Abstract:
A method for modelling the migration of reactant in a subsurface petroleum system is described. The method comprises in part generating a mesh for an area of the petroleum system. The mesh comprises a plurality of nodes, with each node representing a point in space in the area. The method also comprises calculating one or more variables representing one or more physical characteristics at each node in the area and determining the migration of reactant in the petroleum system based on the one or more variables. The method can also handle multiple reactant phases and non-static meshes.

Description:
FIELD OF THE INVENTION 
     The present invention relates generally to the fields of basin and reservoir simulation and the modelling of subsurface oil and gas flow. 
     BACKGROUND 
     Basin modelling is important for oil and gas exploration and production. One of the goals of basin modelling, and often associated reservoir filling studies, is to calculate the migration of non-aqueous reactants (most commonly petroleum, which may represent one or more chemical species) in subsurface deposits of complex geometry. To accomplish this goal, the system under study will usually be described by a grid system, with each cell of the grid being assigned petrophysical properties that describe how they will affect fluid flow. Where a three-dimensional model of a basin or reservoir, consisting of porous or fractured media, (hereafter collectively referred to as a “petroleum system”) is being developed, a mesh of elements is imposed over the space under study. The mesh is usually (but not exclusively) made up of an orthogonal grid which may be cubic in shape. In an orthogonal grid, subsurface structures are represented by points in space in the x-y plane with a common value in the z plane. Each of these points in space is known as a node. If values are stored on the nodes, then values in between the nodes are commonly calculated by interpolation (although they may be held constant throughout the element, depending on the property). There can be millions of grid cells and associated nodes in a subsurface model. The usual technique of prior art systems has been to impose, and to populate the cells of this grid based on the element(s) that contribute most to the volume of each cell. 
     While this conventional approach captures many coarse features of the mesh, the grid can result in a model which is crude when compared the real world, incorporating considerable inherent elements of uncertainty. Such crudeness often distorts or destroys details that are important in determining the correct migration of reactants within the mesh. Thin features with high impedance to migration may be averaged out or omitted entirely in the conversion to the grid. 
     As a result, efforts have been made to provide grids which more closely approximate the features of a subsurface field under review. Several of such known methods used to simulate multi-phase petroleum flow are commonly described as Darcy flow, Flowpath and Invasion Percolation. All of these methods have been successfully applied and their strengths have been well documented, but all are also characterized by weaknesses which have limited their application and the quality of the results. For example, by imposing a regular grid and converting all internal inclined planes into horizontal planes in a staircase configuration, internal watersheds, natural directions of flow, are poorly approximated by such staircase structures, possibly resulting in migration pathways that are inconsistent with the underlying geometry. 
     SUMMARY OF THE INVENTION 
     In accordance with one embodiment, there is provided a method for modelling the migration of reactant in a subsurface petroleum system, the method comprising: generating a mesh for an area of said petroleum system, the mesh comprising a plurality of nodes, and each node representing a point in space in the area; calculating one or more variables representing one or more physical characteristics at each node in the area; determining the migration of reactant in the petroleum system based on the one or more variables. 
     In some embodiments, each node is not restricted to a common depth in the z domain. 
     In some embodiments, the mesh is comprised of a plurality of elements and each element is in one of a filled, a partially filled, or an unfilled state. 
     In some embodiments, the physical characteristics comprise one or more of fluid pressure, water fraction, oil fraction, and gas fraction. 
     In some embodiments, the one or more physical characteristics relate to a single-phase reactant. 
     In some embodiments, the one or more physical characteristics relate to a multi-phase reactant. 
     In some embodiments, the step of calculating one or more variables representing one or more physical characteristics at each node includes: calculating densities and interfacial tension (“IFT”) for a node; computing back pressure for an element containing the node; computing capillary threshold pressure difference for the node; computing maximum depth of reactant below the node; and computing the minimum absolute depth of the node. 
     In another embodiment, there is provided a system comprising: a processor for generating a mesh for an area of a petroleum system, the mesh comprising a plurality of nodes, and each node representing a point in space in the area; a processor for calculating one or more variables representing one or more physical characteristics at each node in the area; and a processor for determining the migration of reactant in the petroleum system based on the one or more variables. 
