One week before ‘Arafah she intended to fast ‘Arafah with the intention of making up one of the days that she did not fast in Ramadan. But when the day of ‘Arafah came, she forgot this intention and she fasted it as a voluntary fast. Now she does not know what that day counts as: is it only a voluntary fast, based on the intention that she had on the day of ‘Arafah, or does it count also as making up the day that she owed, based on the intention that she had one week before ‘Arafah?

Praise
be to Allah.

Firstly:

Having the right intention is one of the
conditions of acts of worship – fasting and others – being valid. No act of
worship is valid unless it is accompanied by the intention, because of the
report narrated by al-Bukhaari (1) and Muslim (1907) from ‘Umar ibn
al-Khattab (may Allah be pleased with him) who said: “Actions are but by
intention, and each one will have but that which he intended.”

An-Nawawi (may Allah have mercy on him)
said:

What this hadith means is that actions are
only reckoned on the basis of the intention, and they do not count if they
are done without an intention. This indicates that purification – including
wudoo’, ghusl and tayammum – is not valid unless it is accompanied by the
intention. The same applies to prayer, zakaah, Hajj, i‘tikaaf (devotional
retreat) and all other acts of worship. End quote.

In the case of all obligatory fasts – which
includes those done to make up for missed obligatory fasts – it is essential
to have the intention from the night before.

The scholars of the Standing Committee said:

Forming the intention during the night,
before dawn breaks, to fast the entire day is obligatory according to
Islamic teaching, because of the hadeeth of Hafsah (may Allah be pleased
with her), from the Prophet (blessings and peace of Allah be upon him), that
he said: “Whoever does not intend to fast before dawn breaks, his fast does
not count.” Narrated by the authors of as-Sunan.

End quote from Fataawa al-Lajnah
ad-Daa’imah (9/147)

Ibn Baaz (may Allah have mercy on him) said:

With regard to the fasts of Ramadan, fasts
observed in fulfillment of vows, expiatory fasts, and fasts done to make up
for fasts missed during Ramadan, it is essential to form the intention the
night before.

End quote.

http://www.binbaz.org.sa/mat/13420

Please see also the answer to question no.
26863 

Thirdly: 

With regard to the intention – in order for
it to be valid – it is essential to be firmly resolved; if there is no firm
resolve, then the intention is not valid. 

Zakariya al-Ansaari said in
Asna’l-Mataalib (1/411): 

With regard to fasting, it is essential to
have a firm, specific intention, as in the case of prayer. All of that must
occur before dawn breaks in the case of obligatory fasts, even those done in
fulfilment of vows, making up missed obligatory fasts or as expiation, even
if the one who is forming the intention is a child, because of the report: 
“Whoever does not intend to fast before dawn breaks, his fast does not
count.” Narrated by ad-Daaraqutni and others; they classed it as saheeh. End
quote. 

Based on the above: 

The fast that was done on the day of ‘Arafah
counts as a naafil (supererogatory) fast only, because this is the intention
that preceded the action. As for the earlier intention of making it an
obligatory fast, it does not count because it was not formed the night
before and that was not the basis of the action when starting the action. 

And Allah knows best.