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PROCEEDINGS, Twenty-Third Workshop eothermalReservoir EngineeringStanfordUniversity, Stanford, California, January 26-28, 1998SGP-TR-158

A GEOMETRIC MODELING FRAMEWORK FOR THE NUMERICALANALYSIS OF GEOTHERMAL RESERVOIRS

Brian HardemanDaniel Swenson

Kansas State University302 Rathbone Hall

Manhattan, KS 66506http://www.mne.ksu.edu/ -geocrack

ABSTRACT

In the past decade, advancements in automatic mesh

generation and topological data structures have made possible the use of a more general and abstractgeometric model for the description of an analysis

 problem. Using a geometric model for describing areservoir results in a conceptual model of thegeothermal system, rather than a simple numerical

simulation. The analyst builds the geometric modelusing true features of a reservoir, such as well boresand known fracture locations. Material propertiesand boundary conditions are then assigned to thesefeatures, not to the underlying finite element mesh orfinite difference grid. This independent storage ofthe problem description makes it simple to runmultiple analyses, potentially using different solution

schemes. Definition of both 2-d and 3-d models can be greatly simplified through this more interactive,intuitive model creation process. State-of -the-artvisualization and manipulation methods assist inconveying the conceptual model of the reservoir andthe assumptions made for performing a simulation.

MOTIVATION

Current reservoir analysis software (GEOCRACK,TOUGH2 TETRAD) requires the user to have in-depth knowledge of the numerical solution schemeused. The problem description, in the form of a finiteelement mesh or finite difference grid, serves as thefundamental carrier of information into thesimulation. This mesh must satisfy not only therestraints dictated by the problem at hand, such asgeometric boundaries and features (reservoir extent,fractures, wellbores), but also must comply withnumeric restrictions of the solution method, such asmaximum or minimum element sizes, grid density,and any special input parameters. This approachrequires the analyst to think ahead when creating themesh, because if any of the desired characteristics are

not met, the mesh or grid must be regenerated,incurring a substantial cost in time and effort.

Relying on the input grid or mesh for problem

information can also cause difficulty when attributes(material properties, boundary conditions, analysisassumptions) or geometry change during the courseof an analysis. Placing all of the problem descriptionin the mesh requires a change to the mesh definition.For example, to change attribute data, one must onlychange a numeric value stored with the meshelements, but this change must occur for everyelement in the region of the change. Whileinteractive tools for querying and manipulating thesevalues can help, the task is still tedious, particularlyfor meshes with a great level of detail and a finediscretization. Even if the interface allows the user to

specify the new value and lets the software determinewhich elements require updating, the program muststill perform repeated searches to find the affectedelements.

A change in geometry is even more serious and userintensive. Because of the close link between thelocation and size of elements and the physical

 boundaries of the problem, a change in geometryrequires regeneration of the mesh. Such changescause problems for finite element meshes, whichmust conform to boundaries, and can also affect finitedifference grids, where the grid density may beclosely related to high solution gradients caused by

 boundary conditions or boundaries of differentmaterials. For many analysis systems based on inputfiles, this would involve recreating the entire grid andreassigning all attributes. Even in systems whichallow interactive modification of the mesh, thesechanges can be very labor intensive, particularly forlarge or detailed grids.

While modifying a two-dimensional solution grid can be tedious, it is relatively straightforward. However,

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direct manipulation of a three-dimensional grid ormesh becomes nearly impossible. Spatialvisualization is difficult, even when using interactiveand graphical modification tools. In addition, thenumber of elements or cells increases greatly in threedimensions, further complicating modification tasks.In order to avoid such low-level manipulations, the

user needs a more intuitive and automated way ofincorporating geometric changes.

The above arguments also apply to situations wherethe solution mesh needs modification independentlyof the defining geometry. Such situations occur as aresult of the numerical techniques used in thesolution. For instance, as the analysis of a particularreservoir evolves through time, areas of hightemperature or  pressure gradients may move todifferent parts of the reservoir, requiring smallerelements in a new area of the mesh. Again, it can betedious for the analyst to manually modify the mesh

to introduce new areas of refinement and assign proper attribute information.

