Integrating analytical well equations for arbitrary trajectories

Randy Hazlett

Associate Professor

McDougall School of Petroleum Engineering

1

Complex Wells Introduce Need for Higher Bandwidth in Well Simulation

Depiction of a complex well crossing multiple cells with heterogeneous transport properties.

Complex wells require long range pressure and flux coupling.

2

Unification of Diffusivity Equation Solutions

Departure from Initial Conditions

Approach to PSS

3

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22

Discrete Well Modeling

BASIS: Time-dependent, uniform flux, partially-penetrating, unit rate, line source solution at arbitrary 3D orientation in a closed rectangular box.

Well performance prediction for any well trajectory as a piecewise linear path representation

Produced by analytic integration in time and space of the point source Neumann function

y

x

(x1, y1, z1)

(x2, y2, z2)

L

a

b

h

z

Patent Pending, Hazlett & Babu

Uniform pressure well BC produced via segmentation Green's function also available Constant BHP solution also in preparation, i.e. decline curve forward model

4

Sample External Boundary Flux for a Complex Well

Depiction of boundary flux to achieve constant pressure for a 50% penetrating slanted well along the diagonal in a cubic cell.

0.55

0.0

FLUX

5

Boundary Flux in Dirichlet Problems

Sensitivities in the pseudo-steady state Dirichlet boundary condition problem as a

function of distance from the interface in a square cell: (a) Boundary flux profiles to

maintain constant pressure; (b) Dimensionless well productivity index.

(a) (b)

6

Linking Related Solutions

Boundary flux in single well, pseudo-steady state problems

and multi-well, steady state problems are related.

+ =

g1(x)

g2(x)=g

1(x) , i.e. g

3(x)=2g

1(x)

g3(x)

sink

g2(x)

Mirror image

source

2 cell problem

7

Linking Related Solutions

Boundary flux profiles for the single offset vertical well pseudo-steady state problem and the

renormalized Dirichlet flux for the corresponding image pair steady state problem with two

rectangular regions without permeability contrast and a sealed exterior BC. 8

Linking Related Solutions

Boundary flux profiles for a centralized horizontal fracture pseudo-steady state problem and the renormalized Dirichlet flux for the corresponding image pair steady state problem with two rectangular regions without permeability contrast and a sealed exterior BC.

9

Proposed Parametric Boundary Flux A combination of uniform pressure and uniform flux boundary conditions

where

g(x) is the renormalized Dirichlet-derived flux, that adapts with changes in the problem, e.g. source type & position

g is the average flux, representing a uniform flux – a constant

gxgxg 1,ˆ

10

Heterogeneous Systems

Boundary flux profiles in PSS between two rectangular regions as a function of permeability

contrast along with an a fitting parameter describing the linear combination of a uniform flux

component and that required to achieve uniform pressure. 11

Transient Systems

Time-dependent line source solutions on the square with Neumann external boundary

conditions with a banded color lookup table fixed to extrema in dimensionless pressure in

the pseudo-steady state case.

tD = 0.001 0.01

0.1 1.0

Mid-axis transport

12

Discrete Fracture Modeling

BASIS: Time-dependent, uniform flux, partially-penetrating, unit rate, plane source solution at arbitrary 3D orientation in a closed rectangular box.

ARBITRARY PLANE SOURCE Produced by analytic integration in time and space of the point source Neumann function along 2 vectors y

x

z

(x1, y

1, z

1)

(x2, y

2, z

2)

L

a

b

h

(x3, y

3, z

3)

13

FuRSST Initiatives

SUMMARY

Integration of large bandwidth well equations for complex wells into solver schemes

Inclusion of high fidelity boundary flux distributions to replace or guide multipoint flux approaches

Assistance with well transient solutions as physics demands

14