Mathematical Modeling of Oil Reservoir Waterflooding Using Fixed Streamtube at Various Values of Viscosity Ratio

Abstract: The speed up of numerical modeling of the oil reservoir waterflooding on high-resolution grids is possible by reducing the dimension of the two-phase flow problem. This problem is posed in fixed streamtubes connecting injection and production wells. The article describes an algorithm for constructing an effective streamtube between a pair of wells in a homogeneous oil reservoir, which guarantees the best approximation of the functions of the total flow rate and water cut of a production well. The obtained functions of the relative width of the streamtube for the periodicity cells of typical well patterns are approximated by piecewise linear functions. An assessment is made of the applicability of the constructed streamtubes for the numerical simulation of two-phase flow with a change in the viscosity ratio of the displacing and displaced phases, which is characteristic of measures to increase oil recovery

Estimation of the heterogeneity of the reservoir fluid inflow to the cross-sectional contour of a vertical well

© 2020, Kazan Federal University. All rights reserved. The reasons for the heterogeneity of the reservoir fluid inflow to the cross-sectional contour of a vertical well can be: a) asymmetry of the external pressure field relative to the well axis as a result of the interference of the surrounding wells; b) heterogeneity of the permeability field near the well, which is a consequence of either the heterogeneity of the absolute permeability field of the reservoir or the mobility function of the multiphase mixture of formation fluids. To simulate filtration in a reservoir over a relatively long time interval, the main interest is constant or long-term factors associated with well spacing and the distribution of absolute permeability. In the work, solutions of two model problems were constructed, which allow a quantitative evaluation of the influence of both factors on the degree of inhomogeneity of the inflow to the well and indicate the conditions under which this effect becomes significant. The obtained estimates are intended primarily for computational schemes of streamline and streamtube methods, which require a high degree of solution detailing near wells

Numerical modeling of local effects on the petroleum reservoir using fixed streamtubes for typical waterflooding schemes

© 2020 The Authors. Published by Georesursy LLC. The difficulty of numerical modeling of areal methods of flows redistribution in the oil reservoir is the need for detailed resolution of local hydrodynamic effects and the fine geological structure of the reservoir, which are centimeter-wide, at inter-well distances of the order of several hundred meters. The dimension of computational grids of traditional 3D models of such resolution, even for impact areas containing a small number of injection and production wells, turns out to be excessively large for design calculations. To overcome these limitations, it is proposed to perform a detailed simulation of the flow in two-dimensional cross sections of the reservoir along fixed streamtubes of variable width between each pair of interacting injector and producer wells. Reducing the dimension of the problem allows the use of high-resolution grids to simulate short-term local effects. In this paper, we present an algorithm for constructing a single fixed streamtube between injector and producer, which provides a minimum error in calculating of flow rate and water cut using a two-phase flow problem of reduced dimension along the streamtube. The algorithm is demonstrated by the example of the two-dimensional two-phase flow problem neglecting capillary and gravitational forces in a homogeneous reservoir of constant thickness for three waterflooding elements corresponding to seven vertical well flooding patterns – standard and inverted four-spot, five-spot and seven-spot, as well as staggered line drive. For these waterflooding elements, efficient streamtubes have been constructed, the relative width of which is approximated by piecewise linear functions. On the example of a staggered line drive or five-spot well patterns, the width of the effective streamtube was parameterized for an arbitrary ratio of the sides of the waterflood element. Presented streamtubes can be used as ready templates for subsequent modeling of geological and technical treatments in the relevant elements of the water flooding of the oil reservoir

