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Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Topic Overview An introduction to standard numerical solution techniques for reservoir flow equations. Next Back html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Introduction Differential equations for mass flow Differential equations for mass flow Numerical Modell Numerical Modell Stability analyses Gridding Difference ApproximationDiscretization Error Reservoir Performance For more information click on the subject you want to learn more about. Reservoir equations Reservoir equations Back

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Discretization Techniques General partial differential equations for reservoir fluid flow must be discretized before they can be treated computationally. The most common techniques are: - finite differences - finite elements We will in in this module learn about the finite difference technique.finite difference Up html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Finite difference approximations are used in most commercial reservoir simulation software to solve fluid flow equations numerically. Main steps in a discretization procedure: - replace differential operators by algebraic ciexpressions - compute approximate solution at given points and iiispecified times Up Finite Differences html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Differential Equations for Mass Flow Mass conservation equations for Black Oil models: Next Where Q l are sink/source term Discretization Techniques

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Reservoir Equations For more information click on the equation you want to learn more about. Discrete equations for Black Oil models for block i,j,k: Next Back html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Water Equation The water equation consists of three parts; a flow term, a well term and an accumulation term. For more information click on the term of the water equation you want to learn more about. Flow term + well term = accumulation term Up Next html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Flow Term for Water The flow term for water consists of three terms, one for each coordinate direction. Next Up For more information click on the term of the equation you want to learn more about. html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Flow Term for Water in x- direction The x-part consists of two terms; one to compute flow to neighbour block in the positive direction and one for flow in the negative direction. Next Up For information on block boundaries, click on the textbox. html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Flow Term for Water in y- direction The y-part consists of two terms; one to compute flow to neighbour block in the positive direction and one for flow in the negative direction. NextBack Up For information on block boundaries, click on the textbox. html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Flow Term for Water in z- direction The z-part consists of two terms; one to compute flow to neighbour block in the positive direction and one for flow in the negative direction. Back Up For information on block boundaries, click on the textbox. html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Well Term for Water NextBack Up Specification are different for production and injection wells. Click here to see how the production term for water is given. water

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Up Well Equations for Black Oil Model P well = pressure in the well

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Up Well Equations for Black Oil Model P well = pressure in the well

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Up Well Equations for Black Oil Model P well = pressure in the well

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Accumulation Term for Water Back Up The change of mass of water in block i,j,k during time  t between step n and n+1 is given by: html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Evaluation on Block Boundaries Back Up html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Oil Equation Up NextBack For more information click on the term of the oil equation you want to learn more about. Flow term + well term = accumulation term The oil equation consists of three parts; a flow term, a well term and an accumulation term. html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Flow Term for Oil The flow term for oil consists of three terms, one for each coordinate direction. Next Up For more information click on the term of the equation you want to learn more about. html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Flow Term for Oil in x- direction Next Up The x-part consists of two terms; one to compute flow to neighbour block in the positive direction and one for flow in the negative direction. For information on block boundaries, click on the textbox. html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Flow Term for Oil in y- direction NextBack Up The y-part consists of two terms; one to compute flow to neighbour block in the positive direction and one for flow in the negative direction. For information on block boundaries, click on the textbox. html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Flow Term for Oil in z- direction Back Up The z-part consists of two terms; one to compute flow to neighbour block in the positive direction and one for flow in the negative direction. For information on block boundaries, click on the textbox. html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Well Term for Oil NextBack Up Specification are different for production and injection wells. Click here to see how the production term for oil is given. oil

