of the fine scale solution , only 20 minutes were required to scale up to a re - gridded model . in contrast , methods previously used required 150 minutes to scale up both the initial model and the re - gridded model . both coarse grids had 87 , 000 active cells . the model size of example 1 is more commonly encountered in current practice . for either size model , optimizing the methods disclosed herein will further improve the advantage in reduced time and cost over presently used methods . using the disclosed methods , it is clear that the greatly reduced time required for re - gridded solutions makes practical a series of manually re - gridded solutions or the application of automatically re - gridded solutions . although the invention has been described in terms of scaling up simulation grids , it should be understood that the methods described herein apply equally well to sets of sample volumes that do not form grids , i . e ., these volumes do not form a non - overlapping partition of the subsurface region . the sample volumes may be selected randomly or according to a regular pattern . the invention allows faster and lower cost determination of statistics from different sets of sample volumes . it should also be noted that the present inventive method does not require that the sample volume be larger than the fine - scale grid cells . the invention works equally for coarse grid cells ( scaleup volumes ) that are smaller than the fine - scale grid cells . although the invention has been described in terms of fluid flow in porous media , it should be understood that simulation of other physical phenomena described by the diffusion equation may also be practiced using the methods described herein . for example , thermal diffusion in solids and molecular diffusion in liquids may be simulated using the inventive method . in those cases , a physical property analogous to permeability may be upscaled from a fine grid calculation to a coarse grid calculation using the steps set out above . the foregoing application is directed to particular embodiments of the present invention for the purpose of illustrating it . it will be apparent , however , to one skilled in the art , that many modifications and variations to the embodiments described herein are possible . all such modifications and variations are intended to be within the scope of the present invention , as defined in the appended claims . 1 . d . stern , “ practical aspects of scaleup of simulation models ,” jpt ( september , 2005 ) 74 . 2 . l . j . durlofsky , “ upscaling and gridding of fine scale geological models for flow simulation ,” proceedings of the 8 th international forum on reservoir simulation ( june 20 - 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