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# PVT and Flow course - Rate Equation (Darcy) Intro

### More By Petro Engineer

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## Rate Equation (Darcy) Intro (20160308 Part 1)

We are going to start by talking about Darcy's law. We're going to look at both for for liquid and gas.
We'll use Darcy's law to derive what I'm going to call rate equations. Darcy's law gives you the velocity of a flow of fluid in a rock, we don't sell velocities, we sell volumes, we sell barrels and we sell standard cubic meters. So rate equations what we're really talking about are volumetric rates. And Darcy's law applies down in the Rock but we don't sell oil and gas in the Rock, it's got to get out of the rock into the production well into the production tubing to the surface.
The rate equations we're interested in are expressed as sellable volumetric rates, not rates down in the reservoir.

So we have to combine Darcy, which talks about the flow in the reservoir rock, with PVT. Because what happens to the oil when it goes from the reservoir to the surface? The oil shrinks. So we need to know how much it shrinks, does it shrink 5% or 50%. The gas doesn't shrink when it goes to the surface, the gas expands. But does it expand 50 times or 300 times the volume? Ee need to know that.

The rate equations that we'll talk about for the most part will be relating surface gas or surface oil or both as a function of the flowing bottomhole pressure. Basically that's what we're interested in. Primarily we're interested in how these rates will vary as we change the flowing bottomhole pressure.
Generally speaking we control the pressure If anywhere at the surface, at least we know what the pressure is at the surface. And ultimately we'd like to know how the gas and the oil rate change as a function of flowing tubing head pressure, because that we can directly control, but most of the equations for most of the rate equations are given as a function of flowing bottom pressure.

The pressure drop between down and the top of the well can vary for many different reasons. Gravity is one of them, because the the length can be kilometersand you've got density of oil or density of water gas and so it's going to have a pressure difference simply because of a static column a fluid. Then, you got friction (hopefully) - if you don't have friction - you're not producing anything. And if you have no friction and you're producing something - you probably designed the size of the tubing wrong - because you can put a monstrous tubing in which means you had to build a drill an even more monstrous hole just so you didn't have any friction. So you're going to have some friction and you're going to always have some gravity, so there's going to be a difference in pressure because of these two reasons. The gravity is mainly dependent on the density of what's going through the pipe - water, oil and gas - we know all about pvt we're expect experts and the friction part will depend on the size of the of the pipe. If you put in a straw then you're going to have too much friction and if you put in a monstrous tubing - you lost your job - because well cost so much money but you have no friction. And then, somewhere in between, the friction is going to be a function mainly of pipe diameter, the velocity according to the Reynolds number and the velocity, the viscosity of the fluids flowing through and tubing roughness.

The other thing about these rate equations that we'll be talking about for the most part is that they will assume what is referred to as steady state flow conditions. Sometimes they refer to it as pseudo steady state (or the new term is boundary dominated flow).

What the pseudo steady state flow condition means is that the rate of pressure drop - how many bar per day or how many PSI per week, or month - is approximately the same everywhere in the volume being drained. But it changes.
Pure steady state is saying that this number will not change with time and equal zere (no pressure drop). And that just never happens usually, that's almost never going to happen - the only way that might happen is that you inject in some wells at the same volumetric rate as you produce.

The rate equations we're going to discuss for the most part in this course will be of this type pseudo steady state or boundary dominated. And this is often a very good assumption when the permeability is high (≥ 10 - 100 mDarcy).

In North America today, where they're producing from Rock that's got like permeability that's 10 to the minus fourth millidarcy, which is kind of a good shale rock permeability, then you don't reach pseudo steady state certainly not in a year maybe in two years, might take 10 years it depends on a few things. I just want to mention up front that there's a time transition to reach pseudo steady state. If it's a very high permeability it might be one day or a week, if it's a lower permeability it might take a month.
This transition time, before it reaches this condition of pseudo steady state, this time is a function of perability, fluid viscosity, area, shape of the area, the compressibility of the system, porosity.

Early on the system behaves as if there are no outer boundaries. And we have a special name for that - transient. But that's not a good word because transient means anything that's time dependent, so this is now called infinite acting.