     Other aspects and features of the present invention will become apparent, to those ordinarily skilled in the art, upon review of the following description of the specific embodiments of the invention. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The invention will now be described in greater detail with reference to the accompanying diagrams, in which: 
         FIG. 1  is a perspective view of a three-dimensional orthogonal grid used to model petroleum systems in accordance with a conventional modelling technique; 
         FIG. 2  is a perspective view of a mesh which can be used with the present invention; 
         FIG. 3  is a flow chart of a method for calculating when a node in an element is under appropriate conditions to transfer reactant to a neighbouring element; 
         FIG. 4  is a flow chart of a method for calculating when a node in an element is under appropriate conditions to will transfer reactant to a neighbouring element where there are multiple reactant phases; 
         FIGS. 5A and 5B  are illustrations of modeled fluid bodies whose elements are continuous; 
         FIG. 5C  is an illustration of a modeled fluid body whose elements no longer form a continuous set; 
         FIG. 6  is a flow chart of a method to transition a body to a new mesh; and 
         FIG. 7  is a schematic diagram of a conventional personal computer which can be used to run software programs embodying the methods of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       FIG. 1  is a perspective view of a three-dimensional orthogonal grid  10  used to model subsurface structures in accordance with a conventional modelling technique. The use of a three-dimensional grid such as grid  10  is well known in the art of migration modelling. Grid  10  is made up of cells C 1 , C 2  and C 3 , though a typical grid used in prior art systems would be made up of hundreds if not millions of such cells. Each one of cells C 1 , C 2  and C 3  is made up of one or more cell faces, some of which are hidden from view. 
     To accurately predict the flow of reactants, a model is built by imposing a grid such as grid  10  on to the petroleum system to facilitate the solution of flow equations by calculating the likelihood of reactant flow between the cells. The goal is to assemble a grid which represents a three-dimensional model of the petroleum system. 
     Variables are calculated for each cell which can include one or more of length, width, thickness, porosity, capillary threshold pressures, absolute permeabilities, elevation, temperatures, pressure(s), and others. In multiphase flow (i.e. oil, gas and water that may be in liquid or vapour states), a pressure for each phase is usually calculated. A petroleum systems model is usually built up by assigning such variables to a plurality of grids  10  over the subsurface structure of interest. This allows the model to characterize reactant flow through the reservoir. 
     However, simulation gridding technology using a grid such as that shown in  FIG. 1  is lacking in its ability to accurately represent a real-world petroleum system. 
     In accordance with the modeling system and method described herein, reactant invasion (i.e. reactant from one element in a mesh to another) is modelled directly on any corner-point mesh that represents the subsurface area under study without resort to an externally-imposed grid of cubic cells such as that shown in  FIG. 1 . As a result, the controlling features of reactant migration are no longer cells or cell faces, but nodes (i.e. corners) of elements in the original mesh. Importantly, this technique allows for the partial filling of the elements, which decreases gridding effects on saturations and volumetrics. These corner-point meshes may be further submeshed for additional accuracy. 
       FIG. 2  is a perspective view of mesh  20 . Elements of mesh  20  are defined in terms of nodes N 1 , N 2 , . . . N 8  that make up the corners of a polyhedral volume. For the sake of clarity, only one element is shown in mesh  20 , though in practise a mesh would be made up of many elements. As compared to grid  10 ,  FIG. 1 , mesh  20  is not necessarily cubic in shape and therefore can more accurately be used to outline the features of a subsurface structure under review. As well, the controlling features of reactant migration are no longer calculated on the basis of cells or cell faces but instead nodes. Each node represents a point in three-dimensional space in the petroleum system under review. Nodes need not have common depth values (“′z”), which are defined as the direction in which the buoyant force is exerted on reactants in the elements. 
