 # Models of reservoirs

Modeling reservoirs and classification of their parameters is carried out in three directions: 1) geometrical, 2) mechanical and 3) linked with the presence of a liquid.

From the geometrical point of view all reservoirs can be subdivided into two big groups: granular (porous) and fractured. The capacity and filtration in a porous reservoir are determined by the structure of porous spaces between grains of rock. The presence of a branched system of fractures is typical for the second group of reservoirs. The density of fractures depends on various factors: structure of rocks, degree of compression, reservoir thickness, structural parameters, and other properties of host medium. Reservoirs of the mixed type are the most widely spread. Their capacity is formed by fractures, cavities and porous spaces. The leading part in a filtration of fluids belongs to the system of the micro fractures interconnecting these voids.

With the purpose of the quantitative description, real complex rocks are modeled by the idealized models.

The fictitious ground is the medium consisting of balls of one size, stacked in all volume of the porous medium in the same manner by eight spheres in comers of rhombohedron (figure 1.1). The sharp angle of rhombohedron varies from 60° up to 90°. The densest stacking of particles occurs when a = 60° and the least dense is when a = 90° (cube). Figure 1.1. Elements of models of a fictitious ground

With the purpose of more exact description of real porous medium, more complex models of a fictitious ground are offered now: with various diameters of spheres, with elements of not spherical form, etc.

The ideal ground is the medium consisting of tubules of the same size, identically stacked by four tubules in the comers of a rhombus. The density of stacking depends on the angle of rhombus opening.

Fractured-porous reservoirs are considered as a set of two different scale porous mediums (figure 1.2): 1) systems of fractures (medium 1) where porous blocks play a role of «grains», and fractures act as torturous «pores», and 2) systems of porous blocks (medium 2). Figure 1.2. The schematic of fractured-porous medium

In the elementary case fractured formation is modeled by one grid of horizontal fractures of some length (figure 1.3). All fractures are equally opened and located equidistantly from each other (a one-dimensional case). In most cases the fractured formation is characterized by presence of two mutually perpendicular systems of vertical fractures (a flat case). Such rock can be represented as model of the reservoir divided by two mutually perpendicular systems of fractures with equal sizes of opening, and with linear size of the block of a rock equal to lm. In spatial case we use the system of three mutually perpendicular systems of fractures (figure 1.4). Figure 1.3. The schematic of the model of a fractured medium with one system of fractures Figure 1.4. The schematic of the model of a fractured formation with three orthogonal grids of fractures

Any change of the forces applied to rocks, causes their deformation, and also change of internal stresses. Thus, the dynamic condition of rocks, as well as fluids, is described by rheological relations. Usually rheological dependences are obtained as a result of the analysis of experimental data of field study or physical modeling. If the volume of emptiness does not change or if it changes so, that this change can be neglected then such medium can be named non-deformable. If there is a linear dependence of volume on pressure, then such medium is “elastic”, its different name is Coulomb’s medium. Sandstones, limestone, basalts fall into the category of such mediums. In elastic bodies after removal of loading force the volume is restored completely and the plot of loading coincides with the plot of unloading. Many rocks are deformed with residual change of volume, i.e. the loading plot does not coincide with unloading plot, and we obtain a loop of a hysteresis. Such rocks refer to plastic ones (clay), flow rocks (non-cemented sand) or destroyable rocks.

Rocks are also divided according to space orientation. Isotropy means identical properties in all directions (isotropy is uniformity in all orientation), anisotropy vice versa means that various changes are observed on separate directions (i.e. the properties are directionally dependent).

Methods ofundergroundhydrodynamics are used to create mathematical filtration models which describe processes of motion and mass-transfer of fluids in a porous formation.

Mathematical filtration models can be divided into the following big groups: balance models and net models.

Balance models. Balance models are characterized by simplicity and low requirements on the information about filtration properties of reservoir and on the information about the wells operating in reservoir. In balance model the whole reservoir is considered as a uniform homogeneous volume with average characteristics. The equation of material balance for the whole reservoir with account for the total value of liquid or gas extracted from reservoir (or injected in it) lies in the basis of a balance model. Gas filtration to separate wells or cross-flows of fluid inside a reservoir is not considered in balance model. From this point of view, balance models are zero-dimensional.

There are various modifications of balance models: zone, block, layered models and their combinations. The specified models arise as a result of splitting a reservoir into separate areas (zones, blocks, layers) with identical filtration properties. Then the equations of material balance, and also the equations of cross-flows for neighboring areas are formulated.

Net models. When constructing net models, the reservoir is broken into a set of cells, the sizes and forms of these cells can vary essentially depending on type of each reservoir, and also depending on a necessary level of accuracy of modeling. To calculate processes of filtration either corresponding differential equations describing hydrodynamic field of the whole reservoir are used, or equations of material balance of filtered fluid are derived for each cell and then the obtained system of the algebraic equations is solved.

The choice of model of reservoir depends on a degree of accuracy which is necessary to achieve at calculations, and on a degree of available detailed information on a reservoir.

Lecture conclusion

The main issues considered in the lection are: modeling concepts, models of filtration flow, and models of reservoirs.

The concepts of «mathematical point» «physical point» are explained. The requirements necessary to meet the demands of adequacy of abstract and physical models to real processes are given. The laws of conservation that are applied in underground fluid mechanics are presented. The main parameters of reservoirs and approaches to reservoir modeling are discussed. Mathematical filtration models and their classification are studied.

Test questions

• 1. What does underground fluid mechanics study?
• 2. What is porous medium?
• 3. What is isotropy?
• 4. What is anisotropy?
• 5. What is fractured medium?
• 6. Describe the models of filtration flow. 