Adaptive meshing, where automatic mesh generationis used in conjunction with an estimation of the errorin the calculated solution, is a powerful approach toaddressing appropriate mesh refinement (Lewis1991). But if the mesh provides the only source of

 problem information, data such as geometry andmaterial properties, must be inferred from the currentmesh before a new one may be generated andassigned the proper information. Storing thegeometry and attributes independent of the meshgreatly simplifies the implementation of an adaptivemeshing scheme.

Another reason for the use of geometric modelingarises when the analyst desires to change thenumerical method of the solution, for instance from afinite difference to a finite element or control-volumefinite element solution. This requires the creation ofanother mesh with all of the proper data attached.  Ahigher -level description of the problem, independentof the actual solution method, could allow the analystto describe the problem’s geometry and input parameters only once and provide this data in anautomated way to the solution mesh or grid actually

used for the solution. The user would only have to provide additional parameters specific to the chosenmethod, as opposed to redefining the entire problemto use the new method.

GEOMETRIC M ODELlNG

Ageometric

model is ahigh-level geometry-based

representation of an object. This representationprovides a natural and convenient location to store allo the necessary information associated with a

numerical analysis problem, such as material properties and boundary conditions. Combining thegeometric model with the parameters of the analysiscreates a problem description unique to the problemat hand, but independent of any given analysismethod.

The geometric model serves as a means ofcommunication between an analyst and the analysissoftware. This model is independent of the solutionmethod, yet contains all of the necessary parameters,making it a natural way to convey information abouta given problem to the simulation software. Thegeometric model contains information about thenature of the problem and can store the resultsobtained from simulations of the problem. In thisway, the analyst need only interact with the abstractdescription of a reservoir, not with the underlyingsolution mesh or grid. This simplifies andstreamlines the job of the analyst and provides an

organizational tool for defining, performing, andinterpreting reservoir simulations. The chore ofdeveloping input files and translation file formats can

 be greatly simplified and automated by the modelingsoftware because all of the problem information iscentralized in the geometric model.

Several different ways exist to implement ageometric model. Each method has different levelsof representational capability and introduces a varietyof benefits and drawbacks. The three primary typesof geometric models are decomposition models,constructive solid geometry, and boundaryrepresentations (Mantayla 1988).

Topology

Boundary representation, a form of geometricmodeling where the explicit storage of boundaryinformation describes an object, provides theunderlying support for the geometric modelingframework discussed herein. The boundary of amodel consists of adjacency information between point sets known as  regions, faces, edges, andvertices. This adjacency information is commonly

referred to as topology.

A  region defines a separate portion of three-dimensional space that may be either a bounded(finite) or unbounded (infinite) subset of R3 A faceis a bounded, two-dimensional subset of R3 thatcorresponds to a surface, and an edge is a

one-

dimensional subset corresponding to a curve. Avertex is a zero-dimensional entity that represents aunique point in space.

The boundary of a region is comprised of a set ofelements of dimension less than three. A typical

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region' boundary would be simply a set of faces.Similarly, faces are bounded by a collection of edges.Every edge is then bound at either end by a vertex.Together, all of the regions, faces, edges, and verticesof the model make up the entire model space of R3This hierarchical representation of an object isdepicted in Figure 1. 

Region

I

1Face

Edge

Vertex

3

0 0

0 0

0 0

0 0

 Figure 1 Topological elements in boundary

 representation modeling.

Storage of the boundaries of all entities in the modelexplicitly maintains the adjacency information of themodel. Because the adjacencies are stored directly, a

 boundary representation supports very efficient queryand traversal methods. This simplifies many tasks,such as display and local modifications of the model.