## Rate Equation (Darcy) Intro (20160308 Part 2)

This time period between when you start the well producing and when you reach boundary dominated or pseudo steady-state behavior, there's a field of petroleum engineering that kind of is dedicated to trying to understand the well and the reservoir properties - permeability and so forth based on the study of pressures and rates during this time. It's referred to as transient flow (or rate) analysis or Pressure Transient Analysis. Historically we used to call it well test analysis.
The well was produced and maybe the rates were changed over a period of days, typically, and during that period of days they assume that they did not reach this complete boundary dominated pseudo steady state behavior, so that the rates and pressures in a sense the behavior of the waves and the in the pond.

That pressure transient analysis does is that it takes the differential equations describing flow, which is Darcy, and a material balance and mathematically combine the two and solve for how the pressure moves out away from the wellbore if you produce at a certain rate. The way they do this, they don't solve with all these parameters we talked about - permeability, viscosity, compressibility, porosity - all these different variables, because if you transform the differential equation into what's called a dimensionless form, you'll see that the well could behave identically even though their permeability are different, because the compressibility are different in the other direction. And there's a term called the diffusivity constant of the rock fluid system which is kind of fundamental to that, but the point is that we work with dimensionless pressure drop and dimensionless quantities and we assume that the well produces at a constant rate and we solve for how the pressure changes at the wellbore. We solve this in a dimensionless form meaning that we don't talk about psi or millibar, we talk about dimensionless pressure drop PD and dimensionless time TD then that same solution can be used for all of the world all of the wells in the world, independent of the permeability or whatever. So we call this dimensionless pressure drop solution PD is a function of TD - dimensionless pressure drop as a function of dimensionless time.
You can solve the same exact set of differential equations where you use the different boundary condition instead of constant rate, which is a the other boundary condition that can (and sometimes is) used is that you say that the pressure of the wellbore is constant, how does the rate change? And you get what's called a dimensionless rate solution QD as a function of dimensionless time.

Mathematically approximately this (it's not exact, but it's pretty close):

```pD(tD)≈1/qD(tD)
```

So the inverse functions, it's just using a different boundary condition: 1) you set the rate, you get the pressure drop; 2) you set the pressure drop you get the rate. And it ends up the solution is almost identical it's just inverse solution.

In reality Pwf changes with time and rate changes of your time, both. That's the reality of what usually happens, you can still use though the same analytical solutions to these differential equations with changing boundary conditions.

### Superposition

Superposition or what's called sometimes convolution. What is superposition?

If you got a well producing a 100 barrels a day and then you change the rate to 200 barrels. Okay, for the first day it produces a hundred barrels per day and for the second day produces 200 barrels per day and then it just stays that 200 barrels per day for a year.
What you can say is that if you put in a well from time zero for the full year that produces that 100 barrels per day and you look at what's the pressure drop due to that, then on your time scale starting at zero.
Then you go and you start a new time scale at one day which is 0 for a new well, we put another well in the same place and we say that well is going to produce at 100 barrels per day also, that it starts a day later okay. And it knows nothing that the reservoirs never been touched before, so it's a fresh reservoir pressure, initial is zero so we put that well on at 100 barrels per day starting at the end of day one then that produces 364 days.
The pressure drop at the end of 365 real days is just the sum of the pressure drop from the first well that went for a hundred days for 365 days plus the pressure drop. There's no pressure drop the first day for the second well and it's producing also 100 barrels from today.
If you add the two pressure drops at the end of 365 you'll get the correct pressure drop for that well that went from 100 to 200 the state at 200.

And if the rate changes every day you just keep adding on a new well, what you have to do is that all the sum of all the rates of all the wells that you've got has to be the actual rate. So if at the end of the second day we shut the well in: you go for 100 barrels a day for one day → two hundred barrels a day for another day → and then zero, the third well that starts at two days. What does it rate? Its rate have to be minus two hundred, because we've got the other two wells from two days on one's doing 100 barrels a day the second wells doing a 100 barrels a day, so to cancel that out because for the next 363 days along shut in okay. 363 days shut in the total rate has to be 0 and if you just take the pressure drop delta P from each of those three wells, add them together at any time from two days on, you'll get the right performance.