     Some elements in a mesh, are defined as “source elements”, act as sources of reactant. These elements accommodate as much of the reactant as they can hold and transfer the rest of it to neighbouring elements, which may in turn transfer the reactant onwards. A fluid body, or accumulation, begins as a single element containing at least some reactant. Sometimes, invading another element will add that element into the body. That is, under certain conditions, invasion of a target element will cause the element to be subsumed into the body which will now contain two or more elements. Thus, a body is a set of one or more elements treated together. 
     The set of nodes grows as the surface area of this set of elements increases, but the backfilling depth is always constrained by the one node which supports the shallowest absolute depth of filling. 
       FIG. 3  is a flow chart of a method for calculating when a node in an element is under appropriate conditions to transfer reactant to a neighbouring element. 
     At step  300 , reactant is added to a mesh element (an emergent fluid body). At given temperature and pressure values, the reactant present in a mesh element has specific values of density and an interfacial tension (“IFT”) exists between it and other reactants (typically water). At step  310 , densities and IFTs are computed for all nodes in a mesh element. 
     When considering whether reactant is to invade from a source element to a target element, two pressures are computed. The first is the back-pressure from the target element due to fluids (water and reactant) in that element. This back-pressure is equal to the excess hydrostatic pressure in the target plus the reactant pressure in the target element. This is calculated at step  320  for the node under review. The reactant pressure is defined as 
                     P   node     =       ∫   meniscus   node     ⁢       G   ⁡     (       ρ   water     -     ρ   reactant       )       ⁢           ⁢     ⅆ   z                 (   1   )               
where ‘G’ is the gravitational acceleration in Newtons/kilogram, ρ is the density in kilograms/cubic meter, and ‘z’ is integrated in meters, to produce a pressure, ‘P’, in Newtons per square meter.
 
     The second pressure contribution is the capillary threshold pressure difference, which is a function of IFT. This provides the amount of excess pressure required to force reactant into a formerly uninvaded element in the absence of any fluid-derived pressures. This is calculated at step  330  for the node under review. The capillary threshold difference is defined by the difference between the threshold pressure of the element the reactant is moving from and the threshold pressure of the element that the reactant is (potentially) moving into. 
     The sum of these two pressures, which is calculated at every node in the element under review, allows a computation of the maximum pressure that can be exerted by reactant at that node before the reactant is forced into the target element touching at that node. This sum is calculated at step  340 . 
     At step  350 , a determination is made as to whether the maximum pressure is negative. If so, the maximum pressure is truncated at zero at step  355 . From this maximum pressure and Equation 1 above, a maximum depth of reactant below the node is calculated at step  360 . This can be described using Equation 2: 
                   z   =       P   breach       G   ⁡     (       ρ   water     -     ρ   reactant       )                 (   2   )               
where P_breach is the pressure required to breach at the node, in Newtons per square meter, ‘G’ is the gravitational constant in Newtons per kilogram, and the p values are densities in kilograms per cubic meter. The resulting value of ‘z’ is supplied in meters.
 
     Knowledge of the depth of the node allows a computation at step  370  of a minimum absolute depth for the node under review. The shallowest of these absolute depths, as computed at all nodes, and provides the maximum filling depth that can be supported by the element under review. Any additional reactant that would fill the element below this depth is, instead, drained out into a target element, as at least one node becomes “leaky” once this depth is exceeded. 
     The above described method is only one example for calculating when a node in an element is under appropriate conditions to transfer reactant to a neighbouring element. One skilled in the art would appreciate that other methods may also be used without departing from the scope of the invention. 
     For example, all IFT values could be approximated as constant throughout the system (or constant per phase), which greatly simplifies the internal implementation, entirely eliminating a heavy data structure. 
     The system could be further simplified by binning node ‘z’ values in discrete increments. This moves half-way to the prior art cubic array-based system, and would be a reasonable approximation in some instances while still maintaining most of the desired behavior, at least on elements that have more than a small slope. That could simplify and speed up calculations, and still constitute an improvement over the use of a cubic array as shown in  FIG. 1 . 