Because the adjacency information simplifies localmodification of a boundary representation,construction and modification operators are designedaround this property. These operators, known as theEuler operators, were irst introduced by Baumgart(1974). The Euler operators guarantee that

modifications keep the model in a consistenttopological state. While simple and robust, the Euleroperators are very low-level tools and are not suitableas the sole means of creating a large and complex

model. However, the use of Boolean set operationsand feature- based modeling in conjunction with theEuler operators provides a convenient and powerhlgeometric modeling interface.

 Non-Manifold Topology

Topology may be divided into two major categories -manifold and non-manifold. Manifold refers to the

 boundaries of the point set elements. A manifold boundary is one that is of dimension one less than theentity that it bounds at every location on the

 boundary. For instance, a traditional solid model hasa manifold boundary if, at every spot on the boundary, the boundary is two-dimensional. This is

equivalent to stating that every point on the boundaryis topologically identical to an open disk in R2Because the boundary must be two-dimensionaleverywhere, each edge must have two and only twofaces incident upon it. These properties have beenused to optimize data structures for manifold models,such as the winged -edge data structure (Baumgart,1974). Every physical object has a manifold

 boundary, so a manifold assumption does not limitthe real objects that a  boundary representation canmodel.

Manifold boundaries simplify some data structure

and manipulation tasks, while not limiting the classof objects that can be represented, yet are still toorestrictive for a truly general geometric modelingsystem. For example, it is very natural to model atruss structure as a set of wireframe edges in space, ora car fender as an open surface (face). Under

manifold boundary assumptions, these situationswould have to be modeled exactly as they are in reallife. The truss elements would have to be modeledexplicitly, each with their own cylindrical face boundary, and the fender would have to be modeledas a complete solid with faces for both the front and

 back and for the thin area around the edges.  A non-manifold boundary allows the bounding elements to

 be a mixture of elements of dimension one less that

the bounded region. This extends therepresentational capability of the boundaryrepresentation from real objects to also handle usefulabstractions that arise in numerical analysis andsimulation.

The geometric modeling framework discussed hereutilizes a fully non-manifold topological database.The data structure, called the Multi-Link data

structure, is a combination of ideas and techniques

from previous work on non-manifold boundaryrepresentation (Weiler 1986, Choi 1989, Rossignac1990).

Feature-Based Modeling

A geometric model is based upon a mathematical,geometric description of an object. This high degreeof abstraction makes developing an interface to themodel based upon the features of the problem domainnatural. In models with a low degree of abstraction,

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such as finite element meshes, the mesh must becreated with the geometry in mind. Nodes andelements must be placed so that the final meshconforms to the material boundaries and allows boundary conditions to be appropriately imposed.For example, to represent a wellbore as a one-dimensional feature, nodes would have to be placed

along the well and elements formed so that edges lieexactly along the well.

Using a geometric model makes a higher level ofabstraction possible. A wellbore may be represented

 by simply inserting an edge at the location of the wellin the model. Intersections with other edges andfaces create vertices, and all information can be propagated to the solution mesh or grid.

To increase abstraction even further, a well may beinserted simply by specifying the geometricdescription of the wellbore and allowing the

geometric model to take care of the details ofrepresentation. Similarly, the user would add afracture or fault to a reservoir model by locating itwithin the model and supplying its strike/dipspecification. A geometric description of a planarsurface patch would be constructed from the input,and a face would be intersected and added to thegeometric model using that geometry. Just as theabstraction of a geometric model insulates a userfrom solution method details, using features todescribe the problem insulates the analyst from theimplementation details of the geometric model.

EXAMPLEA simple example demonstrates the expressive powerand intuitive interface provided by a geometricmodeling system.  A  basic representation of ageothermal system requires only a few operations.The hot dry rock reservoir in Hijiori Japan providesa useful example.

The first step is to provide the dimensions of theoverall boundaries of the reservoir. At Hijiori, theactive region of circulation covers approximately500m by lOOOm in plan view and extends IOOOm indepth. Figure 2 shows the bounding volume of the

reservoir as represented in the geometric modelingframework.