That's superposition. It's a very fundamental concept that it applies basically to all of our flow problems.

### Volumetric Average Pressure

Volumetric average pressure is basically a relationship that says the volumetric average pressure (so you have to integrate the pressure everywhere on a volumetric basis) is going to decrease as a function primarily of how much stuff we've take it out - the cumulative amount of production.

Q is the cumulative volumetric production and it's going to be either surface gas or surface oil. what is Q? Q is just the integral of the rate. And remember, when we talk about rates it's going to be the surface gas or surface oil, always the surface surface rate.
If we have gas then we usually write GP.
if we have oil then we write NP I don't know why it's just the way it is.

The volumetric material balance is basically something that can be used only at boundary dominated, if you've reached the boundary dominated time.

So we're going to take the boundary dominated rate equation and we're going to combine it eventually with this volumetric material balance. We can do that because they're both valid in the boundary dominated period. In this course we're primarily only going to look at the gas.

Why do we want to do this? Whatever you like knowing how this looks into the future, this is what matters. This matters to your salary, it matters to your bosses salary and it matters to your boss's boss's boss's salary and all the stock owners at the company who own the company you're working for. The the value forecast is important and we need to know what it's going to look like in the future. This is basically equal to the rate times the price.

So, the material balance is the integral of rate with time and it doesn't care the profile at all, it just needs to know how much have I taken out, and I'll tell you the average pressure. It then the Darcy equation the rate equation says if I know the average pressure I can calculate the rate for any control drawdown, for any choke size, so we can work back and forth continuously to find the production rate profile versus time and you can look at all kinds of scenarios. The point is that you can start evaluating producing more gas or less gas, for shorter or longer periods of time and the material balance doesn't care what that profile looks like, it gives the feedback - if you've taken so much gas out, I can go back to this equation, you can say well at that point in time I can produce this much from every well today or tomorrow or the next day. That's that's the reason for doing this coupling.

### Arp's Rate Equation

The second option to predict the production profile is to use boundary dominated decline curve analysis.

### Ultimate production

Final, or ultimate production the ultimate recovery factor times the initial volime in place:

``` QP = RF * IFIP
```

QP - totla production,
RF - recovery factor,
IFIP - initial fluid in place.

For gas RF ~ (0.5-0.98). So, if you have a high enough permeability and water moving in, then you can actually deplete almost everything from the reservoir, near one hundred percent. Typically it would be 60-70-80%.
For oil RF ~ (0.05-0.5). Of course if you're Norway they say 60-70%. But it's a goal to reach 60-70%. Typically it is going to be 30-35% at the best. Why? Because oil likes to stick to the rocks, oil and rocks are in love, so the oil just wants to hang onto the rocks interfacial forces. So you have to work to bring the recovery factors above 30-40%, you have to really invest a lot of money usually.

### Reservoir Simulation

I heard the comment: oh, that's all you know simple play game stuff, okay reservoir simulation gives you the right answer.
Okay so what does reservoir simulation? I'm just going to tell you what that is in a nutshell, you won't like it.

Reservoir simulation is taking the reservoir and making it into a bunch of Lego blocks. You just basically discretize it into little Lego blocks and then you set up an equation for transport from one Lego block to the next. And what equation do you think we use in the reservoir simulators to calculate what goes from one block to the next? Darcy. Oh it's the same guy, okay. And what do you think is to constraint on the differential equations which they're solving? For every little grid cell and material balance same deal, that's all they have. It is just that they have the same rate material balance equation for every little Lego block, so it's the same stuff going on. There's no new advanced stuff going on, except that for one reservoir unit as I said we'll be solving here it's like 1 Lego block and what you're saying is that the whole reservoir can be treated as a single Lego block, which is sometimes not a bad assumption.
There's a number fields where you can make that simplifying assumption and there's many many reservoirs where you can't make that assumption because you have multiple reservoir units or the properties vary so much spatially there's many reasons why you need reservoir simulation. But don't get it wrong, you're basically using the same equations - Darcy's law and material balance.