Local Refinement of the Super Element Model of Oil Reservoir

In this paper, we propose a two-stage method for petroleum reservoir simulation. The method uses two models with different degrees of detailing to describe hydrodynamic processes of different space-time scales. At the first stage, the global dynamics of the energy state of the deposit and reserves is modeled (characteristic scale of such changes is km / year). The two-phase flow equations in the model of global dynamics operate with smooth averaged pressure and saturation fields, and they are solved numerically on a large computational grid of super-elements with a characteristic cell size of 200-500 m. The tensor coefficients of the super-element model are calculated using special procedures of upscaling of absolute and relative phase permeabilities. At the second stage, a local refinement of the super-element model is constructed for calculating small-scale processes (with a scale of m / day), which take place, for example, during various geological and technical measures aimed at increasing the oil recovery of a reservoir. Then we solve the two-phase flow problem in the selected area of the measure exposure on a detailed three-dimensional grid, which resolves the geological structure of the reservoir, and with a time step sufficient for describing fast-flowing processes. The initial and boundary conditions of the local problem are formulated on the basis of the super-element solution. This approach allows us to reduce the computational costs in order to solve the problems of designing and monitoring the oil reservoir. To demonstrate the proposed approach, we give an example of the two-stage modeling of the development of a layered reservoir with a local refinement of the model during the isolation of a water-saturated high-permeability interlayer. We show a good compliance between the locally refined solution of the super-element model in the area of measure exposure and the results of numerical modeling of the whole history of reservoir development on a detailed grid

Numerical Simulation of Oil Reservoir Polymer Flooding by the Model of Fixed Stream Tube

The work is devoted to development of high-speed mathematical model for accurately simulating the complex process of polymer flooding of oil reservoir. The proposed technique based on the use of fixed stream tube model. The technique consists of two steps. First, the stream lines and stream tubes from the injection well to surrounding producer wells defined from the solution of the thickness averaged two-dimensional problem of steady state flow. Then to describe the polymer flooding in each stream tube the problems of two-phase (water, oil) three-component (water, oil, polymer) flow are solved on a detail grid with a lateral mesh size to 1 meter, and with the vertical step of about 20 centimeters. Decomposition of the source three-dimensional problem to a series of two-dimensional problems allows to use high-resolution grids to describe the «thin» features of the viscous rim moving process. The viscosity of the aqueous solution of polymer is given as a function of concentration and shear rate. Model allows to make rapid simulations of different scenarios of polymer flooding, assessing the efficiency, and to determine the optimal parameters of development. The article demonstrated examples of numerical simulation of polymer flooding of a layered formation at different polymer injection modes

Modeling of fluid inflow towards multistage hydraulic fractures of infinite permeability using stream tubes

This article is devoted to the problem of a fundamental reduction of the machine time for numerical solution of non-steady state problems of the flow in the vicinity of multistage hydraulic fractures along horizontal wells in a petroleum reservoir. This issue arises when it is necessary to solve inverse problems associated either with the identification of fracture parameters based on the results of their hydrodynamic studies or with their optimization to obtain specified production indicators. As a way to reduce computational costs, we previously proposed replacing the spatial problem of flow in the reservoir with a set of one-dimensional problems along the stream tubes. In this case, the problems for pressure in each fracture are solved taking into account the distributed inflow of reservoir fluid from the stream tubes adjacent to the fracture edges. Decomposition of the spatial problem into a set of one-dimensional problems along the stream tubes reduces the required machine time for numerical simulation of the non-steady state flow by orders of magnitude. The object of this research is the functions of the relative width along the stream tubes distribution and their lengths, which are necessary to calculate the local inflow to the fractures and are the key parameters of the model that determine its accuracy. The parameterization of the functions of length and distribution of the relative width along the stream tubes adjacent to the edges of vertical multistage hydraulic fractures is performed. The well between the fractures is assumed to be non-perforated. The case of fractures of infinite permeability in a homogeneous reservoir is considered, when the problem is reduced to a two-dimensional formulation in a horizontal plane. The results are also applicable without any changes to a stratified heterogeneous formation. Analytical expressions are obtained for the listed properties of stream tubes with a difference between the inner and outer edges of the fractures. For this purpose, analytical solutions of the corresponding model problems are used. An algorithm for the parametrization of stream tubes for the case of fractures differing in length is proposed. The proposed simplification of the spatial model is tested, and the range of values of the initial parameters of the system, which allows for an acceptable level of error of the simplified model for evaluative calculations, is shown