 Next, the wells are added to the model. At Hijiorithree wells of interest extend into the fracturedregion, HDR-1 HDR-2a and HDR -3. For a simpleapproximation, the wells are represented as straightlines. The wells may be input as a sequence of linesegments or as a spline curve if desired for a moreaccurate representation.  A wireframe view of themodel is shown in Figure 3. 

 Figure 2  Boundary of a reservoir.

Fractures are added to the model by inserting planar polygons or infinite planes that are truncated by thereservoir boundary. Two zones of high permeabilityare observed at the Hijiori reservoir. A simple modelof this system that can be used for analysis of basic behavior of the reservoir contains two infinitefractures. Figure 4  shows the basic reservoir systemwith two flowing fractures. The fractures are locatedand oriented by the input of a point on the plane andthe known strike and dip information.

 Figure 3 Geometric model including boundary andwells.

To assist in visualization and conceptualization of thereservoir geometry, the software supports real-timemanipulation and rotation of the model. Thereservoir can be viewed as either a wireframe or solid

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model, and cutting planes can be used to strip away portions and look inside, as shown in Figure 5.  

I

Figure 4 wof iac ture model of  Hijiori.

I

Figure 5 Use of  a cutting plane to view the interiorof a reservoir.

After creation of the geometry, attribute informationrepresenting rock material properties, fracture

opening and permeability, and operating conditionswill be assigned to the elements of the model.Ultimately, this information will be used to generatean appropriate solution grid or mesh. All data is thenavailable to perform an analysis of the reservoir orcreate an input file for an external analysis package.

CONCLUSIONS

Geometric modeling provides a flexible and efficientanalysis interface to serve as a high-level description

of a geothermal reservoir. Mesh- based approachesare simply too detailed and dependent on theassociated numerical method to be generally useful.The specification of a geometric model andassociated problem data are logical first steps in ananalysis and can serve as the medium of

communication between an analyst and the analysissoftware.

The use of geometric modeling, combined withadvanced visualization and manipulation tools, cangreatly simplify the task of preparing complex

 problems for analysis. These advances have the potential to allow reservoir engineers performcalculations previously limited to scientists andcomputer programmers.

ACKNOWLEDGEMENTS

We thank Mark James for long and insightfuldiscussions on non-manifold geometric modeling.This work was supported by the Office ofGeothermal Technologies U.S. Department ofEnergy, and perfornied in the Applied ComputationLaboratory at Kansas State University.

REFERENCES

Baumgart, B. “Geometric Modeling for ComputerVision.” Ph.D. Thesis, Stanford University,1974.

Choi, Young. “Vertex-Based Representation of Non-

Manifold Geometric Models.” Ph.D. Thesis,Carnegie-Mellon University, August 1989.

Lewis, R. W., et. al. “Finite Element Analysis ofHeat Transfer and Flow Problems UsingAdaptive Remeshing Including Application toSolidification Problems.”  International Journalfor  Numerical Methods in Engineering, Vol. 32,

Mantyla, Martti.  An Introduction to Solid Modeling.

Rossignac, Jaroslaw R. and Michael A. O’Connor.

pp. 767-781, 1991.

Computer Science Press, 1988.

“SGC: A dimension-independent model for pointsets with internal structures and incomplete boundaries.” Geometric Modeling fo r Product Engineering. Eds. Wozny, M. J., J. U. Turner,

and K. Preiss. Amsterdam: North-Holland,

Swenson, Daniel, et. al. “GEOCRACK: A CoupledFluid Flow Heat Transfer Rock DeformationProgram for Analysis of Fluid Flow in JointedRock.” Mechanical Engineering Department,Kansas State University, Manhattan, KS.

Weiler, Kevin. “Topological Structures forGeometric Modeling.” Ph.D. Thesis, Rensselaer

Polytechnic Institute, August 1986.

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