Visualization of drained rock volume (DRV) in hydraulically fractured reservoirs with and without natural fractures using complex analysis methods (CAMs)

2019

Energies

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This paper presents insights on flow in porous media from a model tool based on complex analysis methods (CAM) that is grid-less and therefore can visualize fluid flow through pores at high resolution. Elementary pore network models were constructed to visualize flow and the corresponding dynamic bottomhole pressure (BHP) profiles in a well at reservoir outflow points. The pore networks provide the flow paths in shale for transferring hydrocarbons to the wellbore. For the base case model, we constructed a single flow path made up of an array of pores and throats of variable diameter. A passive ganglion (tracer) of an incompressible fluid was introduced to demonstrate the deformation of such ganglions when moving through the pores. The simplified micro-flow channel model was then expanded by stacking flow elements vertically and horizontally to create complex flow paths representing a small section of a porous reservoir. With these model elements in place, the flow transition from the porous reservoir fluid to the wellbore was modeled for typical stages in a well life. The dynamic component of the bottomhole pressure (BHP) was modeled not only during production but also during the drilling of a formation (with either balanced, underbalanced or overbalanced wellbore pressure). In a final set of simulations, the movement of an active ganglion (with surface tension) through the pore space was simulated by introducing a dipole element (which resisted deformation during the movement through the pores). Such movement is of special interest in shale, because of the possible delay in the onset of bubble point pressure due to capillarity. Capillary forces may delay the reservoir to reach the bubble point pressure, which postpones the pressure-drop trigger that would lead to an increase of the gas–oil ratio. The calculation of the estimated ultimate recovery (EUR) with an erroneous assumption of an early increase in the gas–oil ratio will result in a lower volume than when the bubble point delay is considered.

Section: Complex Analysis Methods (Cam)mentioning

confidence: 99%

Section: Upscaling Of Capillarity Effectsmentioning

confidence: 99%

Elementary Pore Network Models Based on Complex Analysis Methods (CAM): Fundamental Insights for Shale Field Development

2019

Energies

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“…Unusual refractions of streamlines near the well system occurred in our prior model code (compare Figures 9f and 13f) when natural fractures were oriented at a high angle to the far-field flow direction. We revisit the examples presented in an earlier study (in Section 4.2), where the effect of natural fractures on the DRV was investigated by using the algorithm presented in Equation (5) [16]. In Section 4.3, the DRV is regenerated using the modified solution of Equation (12).…”

Section: Applicationmentioning

confidence: 99%

Flow in Fractured Porous Media Modeled in Closed-Form: Augmentation of Prior Solution and Side-Stepping Inconvenient Branch Cut Locations

2020

Fluids

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Carefully chosen complex variable formulations can solve flow in fractured porous media. Such a calculus approach is attractive, because the gridless method allows for fast, high-resolution model results. Previously developed complex potentials to describe flow in porous media with discrete heterogeneities such as natural fractures can be modified to expand the accuracy of the solution range. The prior solution became increasingly inaccurate for flows with fractures oriented at larger angles with respect to the far-field flow. The modified solution, presented here, based on complex analysis methods (CAM), removes the limitation of the earlier solution. Benefits of the CAM model are (1) infinite resolution, and (2) speed of use, as no gridding is required. Being gridless and meshless, the CAM model is computationally faster than integration methods based on solutions across discrete volumes. However, branch cut effects may occur in impractical locations due to mathematical singularities. This paper demonstrates how the augmented formulation corrects physically unfeasible refraction of streamlines across high-permeability bands (natural fractures) oriented at high angles with respect to a far-field flow. The current solution is an important repair. An application shows how a drained rock volume in hydraulically fractured hydrocarbon wells will be affected by the presence of natural fractures.

“…However, for cases where the velocity gradient is spatially less convolute, a coarser time-step can be used to reduce the computational time. Further details on complex potentials and stream functions in reservoir flow applications are given elsewhere [2][3][4][5][6][7][8][9]16,17].…”

Section: Particle Tracking and Time-of-flight Contoursmentioning

confidence: 99%

“…Multiple doublets were superposed in a recent study [3] to construct a new analytic element, suitable to model flow in fractured porous media based on complex analysis methods. The algorithm for the analytical element of flow in fractures enabled fundamental models [4,5] and applied models [6][7][8][9] that aim to advance our understanding of flow in fractured porous media.…”

Section: Introductionmentioning

confidence: 99%

Modeling Flow and Pressure Fields in Porous Media with High Conductivity Flow Channels and Smart Placement of Branch Cuts for Variant and Invariant Complex Potentials

2019

Fluids

A long overdue distinction between so-called variant and invariant complex potentials is proposed here for the first time. Invariant complex potentials describe physical flows where a switch of the real and imaginary parts of the function will still describe the same type of physical flow (but only rotated by π/2). Such invariants can be formulated with Euler’s formula to depict the same flow for any arbitrary orientation with respect to the coordinate system used. In contrast, variant complex potentials, when swapping their real and imaginary parts, will result in two fundamentally different physical flows. Next, we show that the contour integrals of the real and imaginary part of simple variant and invariant complex potentials generally do not generate any discernable branch cut problems. However, complex potentials due to the multiple superpositions of simple flows, even when invariant, may involve many options for selecting the branch cut locations. Examples of such branch cut choices are given for so-called areal doublets and areal dipoles, which are powerful tools to describe the streamlines and pressure fields for flow in porous media with enhanced permeability flow channels. After a discussion of the branch cut solutions, applications to a series of synthetic and field examples with enhanced permeability flow channels are given with examples of the streamline and pressure field solutions.