Gas Flow in Gas Reservoirs

8/20/2019 Gas Flow in Gas Reservoirs

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Farshad Asgari

93134020

:SupervisorDr.Khamehchi

Transient Gas low in Unconventional

Gas Reservoirs

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Introduction

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Unlike flow intraditional reservoirs, nanoscale observationsand field data analysis tell us that gas flow in such extremely

low-permeability formations is complicated by flow

condition and many co-existing processes, such as:

1. severe heterogeneity on any scales

2. Klinkenberg effect or known as Knudsen diffusion

3. non-Darcy flow behavior

4. adsorption/desorption5. strong interactions between fluid (gas and water)

molecules and solid materials within tiny pores


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low Model

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A multiphase system of gas and water in a unconventionalreservoir is assumed to be similar to the black oil model,

composed of two phases: gaseous and aqueous phases.

the gas and water components are assumed to be present

only in their associated phases, and adsorbed gas is within the

solid phase of rock.

Each fluid phase flows in response to pressure, gravitational,

and capillary forces according to the multiphase extension of

Darcy law or several non-Darcy laws


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low Model

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In an isothermal system containing two mass components, subjectto flow and sorption, two mass-balance equations are needed to

fully describe the system (β = g for gas and β = w for water).

where " is the density ; v is the volumetric velocity vector ; S is

the saturation; is the effective porosity of formation; t is time;

m is the adsorption or desorption term for gas component (k=g

only) per unit volume of formation; and q is the sink/source termof phase (component) per unit volume of formation.


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low Model

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The flow velocity will be evaluated:

1. using the Darcy’s law with Klinkenberg effects (for gas

flow)

2.Using the nonlinear flow models to describe non-Darcyflow behavior

3. flow condition where flow may not occur until the

pressure or potential gradient reaches a certain threshold

value


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low Model

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Darcy’s Flow: When the Darcy’s law is applicable, thevelocity, v , is defined as:


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low Model

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Non-Darcy’s Flow: In addition to multiphase Darcy flow,non-Darcy flow may also occur between and among the

continua in tight or shale gas reservoirs. The flow velocity, v ,

for non-Darcy flow of each fluid may be described using the

multiphase extension of the Forchheimer equation

where is the effective non-Darcy flow coefficient with a

unit m-1 for fluid " under multiphase flow conditions.


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low Model

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Flow with Threshold Pressure Gradient: Thephenomenon of flow with threshold-pressure-gradient

concept has been observed in laboratory and is commonly

used to describe nonlinear flow behavior in low permeability

reservoirs.Using the effective potential gradient, the flow ofgas or liquid in a low-permeability reservoir is described by,


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low Model

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the effective potential gradient whose scalar component in the xdirection (the flow direction) is defined as,

where G is the threshold (or minimum) potential gradient for thefluid to become mobile.


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low Model

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Adsorption: Natural gas can be present as a free gas phaseor as adsorbed gas on solids in pores. In shales, methane

molecules are adsorbed mainly to the carbon-rich

components, i.e. kerogen .

As observed, adsorbed mass of gas can provide significant

fraction of gas reserves and recovery. As the pressure

decreases with gas production from reservoirs, more

adsorbed gas is released from solid to free gas phase.


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low Model

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In model, the mass of adsorbed gas in formation volume V is described

where is kerogen density, is gas density at standard condition, is

the average volume relative of kerogen in bulk volume, is theadsorbed gas mass in bulk formation volume V. f(P) is the adsorption

isotherm function.

where, is the Langmuir volume (the maximum adsorption capacity at

a given temperature), and is the Langmuir pressure(the pressure at

which the adsorbed gas content is equal to

2 ).


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low Model

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Klinkenberg Effect: In low-permeability formation or underlow pressure condition, the Klinkenberg effect may be toosignificant to be ignored when modeling gas flow in reservoirs,

because of small size pores and low permeability associated.Undersuch flow conditions, absolute permeability for the gas phase is

written as a function of gas pressure as,

Where ∞ is constant, absolute gas-phase permeability under

very large gas-phase pressure (where the Klinkenberg effect isminimized); and b is the Klinkenberg factor, depending on thepore structure of the medium and formation temperature.


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Numerical Model

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In general, the flow model is solved using a numericalapproach. This work follows the methodology for reservoir

simulation, i.e., using numerical approaches to simulate gas

and water flow, following three steps:

1. spatial discretization of mass conservation equations;

2. time discretization;

3. Iterative approaches to solve the resulting nonlinear,

discrete algebraic equations.