     Alternatively, a mathematical equivalence could be obtained by maintaining net pressures within elements. That is, rather than a delta-Pth, one could assign each element an entry pressure which is the sum of water, Pth, and reactant pressures. Mathematically equivalent, but now the concepts would be looking at the difference between elements of a certain scalar field, rather than a delta across nodes. Migration pathways would tend to follow the gradient of this scalar field. Local minima would be infilled, and the pathways would be calculated out to the edges of the mesh. 
     A Monte-Carlo system is also possible that involves computing the probability of a given node being chosen for expulsion, based on a Boltzmann process and an energy function which depends on depth of the node and the back-pressures from adjacent nodes. In this case, the decision is made deterministically, i.e. the minimum-value node is always the one that expels, but a probabilistic approach could also be used, particularly if random variation is not used to condition the rock properties. Such a probabilistic approach would favour the highest node, but would not necessarily expel from the node with the lowest energy. 
     Expulsion from nodes is discussed herein because meshes used in accordance with the present invention may have shared nodes between elements, but the general location from which reactant is expelled can be “the point of highest contact between the elements for the given phase”. 
     Invasion from a source element to a target element can only proceed from nodes that lie within the volume that contains reactant. If the higher nodes in an element cannot support enough depth to allow the element to fill to a lower node, then there will be no invasion from the lower node. This constraint allows elements to be internally partially backfilled, and causes the migration of reactant to favour natural upward flow along inclined elements because, all other conditions being equal, the highest node in the element will have the highest pressure of reactant. 
     In another embodiment, the method described in  FIG. 3  can be extended to multiple reactant phases. This is shown in  FIG. 4  which is a flow chart of a method for calculating when a node in an element is under appropriate conditions to will transfer reactant to a neighbouring element where there are multiple reactant phases. 
     In this embodiment, the phase-separation menisci are recorded in each body. A given phase in a given element is only capable of invading neighbouring elements through a node that contacts that phase. If there are no nodes in contact, or if there is insufficient pressure to force reactant through any of the nodes that are in contact, then the body has not yet reached an equilibrium condition, and one or more phases will be allowed to change significantly in volume as the trapped phase tends to displace the phase(s) capable of invasion into neighbouring volumes. Once an equilibrium state is reached, the menisci move only in response to fluctuations in the density and IFT of the reactant as different chemical constituents are introduced to the body over time. 
     At step  410 , reactant is added to a body. At step  420 , densities and IFTs are computed for all phases of reactant in the body. At step  430 , the phase of highest density is selected. At step  440 , the external node in the selected phase with the lowest required filling depth to breach is located. At step  450 , a computation is made of the filling depth including pressure due to underlying phases. At step  460 , an evaluation is made as to whether all phases have been completed. If no, then at step  465 , the highest density among the remaining phases is selected, and the method is returned again to step  440 .  FIG. 3  basically describes in detail the steps involved in steps  440  and  450  of  FIG. 4 . 
     Once an analysis of all the phases is done, a computation is made of the mass of reactant required to fill the element to computed depths for all phases. At step  480 , surplus reactant is added to bodies contacting at each breaching node. 
     In another embodiment, reactant migration is modeled on the basis of an evolving mesh. In this embodiment, reactant migration is performed on an unchanging mesh for the lifetime of that mesh, and then a new mesh is applied to the reactant in the system. Several important changes are possible as the mesh is updated: (i) One or more elements that were present in the old mesh may be absent in the new mesh, having been “pinched out”. The reactant that was present in these elements should, whenever possible, be recovered and permitted to continue migration; (ii) Due to changes in porosity or water saturation, the volume of reactant that can be held by a given element may change. A body may contain more reactant than can be accommodated by all the elements in the body; (iii) Due to changes in temperature and pressure as rock elements are raised or lowered, the density (and hence volume) and IFT of reactant in an element may change. Any such change will change the total mass of reactant that can be supported within the element or the body; (iv) Due to changes in capillary entry pressure of rock elements, the threshold pressure required to invade neighbouring elements may change; (v) Due to changes in the hydrostatic pressure environment in the subsurface, the fluid-derived pressures at a node may change; and (vi) The geometries of elements that are conjoined into a body may change sufficiently that, at the maximum supported depth of filling of the body, the elements no longer form a continuous set. This is shown in  FIGS. 5A-5C  described below. 