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Accumulation Term for Oil Back Up The change of mass of water in block i,j,k during time  t between step n and n+1 is given by: html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Gas Equation Up Back For more information click on the term of the equation you want to learn more about. The gas equation consists of a flow term for gas and dissolved gas, a well term and an accumulation term for gas and dissolved gas. Flow terms + well term = accumulation terms html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Flow Term for Gas Up Next The flow term for gas consists of three terms, one for each coordinate direction. For more information click on the term of the equation you want to learn more about. html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Flow Term for Gas in x- direction Next Up The x-part consists of two terms; one to compute flow to neighbour block in the positive direction and one for flow in the negative direction. For information on block boundaries, click on the textbox. html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Flow Term for Gas in y- direction NextBack Up The y-part consists of two terms; one to compute flow to neighbour block in the positive direction and one for flow in the negative direction. For information on block boundaries, click on the textbox. (not active yet) html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Flow Term for Gas in z- direction Back Up The x-part consists of two terms; one to compute flow to neighbour block in the positive direction and one for flow in the negative direction. For information on block boundaries, click on the textbox. html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Flow Term for Dissolved Gas Up Back For more information click on the term of the equation you want to learn more about. The flow term for dissolved gas consists of three terms, one for each coordinate direction. Next html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Flow Term for Dissolved Gas in x- direction Next Up The x-part consists of two terms; one to compute flow to neighbour block in the positive direction and one for flow in the negative direction. For information on block boundaries, click on the textbox. html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Flow Term for Dissolved Gas in y- direction NextBack Up The y-part consists of two terms; one to compute flow to neighbour block in the positive direction and one for flow in the negative direction. For information on block boundaries, click on the textbox. html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Flow Term for Dissolved Gas in z- direction Back Up The z-part consists of two terms; one to compute flow to neighbour block in the positive direction and one for flow in the negative direction. For information on block boundaries, click on the textbox. html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Well Term for Gas NextBack Up Specification are different for production and injection wells. Click here to see how the production term for gas is given. gas

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Accumulation Term for Gas and Dissolved Gas Back Up The change of mass of water in block i,j,k during time  t between step n and n+1 is given by: html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Definition of Symbols l = o,w,g s = x,y,z p = i,j,k q l,i,j,k = Q l,i,j,k =  = S l = B l = [k]= k=  l = V i,j,k =  t=  R s = R s =  s Tl s =  s  l s = WI p = p i = p well = Back

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Difference Approximations Taylor series can be used to derive a difference formula for single and double derivates. With these expansion we can deduce: - first order approximation of f ’first order approximation of f ’ - second order approximation of f ’second order approximation of f ’ - second order approximation of f ’’second order approximation of f ’’ Taylor series of f(x+  x) and f(x-  x) are given by: Next Back html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance First Order Approximation of f’ This difference formula is used for discretizing time derivative in the mass equationsdifference formula From the expansion of f(x+Δx) we get an expression for f’(x): Up Click on the box to see how the approximation changes when the step size is halved. From the expansion of f(x-Δx) we get an expression for f’(x): Next html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Difference Formula A first order approximation of u t at the point n+1 is given by: The time axis is divided into points at distance Δt: Up html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance First Order Approximation of f’ From the serie f(x+Δx): From the serie f(x-Δx): Back The step size reduction produces more accurate approximations. html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Adding expansion of f(x+Δx) and f(x-Δx) results in the approximations: Up Second Order Approximation of f’ Next Back Click on the box to see how the approximation changes when the time step is halved. html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance The sum of f’(x) of the series f(x+Δx) and f(x-Δx): Up Second Order Approximation of f’ Back Step size reduction will produce more accurate approximations. html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance The sum of the Taylor series f(x+Δx) and f(x-Δx) is used to deduced a second order approximation of f’’: This approximation is frequently used and the numerator is written:approximation Up Back Second Order Approximation of f’’ html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Difference Approximation U xx can be approximated at each point i by the formula: Back html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Discretization Error The order of a difference approximation can by analysed using Taylor expansions. The discretization error approaches zero faster for a high order approximation then for a low order approximation. Next Back html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Gridding A faulted reservoirWell locationsAn imposed grid Initial fluid distribution Next Back html Click to the picture for sound (not active yet)

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance A Faulted Reservoir Next Up (Not active yet)

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Well Locations Next Back Up (Not active yet)

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance An Imposed Grid Up - The ability to identify saturations and pressures ii at specific locations (existing and planned well i iiiilocations). - The ability to produce a solution with the i iiiirequired accuracy (numerical dispersion and iiiigrid orientation effects). - The ability to represent geometry, geology and iiiphysical properties of the reservoir (external iiiboundaries, faults, permeability distribution iiiincluding vertical layering). - Keep the number of grid blocks small in order to iiimeet requirements of limited money and time iiiavailable for the study. Main criteria for grid selection:

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Initial Fluid Distribution Back Up (Not active yet)

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Stability Analyses StableUnstable Next Back html (Not active yet)

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Stable Next Back Animation of the stable solution html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Unstable Next Back Animation of the unstable solution html

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Reservoir Performance Back Sound not active yet

Topic overview Introduction DevelopersReferences Differential equations Gridding Difference approximation Discretization error Stability Analyses Reservoir equations Reservoir performance Developers Made by students Siril Strømme and Rune Simonsen Stavanger university college Informasjon på min web-side