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Numerical Model

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Discrete Equations: The component mass-balanceEquations are discretized in space using a control-volume or

integrated finite difference concept. Time discretization is

carried out using a backward, first-order, fully implicit finite-

difference scheme.


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Numerical Model

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The discrete nonlinear equations for components of gas andwater at gridblock or node i can be written in a general

form:


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Numerical Model

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where superscript k serves also as an equation index for gasand water components with k = 1 (gas) and 2 (water);superscript n denotes the previous time level, with n+1 thecurrent time level to be solved; subscript i refers to the index

of gridblock or node i, with N being the total number ofnodes in the grid; ∆t is time step size; Vi is the volume ofnode i; i contains the set of direct neighboring nodes (j) ofnode i;, , , and are the accumulation andreaction (absorption or desorption) terms, respectively, at

node i; the component mass ―flow‖ term between nodes i and j, and sink/source term at node i for component k,respectively, defined below.


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Numerical Model

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mass fluxes by advective processes and are described by adiscrete version of Darcy’s law:

mobility term:

Transmissivity:

The flow potential term:


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Numerical Model

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Handling fractured media: to implement fractured modelingapproaches, special attention is needed to calculate fracture-matrix mass transfer. the flow between fractures and matrix is stillevaluated using slide 17; however, the transmissibility for thefracture-matrix flow is given by:

where Afm is the total interface area between fractures and matrixof element i and j (one of them is a fracture and the other is a

matrix blocks); is matrix absolute permeability; and is acharacteristic distance or equivalent length for flow betweenfracture and matrix blocks


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Numerical Model

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Handling non-Darcy flow: Under the non-Darcy flowcondition, the flow term ( ) along the connection (i,

j), between elements i and j, is numerically defined as:


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Numerical Solution

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we use the fully implicit scheme to solve the discretenonlinear Eq slide 15 with a Newton iteration method. Let

us write the discrete nonlinear equation, Eq slide 15, in a

residual form as:


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Numerical Solution

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Eq slide 21 defines a set of 2&N coupled nonlinear equationsthat need to be solved for every balance equation of masscomponents, respectively.

In general, two primary variables per node are needed to use

the Newton iteration for the associated two equations pernode.

The primary variables selected are gas pressure and gassaturation. The rest of the dependent variables, such asrelative permeability, capillary pressures, viscosity anddensities, adsorption term, as well as nonselected pressures,and saturation, — are treated as secondary variables, whichare calculated from selected primary variables.


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Numerical Solution

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In terms of the primary variables, the residual equation, Eqslide 21, at a node i is regarded as a function of the primaryvariables at not only node i, but also at all its directneighboring nodes j. The Newton iteration scheme gives rise

to

where Xm is the primary variable m with m = 1 and 2,respectively, at node i and all its direct neighbors; p is theiteration level; and i =1, 2, 3, …, N. The primary variablesneed to be updated after each iteration.


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Numerical Solution

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The Newton iteration process continues until the residuals,+1 or changes in the primary variables ,+1 over

iteration are reduced below preset convergence tolerances.

At each Newton iteration, Eq slide 23 represents a system of

(2&N) linearized algebraic equations with sparse matrices,

which are solved by a linear equation solver.


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References

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Bybee, K., Horizontal wells in tight gas sands: risk management to maximizesuccess, JPT, 61-63, October, 2008.

Denney, D., Rock type-understanding productivity in tight gas sands, JPT, 53-56, October, 2008.

Forsyth, P. A., Y. S. Wu and K. Pruess, Robust Numerical Methods forSaturated-unsaturated Flow with Dry Initial Conditions in Heterogeneous

Media. Advance in Water Resources 18, p. 25-38, 1995. Klinkenberg, L.J., The Permeability of Porous Media to Liquids and Gases, in

API Drilling and Production Practice, 200 – 213, 1941.

Langmuir Irving, The constitution and fundamental properties of solids andliquids. Journal of the American Chemical Society 38 (11): 2221 – 2295,1916.

Wu, Y.S. and P. Fackahroenphol, ―A Unified Mathematical Model forUnconventional Reservoir Simulation,‖ SPE-142884, presented at the SPEEUROPEC Conference, to be held from 23-26 May 2011 in Vienna, Austria,2011.

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Gas Flow in Gas Reservoirs