       FIGS. 5A and 5B  are illustration of modeled fluid bodies whose elements are continuous in nature with each other. By contrast,  FIG. 5C  is an illustration of a modeled fluid body whose elements no longer form a continuous set. It is preferred if all elements of modeled fluid bodes used in accordance with the present invention be continuous, as shown in  FIGS. 5A and 5B . 
     It is important that the model respond correctly in the face of the changes mentioned above.  FIG. 6  is a flow chart  600  of a method to transition a body to a new mesh. At step  610 , the method is started. At step  620 , the initial mesh is loaded. At step  630 , a new reactant is loaded. At step  640 , computations are performed for the migration of reactants. See  FIG. 3  for further details regarding step  640 . Partial results are then stored at step  650 . 
     At step  660 , an analysis is performed regarding there are any additional meshes. If no, the method ends at step  680 . 
     If there are additional meshes, the reactant contained in the body is recorded at step  662  and a new mesh is loaded at step  664 . At step  666 , a subset of the elements in the body is computed. This consists of all elements in the bodies which exist in the new mesh and for which there are no other elements from the body lying directly below them. This, therefore, is the floor of the body. This ensures that pinched-out elements are excluded from the set, but the reactant that was contained in those elements is not lost. 
     At step  668 , the reactant in the body is reinjected into each element in the floor of the body in an amount proportional to their fractional contribution to the total area of the body in the X-Y plane. This reactant is exposed to new temperature and pressure conditions, and allowed to migrate and invade neighbouring elements as appropriate. As the invasion proceeds and new fluid bodies are formed, the bodies are permitted to merge into larger fluid bodies when the geometry and rock properties allow. If a body consists only of a single element, and that element is pinched out, the reactant formerly held by that body is discarded. 
     At step  670 , the old mesh is discarded and the method resumes again at step  630 . 
     The above described method is only one example for dealing with a new mesh. One skilled in the art would appreciate that other methods may also be used without departing from the scope of the invention. 
     For example, all the elements that were previously in the body could be taken and used to form a new body, giving them the reactant that they had on the previous mesh. This is not ideal because it amounts to preserving the bodies across mesh steps, and can create serious data inconsistencies when the geometry of the mesh changes significantly. 
     Material could be injected into the elements on a per-element basis, rather than into the floor of the bodies. While this may be a more fair system, it is also not ideal because it could produce some undesirable artifacts for bodies that receive no new reactant during the new mesh step and/or bodies whose pore volume increases from one step to the next. 
     Note that, in implementation of  FIG. 6 , many bodies are often only one element thick in the Z-direction, which means that the two approaches (i.e. injecting at the floor of the bodies, versus injecting into each element) are, in fact, equivalent. The implementation shown in  FIG. 6  is preferred when it is possible to employ. 
     As shown in  FIG. 7 , an exemplary system  700  for implementing the invention includes a general purpose computing device  702  in the form of a conventional personal computer or the like, including a processing unit  703 , and a system memory  705 . The personal computer  702  may further include a hard disk drive  704 , a magnetic disk drive  706  for reading from or writing to a removable magnetic disk  708 , and an optical disk drive  710  for reading from or writing to a removable optical disk  712  such as a CD-ROM or other optical media. The drives and their associated computer-readable media provide non-volatile storage of computer readable instructions, data structures, program modules and other data for the personal computer. Other types of computer readable media which can store data that is accessible by a computer can also be used. 
     A user may enter commands and information into the personal computer through input devices such as a keyboard  716  or a pointing device  718 . A monitor  720  or other type of display device is also connected to personal computer  702 . Personal computer  702  may operate in a networked environment using logical connections to one or more remote computers. 
     A user can use computer software running on personal computer  702  to utilize the modeling methods described above. 
     What has been described is merely illustrative of the application of the principles of the invention. Other arrangements and methods can be implemented by those skilled in the art without departing from the spirit and scope of the present invention.